Game Theory and Management

THE TWELFTH INTERNATIONAL CONFERENCE

GAME THEORY AND MANAGEMENT

GTM2018

June 27-29, 2018 St. Petersburg, Russia

ABSTRACTS

Edited by Leon A. Petrosyan and Nikolay A. Zenkevich

St. Petersburg State University St. Petersburg 2018 ST. PETERSBURG STATE UNIVERSITY THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES (Russian Chapter)

GAME THEORY AND MANAGEMENT

The Twelfth International Conference Game Theory and Management

GTM2018

June 27-29, 2018, St. Petersburg, Russia

ABSTRACTS

Edited by Leon A. Petrosyan and Nikolay A. Zenkevich

St. Petersburg State University St. Petersburg 2018

УДК 518.9, 517.9, 681.3.07

GAME THEORY AND MANAGEMENT. Collected abstracts of papers presented on the Twelfth International Conference Game Theory and Management / Editors Leon A. Petrosyan and Nikolay A. Zenkevich. – SPb.: St. Petersburg State University, 2018. – 156 p.

The collection contains abstracts of papers accepted for the Twelfth International Conference Game Theory and Management (June 27-29, 2018, St. Petersburg State University, St. Petersburg, Russia). The presented abstracts belong to the field of game theory and its applications to management. The abstract volume may be recommended for researches and post-graduate students of management, economic and applied mathematics departments.

Computer design: Maria Bulgakova

ТЕОРИЯ ИГР И МЕНЕДЖМЕНТ. Сб. тезисов 12-ой международной конференции по теории игр и менеджменту / Под ред. Л.А. Петросяна и Н.А. Зенкевича. – СПб.: Санкт-Петербургский государственный университет, 2018. – 156 с.

Сборник содержит тезисы докладов участников 12-ой международной конференции по теории игр и менеджменту (27–29 июня 2018 года, Санкт- Петербургский государственный университет, Санкт-Петербург, Россия). Представленные тезисы относятся к теории игр и её приложениям в менеджменте. Тезисы представляют интерес для научных работников, аспирантов и студентов старших курсов университетов, специализирующихся по менеджменту, экономике и прикладной математике.

Компьютерная верстка: М.А. Булгакова

Roland Malhame Polytechnique Montreal (Canada)

Anna Rettieva Institute of Applied Mathematics, RAS (Russia)

Dov Samet Tel Aviv University (Israel)

Larry Samuelson Yale University (USA)

International Program Committee

Leon Petrosyan (Russia) – co-chair Nikolay Zenkevich (Russia) – co-chair Eitan Altman (France) Jesus Mario Bilbao (Spain) Irinel Dragan (USA) Hongway Gao (China) Andrey Garnaev (Russia) Sergiu Hart (Israel) Steffen Jørgensen (Denmark) Ehud Kalai (USA) Andrea Di Liddo (Italy) Vladimir Mazalov (Russia) Shigeo Muto (Japan) Richard Staelin (USA) Krzysztof Szajowski (Poland) Myrna Wooders (USA) David W.K. Yeung (Hong-Kong) Georges Zaccour (Canada) Paul Zipkin (USA) WELCOME ADDRESS

We are pleased to welcome you at the Twelfth International Conference on Game Theory and Management (GTM2018) which is held in St. Petersburg State University and organized by the St. Petersburg State University (SPbSU) in collaboration with the International Society of Dynamic Games (Russian Chapter). The Conference is designed to support further development of dialogue between fundamental game theory research and advanced studies in management. Such collaboration had already proved to be very fruitful, and has been manifested in the last two decades by Nobel Prizes in Economics awarded to John Nash, John Harsanyi, Reinhard Selten, Robert Aumann, Eric Maskin, Roger Myerson, Finn Kydland, Lloyd Shapley, Alvin Roth, Jean Tirole and few other leading scholars in game theory. In its applications to management topics game theory contributed in very significant way to enhancement of our understanding of the most complex issues in competitive strategy, industrial organization and operations management, to name a few areas. Needless to say that Game Theory and Management is very natural area to be developed in the multidisciplinary environment of St. Petersburg State University which is the oldest (est. 1724) Russian classical research University. This Conference was initiated in 2006 at SPbSU as part of the strategic partnership of it’s the Faculty of Applied Mathematics & Control Processes and Graduate School of Management, both internationally recognized centers of research and teaching. We would like to express our gratitude to the Conference’s key speakers – distinguished scholars with path-breaking contributions to economic theory, game theory and management – for accepting our invitations. We would also like to thank all the participants who have generously provided their research papers for this event. We are pleased that this Conference has already become a tradition and wish all the success and solid worldwide recognition.

Leon A. Petrosyan, Co-chair of Program Committee

Nikolay A. Zenkevich, Co-chair of Program Committee

St. Petersburg State University

On competition in the telecommunications market

Petr Ageev 1, Yaroslavna B. Pankratova2 and Svetlana Tarashnina3

1,2,3 St.Petersburg State University, Russia [email protected] [email protected] [email protected]

Keywords: telecommunications market, subgame perfect equilibrium, finite repeated game

In the paper competition between three companies in the telecommunications market is investigated. All firms have different types: the leader, the challenger and the follower. The leader is a company that prevails in the market and acts in three main directions: • expansion of the market by attracting new customers and finding new areas; • increasing its market share in the current telecommunication market; • protecting its business from attacks by using defensive strategies. The challenger is a company that does not lag far behind the leader of the market and tries to become the leader by using attacking strategies. This firm uses strategies aimed to expand its market share, but those that do not cause active opposition to competitors. In Stackelberg's monograph [2] competition in the market is presented by a multistage decision-making model. At the first stage, the decision is made by the leader, and at the next stage, taking into account the decision of the leader, the decision is made by the company-follower. At the same time, when making decisions, each of the firms pursues its goal. In paper [3] authors propose an algorithm for constructing an approximate solution of such a problem. In this paper, we consider a more complicated problem. At the first stage, the leader and the challenger make decisions about which telecommunications services and at what prices to offer subscribers. At this stage, some customers make a choice in favor of the first or second company. At the next stage, the follower, taking into account the choice of competitors, decides on what services it would be better to offer to potential customers. At the same time, the follower tries to keep its customers and, if possible, to attract a part of the competitors’ customers. At this stage, the remaining customers make their own choices.

We assume that each customer must choose one of the telecommunications services (tariff). If a customer decides to stay at his tariff with his company, we believe that he chooses the appropriate service from the relevant company. The leader and the challenger aim to maximize their profits by attracting some of the competitor's customers. The purpose of the follower is to maximize its profit and save the customers. We formalize this problem of competition in the telecommunications market between the leader and the challenger as a nonzero-sum game in normal form. Then we consider a finite repeated game where this nonzero-sum game is realized as the first stage of each repetition. The second stage within a repetition is made by the follower. So, each repetition consists of two stages. As a solution of this game we consider a subgame perfect equilibrium (SPE) [1]. Some examples are given and discussed in the paper. The obtained solution allows each company to develop a long-term strategy to maximize its summing payoff. In the future, it would be interesting to study a strongly time consistency [4] of the solution.

References

[1]. D.W.K. Yeung, L.A. Petrosyan N.A. Zenkevich. Dynamic games and applications in management, 2009, Spb.: GSOM Press, 415 p. [2]. H. von Stackelberg. The theory of the market economy. England: Oxford University Press, 1952. 328 p. [3]. V. L. Beresnev, V. I. Suslov, A Mathematical Model of Market Competition, J. Appl. Industr. Math. 2010, Vol. 4, No 2, 147–157. [4]. L. Petrosyan. Strongly time-consistent diﬀerential optimality principles. Vestnik St. Petersburg Univ. Math. 1993, Vol. 26, No. 4, 40–46.

The Problem of Demand Side Management in Context of Power Companies Competition on The Retail Electricity Market

Natalia Aizenberg1

1Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences (ESI SB RAS), Russia [email protected]

Keywords: adverse selection, Bertrand competition, rates, retail electricity market, bounded rational consumers

The paper is concerned with the coordination of interaction between various types of consumers and a power company to manage electricity consumption by using the adverse selection model based on contract theory. There are several power companies competing in the electricity market according to the Bertrand model [1]. A method is proposed for load curve optimization by different types of consumers and a power company. Moreover, the types of consumers are identified and distinguished. The utility functions constructed for them describe the real situation rather well, and allow the implementation of a system of incentives for load curve optimization (load shifting from a peak time of the day by bounded rational consumer). The prices in rates providing a separating equilibrium are determined. The type of consumers which is characterized by a predictable behavior that can be completely formalized and used to build a price strategy of the power company is called rational type. Such consumers include various enterprises, factories that have a 24-hour working schedule. Their system of power consumption already in the initial stage includes a potential load shift from peak hours and active application of energy saving equipment. There is no need for the power company to additionally stimulate such consumers to behave rationally. The other type includes consumers that are not active when making up their load curves. This manifests itself in a small rate difference between peak and off-peak times. Here we suppose that these consumers can regulate their consumption, i.e. can shift part of their demand from peak to off-peak time. To describe the behavior of such consumers it is necessary to take into account both the costs of load shifting measures and the financial losses in the event that load is not shifted.

We would like to address the problem where each consumer chooses their rate from a set of rates offered by power company. There are reasons why the “adverse selection” [2, 3] can occur. In this case this will happen when consumers of different types choose the same rate schemes, but without their load curve optimization. The formulated statement below pursues the aim to develop a rate scheme where each consumer type selects “their” rate optimally and manages their load. There are several types of consumers that have different load management capabilities.  - a set of n types of consumers each will be denoted by  . There are K consumers in the model. The k -th consumer ( k  1, K ) is defined by the demand u q  q  function determined through the utility function k  tk  and load at time t - tk , as well as by a certain consumer type  . The total utility is calculated as a sum of utilities over

T u q   k  kt  all time periods t 1,T : t 1 . Each consumer has costs related to the payment for R q  received power  k  . The problem solved by each k -th consumer is maximization of a payoff from power purchase.  A share of each consumer can be determined as a ratio of the power consumed by load of type  to all the power sold by the power company. The company does not have full information about the consumer type but has an idea on the share of the consumer types.

 The number of consumers of the  -th type in the market equals m , with

m  K    . A share of each type can be determined as a ratio of the power consumed by load of type to all the power sold by the power company. The company does not have full information about the type of consumer but has an idea on the share of the consumer types. R  The company sets rate for the  -th consumer type according to the total load

  m  m  qkt C qkt  k 1 and function of costs kt , that are calculated for each time period t . In R  the case of the studies considered below, is the rate for consumer of type  .

T   θ  uk qkt  R qk  max, k  1, K,  q  t1   T  m       R  q   C q    0;    k   kt     t1  k 1   T T u q   R  q   u  q   R  q  , k, j [1, K], ,;  k  kt   k   k  kt   k   t1 t1     1; q   0,q   0; R  q   0, R  q   0,    0.  kt kt k k   Here we consider the Bertrand competition problem. This determines the criterion used by the company to set the rates. In this case, consumer is supposed to be able to change power company if it sets the prices higher than the neighboring company. Accordingly, the main criterion for the company offering the rates will be zero profit. The proposed methodology for the determination of rate allows both categories of consumers interacting with a power retailer — fully rational (that do not vary their load) and bounded rational (that change their load curve by shifting part of peak load) — to optimize power supply to them under the rates stimulating their load optimization. We reduce the problem of search for a separating equilibrium to a set of problems that represent either linear equations or maximization of concave functions on a compact set. As a result, we find the rates that fully satisfy consumers, are different for different groups of consumers, and meet the condition of normal profit for retailer.

This research was supported Russian Foundation for Basic Research, № 16-06- 00071.

References

[1]. N. Aizenberg, N. Voropai Price setting in the retail electricity market under the Bertrand competition. Procedia Computer Science. Elsiever 2017. V. 122, 2017, P. 649–656. [2]. P. Bolton, M. Dewatripont, Contract theory, MIT press, 2005, 717p. [3]. N.Aizenberg, N.Voropai and E.Stashkevich A mechanism for encouraging active consumers to optimize the operations of power supply systems. SHS Web Conf. 3rd International Conference on Industrial Engineering (ICIE-2017), vol. 35, no. 01022, p.5, 2017.

Import tariffs and their influence on countries’ welfare

Natalia Aizenberg1, Igor Bykadorov2, and Sergey Kokovin3

1Melentiev Energy Systems Institute of Siberian Branch of the Russian Academy of Sciences (ESI SB RAS), Russia [email protected] 2Sobolev Institute of Mathematics SB RAS, NSU, NSUEM, Russia [email protected] 3National Research University Higher School of Economics, Russia [email protected]

Keywords: international trade, monopolistic competition, variable elasticity, tariffs

In recent decades, the globalization of world trade has significantly reduced tariffs in bilateral and multilateral trade agreements and organizations. Nevertheless, the countries use actively the tariffs and other trade barriers to regulate trade. Among researchers, there is no consensus on the need for tariffs. On the one hand, import and increase of own exports are useful for consumers, since trade promotes specialization and efficiency (this is an argument in favor of reducing barriers). On the other hand, some governments argue that they need to avoid competition from imports and, therefore, to avoid unemployment in certain industries (this is an argument in favor of tariffs). To enrich this discussion, we consider bilateral import tariffs under monopolistic competition with variable mark-ups. The theoretical foundations of our model are standard (see [1], [3]), but the bilateralism of tariffs requires some motivation. Unilateral tariffs have already been extensively studied in [5] with a constant elasticity of substitution (CES) and in [4] with variable elasticity. By contrast, for bilateral tariffs, which are also quite realistic, the theory is not developed enough, whereas in practice, countries usually choose an acceptable level of protection jointly during negotiations. Moreover, in most cases, trade agreements require equal bilateral tariffs. Our setting studies the usual Krugmanian [6] model of trade among many countries with variable elasticity of substitution, with reciprocal import tariffs for every pair of countries. We examine the consequences of introducing the customs tariffs/subsidies of different types:  ad valorem tariff is charged as a percentage of revenue,  specific tariff is charged per each physical unit of goods (in [2] it is labelled as cost tariff).

Studying the model, we find that the type of levied customs fee has a fundamentally different impact on welfare. In the case of an ad valorem tariff/subsidy, under zero transportation costs, welfare always declines. Under a specific tariff (under zero transportation costs), small subsidies have a positive effect. Under non-zero transport losses, their magnitude is important. If transportation costs are very high, then the introduction of any tariff will be beneficial. If transport costs are small, then small ad valorem tariffs are welfare-improving (this is one of the most important results), while specific tariffs reduce welfare. As for subsidies, specific subsidies are beneficial, while ad valorem subsidies reduce welfare. Supported by RFBR grant 18-010- 00728.

References

[1]. Arkolakis C., Costinot A., Rodríguez-Clare A. (2012) New Trade Models, Same Old Gains? // American Economic Review, 102(1): 94–130. [2]. Caliendo L., Feenstra R.C., Romalis J., Taylor A.M. (2015) Tariff Reductions, Entry, and Welfare: Theory and Evidence for the Last Two Decades // NBER Working Paper No. 21768 [3]. Demidova S., Rodriguez-Clare A. (2013) The simple analytics of the Melitz model in a small open economy // Journal of International Economics, 90(2): 266-272. [4]. Demidova S. (2017) Trade policies, firm heterogeneity, and variable markups // Journal of International Economics, 108: 260-273. [5]. Felbermayr G., Jung B., Larch M. (2013) Optimal Tariffs, Retaliation, and the Welfare Loss from Tariff Wars in the Melitz Model // Journal of International Economics 89(2): 13–25. [6]. Krugman P. (1980) Scaled economies, product differentiation, and pattern of trade // American Economic Review, 70(5): 950–959.

Military intervention in the War against Drugs: a Differential Game

Mario Alberto Garcia-Meza1

1Universidad Juarez del Estado de Durango, Mexico [email protected]

Keywords: Differential Game, Oligopoly, Drug War, Warfare Competition, Non- Cooperative Games

The year 2017 presented the highest murder rate in decades for Mexico. These rates of violence can be mostly attributed to the presence of drug cartels in the country. The production and traffic of drugs is a very lucrative activity, and the occupation of certain territories is paramount for the operation of the business. It is, therefore, natural to assume that, in the lack of legal ways to compete, Drug Cartels use violence to dispute territory. The introduction of military activity to suppress Cartels' influence is a natural reaction, which was followed by Mexican government from 2007 and up to date. Nevertheless, the criminal activity of such gangs has all but diminished since. Quite the contrary, the violence in the country has increased since, in particular in cities where the control is left for dispute. In this article we create an oligopoly differential game model of two cartels occupying a territory, with warfare as their control variable and a normalized state variable that indicates their share of occupation of the territory. The Cartels are modeled as profit maximizing agents whose main objective is to increase their control over the territory in order to exploit the profits from it. The government is introduced to the model as a third agent with the capacity of diminishing both agents territory. The game's Nash equilibria shows that the government efforts result in an increase in warfare activity from the Cartels, which diminishes social welfare. The game is a modification of Case's model of a duopoly in competition. Instead of using advertising as a way to compete, the Cartels use warfare methods, which yield an externality to society in the form of violence. This is absorbed by the government in the form of negative goodwill. The goodwill is conceived as a form of asset that diminishes as a function of the violence.

On the infinite dimensional Lyapunov's convexity theorem and applications

Youcef Askoura1

1 LEMMA, Université Paris 2, France [email protected]

Keywords: Lyapunov convexity theorem, Purification of mixed strategies, Nash equilibrium, alpha-core, ambiguity aversion

We consider the following problem motivated by applications in decision and game theory : let (Ω,A) be a measurable space of states of nature and A a compact metric space. Denote by P(A) the set of probabilities on A. By means of additional regularity conditions making meaningful all the following entities, Gilboa and Schmeidler (1989) established under an appropriate axiomatic system that a binary (preference) relation ⪯ on the set of measurable functions δ:Ω→P(A) is characterized by a min-expected utility (called also maxmin expected utility as well) relatively to a function u:Ω×A→IR and a set K⊂P(Ω) of probability measures, in the following meaning :

Note that in the original model of (Gilboa and Schmeidler, 1989), P(A) is replaced by a set Y of finitely supported probabilities and the function u is defined on Y and is linear, a fact equivalent to say in our formulation above that it depends only in its variable a. The set K may be seen as a set of priors of a decision maker on the states of nature. Since he is pessimistic (or averse to the ambiguity), he evaluates a decision δ by computing the minimum of his expected utilities over his set of priors. Since the problem of ambiguity aversion is becoming more and more important in game theory and in other economic problems, the question of the study of these min- expectations among other, in their global aspects is relevant. For instance, since it is admitted that such functions δ above, are somewhat strategies of a last resort, and it is better to provide decisions as simple functions f:Ω→A, it is interesting to know if for every δ, there exists an f (a purification of δ) generating the same min-expected utility value, or at least representing an extremum of the min-expectation which exists generally as δ under weaker regularity assumptions. The purification problem is well known in game theory in

13 case of a one point set K(for instance Balder (2008) and its bibliography), but all remains to be known for a general K. In this work, we try to address precisely this issue. Our results are based mainly in Lyapunov’s convexity theorem for infinite dimensional vector measures as established in (Kluvánek, 1973; Knowles, 1975), and the related results addressed in (Diestel and Uhl, 1977). Roughly speaking, a vector measure G has convex and weakly compact range iff the integral operator relatively to G is not an injection from the spaces of bounded functions on measurable subsets of Ω to the range of G. Closely related approaches use Maharam types and saturated measure spaces, consisting in constraining more the underpinning measure space of the state of nature. By this ways, infinite dimensional Lyapunov convexity theorems and purification processes are obtained in (Greinecker and Podczeck, 2013; Khan and Sagara, 2013, 2015, 2016) and applied to the integration of correspondances, equilibria of exchange economies, and control systems. The incorporation of the notion of ambiguity in game theory can be addressed in different ways. Our applications consider models close to that in (Kajii and Ui, 2005). That is, we endow each player i with a set of probability measures K on the set of states i of nature and a utility depending in the state of nature and the action profil of all players. Then, each player uses accordingly a min-expected utility computed as above.

References

[1]. Balder, E.J., Comments on purification in continuum games, Int J Game Theory 37(2008),73-92. [2]. Diestel, J., Uhl, J.J., 1977. Vector measures. Mathematical Surveys, Number 15. American Mathematical Society. [3]. Greinecker, M., Podczeck, K., Liapounoff’s vector measure theorem in Banach spaces and applications to general equilibrium theory, Econ. Theory Bull. 1 (2013);157-173. [4]. Gilboa, I., and Schmeidler, D., Maxmin Expected Utility with non-Unique Prior, Journal of Mathematical Economics 18(1989), 141-153. [5]. Kajii, A., Ui, T., Incomplete information games with multiple priors, The Japanese Economic Review 56(2005), 332-351. [6]. Khan, M., Sagara, N., Maharam-types and Lyapunov’s theorem for vector measures on Banach spaces. Illinois J. Math. 57 (2013), 145-169. [7]. Khan, M.A., Sagara, N., Maharam-types and Lyapunov’s theorem for vector measures on locally convex spaces without control measures, J. Convex Anal. 22(2015), 647-672. [8]. Khan, M.A., Sagara, N., Relaxed large economies with with infinite-dimensional commodity spaces : the existence of Walrasian equilibria , Journal of Mathematical Economics 67(2016), 95-107.

Strategic “Mistakes”: Implications for Market Design Research

Georgy Artemov1, Yeon-Koo Che 2 and Yinghua He3

1The University of Melbourne, Australia [email protected] 2Columbia University, United States [email protected] 3Rice University, United States [email protected]

Keywords: strategic mistakes, payoff relevance of mistakes, robust equilibria, truthful- reporting strategy, stable-response strategy, stable matching, preference estimation

Using a rich data set on Australian college admissions, we show that even in strategically straightforward situations, a non-negligible fraction of applicants adopt strategies that are unambiguously dominated; however, the majority of these `mistakes' are payoff irrelevant. We then propose a new equilibrium solution concept that allows for mistakes. Applying it to a strategy-proof mechanism in which colleges strictly rank applicants, we show that equilibrium strategies need not be truth-telling, but that every equilibrium outcome is asymptotically stable. Our Monte Carlo simulations illustrate the differences between the empirical methods based on truth-telling or outcome stability, revealing that the latter is more robust to potential mistakes. Taken together, our results suggest that strategy-proof mechanisms perform reasonably well in real life, although applicants' mistakes should be carefully taken into account in empirical analysis.

Policy Design under Collusion

Stefano Barbieri1, Benjamin Sperisen2, K. Brent Venable3 and Zizhan Zheng4

1,2,3,4 Tulane University, United States [email protected], [email protected] [email protected], [email protected]

Keywords: dynamic games, algorithm, policy design, collusion

We develop an algorithm for designing optimal policy (e.g. for a government) in the presence of agents who cooperate together as best they can. How can a policy designer either prevent these agents from colluding to undermine policy goals, or incentivize cooperation that furthers the goals? Possible settings include auction design under bid- rigging, law enforcement against organized crime operating in illegal markets, anti-trust policy in legal markets, or maintaining an international alliance of countries with competing interests. For example, a government wanting to produce a public good might outsource through a competitive procurement auction where the private firms repeatedly bid on contracts offered over time. However, the firms could collude on high bids, obtaining a worse price for the government. The government could deter collusion by investigating and fining the colluding firms, but this might be costly. How should the government decide which projects to outsource, how to design these auctions, and when to investigate bid-rigging? Consider a repeated game played by the “government” and a number of other agents (“players”). The government announces its strategy (a “policy”) in advance, mapping future histories to actions it will play. The government's policy can be interpreted as specifying a dynamic game played by the players. We assume the players “collude” in the sense that they play the player-optimal strongly symmetric equilibrium of that dynamic game. How should this dynamic game be designed to maximize the government's payoff given this collusion? We construct an algorithm for computing a policy that is approximately (arbitrarily close to) optimal. We first show that the recursive “self- generating sets of payoffs” characterization of repeated game equilibria (of Abreu, Pearce, and Stachetti, 1990) can be extended to the policy choice setting as “self-generating collections of payoff sets.” These collections are convex sets in 4-dimensional space

16 whose coordinates are the minimum and maximum payoffs for the players and the corresponding payoffs for the government, i.e. the payoff the government receives when players play the best and worst possible enforceable actions. We show how to generalize the notion of “polytope outer and inner bounds” (from the algorithm of Judd, Yeltekin, and Conklin, 2003) to payoff collections. Our algorithm starts from the singleton collection of feasible payoffs. Iteratively applying our generating operator generates a sequence of “contracting” collections that converge to the equilibrium collection E. Under perfect monitoring, the computation of these generated collections is shown to be representable as a mixed integer linear program. Under imperfect monitoring, the generated collections can be computed as a quadratic program. Each element of E is the equilibrium payoff set for a particular policy, and the optimal policy corresponds to the element of E with the highest government payoff corresponding to the best action available to the players. We illustrate this in a simple example where the government chooses to “attack” or “reward” two players who play a prisoner's dilemma, where the government's action shifts the payoffs by a constant. The government's payoff is the negative sum of the players' payoffs. Thus, the government seeks to minimize the players' payoffs, given that they collude on the best strongly symmetric equilibrium available under the policy. For an intermediate range of discount factors, the algorithm shows the optimal policy can achieve the lowest feasible symmetric payoff by incentivizing deviations to undermine cooperation.

Myopic and Farsighted Players in the Local Public Goods Game

Péter Bayer1, Jean-Jacques Herings2, Ronald Peeters3 and Frank Thuijsman4

1,2,4Maastricht University, Netherlands [email protected] 3Otago University, New Zealand

Keywords: Networks, Public goods, Farsighted

Our research focuses on long-run sophisticated behavior in a dynamic version of the local public goods game Bramoullé and Kranton (2007). Initially we consider the case of one farsighted player against a myopic population. As in Ellison (1997), examining optimal behavior of a single sophisticated agent against a naive population is a natural and necessary first step before a sophisticated population can be characterized in any detail. By far the simplest and the most studied strategy in public goods games is free- riding on naive neighbors, where a more sophisticated player can withhold his contributions to the public good and thereby force his opponents to contribute more. However, there are a number of simple network structures where free-riding is no longer the only viable strategy (Eshel et al., 1998). This project aims to identify optimal farsighted behavior in the local public goods game on a general class of networks. Ours is the first paper to consider heterogeneous time horizons in this gameclass. Through this effort we uncover the characteristics of networks in applied settings where free-riding is/is not a viable strategy. In addition, we identify the possible set outcomes and the implications for the production of local public goods.

Let I={1,…,n} denote the set of players with a fixed network structure, (gij)i,j∈I with g =g . If g =1 then players i and j are connected, while g =0 means that they are ij ji ij ij

1 2 not. Let Xi={0, , , …}, with positive integer d be the set of possible efforts. The players 푑 푑 are engaged in a local public goods game with payoff functions π given by i

where the benefit functions f are increasing and concave. i

We study a recurrent setting of the game in discrete time. In every period t an t active player i is chosen with a uniform probability distribution from all players. The active player updates his effort level while that of every other player remains the same for t the next period. Thus, a series of effort profiles (x ) produced by the players’ individual t∈N decisions will satisfy the admissibility property t t xt+1,−i =xt,−i (2) for every t∈N. Player 1 is farsighted and maximizes discounted expected payoffs with discount factor δ∈(0,1). All other maximize immediate payoffs. Our initial results characterize some useful characteristics of the equilibrium strategies of this game.

Proposition 1 Fix a network structure (gij)i,j∈I. 1. In every Subgame Perfect Equilibrium (SPE), the players 2,…,n are myopic best-responders. 2. There exists a Stationary Subgame Perfect Equilibrium (SSPE) of this game. 3. Every SPE has a utility-equivalent SSPE for the farsighted player. The first part of Proposition follows naturally. Since a myopic best-response strategy is stationary, once the optimal behavior of players 2,…,n is pinned down, the game reduces into a Markov Decision Process for player 1. This fact guarantees the existence of an optimal stationary policy, and an SSPE naturally follows. The third part follows by the fact that every optimal policy yields the same utility for player 1. With the existence of optimal stationary policies are established, outcomes of the game can be studied. We characterize the outcomes of the game as steady-states of SSPE strategies. ∗ Definition 1 Let (s ) be an SSPE strategy profile. A profile of efforts i∈I x=(x ) is called a steady-state, if for every player i, s∗,i(x)=x . i i∈I i It is clear that in every steady-state, players 2 to n are playing their myopic best- response to the others’ effort levels. We establish convergence to the set of steady-states. Proposition 2 For any initial profile of efforts, any SSPE will lead to a steady- state with probability one. The main significance of this result is showing that the game will not cycle indefinitely in the space of action profiles. This means that the set of steady-states reveal

19 the key properties of farsighted strategies, as well as predict the possible outcomes of the game.

References

[1]. Bayer, P., Herings, P., Peeters, R. and Thuijsman, F., 2017. Adaptive Learning in Weighted Network Games. GSBE Research Memorandum No. 25. [2]. Bramoullé Y, Kranton R. (2007). Public goods in networks. Journal of Economic Theory 135: 478-494. [3]. Ellison G (1997). Learning from personal experience: One rational guy and the justification of myopia. Games and Economic Behavior 19: 180-210. [4]. Eshel I, Samuelson L, Shaked A (1998). Altruists, egoists, and hooligans in a local interaction model. American Economic Review 88: 157-179. [5]. Fudenberg D, Levine D (1998). The Theory of Learning in Games, MIT Press, Cambridge.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МЕНЕДЖМЕНТ

Главный редактор Н. П. Дроздова

Издательство Санкт-Петербургского государственного университета

A method of solution of the Stackelberg game of resource allocation

Grigory Belyavsky1, Natalya Danilova 2 and Guennady Ougolnitsky3

1,2,3 Southern Federal University, Russia [email protected] [email protected] [email protected]

Keywords: differential Stackelberg games, random search, resource allocation

Consider a Stackelberg game in the form

J(u,v)  max; Ji (u,v)  max, i  1,...,r . The functions u and vi belong to the u vi

Banach space B 0 ,1 of bounded functions f with the uniform norm:

f sup f t . It is supposed that the leader reports to the followers her strategy u , 01t after that the followers choose their strategies from a set Nu  , for example, the set of Nash equilibria which is known to the leader. As a result, the guaranteed payoff of the leader is determined by the calculation of maxmin, Juv0   . Suppose that the leader u vNu  

, can calculate the functional vuAJu  v rgmin, 0   where vN u 

uB0,1 . Thus, her problem consists in the calculation of , where i     max Ju0   u

JuJuu00   ,   . Assume that JuJwLuw00     . This constraint ensures a possibility of the usage of approximate methods for the leader's problem. Its sufficient conditions are: (1) JuvJu,, wLuuLvw , 020121    uvr

u   u  L u  u .  2  1 r  2 1 Consider an example. In each instant of time the resource ut  is allocated between the followers according to the proportional principle, after that each follower

 solves the problem: uvi . The respective game in normal form has a max iiv  vi 0 v j j

Nash equilibrium if the cost functions iiv  are strictly concave and differentiable. The equilibrium is determined by the system of equations:

2 di (zi ) di (0) di (0) s  u(s  zi ) if s  u; z  0 if s  u; z j  s dvi dvi dvi  j

The system has the unique solution ( z u, s u ). Define the function

2 d Fxywuxyuxw ,,,     ; then Fszzui,,,0, iii  . As a dvi d result we receive the system of linear equations relative to the derivatives zu  : du i

2 dz1  dzdz    ssij2 iiii 2, dz du 1  2 AsIz , where dz ,  ,  dvi dvi . du  ... I  ... aij,   du    1  dzii  dzl  2,su ij    dv du  i  2 2 dzdziiii    T ssij 2 2, Here ABuEuII  , where  dvi dvi , E is a bij,    dzii  2,sij   dvi

1 1 unit matrix. Denote HBuE  , gsIz, then the equation Ax g has

H1 g, I  the solution x H1  g u I . 1, HII1   Thus the Euclidean norm of the vector x has the form

H1 g, I 1   xHgru, (2) 1, HII1  

1 1 where H is a spectral norm of the matrix. The matrix H can be represented as a

 k  k 1 11k  1 11 power series: HB  1 и HB  . For the spectral norm uuk 0  uuk 0  of the matrix B the inequality holds: 2 dzdzjjjj    2 . Denote the right hand side of the Bssrmax2max 2 jjdv j dv j inequality by L then if uL 1 (3) H 1  uL 1 In particular, this fact ensures the existence of the matrix H . Suppose also that LuM Then the combination of (2), (3) and (4) gives that the Euclidean norm of the solution x . In the problem of resource allocation the leader's functional depends on s . Therefore, it is necessary to find an estimation of the value ds . The Cauchy- du ds Bunyakovsky inequality provides this estimation:  rx that implies du

uuLuu 2121    .

For the solution of the problem a random search method is used. Let the set of feasible solutions satisfies to the Lipshitz condition:  u tu  s   LuBtsts0,1:,sup,01,01,     ts The following statement holds: the sequence of functions

n u n u uu  Ii n  in   ,/ , 0,1,..., ; i  1,0,1 tends to nii 0  ti n i1 u in the uniform norm. So, consider the following approximate problem: n maxmax,J uJ ,  u  ,   which is solved by a genetic 000     i i1 uL 0,1 u0 ,, algorithm or the simulated annealing method. In the case of an additive game a piecewise- constant approximation may be used with consideration of the Lipshitz condition.

Gains and Looses Effects of Trade under Monopolistic Competition: the case of nonlinear production costs

Igor Bykadorov1

1Sobolev Institute of Mathematics SB RAS, NSU, NSUEM, Russia [email protected]

Keywords: international trade, monopolistic competition, welfare, autarky

In New Trade theory, gains from trade are an evergreen topic. It recently generated a vivid discussion after influential papers [3], [4], which puzzled theorists with surprisingly low estimated gains. In [6] (see also [7]) we consider a version of Krugman's general one-sector monopolistic competition model, with unspecified additive utilities, and without outside good. Labor markets do clear, and trade is balanced. Homogeneous firms use one production factor (labor) facing uniform fixed and marginal costs, consumers are identical. Expectedly, welfare locally increases in each country with liberalization at least near free trade, in the end of this process. More unexpected is the effect in the beginning of liberalization, near “complete autarky.” We prove that during the initial stage of trade liberalization, welfare deteriorates in each country, for any additive utility having a choke- price: quadratic, CARA, etc. From the policy viewpoint, our “harmful trade” seems to favor protectionism, but in fact, it only suggests not to liberalize trade gradually, to jump over the initial losses towards sufficiently massive trade. Comparing the main result of [6] with the recent literature, let us mention similar effect in studies of import tariffs. For example, [10] uses a version of Melitz-Ottaviano model with variable markups and supports some degree of protectionism; [1] proves that small ad valorem and specific tariffs are beneficial when combined with sufficiently high transport costs (comparable with prohibitive ones). This study differs from [6] not only in their settings, but also in the mechanism behind welfare losses from trade. Note that in [2], [5], [8] and [9] the specific case of nonlinear production costs (with investments in R&D) are considered. We generalize the results of [6] on the case of general nonlinear production costs. It turned out that the results of [6] can be generalized to the case of convex production costs, and partly to the case of concave production costs.

Supported in part by RFBR, grants 16-01-00108, 16-06-00101 and 18-010- 00728. References

[1]. Aizenberg N., Bykadorov I., Kokovin S. (2017) Beneficial welfare impact of bilateral tariffs under monopolistic competition // Abstracts of the Tenth International Conference Game Theory and Management (GTM2017, June 28-30, 2017), Eds. L.A. Petrosyan and N.A. Zenkevich. – Saint Petersburg: Saint Petersburg State University. – P. 5-7. [2]. Antoshchenkova I.V., Bykadorov I.A. (2017) Monopolistic competition model: The impact of technological innovation on equilibrium and social optimality // Automation and Remote Control 78(3): 537-556. [3]. Arkolakis C., Costinot A., Donaldson D., Rodríguez-Clare A. (2018) The Elusive Pro- Competitive Effects of Trade // The Review of Economic Studies, forthcoming. [4]. Arkolakis C., Costinot A., Rodríguez-Clare A. (2012) New Trade Models, Same Old Gains? // American Economic Review 102(1): 94-130. [5]. Bykadorov I. (2017) Monopolistic Competition Model with Different Technological Innovation and Consumer Utility Levels // CEUR Workshop Proceedings 1987: 108-114. [6]. Bykadorov I., Ellero A., Funari S., Kokovin S., Molchanov P. (2016) Painful Birth of Trade under Classical Monopolistic Competition // National Research University Higher School of Economics, Basic Research Program Working Papers, Series: Economics, WP BRP 132/EC/2016. [7]. Bykadorov I., Ellero A., Funari S., Kokovin S., Molchanov P. (2017) Negative Pro- Compepetive Effects of trade under Monopolistic Competition // Abstracts of the Tenth International Conference Game Theory and Management (GTM2017, June 28-30, 2017), Eds. L.A. Petrosyan and N.A. Zenkevich. – Saint Petersburg: Saint Petersburg State University. – P. 34-36. [8]. Bykadorov I., Gorn A., Kokovin S., Zhelobodko E. (2015) Why are losses from trade unlikely? // Economics Letters 129: 35-38. [8]. Bykadorov I., Kokovin S. (2017) Can a larger market foster R&D under monopolistic competition with variable mark-ups? // Research in Economics 71(4): 663-674. [8]. Demidova S. (2017) Trade policies, firm heterogeneity, and variable markups // Journal of International Economics 108: 260-273.

VESTNIK OF SAINT PETERSBURG UNIVERSITY. MANAGEMENT

Editor-in Chief Natalia P. Drozdova

St. Petersburg University Press

Intra-group heterogeneity in endogenous-policy contests

Daniel Cardona1, Jenny de Freitas 2 and Antoni Rubi-Barcelo 3

1,2,3Universitat de les Illes Balears, Spain [email protected] [email protected] [email protected]

Keywords: political process, conflict, group contests, endogenous claims, intra-group heterogeneity

The choice of the common policy platform by the members of a group in a political competition naturally displays a tension between selecting the policy that maximizes the probability of winning or selecting the most preferred one. If, additionally, this group is composed by heterogeneous agents, an internal conflict among its members also comes into play. This occurs, for instance, when a political party internally chooses an alternative (either a policy or a candidate) to face the opponents' choice in a succeeding dispute and also in conflicts among industries, lobbies or interest groups. This study aims to shed some light on the effect of intra-group heterogeneity on this classical trade-off of the policy choice when the subsequent competition is modeled as a contest. We consider the canonical environment of collective choice where the policy space is one-dimensional and agents have single-peaked preferences. These agents are organized into two groups: Defenders of the status quo and challengers. Without loss of generality, it is assumed that challengers prefer policies to the left of the policy space. These agents are heterogeneous regarding their most preferred policy or peak. Contrarily, status-quo defenders are homogeneous and they all prefer the extreme-right policy. A two- stage game is played: First, challengers set a common target-policy. Second, a contest between challengers and status-quo defenders determines which of the two policies is finally implemented. In our baseline model, the winning probability of a group is determined by a contest success function (hereafter CSF) that depends on the relative size of the group's aggregate effort. Agents select their effort individually and non- cooperatively and the cost-effort function is convex. For most part of the paper, we shall assume that the target-policy of the leftist group is selected by a representative. The alternative setting in which this policy is collectively selected by the group is also considered. In these cases, we focus on processes satisfying the Condorcet Criterion, procedures that select the Condorcet winner when it exists.

Optimal Sales Mechanism with Outside Options and Withdrawal Rights

Dongkyu Chang1

1City University of Hong Kong, Hong Kong [email protected]

Keywords: bargaining, commitment, delay, mechanism, outside option, sales negotiation, option contract, upfront payment

In sales negotiations, parties often face a choice between haggling and promptly breaking off the negotiation. There is clearly a trade-off. They can haggle in search of a delayed agreement. Howev-er, it is sometimes better for both parties to immediately break off the negotiation and opt for out-side options. This trade-off is resolved in various manners. For instance, a delayed agreement is normal in real estate sales negotiations, whereas retailers typically adopt a fixed-price mechanism with no room for haggling. The aim of this paper is to address the following questions: what makes negotiating parties choose a delay rather than an immediate breakdown of the negotiation, or vice versa? How does the negotiating parties’ effort to avoid breakdown or delay affect the dynamics of the offer, the divi-sion of the surplus, and other features of the negotiation process and outcome? Suppose that a buyer and a seller negotiate the terms of trade for a single indivisible good. The buyer (informed party) privately learns her type prior to the negotiation, which determines both the value of the good and the value of her outside option. The seller (uninformed party) can choose and commit to any mechanism, thus allowing him to choose a time-dependent price path, arrange auxilia-ry transfers between two parties that may depend on the buyer’s reports, break off the negotiation at any time, and so on. Sales negotiations with the seller’s full commitment power have long been studied in the eco-nomics literature. The novelty of this paper’s model is twofold. First, the buyer’s outside option randomly arrives during the negotiation, and more importantly, the value of the outside option de-pends on her type (type-dependent outside option). Second, the buyer can withdraw from the nego-tiation table at any time during the negotiation (withdrawal rights), in which case she can still enjoy the outside option. Implicitly, it is impossible or prohibitively costly to sign any contract that abro-gates the withdrawal right.

The main results of the paper identify when and how a delay occurs in profit- maximizing (hereafter, optimal) mechanisms rather than an immediate outcome. The seller strictly prefers delaying to immediately breaking off the negotiation if (i) the value of the outside option is positively related to the buyer’s value of the good, and (ii) outside options are highly dispersed among buyer types. Oth-erwise, it is optimal for the seller to commit to a fixed-price mechanism, such that the buyer would immediately purchase the seller’s good or break off the negotiation. The occurrence of a delay in optimal mechanisms shows that the the no-haggling result does not hold if outside options are heterogeneous (type-dependent) across different buyer types. The com-parative statics with respect to outside options also provides an explanation for why bargaining de-lays remain the norm in markets for real estate, automobiles, inputs for manufacturing, and so forth; buyers in these markets have different outside options owing to their financial status and/or infor-mation, and thus the seller can make a higher profit by delaying. In contrast, a fixed-price mecha-nism is optimal for typical retail transactions in which buyers have similar outside options. In addition to a delay in the transaction, the optimal mechanisms also feature the buyer’s upfront payment. The buyer would be compensated for the upfront payment by the corresponding price dis-count when she purchases the good in the future. However, if she withdraws from the table without trading, the buyer would lose the money paid as the upfront payment. Hence, the upfront payment scheme prevents negotiation breakdown from occurring during the negotiation, which results in less information rent to the buyer and more profit to the seller. The upfront payment for the price discount in the future makes the optimal mechanisms resem-ble call option contracts. Indeed, the seller can achieve the optimal profit level by offering a menu of European call options, which is widely used in financial markets as well as other production con-tracts. Moreover, if there are multiple optimal mechanisms, all of them feature the option-like struc-ture with a delay in transaction (maturity date in the future) and an upfront payment (premium). The strict optimality of option-like contracts stands in contrast to the case without outside options, in which a posting-price mechanism (with a time-varying price path) is optimal with or without the seller’s commitment power. The result also contrasts with the revenue management problem in which committing to a posting-price mechanism is often sufficient to achieve the optimal profit lev-el.

The paper’s findings also have an implication for bargaining theory. To incentivize some buyer types to delay, the optimal mechanisms decrease the price over time. Interestingly, in this environ-ment, a declining price path requires more commitment power for the seller than does a fixed price. In a recent paper published in American Economic Review in 2014, Board and Pycia consider the bargaining problem without the seller committing to any mechanism but otherwise identical to this paper’s model. They show that the seller insists on a fixed price indefinitely in an (essentially) unique equilibrium. The seller could earn more profit by decreasing the price over time, but this strategy violates sequential rationality. This observation contrasts with the prediction of the Coase conjecture and hence provides a new perspective on the role of commitment in bargaining (or equivalently, a durable-good monopoly). The bargaining literature has emphasized the seller’s inability to commit to a fixed price and its ad-verse effect on his profit. However, combined with the observation of Board and Pycia, this paper’s results demonstrate that the Coase conjecture is overturned when buyers have type-dependent out-side options and withdrawal rights; it is the seller’s inability to commit to a declining price path that harms the seller without sufficient commitment power.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА, ИНФОРМАТИКА, ПРОЦЕССЫУ ПРАВЛЕНИЯ

Главный редактор Л.А. Петросян

Издательство Санкт-Петербургского государственного университета

Horizontal Misleading Advertising

Stefano Colombo1, Elias Carroni2 and Luigi Filippini3

1,3 Università Cattolica del Sacro Cuore, Italy 1 [email protected] 2University of Bologna, Italy [email protected] [email protected]

Keywords: Misleading advertising, Horizontal differentiation, Behavior-based price discrimination

Post-purchase satisfaction surveys commonly reveal that some consumers are more pleased than expected about their purchases, whereas some others are less satisfied in relation to their prior expectations (Casti, 1995). There are many reasons for this dissonance between pre-purchase expectations and post-purchase satisfaction. On one hand, consumers’ limited rationality may prevent them from forming correct expectations about the product they are going to purchase, so that the resulting level of satisfaction does not match their expectations (Anderson, 1973). In this case, the dissonance between expectations and satisfaction originates from innate bounded cognitive ability. On the other hand, consumers’ expectations are also influenced by marketing, so that incorrect expectations might be deliberately induced by firms (Woodruff et al., 1983, Meyer and Schwager, 2007). A typical example is false (or misleading) advertising: as long as consumers trust it, at least partially, it generates incorrect expectations and, hence, induces a mismatch between expectations and post-purchase satisfaction. In this case, the dissonance between expectations and satisfaction is the result of firms’ strategic choices. In this paper, we consider the strategic incentives of competing oligopolistic firms in creating false expectations in consumers. In particular, we introduce the notion of horizontal misleading advertising to describe a situation where a firm advertises horizontal characteristics that may differ from the true characteristics of its product. As a consequence, some consumers believe that the product is closer to their needs than it really is, whereas some others believe that the product is further from their needs. We use the term ‘horizontal’ misleading advertising to distinguish it from vertical misleading advertising, which has recently been considered by marketing scholars and economists (see for example Gardete, 2013, Rhodes and Wilson, 2016). Vertical misleading advertising describes a situation where commercials are aimed at making consumers expect a product to be of higher quality than it really is. As a result, after the purchase, all

30 consumers experiment a degree of satisfaction that is lower than expected. Unlike horizontal misleading advertising, the vertical spirit of this approach makes it unsuitable for those situations where, post-purchase, some consumers experience a positive surprise, whereas others experience a negative surprise. We study horizontal misleading advertising by providing a model investigating the incentives for two competing firms to provide information regarding the horizontal characteristics of their products when consumers can be identified and firms can engage in dynamic price discrimination. Following the Hotelling (1929) model, consumers’ horizontal bliss points are distributed on a unitary interval. We study two market configurations: in the first, labelled mass-product configuration, both firms sell a product that fits the average location; in the second, labelled niche-product configuration, each firm sells a product matching one of the extreme locations. In both market configurations, firms track the identity of first-period own buyers and, thus, the second-period prices are allowed to be based on the past buying decision (“behavior-based price discrimination”, or BBPD, Fudenberg and Tirole, 2000, Fudenberg and Villas-Boas, 2007). In such a framework, some consumers switch in the second period: that is, some people belonging to the “turf” of a firm patronize the rival which offers them a lower (“poaching”) price. Beyond prices, the competing firms commit, at the beginning of the game, to a certain advertising strategy. Advertising is interpreted as a claim on the horizontal positioning of a firm on the product spectrum. Consumers are assumed to partially rely on the advertising content, which can be believed as either (partially) misleading or true. Once a given consumer buys a product whose horizontal positioning is known only through advertising, he becomes aware about the real positioning of that product. We show that there is a trade-off for firms in providing a false rather than a truthful commercial. On the one hand, in the niche-product (respectively, mass-product) configuration, firms may have incentives to truthfully (misleadingly) advertise their maximally (minimally) differentiated locations to reduce the fierceness of competition: we refer to this as the differentiation effect. On the other hand, reporting a false (true) location in response to a rival’s truthful (misleading) advertisement could be a way to improve the competitiveness of a firm in the switching phase: we refer to this as the turf effect. Our findings are along different lines. In the mass-product configuration, the equilibrium advertising strategy implies that firms prefer to advertise niche products, in order to be perceived as maximally differentiated. That is, they choose misleading advertising. The niche-product configuration gives much richer results. In the first 31 instance, when consumers trust advertising, a low discount factor results in a symmetric equilibrium in which product characteristics are truthfully outlined by advertising. This comes from the fact that, when the switching phase is not so important, the need to be differentiated in the first period to reduce price competition, prevails: that is, the differentiation effect dominates. In the second case, when the future is sufficiently important, then asymmetric equilibria - with one firm opting for misleading and another for truthful advertising - arise if consumers trust advertising enough. In the third case, when the discount factor is very high and consumers trust firms’ claims very little, both firms choose a misleading advertisement strategy, as the turf effect dominates. Finally, we extend our model to the case of word-of-mouth and dynamic advertising. The first extension has the objective of taking into account an important stylized fact: consumers talk to each other (Godes, 2017). This word-of-mouth scenario informs the consumers about the characteristics of each product independently of their individual purchase behavior, as non-buyers receive information from past buyers. We find that whereas the results in the mass-product configuration are not affected, in the niche-product configuration the parameter region in which symmetric truthful advertising emerges as an equilibrium enlarges when word-of-mouth occurs. Moreover, it is shown that word-of-mouth reduces the equilibrium profits of the firms, as the possibility for consumers to communicate with each other about their previous purchase experience induces firms to compete more fiercely in the first period. The second extension aims at considering the possibility for one firm to change its advertising strategy over time. We show that it may be optimal for firms to adopt different advertising strategies over time. Indeed, firms might be better off choosing the advertisement policy that, at any period and given the rival’s choice, best exploits the differentiation effect and the turf effect.

Characterizing NTU-Bankruptcy Rules using Bargaining Axioms

Bas Dietzenbacher1 and Hans Peters2

1Tilburg University, Netherlands [email protected] 2Maastricht University, Netherlands

Keywords: NTU-bankruptcy problem, axiomatic analysis, bargaining theory

This paper takes an axiomatic bargaining approach to bankruptcy problems with nontransferable utility by characterizing bankruptcy rules in terms of properties from bargaining theory. In particular, we derive new axiomatic characterizations of the proportional rule, the truncated proportional rule, and the constrained relative equal awards rule using properties which concern changes in the estate or the claims.

Periodicals in Game Theory

CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT

Volumes 1–11

Edited by Leon A. Petrosyan Nikolay A. Zenkevich

St. Petersburg State University

On partisan games in secretary problem

Sergei Dotsenko1

1National Taras Shevchenko university of Kiev, Ukraine [email protected]

Keywords: secretary problem, partisan games, Nash equilibrium

Preliminaries on the classical secretary problem. In [1],[2] the best choice problem (also known as secretary problem) is formulated as follows. Let somebody randomly browse n objects and should choose the best one among them. The quantities of the objects are compatible with each other. Random acquaintance means that initially all the n! permutations, that specify the order of browsing are equiprobable. After the acquaintance with the object it is necessary either to choose it or to reject it and to continue browsing: it is not allowed to return to already browsed objects. In what follows, by the best object we will call the object favorable for all n, and the maximal is the best object among considered n ones. It is obvious, that during the search it is necessary to analyze the expediency of choosing some object only if it maximal. It is proved, that to choose the best object among n with maximal probability, one should adhere the following strategy: first to skip all elements with indices

1,...,kn 1 and then to choose the first maximal element, whose index is not less, than

k , where is defined from double inequality 1 1 1 1 . n  ... 1  ... It turns kn n 1 kn 1 n 1

k 1 out, that n  as n   and the probability of choosing the best object when using the n e described object tends to 1/e. The four games based on classical secretary problem are considered: the censorship game, the prompt game, the extortion game, the guessing game.

References

[1] P.Borosan, M.Shabbir. A survey of secretary problem and its extensions. Preprint, 2009. www.researhgate.net. [2] T.Ferguson. Who solved the secretary problem? Statistical science 4(3), 282-289 (1989). [3] S.Dotsenko, O.Marynych. Censorship game in optimal choise problem. Cybernetics and system analysis, , vol. 49, №5, September, 2013. [4] S.Dotsenko, O.Marynych. Hint, extortion an guessing games in the best choice problem. Cybernetics and system analysis, vol. 50, № 3, May 2014.

Mult-Unit Common Value Auctions and the Winner's Curse – A Theoretical and Experimental Analysis

Karl-Martin Ehrhart 1, Marie-Christin Haufe 2 and Jan Kreiss 3

1,3Karlsruhe Institute of Technology, Germany [email protected] [email protected] 2Takon GmbH, Germany [email protected]

Keywords: Common Value Auction, Winner's Curse, Multi-Unit Auction, Learning, Experimental Economics, Gouvernemental Procurement

Common values and common cost components respectively are a frequent characteristic of goods that are procured in governmental tenders. That is, part of or the complete costs for the goods to be supplied are the same for all bidders, e.g. the same costs for resources or materials. A very recent and relevant application for governmental procurement auctions with this characteristic are auctions for renewable energy support where the support payment is determined in a competitive bidding process. Such auctions are implemented in many countries, especially in Latin America and Europe where auctions are obligatory for member states of the EU (European Commission, 2014; IRENA, 2017). Usually those auctions are conducted as multi-unit procurement auctions implementing the two most common pricing rules, pay-as-bid and uniform pricing. Auction-theoretically, common costs are often accompanied by the winner's curse, i.e., the winner of the auction might experience a loss (Thaler, 1988). In the case of governmental procurement auctions the winner’s curse is particularly unintended. The government tries to increase the overall economic benefit which might be at risk if projects are either not realised due to the winner’s curse or the competition diminishes as participants either quit or even declare bankruptcy due to the ruinous conditions. We analyse theoretically and by conducting an experiment how the two pricing rules influence the auction outcome and the winner's curse given different competition levels and pricing rules. The experimental results contrast with the theoretical expectations. In general, the bidders perform worse than expected. The average profit is lower than expected and the probability of the winner’s curse is (much) higher than theoretically predicted. That holds for both pricing rules and for each competition level. Moreover, the expected differences between both pricing rules cannot be observed. There are only minor

35 deviations between both pricing rules with respect to profit and winner’s curse. However, as expected, the profit increases and the probability of the winner’s curse decreases with an increasing k. When analysing the bidding behaviour, we find that the bidders underbid the equilibrium bidding strategy. That is, at average the bidders bid less than we predicted according to our theoretical findings. As a result, there is a negative correlation between deviation from the equilibrium bidding behaviour and the profit, i.e., the more the actual bidding behaviour deviates from the equilibrium bidding strategy the lower is the profit of the bidder in our experiment. The only significant influencing factor on the bidding behaviour is the competition level. The lower the competition level, i.e., the higher k, the lower is the difference between actual and equilibrium bidding behaviour. This holds for both pricing rules. As bidders in our experiment participated in 40 auction rounds we also tested for learning patterns. In general, the bidders do not adapt their bidding behaviour over time. That is, the bidders bid equally in the first auction round as in the last. However, we could identify a comprehensive pattern. The bidding behaviour in one round depends on the auction outcome of the previous round. If the bidders had a bad experience, i.e., they suffered from the winner’s curse, they increase their relative bid in the next round. There is the opposite effect given a good experience in the previous round. On average, both effects balance out and, thus, there is no overall improvement regarding the bidding behaviour over the time of the experiment. We prove that also in the multi-item counterparts of the standard common value auctions, the pay-as-bid pricing rule is more expensive for the auctioneer than the uniform pricing rule. However, in the particular case of uniformly-distributed common costs, the probability of the winner’s curse is also much lower. That might be of particular interest in governmental procurement auctions with high common cost components where the government aims for a long-lasting and healthy competition. Depending on the application, it might even be of higher importance than a particularly low auction outcome. The comparison to our experimental results is alarming. The differences in the results of both conducted pricing rules are only minor. That is, no pricing rule is preferable regarding either goal of the auctioneer, a low price or a low occurrence of the winner’s curse. Moreover, we cannot identify a long-term learning and improvement of bidding behaviour. The average results are equally bad in the first as in the last auction round. Our

36 analysis show that the bidding behaviour especially depends on the outcome of the previous auction round. But all effects balance out. For the auctioneer our results show that a prevention of the winner’s curse is hard given (high) common cost components. It can only be achieved partly through a low competition level which in contrast increases the prices. The decision regarding the pricing rule is not difficult, as both rules perform similar.

References

[1]. Chen, D. L., Schonger, M., & Wickens, C. (2016). oTree - An open-source platform for laboratory, online, and field experiments. Journal of Behavioral and Experimental Finance, 9, pp. 88-97. [2]. European Commission. (2014). Guidelines on State aid for environmental protection and energy 2014-2020 (2014/C 200/01). Retrieved from http://eur-lex.europa.eu/legal- content/EN/TXT/?uri=CELEX: [3]. IRENA. (2017). Renewable Energy Auctions: Analysing 2016. Abu Dhabi: IRENA. [4]. Kagel, J. H., & Levin, D. (1986, Dec). The Winner's Curse and Public Information in Common Value Auctions. The American Economic Review, 5(76), pp. 894-920. [5]. Milgrom, P. R., & Weber, R. J. (1982). A Theory of Auctions and Competitive Bidding. Econometrica, 50(5), pp. 1089-1122. [6]. Thaler, R. H. (1988). Anomalies: The Winner's Curse. Journal of Economic Perspectives, 2(1), pp. 191-202.

Periodicals in Game Theory

CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT

Volumes 1–11

Edited by Leon A. Petrosyan Nikolay A. Zenkevich

St. Petersburg State University

A location-price game for locating a competitive facility in the plane

José Fernández 1, Said Salhi2 and Boglárka G.-Tóth 3

1,2University of Murcia, Spain [email protected] [email protected] 3UCAM, Spain [email protected]

Keywords: Location-price game, Continuous facility location, Delivered pricing

The problem of selecting the optimal or `best' location of the facilities and the setting of the `right' price of a product are two of the main drivers in a supply chain. The profit each firm gets is affected not only by the location of its facilities and the price that the firm sets in the market, but also by the location of its competitors and the prices they set. The maximization of profit for each competing firm can be seen as a location-price game, which has been studied since the work of Hotelling (1929). Most existing literature deals with a linear market, which is in part due to the complexity of solving the associated location problems in other location spaces such as the plane or the network. This paper deals with the competitive location problem on the plane where two firms aim to locate optimally one facility each while maximizing their market share. Most models in this context consider a refinement of the Nash equilibrium by using a two-stage process where the choice of the location is usually prior to the decision on the price. In the first stage, firms solve the location problem and given the outcome of the first stage, they then choose the price of the product in the second stage. The corresponding two-stage solution is called a subgame perfect Nash equilibrium. It captures the idea that when firms select their location, they all anticipate the consequences of their choice on the price. The division into two stages is motivated by the fact that the location decision usually requires a massive investment. This is a strategic decision which cannot be easily altered. On the other hand, the price decision is a short term or/and operational type decision which can be adjusted from one week to the next without too much inconvenience. This two-stage process is also adopted in this paper. Note that in this context firms are supposed to decide simultaneously the location and the prices. This is in contrast to Stackelberg competition models, where a firm (the leader) decides first the

38 location and then the prices for its facilities, taking into account that the other firm (the follower) will react by subsequently locating and selecting prices for its own facilities, knowing the previous decision of the leader,see Redondo et al. (2013}. The existence of a price equilibrium in the second stage of the game depends, among other factors, on the price policy adopted. There always exists a price equilibrium when each firm charges a specific price in each market area, which includes the transport costs (delivered pricing policy) (see for instance Dorta et al. 2005). In a duopoly with completely inelastic demand and constant marginal production costs, Lederer and Thisse (1990) show that a location equilibrium exists and that any global minimizer of the social cost is an equilibrium. This is the total delivered cost if each customer was served with the lowest marginal delivered cost. In this study, we assume that markets are aggregated at $n$ demand points, where a given homogeneous price-inelastic product will be sold by the competing firms. The firms manufacture and deliver the product to the customers, who always opt to buy from the firm that offers the lowest price. This is the case for instance of firms which have to serve clients who are within a distance such that the transportation costs are similar or higher than the production cost of the product to be delivered. In particular, we consider two firms locating a single facility each on the continuous space with the presence of constant marginal production costs. In this game the number of available actions for each player (firm) is infinitely many (as the location of the facilities vary in (a subset of) the plane and the prices may also be chosen continuously). The payoff function for a firm is the profit it gets, which is determined by the segment of the market served by each firm. This market depends on the prices, which in turn depend on the location of the facilities, as it will be shown. We first show that, for any given location, a price equilibrium exists, and then, using the equilibrium prices, we reduce the location-price game to a location game. The location game is then solved by minimizing the social cost. As this is a global optimization problem, we develop a branch-and-bound (B\&B) approach based on interval analysis for small/medium size problems. A new iterative Weiszfeld-like formula is developed and used as part of an adapted version of an `allocate-locate-allocate' scheme heuristic to cope with larger problem.

Complex Mechanisms of Reflexive Control in the Game of Collective Actions

Denis Fedyanin1

1Institute of Control Sceince, Russia [email protected]

Keywords: network, reflection, uncertainty, collective actions, deGroot model, Threshold model, complex mechanisms

Let us continue investigations of a game of collective actions that is described in [1]. There is a set of agent N={1,…, n} and with real not negative strategies and utility functions fi = xi(Σjxj-A) - xixi/ri, where 0< ri <1. This game looks like Cournot oligopoly [2]. Example of it is a game with a set of agent N={1,…, n} and with real not negative strategies and utility functions fi = xi(A-Σjxj) - xixi/ri,. There are some important differences that makes the game of collective actions looks like combination Cournot oligopoly and game theoretical modification of Granovetter [3] not just Cournot oligopoly. Breer Threshold model [4] is the one where utility functions are fi = xi(Σjxj-Ai) and set of actions is restricted to binary values – strategy is equal either 0 or 1. Anyway we can apply all ideas below for Cournot oligopoly as well but we haven’t applied it yet. There is a well-known way to find Nash equilibrium for the game of collective actions. It is to compose and solve a system of equations where strategy of each player is equals her best response that are xi=BRi(x-i)=2ri/(1-ri) (Σjixj - A). In this paper we propose to consider A as an uncertain parameter for agents and they have to make some suggestion about it. Their suggestion could be different. So player

I could believe that all utility functions are fi = xi(Σjxj-Ai) - xixi/ri. It coincide with the Nash equilibrium with certain value of parameter A, if there is A=Ai for any i common knowledge that A= Ai. If agents have no communication among them then they should compose and solve a system of equations. Strategy of each player is equals her best response that are xi=BRi(x-i)=2ri/(1-ri) (Σjixj - Ai) and for all other agents agent i make a best response that is based on his own believe. Thus from the i player’s point of view he should compose and solve system for following best responses xj=BRi(x-i)=2rj/(1-rj) (Σkjxk - Ai) for each j. There could be a communication between agents but then trust each one only partially and they can communicate according de Groot model [5]. There is not difference

40 if an existence of such communication is a common knowledge among all agents or it is not. Let their influences are xj then one should compose and solve system xi=BRi(x- i)=2ri/(1-ri) (Σjixj - Σjwj Ai) for each i. If there is a communication with no trust at all then all agents become stubborn and other opinion doesn’t change their opinions though they have to be taking into account. There is not difference if an existence of such communication is a common knowledge among all agents or it is not. The important information is that Ai is a common knowledge and that all agents are stubborn in our sense. Thus from the i player’s point of view he should compose and solve system for following best responses xj=BRi(x-i)=2ri/(1- ri) (Σjixj - Ai) for each i. We have composed and solved all these systems and found direct expressions for equilibria for all these alternatives. Another alternative is that as before there is a communication with no trust at all then all agents become stubborn and other opinion doesn’t change their opinions though they have to be taking into account. The only difference is that Ai is not a common knowledge. It makes thing much more complicated since we should take into account opinion of an agent about opinion of another agent … and so on… of another agent on a value Ai. There is a way to solve it [6-8] but we should specify some additional values so we do not include these results in this paper. Another topic for future research is to consider that agents doesn’t make themselves choose opinion about Ai as one not negative real number. Agents could think about it at in two different manners as least.

Agents can choose not a one number and a range or a number of numbers like “Ai is 3 or in a range from 1 to 2”. There is a way how to handle this and the most interesting results occur when we allow agents play this game several times. It will help us to use methods from [9]. Alternatively agents can choose a probability distribution of Ai. It will lead us to a Bayesian games [10] and modifications of de Groot model [11]. The research is supported by the grant 16-07-00816 of Russian Foundation for Basic Research. References

[1]. Fedyanin D.N., Chkhartishvili A.G. On a model of informational control in social networks // Automation and Remote Control. 2011. Vol. 72, No. 10. С. 2181–2187. [2]. Cournot, A. Reserches sur les Principles Mathematiques de la Theorie des Richesses, Paris: Hachette, translated as Research into the Mathematical Principles of the Theory of Wealth, New York: Kelley, 1960.

[3]. Granovetter, Mark, “Threshold Models of Collective Behavior,” American Journal of Sociology, (1978), 83, 489–515. [4]. Breer V.V., Novikov D.A., Rogatkin A.D. Mob Control: Models of Threshold Collective Behavior. Series: “Studies in Systems, Decision and Control”. – Heidelberg: Springer, (2017). 134 p. [5]. DeGroot M.H. Reaching a Consensus // Journal of American Statistical Assotiation. 1974. №69, P.118-121. [6]. Aumann R.J. Interactive epistemology I: Knowledge // International Journal of Game Theory. August 1999, Volume 28, Issue 3, pp 263–300. [7]. Novikov D., Chkhartishvili, A. Reflexion & Control: Mathematical models.// Series: Communications in Cybernetics, Systems Science and Engineering (Book 5). CRC Press. March 10, 2014. P. 298 [8]. Shoham Y., Leyton-Brown K.. 2008. Multiagent Systems: Algorithmic, Game- Theoretic, and Logical Foundations. Cambridge University Press, New York, NY, USA. [9]. Fedyanin D. Threshold and Network generalizations of Muddy Faces Puzzle / Proceedings of the 11th IEEE International Conference on Application of Information and Communication Technologies (AICT2017, Mosсow). М.: IEEE, 2017. V.1. С. 256- 260. [10]. Harsanyi, J. C., "Games with Incomplete Information Played by Bayesian Players, I-III." Management Science 1967/1968. 14 (3): 159-183 (Part I), 14 (5): 320-334 (Part II), 14 (7): 486-502 (Part III). [11]. Sarwate A.D., Javidi T. Distributed learning from social sampling, 46th Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, 21-23 March 2012, IEEE, 2012, pp. 1-6.

Journals in Game Theory

GAMES AND ECONOMIC BEHAVIOR

Editor-in-Chief Ehud Kalai

ELSEVIER

Differential-Game-Based Driver Assistance System for Fuel-Optimal Driving

Michael Flad1, Julian Ludwig2 and Soeren Hohmann3

1,2,3Karlsruhe Institute of Technology, Germany 1 [email protected] [email protected] [email protected]

Keywords: Differential Game, Advanced Driver Assistance System, Human-Machine Interaction, Moving Information Horizon

We would be pleased to present our work of a cooperative driver assistance system on your conference. The systems supports the driver to achieve a fuel optimal driving behaviour. Interaction between driver and assistance system is modelled as a differential game. We would like to present the differential game that model the system human interaction, the mathematical concept to use this to design an ADAS and the concept we use to calculate the solutions of the resulting game problems.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА, ИНФОРМАТИКА, ПРОЦЕССЫ УПРАВЛЕНИЯ

Главный редактор Л.А. Петросян

Издательство Санкт-Петербургского государственного университета

Mixed and behavior strategies in finitely additive decision problems

János Flesch1, Dries Vermeulen2 and Anna Zseleva3

1,2Maastricht University, Netherlands [email protected] [email protected] 3Higher School of Economics, St Petersburg, Russia [email protected]

Keywords: Mixed strategies, behavior strategies, finitely additive probability measures, equivalent strategies, Kuhn's theorem

The decision problem. We consider the following decision problem. Let A denote a set of actions, having at least two elements, and let P denote the set of all infinite sequences of elements of A. Let u:P→R denote a bounded payoff function. At every period t=1,2,…, the decision maker chooses an action a ∈A, knowing the history consisting of t his previously chosen action a1,…at-1.This induces an infinite sequence (a1,a2,…) of actions, which we call a play. The decision maker’s payoff is given by u(a1,a2,…). Mixed and behavior strategies. In the usual setup when probability measures are countably additive, a fundamental theorem of Kuhn (Maschler at all (2013), Theorem 6.15.) implies that for every mixed strategy of the decision maker there exists an equivalent behavior strategy. That is, this behavior strategy induces the same probability measure on the set of plays, and therefore it induces the same expected payoff, provided that the payoff function is Borel measurable. In this paper we examine to which extent this remains valid when probability measures are only assumed to be finitely additive. From now on, we call finitely additive probability measures charges. For more on charges, see Finetti (1972), Savage (1972) or Bingham (2010). A pure strategy is a function that specifies an action at each history. A mixed strategy is a charge on the set of pure strategies, with the interpretation that the decision maker should choose a pure strategy according to this charge. Since each pure strategy induces a unique play, each mixed strategy induces a charge on the set of plays. A behavior strategy gives recommendations to the decision maker in an essentially different fashion. A behavior strategy prescribes a charge on the action space at each history. The interpretation is that, at each history that arises during the play of the

44 decision problem, the decision maker should choose an action according to the corresponding charge. For a behavior strategy, it is not immediately clear how to define the charge it induces on the set of plays. Various alternatives have been considered in the literature, motivated by mainly conceptual but to some extent also technical reasons, see for example Dubins and Savage (2014) and Purves and Sudderth (1976). A classical approach to this question is given in Dubins and Savage (2014). Their focus is on the the algebra of all clopen (closed and open) sets of plays, when the set of actions is endowed with the discrete topology and the set of plays is endowed with the product topology. Then, for each behavior strategy they define a natural charge on this algebra. The approach of Dubins and Savage (2014) is rather minimal in the sense that essentially all later papers in the literature agree to define the induced charge on this algebra exactly the same way, and only ask the question how to define it on larger algebras. The general approach to define the charge induced by a behavior strategy on the set of plays goes as follows. First, fix an algebra on the set of plays. One could think of this algebra as the collection of sets which we think should obtain a probability under each behavior strategy. Then, given this algebra, specify for each behavior strategy a certain charge on this algebra. This enables us to define equivalence between mixed and behavior strategies. We say that a mixed strategy and a behavior strategy are equivalent whenever they generate the same charge on the chosen algebra. Our contribution. In this paper, we show the following three results. 1. With respect to any algebra on the set of plays and any definition of the charge induced by a behavior strategy, each behavior strategy has an equivalent mixed strategy. 2. If the action space is finite, then with respect to the approach by Dubins- Savage2014 on the algebra of all clopen sets of plays, every mixed strategy has an equivalent behavior strategy. So, Kuhn’s theorem extends in this case. 3. The approach by Dubins-Savage2014 algebra is maximal with this property. That is, with respect to any algebra that is strictly larger than the algebra of all clopen sets of plays, there is a mixed strategy that has no equivalent behavior strategy. In fact, for every such algebra and mixed strategy, there is a payoff function such that this mixed strategy is the unique optimal strategy. That is, behavior strategies stricly underperform compared to mixed strategies.

4. If the action space is infinite, then even with respect to a weaker approach than that of Dubins-Savage2014 there is a mixed strategy without an equivalent behavior strategy. References

[1]. Dubins, LE and Savage, LJ, edited and updated by Sudderth, WD and Gilat, D (2014) How to gamble if you must: inequalities for stochastic processes. Dover Publications, New York. [2]. Bingham, NH (2010) Finite additivity versus countable additivity. Electronic Journal for History of Probability and Statistics, vol 6, no 1. [3]. de Finetti, B (1972) Probability, Induction and Statistics. Wiley, New York. [4]. Maschler, M, Solan, E and Zamir, S (2013) Game Theory. Cambridge University Press. [5]. Purves, R and Sudderth, W (1976) Some finitely additive probability. Annals of Probability, 4, 259-276. [6]. Savage, LJ (1972) The foundations of statistics. Dover Publications, New York.

Journals in Game Theory

INTERNATIONAL GAME THEORY REVIEW

Editors

David Yeung Leon Petrosyan Hans Peters

WORLD SCIENTIFIC

Decoding of public signals about the state of a game with incomplete information

Misha Gavrilovich1, and Victoria Kreps2

1HSE, Russia [email protected] 2Senior researcher, Russia [email protected]

Keywords: Zero-sum game, incomplete information, random signal, finite automata

We consider two-person zero-sum matrix game A(p) with incomplete information on both sides given by two square payoff matrices A and A (Aumann, 1 2 Maschler, 1995). Before the game starts a chance move determines the ”state of nature” k∈K={1,2} and therefore the payoff matrix A : with probability p matrix A is played and k 1 with probability 1−p matrix A is played. Both players know the probability p and do not 2 know the result of the chance move, so they do not know what matrix game is played. A random public signal f is coming on the state of this game: f is a random injective function of the state of the game, f(k), k=1,2. The codomain of the function f is a set S of binary strings (i.e. strings consisting of symbols 0’s and 1’s). A signal f is determined by a

f,1 f,2 푓 푓 푓 푓 decomposition S into two nonempty subsets S and S as 푆 = 푆1 ∪ 푆2 , 푆1 ∩ 푆2 =ø

The signal f picks strings in Sf,k randomly and uniformly with probability 1/card(Sf,k), for k=1,2 (uniform probability distributions over Sf,1 and over Sf,2). Here card(⋅) denotes the number of elements of the finite set. Function f is known to both players. To decode the signal and to learn the state of the game each player chooses a finite automaton. Computational capacity of Players are limited, see Neyman (1998), Hernández, Solan (2016). Player 1 chooses an arbitrary automaton of size at most m. Player 2 chooses an automaton of size at most n, where n

L Here 푆 = {0,1}퐿 has 2L elements and there are 22 different signalling structures

47 corresponding to such S. It is supposed that the number of elements of Sf,1 is proportional to probability p of state 1. There are 2Lp2L different decompositions of this form (different signals) corresponding to a fixed probability p. This class of signals is denoted by F(p). In Gavrilovich, Kreps (2018) is given the following low bound for the size of automata available to Player 1 which allows him “to compute” the state of the game for any signal f∈F(p). Proposition. Let probability p be fixed. If 푚 ≥ 푚(푝) = min(푝, 1 − 푝) 퐿2퐿 then for any signal f∈F(p) Player 1 can choose a finite automaton which distinguishes perfectly 푚,푛 the state of the game 풜푓 (푝). Here we give an essentially better estimate for this size. Theorem 1. Let probability p be fixed and min(푝, 1 − 푝) ≥ 1/2퐿. If the size m of automata available to Player 1 greater or equal to

퐿 퐿−퐸푛푡(−푙표푔2(min(푝,1−푝)) 푚̃(푝) = min(푝, 1 − 푝) 퐸푛푡(−푙표푔2(min(푝, 1 − 푝)))2 + 2 + 1 then for any signal f∈F(p) Player 1 can choose a finite automaton which 푚,푛 distinguishes perfectly the state of the game 풜푓 (푝). Here Ent(x) is the integer part of x. It is easy to see that if L≥3 then ̃m(p)

̃m(1/2)=2L+1

퐿+푙표푔2(− min(푝,1−푝)푙표푔2 min(푝,1−푝)) 푛(푝) = 2 /(퐿 + 푙표푔2(− min(푝, 1 − 푝) 푙표푔2 min(푝, 1 − 푝))) then there exists a signal f∈ such that any automaton of size n could not distinguish f perfectly. Support from Basic Research Program of the National Research University Higher School of Economics is gratefully aknowledged. This study was partially supported by the grant 16-01-00124-a of Russian Foundation for Basic Research.

References

[1]. Aumann R. and M. Maschler, 1995. Repeated Games with Incomplete Information. The MIT Press: Cambridge, Massachusetts - London, England. [2]. Neyman A., 1998. Finitely repeated games with finite automata. Math. Oper. Res., 23, 513-–552.

[3]. Hernández P., E. Solan, 2016. Bounded computational capacity equilibrium. Journal of Economic Theory, 163, 342-–364. [4]. Gavrilovich M., V. Kreps, 2018. Games with symmetric incomplete information and asymmetric computational resources. International Game Theory Review, Vol. 20, No. 2.

Journals in Game Theory

INTERNATIONAL JOURNAL OF GAME THEORY

Editor Shmuel Zamir

SPRINGER

Banking oligopoly market segmentation based on modified Cournot oligopoly model

Vadim Glukhov 1, and Olga Kuznetsova2

1,2Samara National Research University, Russia [email protected] [email protected]

Keywords: Cournot oligopoly model, banking market, market segmentation, advertising costs Nowadays market researches are very interesting and popular topic in different areas. Researchers are studying how firms compete with each other in perfect competition and find equilibrium on product markets. Also, there are several developments about oligopoly on different types of markets: telecommunication market [1], electricity market [2] and heat market [3]. But there is another area that is not so explored as others: banking oligopoly. Researches elaborated several banking models based on Markov chains [4], Cournot-Bertrand oligopoly model [5], Cournot oligopoly model [6] and some other models and methods [7]. The main purpose of the research is to investigate impact of indirect costs on the segmentation of banking services market in oligopoly situation. We mean advertising as indirect costs because these costs do not participate in product value chain. Quantity supplied by firm as a depended variable from advertising is a feature of this research. We are considering modified Cournot oligopoly model. In this model, we have an additional variable which influence on quantity supplied by firm. This variable is the advertising cost. Previously there was developed a regression model on dependence of quantity supplied by firm on advertising costs [8]. A subsequent study determines the impact of advertising costs on the structure of the banking services market under oligopoly conditions. This model is universal and can be applied in other markets. In the research for the modified model of Cournot's oligopoly, the Cournot equilibrium, and Stackelberg equilibrium and disequilibrium were also investigated.

References [1]. Biryukova I.A., Geraskin M.I. Structural analysis of the oligopoly market based on the model of the reflective game on the example of the telecommunications market in Russia // Aktual'nye problemy ekonomiki i prava. 2017. Т. 11. №4 (44). С. 66-81.

[2]. David M. Newbery, Thomas Greve. Energy economics. The strategic robustness of oligopoly electricity market models. 2017. V. 68. Pp. 124-132 [3]. Andrey V.PenkovskiiValery A.StennikovOleg V.KhamisovEkaterina E.MednikovaIvan V.Postnikov. Energy Procedia. Search for a Market Equilibrium in the Oligopoly Heat Market. 2017. V. 105. Pp. 3158-3163. [4]. Bubnova G.V., Seslavina E.A. Mathematical models of the dependence of the final sales volume on the effectiveness of advertising // Prikladnays informatika. 2008. №2. С.118- 123. [5]. Haraguchi, J. & Matsumura, T. J Econ (2016) 117: 117. https://doi.org/10.1007/s00712-015-0452-6 [6]. Cetorelli, Nicola and Peretto, Pietro F., Oligopoly Banking and Capital Accumulation (December 2000). FRB of Chicago Working Paper No. 2000-12; Duke Department of Economics Research Paper No. 19. [7]. Florian Leon. Measuring competition in banking: A critical review of methods. 2014.12. 2015. [8]. Glukhov V.N., Kuznetsova O.A. Econometric modeling of the bank's profit depending on its investments in information and communication technologies and advertising using the programming language R // Prikladnaya informatika i voprosy upravleniya. 2017. №1. С. 81-86.

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

Game-theoretical Problem of Pollution Control with Application to Ecological Management in Irkutsk Region of Russia

Ekaterina Gromova 1, Anna Tur 2 and Alla Vikulova 3

1Adelson School of Entrepreneurship, Israel [email protected] 2Haifa University, Human Service Department, Israel [email protected] 3Technion, Israel [email protected]

Keywords: pollution control, ecological management, cooperative differential games, random time horizon, time-consistency, imputation distribution procedure

Game theory is a common tool for solving many economic problems. One particular application deals with environmental problems [2,3,5]. The main issue here is the control of emissions. Harmful atmospheric emissions are a major factor affecting a variety of processes and objects, including the Earth climate. The importance of this problem is being actively discussed, in particular during the 21st Session of the Conference of the Parties to the United Nations Framework Convention on Climate Change (COP21/CMP11), which took place in Paris in December 2015. One of the concepts of participation in an international agreement that promotes normalization of the climate is that each country should make its contribution, adopted at the national level, as a public knowledge. Currently, there are a large number of publications on this topic in the field of game theory, which, nevertheless, mostly are of a theoretical nature, and the given examples are merely illustrative. In contrast to that, in this contribution we consider the application of a game-theoretical model [6] to the real ecological situation in Irkutsk region of Russia. Air pollution in cities and towns of the region is a consequence of emissions from thermal power, petrochemical, coal, and woodworking plants, motor vehicles, etc. After studying various data on the volume of emissions of pollutants into the atmosphere [7], we decided to pay attention to environmental problems in the Irkutsk region. The most difficult environmental situation from all the cities in the Irkutsk region is observed in Bratsk. In the list of 100 the most polluted towns of Russia in 2011 year Bratsk takes the second place [11]. According to [9, 10], there are three main sources of air pollution in

Bratsk: JSC "Bratsk Aluminum Plant", OJSC “Ilim Group” in Bratsk, units of JSC "Irkutskenergo" (Thermal Power Station-6 and others). These three sources summarily emit 93% of the total volume of pollutants to the atmosphere [9]. In this paper we present the data on the net profit of each of the companies and the air pollution corresponding to the situation for 2011 [10, 12, 13, 14]. According to these real data, the parameters of the differential game are calculated. The game is solved both in the cooperative and noncooperative versions. Finally, we draw conclusions about the measures that would contribute to the normalization of the environmental situation in the region.

References

[1] L. Petrosyan, N. Danilov, Stability of solutions of non-antagonistic differential games with transferable payoffs, Vestnik LGU, ser. 1, 1979, 1, pp. 52—59. [2] E. Dockner, S. Jorgensen, N. V. Long and G. Sorger, Differential Games in Economics and Management Science. Cambridge: Cambridge University Press, 2000. [3] L. Petrosjan and G. Zaccour, Time-consistent Shapley value allocation of pollution cost reduction, Journal of Economic Dynamics and Control, 27 (3), pp. 381—398, 2003. [4] D. W.K. Yeung and L. A. Petrosyan, Subgame Consistent Economic Optimization. New York: Birkhauser, 2012, 395 p. [5] L. A. Petrosyan, N. A. Zenkevich and E. V. Shevkoplyas, Game Theory, 2012, Spb.: BHV Press, 424 p. (in Russian). [6] E. V. Gromova, A. V. Tur and L. I. Balandina, A game-theoretic model of pollution control with asymmetric time horizons, "Contributions to Game Theory and Management, vol. 9, pp. 170—179, 2016. [7] RBC rating of companies in terms of environmental pollution, URL: http://rating.rbc.ru/article.shtml?2006/01/25/4627399 (as on 26.02.2016). [8] Norms of payment for air emissions of pollutants by stationary sources, URL: http://base.consultant.ru/cons/cgi/online.cgi?req=doc;base=LAW;n=172885 (as on 20.03.2016). [9] N.N. Yushkov, M.R. Erofeeva, A.D. Sinegibskaya, Influence of the economic entities operating on the territory of Bratsk on the state of environmental components, Systems. Methods. Technologies, 3 (27), pp. 128—138, 2015 (in Russian) [10] L.A. Kaverzina, State of Municipal Formation Bratsk Economy in Post-Crisis Period, Financial analytics, 48(186), pp. 24—34, 2013 (in Russian) [11] State report “On the state of the environment in Russian Federation and its protection”, 2011 (in Russian) http://www.mnr.gov.ru/docs/o_sostoyanii_i_ob_okhrane_okruzhayushchey_sredy_rossiy skoy_federatsii/130175/ [12] State report “On the state of the environment in Irkutsk region and its protection”, 2011 (in Russian) http://irkobl.ru/sites/ecology/working/woter/aukzion/doklad2011.pdf [13] Financial statement of OJSC "RUSAL Bratsk", 2011 (in Russian) https://braz-rusal.ru/ [14] Annual report of OJSC “Ilim Group”, 2011 (in Russian) http://www.ilimgroup.ru/aktsioneram/raskrytie-informatsii/godovji-otchet/

Assignment games with externalities revisited

Jens Gudmundsson1, and Helga Habis 2

1Department of Economics, University of Copenhagen, Denmark [email protected] 2 Corvinus University of Budapest, Hungary [email protected]

Keywords: Two-sided matching, Assignment games, Externalities, Stability We study assignment games with externalities. The value that a firm and a worker create depends on the matching of the other firms and workers. We ask how the classical results on assignment games are affected by the presence of externalities. The answer is that they change dramatically. Though stable outcomes exist if agents are “pessimistic”, this is a knife-edge result: we show that there are problems in which the slightest optimism by a single pair erases all stable outcomes. If agents are sufficiently optimistic, then there need not exist stable outcomes even if externalities are vanishingly small. The negative result persists also when we impose a very restrictive structure on the values and the externalities. Furthermore, stability and efficiency no longer go hand in hand and the set of stable outcomes need not form a lattice with respect to the agents’ payoffs.

МАТЕМАТИЧЕСКАЯ ТЕОРИЯ ИГР И ЕЕ ПРИЛОЖЕНИЯ

Главный редактор: Л.А. Петросян Зам. главного редактора: В.В. Мазалов Ответственный секретарь: Н.А. Зенкевич Выпускающий редактор: А.Н. Реттиева

Dynamic search model of two moving objects on a graph

Vasily Gusev1

1Institute of Applied Mathematical Research of Karelian Research Centre, Russia [email protected]

Keywords: search theory, game theory, search model, search on the graph, dynamic search, multi-stage game

Consider a multistage non-cooperative search game with three players (two mobile objects and a searcher) on a tree graph. The mobile objects are hiders numbered as first and second. Both mobile objects take start from the root vertex of the graph. We assume the searcher is watching the graph from above. At one stage each mobile object moves independently from one another from vertex i (mobile object’s position) to vertex j with some probability. At one stage it is possible to move only across one edge out of the set E. The searcher is assumed to know the initial position of the mobile targets. Each next position of the hiders is unknown to either the searcher or the other mobile object. The game proceeds from step k to step k-1. The mobile objects do not return to the vertexes they have already been to. Let us determine the players’ payoff functions at stage 1.

−훼푖 휑푖 퐻1(1, 푝(푔), 휑(푔)) = ∑푖∈푐ℎ(푔) 푝푖(1 − 푒 )

−훼푗 휑푗 퐻2(1, 푝(푙), 휑(푙)) = ∑푗∈푐ℎ(푙) 푝푗(1 − 푒 )

−훼푖 휑푖−훼푗 휑푗 퐻3(1, 푝(푔), 푞(푙), 휑(푔), 휑(푙)) = ∑푖∈푐ℎ(푔) ∑푗∈푐ℎ(푙) 푝푖 푞푗(1 − 푒 ) p(g)∈P(g),q(l)∈Q(l),ϕ(g)∈Ψ(g),ϕ(l)∈Ψ(l),

where 퐻1(∙), 퐻2(∙), 퐻3(∙) is the probability of detecting the first, the second and at least one mobile object, respectively.The first mobile object moves from vertex g to a new vertex i,(g,i)∈E with probability 푝푖 ≥ 0, ∑푖∈푐ℎ(푔) 푝푖 = 1. Similarly, the second mobile object moves from vertex l to j,(l,j)∈E with probability 푞푗 ≥ 0, ∑푗∈푐ℎ(푙) 푞푗 = 1

−α ϕ In most studies the detection probability of a target was equaled to 1−e i i. If the amount of the resource is sent to infinity, such detection probability will tend rapidly to 1. In practice, however, where the resource amounts are great, the methods and algorithms of tracking a mobile object may not be as simple, i.e. other functions setting the capture probability should be considered instead of the exponential detection probability. Let f(α ϕ ) be the probability of not detecting an object at vertex i. The higher the i i ϕ , the lower the probability of not detecting the object. We therefore believe that f(α ϕ ) is i i i a strictly decreasing function. If the resource amount allocated to vertex i equals zero, the non-detection probability equals 1, i. e. f(0)=1.. The property of probability dictates that the inequality f(α ϕ )>0 must hold. The value of the function f(α ϕ )≠0∀ϕ because f(α ϕ ) i i i i i i i is strictly decreasing. Let

Theorem Let f(x) be a differentiable on the interval [0;+∞), convex downward, strictly decreasing function. Then, for the zero-sum game Γ=〈I,II;P,Ψ;H(P,Ψ)〉 the value of the game and the equilibrium point have the form

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

Refund of Uncertainty-Reducing Prequalification Measures in Procurement Auctions

Ann-Katrin Hanke1, Karl-Martin Ehrhart2 and Marion Ott3

1,2 Karlsruhe Institute of Technology, Germany [email protected] [email protected] 3RWTH Aachen University, Germany [email protected]

Keywords: Procurement Auction, Mechanism Design, Cost Uncertainty, Sunk Costs, Refund In procurement auctions, the auctioneer (i.e., the buyer) often requires, as prequalification, that the bidders must achieve a certain stage of project preparation before being allowed to participate in an auction. The most prominent example are the spreading auctions for renewable energy support, which are conducted worldwide (Wigand, Förster, Amazo, & Tiedemann, 2016). In these auction, potential bidders usually are obliged to perform certain physical prequalifications to ensure the sincerity of their project and because most of the required activities are needed anyway to realize the project (del Rio, Haufe, Wiegand, & Steinhilber, 2015). The effects of these requirements are twofold. On the one hand, prequalifications usually generate valuable information that reduces the bidders’ uncertainty regarding their project costs, which, in turn, increases the probability of realizing their project (Kreiss, Ehrhart, & Haufe, 2017). This is one of the main advantages of the introduction of mandatory prequalifications. On the other hand, prequalification activities cause costs. The situation can be compared to auctions with entry costs (Levin & Smith, 1994). This introduces the risk of participation costs (Tan & Yilankaya, 2006), as they have to pay for project preparations before knowing if they are actually allowed to realize their projects. In this case, the decision to participate and, thus, to carry out the required preparation activities depends on the project developer’s knowledge about the competitiveness of his project (Samuelson, 1985). If he decides to participate, he will gain further knowledge that improves the accuracy of his cost calculations by performing the required project preparations. In this paper we examine the model of a procurement auction, in which bidders are required to perform project preparations that cause costs but reduce uncertainty. This requirement reduces the expected number of bidders and involves the risk of inefficiency by a possible exclusion of the strongest bidder. We present and compare different

57 measures (scenarios) for the auctioneer to counteract these effects. In general, all measures include the refund of a share of the preparation costs, which the auctioneer pays back to some or all bidders. The four considered scenarios are the benchmark case with no payback at all, the scenario where the winner is further rewarded by the repayment, the scenario where only the non-awarded bidders are compensated, and the case where every participant is paid back his share, which is equivalent to lowered prequalification costs. A similar approach can also be found in (Courty & Li, 2000), where price discrimination by refund is examined in the light of mechanisms. We show that the introduction of compensation payments increases the bidders’ willingness to participate, and thus the expected efficiency rises as well as the certainty about future costs. As a main feature we will compare the different expected costs for the auctioneer to choose the design in which he expects the lowest costs. Notable, it is always optimal for a cost minimizing auctioneer to conduct an auction where only the awarded bidder is further compensated, since the positive effect of a lower probability for paying a high reservation price outweighs the negative effects of the payback costs. Comparing the other cases, up to a certain level of compensation, it is preferable to execute an auction with refund for all participants than to pay only the non-awarded bidders. Once this level is exceeded, the negative effect of additional compensated bidders overweighs and, thus, it is better for the auctioneer to only refund non-awarded bidders. The only exception is a complete refund of the preparation costs, where in both cases all potential bidders participate and, thus, have to be compensated. This is because the awarded bidder, who is not compensated, can expect a higher payment since bidders bid less aggressive in the non-awarded bidder scenario. The optimal share of the preparation costs that are refunded depends on the implemented measure. In the awarded bidder’s scenario, the expected auctioneer’s costs decrease when the compensation increases. This leads to an optimal approach of compensating all project preparation costs. If only non-awarded bidders or all bidders are compensated, the expected costs increase monotonously with the share of refunded costs, and thus the smaller the chosen share, the lower the expected costs. In conclusion we advise accordingly to the preferred characteristics of an auction, such as level of competition, efficiency and minimal costs.

References [1]. Courty, P., & Li, H. (2000). Sequential Screening. Review of Economic Studies, pp. 697-717.

[2]. del Rio, P., Haufe, M.-C., Wiegand, F., & Steinhilber, S. (2015). Overview of Design Elements for RES-E Auctions. http://auresproject.eu/files/media/documents/design_elements_october2015.pdf. [3]. Kreiss, J., Ehrhart, K.-M., & Haufe, M.-C. (2017). Appropriate Design of Auctions for Renewable Energy Support - Prequalifications and Penalties. Energy Policy, pp. 512- 520. [4]. Levin, D., & Smith, J. L. (1994). Equilibrium in Auctions with Entry. The American Economic Review, pp. 585-599. [5]. Samuelson, W. F. (1985). Competitive Bidding with entry costs. Economics Letters, pp. 99-117. [6]. Tan, G., & Yilankaya, O. (2006). Equlibria in second price auctions with participation costs. Journal of Economic Theory, pp. 205-2019. [7]. Wigand, F., Förster, S., Amazo, A., & Tiedemann, S. (2016). Auctions for Renewable Energy Support: Lessons Learnt from International Experiences. http://auresproject.eu/sites/aures.eu/files/media/documents/aures_wp4_synthesis_report. pdf.

Periodicals in Game Theory

ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES

Volumes 1–14

Series Editor Tamer Basar

Birkhäuser, Boston

Optimal Bundling Strategies for Complements and Substitutes with Heavy-Tailed Valuations

Rustam Ibragimov1, Artem Prokhorov2 and Johan Walden3

1Imperial College Business School, United Kingdom [email protected] 2U Sydney Business School, Australia [email protected] 3University of California, Berkeley, United States [email protected]

Keywords: Optimal bundling strategies, multiproduct monopolist, interrelated goods, substitutes, complements, heavy-tailed valuations, robustness

We develop a framework that allows one to model the optimal bundling problem of a multiproduct monopolist providing interrelated goods with an arbitrary degree of complementarity or substitutability. Using the framework, we derive characterizations of the optimal bundling strategies in the case of heavy-tailed valuations of the products by the consumers. The results show that patterns in the optimal bundling strategies are the opposites of one another, depending on the degrees of heavy-tailedness of consumers' valuations and the degrees of complementarity and substitutability among the goods provided. For substitutes with moderately heavy-tailed valuations, the seller's optimal bundling strategies are the same as in the case of independently priced goods with thin- tailed (log-concavely distributed) or moderately heavy-tailed valuations discussed in the previous literature. That is, the seller prefers separate provision for substitute goods with high marginal costs and provision in a single bundle for substitutes with low marginal costs. These conclusions are reversed for complements with suciently (extremely) heavy- tailed valuations. In such a case, seller's optimal strategy is to provide complement goods with low marginal costs separately, and as a single bundle under high marginal costs. As discussed in the paper, the conclusions may help to explain several bundling strategies commonly observed in real-world markets.

Cooperative Solutions of Working Capital Cost Game in Distributive Supply Network

Anastasiia Ivakina1 and Nikolay Zenkevich2

1,2Graduate School of Management, St. Petersburg State University, Russia [email protected] [email protected]

Keywords: Working Capital Management, Supply Chain Finance, Cooperative Game with Coalitional Structure

Working capital management (WCM) is increasingly recognized as important means of liquidity and profitability improvement (Deloof, 2003; García-Teruel and Martínez-Solano, 2007; Johnson and Templar, 2011; Viskari et al., 2011; Viskari et al., 2012; Viskari and Karri, 2012; Talonpoika et al., 2016), specifically in terms of globalization and growing competition between supply chains. The physical product and information flows have long been addressed by researchers and practitioners, unlike the upstream and downstream flows of money (Hofmann and Kotzab, 2010; Protopappa- Sieke and Seifert, 2017), although rising financial risk in SCs stimulated management to recognize that the financial side of supply chain management (SCM) is a promising area for improvements. Nevertheless, companies still focus on their individual SC issues and take their own interests into account rather than understanding the whole SC and coordinating with their partners (Wuttke et al., 2016, 2013). Authors address this distinct gap and show that WCM optimizes planning, managing, and controlling of SC cash flows by developing cooperative game of working capital management aimed at minimizing total financial costs associated with each SC stage. The model is further verified on the grounds of the combination of mathematical modeling and case study of Russian collaborative SC from ICT industry. The suggested model introduces holistic perspective to WCM and provides financial illustration for the motivation of SC partners to cooperate in order to simultaneously achieve target levels of working capital investments and improve individual financial performance through collaborative actions as a result strengthening whole SC competitiveness and value. We develop a model that analyses working capital management process for 3- stage supply network. The focal network is a distributive supply network consists of M suppliers, one distributor and N retailers connected through material, information and financial flows. The members of the network can form various coalitions with the

61 distributor. Each member’s working capital position is constrained by liquidity and profitability requirements. As such, they face the need to control and manage financial costs associated with each stage. We construct characteristic function of each coalition as a minimum value of the sum of financial costs associated with working capital allocation. For this cooperative game with coalitional structure we investigate such optimal imputations as C-core, Shapley, Owen and Aumann-Dreze values and provide their comparative analysis. Theoretical results are illustrated with the numeric example of a supply network from ICT industry.

References

[1]. Deloof, M. (2003). “Does working capital management affect profitability of Belgian firms?”. Journal of Business Finance and Accounting, 30(3-4), 573–587. [2]. García-Teruel P.J. and Martínez-Solano, P. (2007). “Effects of working capital management on SME profitability”. International Journal of Managerial Finance, 3(2), 164-177. [3]. Hofmann, E., and Kotzab, H. (2010). “A Supply Chain-Oriented Approach of Working Capital Management”. Journal of Business Logistics, 31(2), 305–330. [4]. Johnson, M. and Templar, S. (2011). “The relationships between supply chain and firm performance: the development and testing of a unified proxy”. International Journal of Physical Distribution & Logistics Management, 41(2), 88–103. [5]. Protopappa-Sieke, M., and Seifert, R. W. (2017). “Benefits of working capital sharing in supply chains”. Journal of the Operational Research Society, 68(5), 521-532. [6]. Talonpoika, A. M., Kärri, T., Pirttilä, M., and Monto, S. (2016). “Defined strategies for financial working capital management”. International Journal of Managerial Finance, 12(3), 277-294. [7]. Viskari, S., and Kärri, T. (2012). “A model for working capital management in the inter- organisational context”. International Journal of Integrated Supply Management, 7(1-3), 61-79. [8]. Viskari, S., Pirttilä, M., and Kärri, T. (2011). “Improving profitability by managing working capital in the value chain of pulp and paper industry”. International Journal of Managerial and Financial Accounting, 3(4), 348-366. [9]. Viskari, S., Ruokola, A., Pirttilä, M., and Kärri, T. (2012). “Advanced model for working capital management: bridging theory and practice”. International Journal of Applied Management Science, 4(1), 1-17. [10]. Wuttke, D. A., Blome, C., and Henke, M. (2013). “Focusing the financial flow of supply chains: An empirical investigation of financial supply chain management”. International Journal of Production Economics, 145(2), 773–789. [11]. Wuttke, D.A., Blome, C., Sebastian Heese, H., Protopappa-Sieke, M., 2016. Supply Chain Finance: Optimal Introduction and Adoption Decisions. International Journal of Production Economics 178, 72–81.

Time-Consistency of Nash Bargaining Solution for a Special Class of Two-stage Network Games

Junnan Jie1 and Leon Petrosyan2

1St.Petersburg State University, China [email protected] 2St.Petersburg State University, Russia [email protected]

Keywords: network, time-consistency, Nash bargaining solution

In this paper, a Special Class of two-stage network games are studied. At first stage each player choose his partners -the players with whom he wants to form links. Choosing partners and establishing links,players,thereby,form a network. Having formed the network,each player chooses control influencing his payoff on second stage. Cooperative setting is considered. In cooperative case, we use Nash Bargaining Solution as a solution concept. It is demonstrated that the Nash Bargaining Solution satisfy the time consistency property. The proof of the Nash Bargaining Solution satisfy the time consistency property is illustrated.

VESTNIK OF SAINT PETERSBURG UNIVERSITY. MANAGEMENT

Editor-in-Chief Natalia P. Drozdova

St. Petersburg University Press

Endogenous graph formation in TU games with major player

Anna Khmelnitskaya1, Elena Parilina2 and Artem Sedakov3

1Imperial College Business School, United Kingdom [email protected] 2U Sydney Business School, Australia [email protected] 3University of California, Berkeley, United States [email protected]

Keywords: Graph games, Myerson value, Average tree solution, Centrality rewarding, Shapley value, Graph formation, Subgame perfect equilibrium

Our main goal is to provide comparative analysis of several procedures for endogenous dynamic formation of the cooperation structure for TU games with major player [5, 6]. In the paper we consider both approaches to endogenous graph formation, of Aumann and Myerson [1], and of Petrosyan and Sedakov [7]. For the evaluation of the pros and cons when adding of a new link is in question, along with the Myerson value [4] we consider also the average tree solution introduced by Herings, van der Laan, Talman and Yang in [2] and the centrality rewarding Shapley and Myerson values, recently introduced by Khmelnitskaya, van der Laan and Talman in [3].

References

[1]. Aumann, R., Myerson, R. [1988] Endogenous Formation of Links Between Players and Coalitions: An Application of the Shapley Value. In: The Shapley Value: Essays in Honor of Lloyd S. Shapley, Roth, A. (ed.), Cambridge University Press, 175–191. [2]. Herings, P. J. J., van der Laan, G., Talman, A.J.J., Yang, Z. [2010] The average tree solution for cooperative games with communication structure. Games and Economic Behavior, 68, 626–633. [3]. Khmelnitskaya, A.B., van der Laan, G., Talman, A.J.J. [2016] Centrality rewarding Shapley and Myerson values for undirected graph games, Memorandum 2057 (September 2016), Department of Applied Mathematics, University of Twente, Enschede, The Netherlands, ISSN 1874-4850. [4]. Myerson, R. [1977] Graphs and cooperation in games. Mathematics of Operations Research, 2, 225–229. [5]. Parilina, E., Sedakov, A. [2014] Stable cooperation in graph-restricted games. Contributions to Game Theory and Management, 7, 271–281. [6]. Parilina, E., Sedakov, A. [2016] Stable Cooperation in a Game with a Major Player. International Game Theory Review, 18(2), art.no. 1640005. [7]. Petrosyan L. A. and Sedakov A. A. [2014] Multistage network games with perfect information. Automation and Remote Control, 75(8), 1532–1540.

The average tree value for graph games with main players and application to hub and spoke models

Anna Khmelnitskaya1, Dolf Talman2 and Guang Zhang3

1Imperial College Business School, United Kingdom [email protected] 2Tilburg University, Netherlands [email protected] 3Shanghai University, China [email protected]

Keywords: TU games, communication structure, average tree solution, hub and spoke cooperation structure

We consider cooperative games with cooperation structure represented by an undirected communication graph, in which several players playing more important role are selected a priori as the main players. We introduce a solution concept for graph games with main players which generalizes the average tree solution for graph games and takes into account that the main players should be rewarded better than the others. We provide its axiomatic characterization for cycle-free graph games and for non-cycle-free graph games with single main player on the subclass of graph games for which the graphs are cycles. We also show that in case when each ordinary player is connected with only one main player the average tree value for graph games with main players can be represented as a two-step distribution procedure, in which first the total rewards from cooperation are distributed among the unions determined by each of the main players together with all ordinary players related to him, and then the total payoff of each union is distributed among its members. As an application of the new solution concept for graph games with main players, we consider an allocation problem for cooperative games with hub and spoke cooperation structure.

Decision-making in a hybrid two-stage dynamic control system

Anatolii Kleimenov1

1Krasovskii Institute of Mathematics and Mechanics of the UrB of the RAS, Russia [email protected]

Keywords: optimal control problem, non-antagonistic positional differential game, Nash solution

Consider a two-step decision-making problem for a control system, the dynamics of which is described on the given segment by ordinary differential equations (see, for example, [1, 2]). The maximal number of participants of the control process (players) is two. Positional strategies used by players as actions, as well as motions generated by these actions, are defined as in [3-5]. In the first stage of the process (from the initial moment of time t0 to some given time moment T , t0  T   ), the right-hand side of the equations is assumed to contain the control action u , u(t)P of only the first player (P1). In this case, P1 solves the optimal control problem with the given terminal cost functional I1  1 (x(T)) on the segment [t0 ,T] .

Further, at the time moment T , P1 must decide whether the second player (P2) will participate in the control process on the remaining interval [T,] or not. P2 controls the choice of control v , v(t)Q and has his own terminal cost functional

I 2   2 (x()) . We assume that for permission to participate in the control process, P2 must pay P1 a side payment of K units. Two possibilities can be realized. The first possibility, when P1, having received K units of payment, allows P2 to enter the control process and plays together with P2 a non-antagonistic positional differential game; in this case it is natural to take P(NE) - solution [5] as a solution of the game. The second possibility is realized, when P1 does not allow P2 to enter the control process. In this case, P1 continues to solve the problem of optimal control with the terminal cost functional I1  1 (x()) on the segment [T,].

In the report an example of a control process with dynamics of simple motion on a plane and in the presence of a phase constraint is proposed. The optimal control problem in the first step:

2 x u , x,u  R , u  2 , 0  t  T , x(0)  x0 , I1  1 (x(T)) . The two-person non-antagonistic game in the second step:

2 x u  v , x,u,v R , u   , v   , T  t   , x(T )  xT ,

I1  1 (x()) , I 2   2 (x()) . There is following phase restriction in this example. The trajectories of both the above systems are forbidden from entering the interior of a certain given quadrilateral S . The task was calculated for different parameter values, including the parameters of the vertices of the quadrilateral . In particular, for certain values of the parameters, situations were realized in which it makes sense to solve the problem by applying BT -solutions [6-9]. In the formalization of -solutions, the so-called abnormal types of behavior of players are used, which differ from the usual one, focused on maximizing his own functional.

References

[1]. Petrosyan, L.A., Zenkevich, N.A., and Shevkoplas, E.V., Game Theory. BXV, St- Petersburg, 2012 (in Russian). [2]. Peter M. Kort, and Stefan Wrzaczek. Optimal firm growth under the threat of entry. Eur J Oper Res., 2015 Oct 1; 246(1): pp. 281–292. [3]. Krasovskii, N.N., and Subbotin, A.I., Game-Theoretical Control Problems. Springer, 1988. [4]. Krasovskii, N.N., Control of a Dynamical System. Nauka, Moscow, 1985 (in Russian). [5]. Kleimenov, A.F., Non-antagonistic Positional Differential Games. Nauka, Ekaterinburg, 1993 (in Russian). [6]. Kleimenov, A.F., Solutions in a Non-antagonistic Positional Differential Game, J. Appl. Math. Mechs., 1997. V.61, 717-723. [7]. Kleimenov, A.F., and Kryazhimskii, A.V. Normal Behavior, Altruism and Aggression in Cooperative Game Dynamics, Interim Report IR-98-076, Laxenburg: IIASA, 1998, 47 pp. [8]. Kleimenov, A.F., Altruistic behavior in a non-antagonistic positional differential game, Automation and Remote Control, 2017, Vol. 78, No. 4, pp. 762–769. [9]. Kleimenov, A.F. The use of altruistic and aggressive types behavior in the non- antagonistic positional differential two-person game on the plane. Trudy of Institute of Mathematics and Mechanics, Ural Brunch of the RAS, 2017, V.23, No. 4, pp. 181-191.

Evolutionary games under incompetence

Maria Kleshnina1, Jerzy Filar2, Vladimir Ejov3, and Jody McKerral4

1,2The University of Queensland, Australia [email protected] [email protected] 3,4Flinders University, Australia [email protected] [email protected]

Keywords: Evolutionary game theory, Incompetence, Matrix games, Replicator dynamics

The adaptation process of a species to a new environment is a significant area of study in biology. As part of natural selection, adaptation is a mutation process which improves survival skills and reproductive functions of species. Here, we investigate this process by combining the idea of incompetence with evolutionary game theory. In the sense of evolution, incompetence and training can be interpreted as a special learning process. With focus on the social side of the problem, we analyze the influence of incompetence on behavior of species. We introduce an incompetence parameter into a learning function in a single-population game and analyze its effect on the outcome of the replicator dynamics. Incompetence can change the outcome of the game and its dynamics, indicating its significance within what are inherently imperfect natural systems.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА, ИНФОРМАТИКА, ПРОЦЕССЫ УПРАВЛЕНИЯ

Главный редактор Л.А. Петросян

Издательство Санкт-Петербургского государственного университета

A simple dynamic climate cooperation model

Eugen Kovac1 and Robert Schmidt2

1University of Duisburg-Essen, Germany [email protected] 2Humboldt University Berlin, Germany [email protected]

Keywords: climate treaty, coalition, dynamic game, coordination, delay

In the past decades, international environmental agreements (IEAs) became an important instrument to tackle negative externalities caused by economic activity. While many of these agreements involve a large number of countries (Barrett 2003), economic theory has struggled to provide a sound explanation for this empirical observation. From an economic perspective, environmental cooperation resembles the provision of a public good and, therefore, it suffers from the free-rider problem. In line with this, a recent strand of literature that analyzes participation in international environmental agreements finds that due to the free-rider incentive, usually only small coalitions form, and especially so when the potential gains from cooperation are large (Barrett 1994). Kolstad and Toman (2005) refer to this controversy between theory and the empirical observations as the “Paradox of International Agreements”. In this paper we present a novel theoretical framework that sheds new light on the issue of coalition formation. Our findings can help to resolve the Paradox of International Agreements, and provide some normative implications for the design of successful negotiations. In line with much of the existing literature, we assume that there is only one (long-term) agreement that can be signed and we adopt an open membership approach, i.e., each country is free to join the agreement. However, most climate coalition formation models assume that there is only a single participation stage, so that countries can decide only once and for all if they join the coalition or not. In such a case, countries always sign an agreement, but generally with few members. When joining the agreement, each country faces a trade-off between being a signatory of a larger agreement vs. being an outsider under a smaller coalition. Departing from the bulk of the literature, we present a dynamic model, where countries may suspend the current negotiations and continue negotiating in the next period until an agreement is signed. Introducing dynamics, changes each country’s trade-off. If no long-term climate contract is signed today, then there is a delay and a new round of negotiations starts in the next period. Such a delay is costly in

69 the short-run, but may be profitable in the long-run if the countries anticipate that a better agreement can be signed in the future. Surprisingly, we find that this simple modification of an otherwise standard coalition formation game leads to fundamentally different results. As the main result, we show that large coalitions that achieve substantial welfare gains can be stable under mild conditions. At the heart of our analysis lies an endogenous threshold effect: coalition members only sign an agreement today if the resulting welfare is at least as large as their expected welfare under delay. This requires a sufficiently large number of participants. The corresponding equilibria have the property that if a single country deviates, no agreement is signed in the current period and negotiations are delayed. Countries thus join the coalition and sign a long-term agreement in order to prevent inefficient delay. In static models, by contrast, the incentive of a country to join stems typically from the additional abatement efforts carried out by other coalition members induced in a larger coalition. The resulting free-rider incentive, thus, explains the pessimistic conclusion from the static literature, that stable coalitions are typically small: An individual country is usually better off staying outside of the coalition to enjoy the benefits from other countries’ efforts, while contributing little to the global public good of climate stability and avoiding the associated costs. Our dynamic model shows that this pessimistic prediction depends heavily on the one-shot nature of the negotiation process in static models: while a unilateral deviation in a static model leads to the signature of a smaller agreement (with one member less), in our dynamic model a deviation by a country supposed to be a coalition member leads to a period of delay. The incentives to join or stay out are, thus, significantly different from those in a static model. Paying more attention to the dynamics of reaching an agreement is, therefore, crucial to gain a deeper understanding of the trade-offs involved in countries’ decisions whether or not to cooperate.

Best-Response Dynamics in Network Games with Non- reciprocal Interaction

György Kozics1 and Péter Bayer2

1Central European University, Hungary [email protected] 2Maastricht University, Netherlands [email protected]

Keywords: Learning, Networks, Potential games, Public goods, Non-reciprocal interaction

We study non-reciprocal interactions on network games (Bramoullé and Kranton, 2007), also known as local public goods games, a class of games that has received increased recent attention. One of the main reasons for this attention is that the richness of the possible network structures allows for a wide range of applicability. Bramoullé et al. (2014) mentions applications such as R&D within interlinked firms (König et al., 2014), crime in social networks (Ballester et al., 2006), and peer effects with spatial interactions (Blume et al., 2010). Bayer et al. (2017) mentions Cournot oligopolies and defense expenditures between allies (Sandler and Hartley, 2007). All existing models of network games assume a form of reciprocal interaction between pairs of players, which allows for a number of elegant analytic properties. However, in the above applications reciprocity is often not guaranteed. A relaxation of this assumption contributes to the economic literature by providing a robustness check to existing results. Additionally, it opens the door for a number of other applications such as pollution models (Leontief, 1970) where non-reciprocal interactions are more natural to be assumed. To our knowledge, this project is the first to consider non-reciprocal interactions in the local public goods game. Our objective is to find the implications of this generalization to the properties of learning processes. We strive to identify the set of network games where best-response cycles exist/do not exist. The remainder of this extended abstract is devoted for the outline of our model and early results.

References

[1]. Bayer P., Herings PJJ., Peeters R., Thuijsman F., 2017. Adaptive Learning in Weighted Network Games. GSBE Research Memoranda No. 025, GSBE, Maastricht University. [2]. Ballester, C., Calvó-Armengol, A., and Zenou, Y., 2006. Who’s who in networks. Wanted: The key player. Econometrica, 74, 1403-1417.

[3]. Blume, L.E., Brock, W.A., Durlauf, S.N., and Ioannides, Y.M., 2010. Identification of social interactions. In Handbook of Social Economics, edited by Jess Benhabib, Alberto Bisin, and Matthew O. Jackson, 853-964. Amsterdam: North Holland. [4]. Bervoets, S., Faure, M., 2016. Best-response in pure public good games on networks. Working paper. [5]. Bramoullé Y., Kranton R., 2007. Public goods in networks. Journal of Economic Theory 135: 478-494. [6]. Bramoullé Y., Kranton R., D’Amours M., 2014. Strategic Interaction and Networks. American Economic Review 104: 898-930. [7]. König M., Liu X., Zenou Y., 2014. R&D networks: Theory, empirics and policy implications. Unpublished manuscript. [8]. Leontief, W., 1970. Environmental repercussions and the economic structure: An input- output approach. The Review of Economics and Statistics 5: 262-271. [9]. Monderer D., Shapley L., 1996. Potential Games. Games and Economic Behavior 14: 124-143. [10]. Sandler, T., and Hartley, K., 2007. Handbook of Defense Economics: Defense in a globalized world. Elsevier. [11]. Voorneveld M., 2000. Best-response potential games. Economic Letters 66: 289-295.

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

Shift of the Dynamical Game Trajectory from Competitive Equilibrium to Cooperative Solution

Nikolay Krasovskii1 and Alexander Tarasyev2

1,2Krasovskii Institute of Mathematics and Mechanics UrB RAS, Russia [email protected] [email protected]

Keywords: dynamical game, Nash equilibrium, Pareto maximum, market equilibrium, equilibrium search algorithms

In the paper, we consider the dynamical game [1,3,7,9] in which players (governments of neighboring countries) implement trade of quotas to reduce greenhouse gas emissions [6]. A definition of the market equilibrium is introduced for combining properties of Nash equilibrium and Pareto maximum. We prove the existence theorem for the market equilibrium. An algorithm for searching the market equilibrium is proposed basing on the approach [3,10]. This algorithm shifts competitive Nash equilibrium to cooperative Pareto maximum [4,5,6]. The algorithm is interpreted in the form of a repeated auction, in which an auctioneer does not have information about cost functions and ecological benefit functions from emission reduction for auction participants (countries). Participants of the auction do not have information about cost functions and ecological benefit functions of other players. In each round of auction participants are offered with individual rates on emission reduction. Players provide the maximization of their utility functions based on offered rates and send to the auctioneer their best replies – the optimal emission reduction in the current period. In such formulation, we propose the strategy of the auctioneer, that allows to reach the market equilibrium. The considered auction describes the process of learning in the repeated game with lack of information [1,3,7,8]. The designed algorithm is implemented in the MATLAB software environment. The proposed approach is oriented on construction of balanced equilibrium trajectories for economic development in which optimal proportions between investments in economy and environment protection are elaborated [2]. For the computer experiment we examine a game situation between countries of the European Union and the Russian Federation. In the framework of cooperation with the Ecosystems Services and Management department of the International Institute of Applied System Analysis (IIASA, Austria) the real data processing is implemented for cost functions and ecological benefits. Basing on this data, function parameters are calibrated.

Obtained results are presented on Fig. 1. Here we depict the location of Nash equilibrium (NE), the set of Pareto maximum points (PM), best reply curves of competitors (BR1) and (BR2), the point of the market equilibrium (ME) at their intersection, the initial point (IP) and the trajectory (TA) of the algorithm converging to the market equilibrium.

Fig. 1. Shift of Dynamical Game Trajectory from Nash Equilibrium to Pareto Maximum (Market Equilibrium). The paper is supported by the Russian Foundation for Basic Research (Project No. 18-01-00264_a). References

[1]. Basar T., Olsder G.J. Dynamic Noncooperative Game Theory. London: Academic Press, 1982. 519 p. [2]. Krasovskii A.A., Taras’ev A.M. Dynamic Optimization of Investments in the Economic Growth Models // Automation and Remote Control. 2007. Vol. 68. Issue 10. Pp. 1765- 1777. [3]. Krasovskii A.N., Krasovskii N.N. Control Under Lack of Information. Boston etc.: Birkhauser, 1995. 322 p. [4]. Krasovskii N.A., Tarasyev A.M. Search of Maximum Points for Vector Criterion with Decomposition Properties // Proceedings of IMM UrB RAS. 2009. Vol. 15. No. 4. Pp. 167-182. [5]. Krasovskii N.A., Tarasyev A.M. Decomposition Algorithm of Searching Equilibria in a Dynamic Game // Mathematical Game Theory and Application. 2011. Vol. 3. No. 4. Pp. 49-88.

[6]. Kryazhimskii A., Nentjes A., Shybaiev S., Tarasyev A. Modeling Market Equilibrium for Transboundary Environmental Problem // Nonlinear Analysis. 2001. Vol. 47. No. 2. Pp. 991-1002. [7]. Kryazhimskii A.V., Osipov Yu.S. On Differential-Evolutionary Games // Proceedings of Math. Institute of RAS. 1995. Vol. 211. Pp. 257-287. [8]. Mazalov V.V., Rettieva A.N. Asymmetry in a Cooperative Bioresource Management Problem // Large-scale Systems Control. 2015. No. 55. Pp. 280-325. [9]. Petrosyan L.A., Zenkevich N.A. Conditions for Sustainable Cooperation // Contributions to Games Theory and Management. 2009. Vol. 2. P. 344-354. [10]. Subbotin A.I., Tarasyev A.M. Conjugate Derivatives of the Value Function of a Differential Game // Doklady AN SSSR. 1985. Vol. 283. No. 3. Pp. 559-564.

РОССИЙСКИЙ ЖУРНАЛ МЕНЕДЖМЕНТА

Главный редактор А. В. Бухвалов

Издательство Санкт–Петербургского государственного университета

On the value positivity for matrix game: an economic application

Victoria Kreps1

1NRU Higher School of Economics, Russia [email protected]

Keywords: value of matrix game, uniqueness of competitive equilibria, positive-definite matrix, copositive matrix

The problem of factor price equalization in the theory of international trade represents a typical and most stimulating motivation for the study of global univalence in terms of the Jacobian matrix A of production function. This problem is connected to the problem of uniqueness of competitive equilibria what is equivalent to the absence of nontrivial (non-zero) solutions of the system of linear inequalities Ay≤0, y≥0,(1) where A=[a(i,j)] is a real n×n-matrix and 풚 = (푦1, … , 푦푛) is n-dimensional column- vector. The result of Gale and Nikaido (1965) states that positivity of principal minors of an n×n-matrices A is sufficient for the system (1) to have a trivial solution y=0 only. Note that it is sufficient to check the absence of nontrivial solutions of system (1) for y∈Δn: 푛 푛 ∆ = {풚: 푦푖 ≥ 0, ∑푘=1 푦푘 = 1}. So without loss of generality y is an n-dimensional probability vector. Consider 퐴 = [푎(푖, 푗)] ∈ 푅푚×푛 as a payoff matrix of a m×n-matrix game A in which minimizing player chooses a column. It is easy to see that game A has positive value val A>0 if and only if the system (1) has the zero solution only. Indeed, if the system (1) ∗ ∗ ∗ ∗ has the zero solution only then for an optimal strategy of Player 2 d 풚 = (푦1, … , 푦푛), 푦푗 ∈ 푛 푛 ∗ ∆ , there exists i, 1≤i≤n such that ∑푗=1 푎푖푗푦푗 > 0

Thus, the best reply of Player 1 on y* guarantees him a positive payoff. So val A>0. And on the contrary: if y*∈Δn is a nontrivial solution of system (1), then the strategy y* of Player 2 provides him a non-positive payoff for any strategy of Player 1. It follows that val A≤0. Here we give an inductive criterion for square matrix games to have positive values. Denote M(m,n) the set of all real m×n-matrices A for which matrix game A has a positive value (the system (1) has the zero solution only). It is easy to verify that if det A≠0

76 and A−1≥0 (all elements of matrix A−1 are non-negative) then A belongs to M(n,n). For related result see Olech, Parthasarathy and Ravindran (1991). Proposition 1. Let A be a real n×n-matrix such that for all i=1,…,n the (n−1)×n- matrix 퐴푖 ∉ 푀(푛 − 1, 푛). Matrix A∈M(n,n) if and only if there exists A−1 and A−1≥0. Consider square matrices A with non-positive elements out of the main diagonal (a ≤0 for i≠J). For this class of matrices we show that the criterion of value positivity of ij matrix games has the following simple form. Proposition 2. Let for all elements of square matrix A all elements out of the main diagonal are non-positive (a ≤0 for i≠J). The matrix game A has a positive value ij val A>0 if and only if there exists det A≠0 and A−1≥0. Remark 1. It is easy to see that if a ≤0 for i≠j and val A>0 then a >0 for i=1,…,n. ij ii For square matrices with non-positive elements out of the main diagonal (a ≤0 ij for i≠j) the following theorem demonstrates that positivity of principal minors of a n×n- matrices A is a necessary and sufficient condition for the system (1) to have a trivial solution y=0 only. Theorem 1. Let A be a square n×n-matrix with a ≤0 for all i≠j. Matrix game A ij has a positive value, val A>0, if and only if all principal minors of A are positive. Thus for this class of matrices the Gale-Nikaido sufficient condition is a necessary one. Here we consider symmetric matrices A, a =a , i=1,…,n, j=1,…,n. For ij ji symmetric matrix according to Silvester criterion, see for example Gilbert (1991), the Gale-Nikaido condition means that A is a positive-definite matrix, i.e. 풙푇퐴풙 > 0 for every non-zero 풙 ∈ 푅푛 (2)

Here xT denotes the transpose of x. In linear algebra symmetric matrix A is called a copositive matrix if the inequality (2) holds for every non-zero nonnegative vector x∈Rn,+, where Rn,+ is the positive orthant. The collection of all copositive n×n-matrices is a proper cone; it includes as a subset the collection of real positive-definite n×n-matrices. Copositive matrices find applications in economics and statistics, see Berman, Plemmons (1979). Theorem 2. If matrix A is copositive then matrix game A has a positive value. Note that the contrary assumption is not true. For example 2×2-matrix A with a =−10, a =a =1, a =2 has positive value but it is not copositive. 11 12 21 22

In Kreps(1984) the following criterion for square symmetric matrix to be a copositive one is given. Denote A(i …i ) the k×k-submatrix of matrix A obtained by 1 k intersection the rows and columns of A with the same numbers i1 … 푖푘. Theorem. Symmetric n×n-matrix is copositive if and only if for any submatrix A(i …i ) of A, k=1,…,n, 1 k a) either det A(i …i )>0; 1 k b) or in matrix A(i …i ) there exists a row i , 1≤l≤k such that ∑푘 휑 퐴(i … 푖 ) ≤ 0 1 k l 푗=1 푖1푗 1 푘 where φ A(i …i ) is a cofactor of element a in matrix A(i …i ). i j 1 k i j 1 k l l Combination of this criterion and Theorem 2 results in the following sufficient condition for symmetric matrix game to have a positive value. Corollary. If for symmetric matrix A the conditions a) and b) hold then value of matrix game A is positive (the system (1) has a trivial solution only). Support from Basic Research Program of the National Research University Higher School of Economics is gratefully acknowledged. This study was partially supported by the grant 16-01-00124-a of Russian Foundation for Basic Research.

References

[1]. Gale D., H.Nikaido (1965): The Jacobian Matrix and Global Univalence of Mappings. Mathematische Annalen 159, 81-93. [2]. Olech O., T.Parthasarathy, and G.Ravindran (1991): Almost N-matrices and its Applicationsto Linear Complementarity Problem and Global Univalence. Indian Alg. Appl., 145, 107-125. [3]. Berman A., R. Plemmons (1979): Nonnegative Matrices in the Mathematical Sciences. Academic Press. [4]. Gilbert G. (1991): Positive definite matrices and Sylvester’s criterion. The American Mathematical Monthly, Mathematical Association of America, 98 (1): 44-46. [5]. Kreps V. (1984): On quadratic forms non-negative over an orthant. USSR Computational Mathematics and Mathematical Physics, 24:2, 105-109.

Scenario analysis of the impact of information on tax collections

Suriya Kumacheva1, Elena Gubar2, Ekaterina Zhitkova3 and Galina Tomilina4

1,2,3,4Saint Petersburg State University, Russia [email protected] [email protected] [email protected] [email protected]

Keywords: tax control, tax evasions, evolutionary process, structured network, economic agents, risk-statuses

Nowadays information is a very important tool in many areas of economic system, such as marketing, management, strategic analysis as well as tax control. The tax control is one of the most important elements of fiscal system. Many studies are dedicated to the modeling of tax control in the framework of game-theoretical attitude. For example, in the previous papers the “threshold value” of the probability of tax auditing (which is the critical value for the taxpayers to evade or not) was obtained. However, in the real life the procedure of tax auditing with such probability is very expensive while the tax authority’s budget is strongly restricted. Therefore it is necessary to stimulate taxpayers to pay taxes in accordance to their true level of income. In the present work as a way of such stimulating we consider the dissemination of information about future audits among the population of taxpayers. We suppose the process of information spreading to be similar to the evolutionary process in the structured network of economic agents. The connections between the nodes of such networks present social links of taxpayers, which have three different levels of real and declared income – L (low), M (medium) and H (high). Thus, each taxpayer has different ways to evade taxation. Additionally we assume that every taxpayer has one of three possible risk- statuses (risk averse, risk neutral and risk preferred) and change it depending on the external conditions. We investigate networks with different structure and two algorithms of information spreading: the algorithm based on the Markov processes and the proportional imitation rule. All these approaches form the series of numerical simulations which can be merged into several scenarios. Thus we estimate the impact of information spreading

79 to the population of taxpayers varying the combination of parameters of the system such as topology of the network, the rule of imitation, the preferences of taxpayers, etc. To support our theoretical hypothesis we use data of economical and psychological statistics in all numerical experiments.

МАТЕМАТИЧЕСКАЯ ТЕОРИЯ ИГР И ЕЕ ПРИЛОЖЕНИЯ

Development of Strategic thinking model for limited resource allocation in business games by modified Groves-Ledyard mechanism

Olga Kuznetsova1 and Natalya Dodonova2

1,2 National Research Samara University, Russia [email protected] [email protected]

Keywords: business game, utility function, resource allocation mechanisms, strategic thinking model, research, fuzzy logic

The article deals with the development of a player behavior model based on fuzzy logical inference. The behavior rules’ set (knowledge base) formation is the main problem. The clustering is decides this problem. This approach is universal. We decide problem creation models that describe human behavioral for limited resources allocation in business games. It is zero-sum game. During game players use some behavioral models. It is intresting which strategic thinking model describes human behavior. Strategic thinking models that don't take into account the factor of reflection are well known in behavioral game theory [1, 2]. On the other hand, strategic thinking models that take into account the factor of reflectionis are being developed. These models consider the reflection of different levels [3]. Paper [4] describes first level of a reflection for game with reverse priorities mechanism. That model is approximating and good for forecast. This paper describes process creation the same model for game with modified Groves-Ledyard mechanism. We offer a new approach to the development fuzzy logec model. It is based on a rules that defined in the process of clastering [5] experiments data [6]. Imitation experiments were conducted. We had three groups:  players used arbitrary behavior models;  players used our model only;  each player used special model. Part of experiments was conducted with automatic sistem, other was conducted with players. Each game consists of ten times. The hypothesis about the adequacy of the proposed model is put forward on the research results. This work is partially supported by the RFBR, grant 17-07-015550 A.

References

[1]. Wright, J.R., Leyton-Broun, K. (2013). Predicting human behavior in unrepeated, simultaneous-mmove games. arXiv preprint arXiv: 1306.0918 [2]. Stahl, D.O. Wilson, P.W. (1995). On players’ models of other players: Theory and experimental evidence. Games and Economic Behavior, 10(1), pp.218-254. [3]. Korepanov V. (2017). Experiments on the influence of strategic thinking models on resource allocaton mechanisms. Game Theory and Management. Collected abstracts of papers presented on the Eleventh International Conference Game Theory and Management / Editor Leon A. Petrosyan and Nikolay A. Zenkevich. - Spb.: Graduate School of Management SpbU. - pp. 80-81. [4]. Kuznetsova O., Dodonova N. (2017). Reseach of decision making model for allocation scarce resources in business games. Experiments on the influence of strategic thinking models on resource allocaton mechanisms. Game Theory and Management. Collected abstracts of papers presented on the Eleventh International Conference Game Theory and Management / Editor Leon A. Petrosyan and Nikolay A. Zenkevich. - Spb.: Graduate School of Management SpbU. - pp. 92-93. [5]. Kuznetsova O., Dodonova N., Gluhov V. (2017). Process Description of Clustering Experimental Games on Allocation of Limited Resources by the Modified Groves- Ledyard Mechanism. Proceedings of the Second Workshop on Computer Modelling in Decision Making co-located with the VI International Youth Research and Practice Conference on Mathematical and Computer Modelling in Economics, Insurance and Risk Management (MCMEIRM 2017). - pp. 43-49. [6]. Korgin, N., Korepanov, V., (2016) An Efficient Solution of the Resource Allotment Problem with the Groves–Ledyard Mechanism under Transferable Utility. Automation and Remote Control, (2016) Volume 77, Issue 5, pp 914-942.

Periodicals in Game Theory

CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT

Volumes 1–11

Edited by Leon A. Petrosyan Nikolay A. Zenkevich

St. Petersburg State University

Product differentiation in the insurance market: equilibria analysis

Denis Kuzyutin1, Maria Bartel2 and Nadezhda Smirnova3

1Saint-Petersburg State University,National Research University Higher School of Economics, Russia [email protected] 2National Research University Higher School of Economics, Russia [email protected] 3National Research University Higher School of Economics,Saint-Petersburg; Saint-Petersburg State University [email protected]

Keywords: vertical product differentiation, insurance market, equilibria, subgame perfect equilibrium, Stackelberg equilibria

Vertical product differentiation is a useful strategy for avoiding fierce price competition that can be used in the markets of different structure [e.g. 1, 2, 3]. We use the vertical differentiation framework to explore the quality – price competition in the insurance market [4, 5, 6]. The competition between insurance firms takes place in a two-stage game. At the first stage they decide on the quality (the list of insured events, the amount of insurance compensation and the level of claims handling procedure) 푞 , 푖 = 1,2 to offer (let 0≤q≤q

Nash equilibria) or sequentially (the case of Stackelberg equilibria) their prices 푝푖, 푖 = 1,2 (insurance rates). The heterogeneity of the consumers is charactirized by a parameter 푡 ∈ [0, 푡],̅ 푡̅ ≤ 1 which is equal the insured event probability (from the one hand) and shows the consumer’s willingness to pay for quality increasing (from the other hand). The consumer t utility function (in the case of optional insurance) is as follows:

푈(푡) = max {푡푞1 − 푝1, 푡푞2 − 푝2}. To derive the expected profit functions of the insurance firms we use different assumptions about firms’ cost structure: • the insurance compensation and the cost of claims handling procedure (variable costs if the insured event takes place) may depend on the quality level (and be equal to vq i , v>0 ), or may not depend on quality [6]. Parameter v can be interpreted as the average amount of insurance compensation plus the average cost of claims handling procedure;

• the costs of quality improvement (fixed costs) may be zero, linear or quadratic functions of q. We managed to derive Nash and Stackelberg equilibrium prices and qualities (analytically in explicit form or numerically) and compare the results of equilibria analysis for different costs structure.

References

[1]. Gabszewics, J., Thisse, J. 1979. Price competition, quality and income disparities. J. of Economic Theory, 20, - pp. 340–359. [2]. Shaked, A., Sutton, J. 1982. Relaxing price competition through product differentiation. Review of Economic Studies, 49, - pp. 3–14. [3]. Tirole J. 1988. The Theory of Industrial Organization. The MIT Press. Cambridge, MA. [4]. Okura, M. 2010. The vertical differentiation model in the insurance market. Intern. J. of Economics and Business Modeling, 1(2), - pp. 12–14. [5]. Kuzyutin D., Zhukova E., Borovtsova M. 2007. Subgame Perfect Nash Equilibrium in a Quality-Price Competition Model with Vertical and Horizontal Differentiation. Contributions to Game Theory and Management. SPb.: SPbSU. - pp. 270-276. [6]. Kuzyutin D., Nikitina M., Smirnova N., Razgulyaeva L. 2015. The vertical differentiation model in the insurance market: costs structure and equilibria analysis. Contributions to Game Theory and Management. SPb.: SPbSU. Vol. 8. - pp. 176-186.

Journals in Game Theory

INTERNATIONAL JOURNAL OF GAME THEORY

Editor Shmuel Zamir

SPRINGER

Construction of M-strongly time-consistent subcore in the game with spanning forest

Yin Li1 and Puneet Tomar2

1Saint Petersburg State University, Russia [email protected] 2Saint Petersburg State University, India [email protected]

Keywords: minimum cost spanning forest, cooperative game, dynamic games, time consistency

In the paper the multistage N-person and N'-suppliers in minimum cost spanning forest game is considered. The cooperative behavior of players is defined. Selecting strategies, players build a minimum cost spanning forest at each stage. On the bases of values of characteristic function, a new characteristic function V' is constructed for each coalition in subgames. The function V' is defined as in [L. A. Petrosjan, Y. B. Pankratova, 2017]. With the help of this newly constructed characteristic function, an analogue of the core is defined. This analogue of the core can be considered a new optimality principle. It is proved that this constructed optimality principle has the property of M-strongly time- consistent in the game with spanning forest.

Journals in Game Theory

GAMES AND ECONOMIC BEHAVIOR

Editor-in-Cheif Ehud Kalai

ELSEVIER

Nash equilibrium in supply chains with hourglasses structure

Yulia Lonyagina1, Margarita Gladkova2 and Natalia Nikolchenko3

1Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, Russia [email protected] 2Graduate School of Management, St. Petersburg State University, Russia [email protected] 3Graduate School of Management, St. Petersburg State University, Russia [email protected]

Keywords: supply chain, hourglasses structure, multi-step hierarchical game, complete information, Nash equilibrium

In this article a model of competition in supply chains with hourglasses structure is analyzed. The production and distribution systems of some products may be presented as supply chains with such structure.

Consider the tree-like graph 퐺1 = (푋1, 퐹1) where 푋1 is a set of nodes and 퐹1 is a function of alternatives. The root node of this tree is named by 푥∗. Suppose that the graph 퐺2 = (푋2, 퐹2) follows the conditions: ∗ ∗ There is unique node 푥 ∈ 푋2 such that 퐹2(푥 ) = ∅; ∗ For all 푥 ∈ 푋2\푥 : |퐹2(푥)| = 1, where |퐹2(푥)| means a cardinality of the set

퐹2(푥). Let the graph G=(X, F) to have a hourglasses structure if

푋 = 푋1 ∪ 푋2; 퐹 (푥), 푥 ∈ 푋 \푥∗; 퐹 = { 1 1 퐹2(푥), 푒푙푠푒푤ℎ푒푟푒. 푖 Assume that every node 푥푗 , 푖 = 1, … , 푙, 푗 = 1, … , 푚푖 of a supply chain

푖 푛푖푗 consists of a finite set of elements {푥푗푘}푘=1 for which the set of numbers is defined

푛푖푗 {푣푖푗푘}푘=1, 푣푖푗푘 ≥ 0, where 푛푖푗 is a number of elements. This set of elements is a group of competitive firms that are producing and consuming the homogeneous product as well as having the different production costs (the production power is unrestricted). For each firm

푖 푛푖푗 푥푗푘 let 푞푖푗푘 ≥ 0 be the production quantity of this firm and 푄푖푗 = ∑푘=1 푞푖푗푘 be the

푖 푛푖푗 푖 integrated quantity that was produced by all firms {푥푗푘}푘=1 in the node 푥푗 . Let 푝푖푗 be the price that firm use to sell the unit of the produced good. It is assumed that

푝푙푗 = 푎푙푗 − 푏푙푗푄푙푗,

where 푎푙푗, 푏푙푗 > 0.

The set ({푞 } , {푝 } ) defines the commodity flow d in the supply chain. 푖푗푘 푖,푗,푘 푖푗 푖,푗 Let the set D is the set of all feasible flows in a supply chain, where flow d is named

푛푖푗 feasible if the condition 푄푖푗 = ∑푘=1 푞푖푗푘 is satisfied and 푝푙푗 > 0, 푄푙푗 > 0, 푗 =

1, … , 푚푙. For each firm define the profit function as following:

푞1푗푘(푝1푗 − 푣1푗푘), 푖푓 푖 = 1;

휋푖푗푘 = {푞푙푗푘(푎푙푗 − 푏푙푗푄푙푗 − 푝푟ℎ − 푣푙푗푘), 푖푓 푖 = 푙;

푞푖푗푘(푝푖푗 − 푝푟ℎ − 푣푖푗푘), 푒푙푠푒푤ℎ푒푟푒; 푖 푟 where 푝푟ℎ: 푥푗 ∈ 푆ℎ . 푖 The set 푈푖푗 = {푢푖푗} the strategy of player 푥푗 will be considered as the set of all the possible vectors 푢푖푗 ∈ 퐷, where:

푖 (푞푖푗1, … , 푞푖푗푛푖푗, 푝푖푗) ∈ 퐷, 푥푗 ∈ 푁, 푖 = 1, … , 푙 − 1, 푗 = 1, … , 푚푖푗 푢푖푗 = [ 푙 (푞푙푗1, … , 푞푙푗푛푙푗) ∈ 퐷, 푥푗 ∈ 푁, 푗 = 1, … , 푚푖푗. We assume that each of the supply chain participants is acting independently from each other and exclusively in favor of his own interests. The optimal solution for such supply chain is Nash equilibrium in a multi-stage hierarchical game with the complete information  = 〈푁, {푈 } , {휋 } 〉 on the graph G. 푖푗 푖,푗 푖푗푘 푖,푗,푘

References

[1]. Lee, H. L.; Padmanabhan, V.; Seungjin, W. 1997. The bullwhip effect in supply chains, Sloan Management Review Spring: 93–102. [2]. Esmaeili, M.; Aryanejad, M.; Zeephongsekul, P. 2008. A game theory approach in seller–buyer supply chain, European Journal Of Operation Research 195(2): 442–448. [3]. Cho S-H. 2014. Horizontal mergers in multitier decentralized supply chains. Management Science. 60, 356-379. [4]. Adida E., DeMiguel V. 2011. Supply chain competition with multiple manufacturers and retailers. Operation Research. 59 (1), 156-172. [5]. Corbett C., U. S. Karmarkar. 2001. Competition and structure in serial supply chains with deterministic demand. Management Science. 47, 966-978. [6]. Zhou, D., Karmarkar, U. S., Jiang, B. (2015). Competition in multi-echelon distributive supply chains with linear demand. International Journal of Production Research, 53(22), 6787–6807. [7]. Carr M. S., U. S. Karmarkar. 2005. Competition in multi-echelon assembly supply chains. Management Science. 51, 45-59. [8]. Zenkevich E., Lonyagina Y., Fattakhova M. 2017. Contributions to game theory and management. 10, 350-374. [9]. Petrosyan, L. A., Zenkevich, N. A. and E. V. Shevkoplyas (2014). Game theory. 2nd Edition. BCV-Press, Saint-Petersburg, 432 p.

Determining stationary Nash equilibria for average stochastic positional games

Dmitrii Lozovanu1 and Stefan Pickl2

1Insitute of Mathematics and CS of Modova Academy of Sciences, Moldova [email protected] 2Universität der Bundeswehr, Germany [email protected]

Keywords: Average stochastic positional game, Stationary strategy, Nash equilibria

We consider a class of stochastic games with average payoffs in which the set of states is divided into several disjoint subsets such that each subset represents the position set for one of the players and each player controls the Markov process only in his position set. In such a game each player chooses actions in his position set in order to maximize his average reward per transition. We show that for the considered class of games there exist Nash equilibria in stationary strategies. This class of games extends the deterministic positional games from [1, 4], and we call it average stochastic positional games. An average stochastic positional game is determined by the following elements : - a state space X (which we assume to be finite); - a partition X=X ∪X ∪…∪X where X represents the position set of 1 2 m i - player i∈{1,2,…,m}; - a finite set A(x) of actions in each state x∈X; - a step reward 푓푖(푥, 푎) with respect to each player i∈{1,2,…,m} in each - state x∈X and for an arbitrary action a∈A(x);

- a transition probability function 푝: 푋 × ∏푥∈푋 퐴(푥) × 푋 → [0,1] that gives 푎 - the probability transitions 푝푥,푦 from an arbitrary x∈X to an arbitrary y∈Y 푎 - for a fixed action a∈A(x), where ∑푦∈푋 푝푥,푦 = 1, ∀푥 ∈ 푋, 푎 ∈ 퐴(푥) - a starting state x ∈X. 0 The game starts at the moment of time t=0 in the state x where the player 0 i∈{1,2,…,m} who is the owner of the state position x (x ∈X ) chooses an action a ∈A(x ) 0 0 i 0 0

1 2 푚 and determines the rewards 푓 (푥0, 푎0), 푓 (푥0, 푎0), …, 푓 (푥0, 푎0), for the corresponding players 1,2,…,m. After that the game passes to a state y=x ∈X according to a probability 1 distribution 푎0 . At the moment of time t=1 the player k {1,2,…,m} who is the owner 푝푥0,푦 ∈

88 of the state position x (x ∈X ) chooses an action a ∈A(x ) and players 1,2,…,m receive 1 1 k 1 1

1 2 푚 the corresponding rewards 푓 (푥1, 푎1), 푓 (푥1, 푎1), …, 푓 (푥1, 푎1). Then the game passes to a state y=x ∈X according to a probability distribution 푝푎1 and so on indefinitely. Such 2 푥1,푦 a play of the game produces a sequence of states and actions 푥0, 푎0, 푥1, 푎1, … , 푥푡, 푎푡, …. 1 2 푚 that defines a stream of stage rewards 푓 (푥푡, 푎푡), 푓 (푥푡, 푎푡), …, 푓 (푥푡, 푎푡), t=0,1,2…. The average stochastic positional game is the game with payoffs of the players

where Fi,xo expresses the average reward per transition of player i. Each player in this game has the aim to maximize his average reward per transition. We consider average stochastic positional games when the players use stationary strategies of choosing the actions in the states. A stationary strategy of player i is a mapping si that provides for every x∈X a probability distribution over A(x). So, a i stationary strategy for player ∈{1,2,…,m} is given if for an arbitrary x∈X and an arbitrary i 푖 푖 a∈A(x) are given 푠푥,푎 ≥ 0 such that ∑푎∈퐴(푥) 푠푥,푎 = 1. We show that for determining a stationary Nash equilibrium in an average stochastic game can the following auxiliary game in normal form

푖 푖 푖 〈{푆 }푖=1̅̅,̅푚̅̅, {퐹 (푠)}푖=1̅̅,̅푚̅̅〉, where 푆 for i∈{1,2,…,m} are determined by the set of solutions of the system can be used

(1)

that determines the set of stationary strategies of player i. Each Si is a convex compact set and an arbitrary extreme point corresponds to a basic solution si of system 푖 (1), where 푠푥,푎 ∈ {0,1}, ∀푥 ∈ 푋푖, 푎 ∈ 퐴(푥) i.e each basic solution of this system corresponds to a pure stationary strategy of player i. On the set S=S1×S2×…×Sm we define m payoff functions

where q for x∈X are determined uniquely from the following system of linear equations x

for an arbitrary fixed profile 푠 = (푠 , 푠 , … , 푠 ) ∈ 푆. Here Θ for y∈X represent 1 2 푚 y arbitrary values such that 휃푦 > 0, ∀푦 ∈ 푋 and ∑푦∈푋 휃푦 = 1 . The functions 푖 퐹 (푠1, 푠2, … , 푠푚) i=1,2,…,m, represent the payoff functions for the average stochastic 푖 푖 game in normal form 〈{푆 }푖=1̅̅,̅푚̅̅, {퐹 (푠)}푖=1̅̅,̅푚̅̅〉. For this game we proved the following theorem.

푖 푖 Theorem 1: The game 〈{푆 }푖=1̅̅̅,̅푛̅, {퐹 (푠)}푖=1̅̅̅,̅푛̅〉 possesses a Nash equilibrium 푠∗ = (푠1∗, 푠2∗, … , 푠푚∗) ∈ 푺̅ which is a Nash equilibrium in mixed stationary strategies for the average stochastic positional game with an arbitrary starting state x∈X. Note that in general, for an average stochastic game with m≥3 players a Nash equilibrium in stationary strategies may not exist (see [2]), however for the positional games stationary Nash equilibria always exist. In [3, 4] the existence of Nash equilibria in pure stationary strategies for two player average stochastic positional games and for the games when each profile of strategies of the players induces a Markov unichain is proven.

References

[1]. Ehrenfeucht A., Mycielski J. Positional strategies for mean payoff games. Int. J. of Game Theory, 8, 1979, 109-113. [2]. Flesch J., Thuijsman F., Vrieze K. Cyclic Markov equilibria in stochastic games, International Journal of Game Theory, 26, 1997. 303–314. [3]. Lozovanu D., Pickl S. Nash equilibria conditions for stochastic positional games. Contribution to game theory and management. St.Petersburg Univ., 8, 2014, 201-213. [4]. Lozovanu D., Pickl S.Determining the optimal strategies for zero-sum average stochastic positional games. Electronic Notes in Discrete Mathematics, 55, 2016, 155-159.

Impact of Short-Selling on Security Trading and Pricing

Chenghu Ma1

1School of Management, Fudan University, China [email protected]

Keywords: short selling, security lending, market segmentation, imperfect competition

We develop models to address the interaction between the security lending market and the security trading market. Within an idealistic perfect competitive security lending market, we show that short selling restrictions tend to cause endogenous price uncertainty. This is particularly true in the presence of derivative trading, even when investors have homogeneous beliefs. Furthermore, when investors can reach consensus on the volatility structure, but disagree on the expected payoffs, there will be an “upward bias” plus “price uncertainty” from the fundamental CAPM. The degree of price uncertainty and the price gap from the fundamental CAPM must increase in the margin ratio and the risk free interest rate. In a more realistic market environment with market segmentations among investors for either security lending or short selling (or for both), while maintaining the assumption on the perfect competitive markets, we found that: (a) When supply side of the security lending market becomes highly competitive, the lending fee can be arbitrarily close to zero, and the equilibrium price will bias upward from the equilibrium price under no market segmentations. In this circumstance, both the equilibrium price and the short interest vary with the degree of market segmentation and the dispersion in beliefs (among all investors). (b) A positive lending fee will be realized when competition among potential lenders is sufficiently less intensive. In this case, as the negative opinion of short sellers is totally offset by the opposite view of security lenders, it results in a sticky equilibrium security price that fully reflects the perception of those optimistic non-lenders. Again, the sticky price is strictly above the equilibrium under no market segmentations and no short selling restrictions. As an alternative extension to the basic model we consider the case when only few large financial institutions are permitted to act as security lenders, and when the security lenders play a two-stage game in the security lending market: In the first stage, as price-takers, they purchase stocks in the security trading market; and, in the second, they compete in security lending fees to attract demands of short sellers, assuming that security

91 lenders can not lend out shares exceeding the stocks purchased in the security market. Equilibrium security price, lending fee, along with the short interest, are obtained, which shed light on the impact of this particular type of short selling restrictions on security trading and asset price.

RUSSIAN MANAGEMENT JOURNAL

Editor-in-Chief Alexander V. Bukhvalov

St. Petersburg University Press

A Myerson value for TU-games restricted to graphs with weigted nodes

Conrado Manuel1 and Daniel Martin2

1,2Universidad Complutense de Madrid, Spain [email protected] [email protected]

Keywords: TU-game, Myerson value, social netwok, bargaining abilities

In this communication we introduce a value for players in a TU-game with cooperation restricted by means a graph which represents a social network. Moreover players have different bargaining abilities. We use ideas "a la Myerson" and we characterize the obtained value.

МАТЕМАТИЧЕСКАЯ ТЕОРИЯ ИГР И ЕЕ ПРИЛОЖЕНИЯ

Myerson value and marginality

Conrado Manuel1 , Eduardo Ortega2 and Monica Del Pozo3

1,2Universidad Complutense de Madrid, Spain [email protected] [email protected] 3Universidad Carlos III de Madrid, Spain [email protected]

Keywords: Myerson value, within groups Myerson value, between groups Myerson value, marginality

In this communication we deal with several types of marginal contributions for players in a TU-game with communication restricted by a graph, the classical marginal contributions but also the link-marginal contributions and the player and link marginal contributions. Accordingly with them we introduce three types of marginality that we use to characterize the Myerson value, the within groups Myerson value and the between groups Myerson value.

VESTNIK OF SAINT PETERSBURG UNIVERSITY. MANAGEMENT

Editor-in-Chief Natalia P. Drozdova

St. PetersburgUniversity Press

Constructing a characteristic function in cooperative differential games with negative externalities

Ekaterina Marova1 and Ekaterina Gromova2

1,2Saint-Petersburg State University, Russia [email protected] [email protected]

Keywords: Cooperative differential games, Characteristic function, Negative externalities

In this paper we consider a model of cooperative differential game with prescribed and random duration as well for which players have negative externalities [10]. Nowadays there exist different ways of calculation of characteristic function in cooperative games. Some of them are described in [10] with a modern view on the subject for a static formulation. In a dynamic formulation construction of so-called α-, δ- characteristic functions (see, correspondingly, [5], [9]) was analyzed in [1] and a new approach for construction ζ-characteristic function was introduced (see also [6] for the first reference in Russian). One of the important properties of characteristic function is superadditivity [4]. Presence of this property of characteristic functions provides some advantages for the next decision making in cooperative game [2, 5, 6]. In the first part of the paper we consider a wide class of linear-quadratic differential game with negative externalities applied to the problem of pollution control [2]. We calculated α-, δ- and ζ- characteristic functions and analyze their properties and relations. We show that α- and ζ- characteristic functions are superadditive for the considered game which concludes a correctness of the results. Furthermore we prove that δ- characteristic function is superadditive under some additional constraints on the parameters of the linear-quadratic game. Moreover we prove the following inequalities for the considered linear-quadratic differential game: 푉훿(푆, 푥(푡), 푇 − 푡) ≥ 푉훼(푆, 푥(푡), 푇 − 푡) ≥ 푉휁(푆, 푥(푡), 푇 − 푡), 푆 ⊂ 푁 Finally we investigate the relations of the obtained characteristic functions with constructing a new characteristic function that provide strong time-consistency of subset of the core of the game [8].

The second part of this paper is focused on the problem of constructing a characteristic function in the same differential game but with a random duration (see [7] and further works). α-, δ- and ζ- characteristic functions are constructed. The inequalities mentioned above are also proved for this game. The detailed analysis of the differences of characteristic functions in games with prescribed and random duration is done.

References

[1]. Gromova, E.V., Petrosyan, L.A. On an approach to constructing a characteristic function in cooperative differential games, Autom Remote Control (2017) 78: 1680. https://doi.org/10.1134/S0005117917090120 [2]. Gromova, E.V., The Shapley value as a sustainable cooperative solution in differential games of 3 players, Recent Advances in Game Theory and Applications, Birkhauser, Springer Int. Publishing, 2016, P. 67–89. [3]. Gromova, E.V., Malakhova, A.P., Marova, E.V.On the superadditivity of a characteristic function in cooperative differential games with negative externalities, Conference: 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), 2017. [4]. J. von Neumann, O. Morgenstern, Theory of Games and Economic Behavior, Princeton University Press, 1953. [5]. Petrosjan L.A., Danilov N.N., Cooperative differential games and their applications, Tomsk University Press, 1982, Tomsk. [6]. Petrosyan, L.A. and Gromova, E.V., Two-level Cooperation in Coalitional Differential Games, Tr. Inst. Mat. Mekh. UrO RAN, 2014, vol. 20, no. 3, pp. 193–203. (in Russian) [7]. Petrosyan L.A., Shevkoplyas E.V. (2000) Cooperative differential games with random duration Vestnik Sankt-Peterburgskogo Universiteta. Ser 1. Matematika Mekhanika Astronomiya 4: 18–23. [8]. Petrosjan L.A., Pankratova Y. B., Construction of strongly time-consistent subcores in differential games with prescribed duration, Tr. Inst. Mat. Mekh. UrO RAN, V. 23, no. 1, 2017, P. 219–227. (in Russian) [9]. Petrosjan L.A., Zaccour G., Time-consistent Shapley value allocation of pollution cost reduction, J. of Economic Dynamics and Control, V. 27, no. 3, 2003, P. 381–398. [10]. Reddy P., Zaccour G., A friendly computable characteristic function, Math. Social Sci., vol. 82, no. C, pp. 18–25, 2016.

Network Structure, Equilibria, and Adjustment Dynamics in Network Games with Nonhomogeneous Players

Vladimir Matveenko1, Alexei Korolev2 and Maria Garmash3

1,2,3National Research University Higher School of Economics, Russia [email protected] [email protected] [email protected]

Keywords: Network, Nash equilibrium, Nonhomogeneous players, Network formation, Adjustment dynamics

Diversity and heterogeneity have become an important aspect of social and economic life (international working teams is a typical example). Correspondingly, along with accounting for position of agents in the network, an important task is to account for heterogeneity of agents as a factor defining differences in their behavior and wellbeing. We demonstrate that several centrality measures in networks (undirected graphs), the importance of which was revealed in network games (degrees, eigenvalue centrality, Katz-Bonacich centrality, alpha-centrality, diffusion centrality measures), relate in fact not to individual nodes of network, but to disjoint classes of nodes, which we call types. To develop a concept of types with nonhomogeneous players, we use the game model of production with knowledge externalities [1-3], which considers situations in which in period 1 agents in network, at the expense of diminishing current consumption, make investments of some resource (such as money or time) with the goal to increase consumption in period 2. The latter depends not only on own investment but on investments by the neighbors in the network. The total utility of each agent in network depends on her consumption in the two time periods. Three ways of agent’s behavior are possible: passive (no investment), active (a part of endowment is invested), and hyperactive (the whole endowment is invested). The main attention is devoted to the concept of Nash equilibrium with externalities, similar to the one used in [4, 5] (under which the players are less free in changing their behavior in equilibrium than under common definition of Nash equilibrium). We prove that the player’s utility depends monotonously on her environment and find the dependence of the investment on the externality received by the player. With homogeneous agents, the equilibrium behavior is totally defined by agent’s (or more precisely, type’s) position in the network, described by

97 the alpha-centrality measure, whose parameters are built on the basic parameters of the model: utility satiation a and productivity b . With nonhomogeneous players, a type i 1,2,...,S is a subset of the set of nodes, such that (1) all nodes of type i have the same productivity bi , (2) have the same numbers of neighbors, tij , j 1,2,...,S , and (3) the division of the set of nodes in types is minimal. Thus, a network is characterized by a vector of productivities

b  (b1,b2 ,...,bS ) and a type adjacency S  S matrix T  (tij ). We introduce a continuous adjustment dynamics described by a system of differential equations, which starts from an initial state in which players of the same type have the same strategies. We study in details the case of S  2 ; it corresponds the situation when two regular networks of degrees t11, t22 , initially in equilibrium, unify in such way that each

st nd nd 1 type player establishes t12 links with 2 type, and each 2 type player establishes t21 links with 1st type. A special case of connection of complete networks with formation of a new complete network was considered in [6]. We find conditions under which the initial equilibrium holds after unification, and conditions under which the equilibrium changes. In particular, we study how the behavior of nonadopters (passive agents) changes when they connect to adopters (active or hyperactive) agents. For instance, let initially type 1 be active and type 2 passive. Then, if the 1st type productivity is relatively high,

b2  [b1 (t11 1)  2a]/t21 , then the game stays in the initial (dynamically unstable) equilibrium. If the 2nd type relative productivity is somewhat higher, then the transition process leads to the stable equilibrium with hyperactive 1st type and active 2nd type

nd players. And if 2 type productivity is high, b2  2a /(t22 1) , then the transition process leads to the stable equilibrium with all hyperactive players. Such results can be useful in analysis of real situations of networks unification. The research is supported by the Russian Foundation for Basic Research (project 17-06-00618).

References

[1]. Matveenko V.D, Korolev A.V. (2016) Equilibria in a network game with production and knowledge externalities. Mathematical Game Theory and its Applications, Vol. 8 (1), pp. 106-137 (In Russian).

[2]. Matveenko V.D, Korolev A.V. (2016) Typology of networks and equilibria in network game with production and externalities of knowledge. Mathematical Game Theory and its Applications, Vol. 8 (1), pp. 106-137 (In Russian). [3]. Matveenko V., Korolev A. (2017) Knowledge externalities and production in network: game equilibria, types of nodes, network formation. Int. J. of Computational Economics and Econometrics. Vol.7 (4), pp.323 – 358. [4]. Lucas, R. E. (1988) On the mechanics of economic development. J. of Monetary Economics.Vol. 22, pp. 3–42. [5]. Romer, P. M. (1986) ‘Increasing returns and long-run growth’, J. of Political Economy. Vol. 94, pp. 1002–1037. [6]. Matveenko V. D., Korolev A. V., Zhdanova M.O. (2017) Game equilibria and unification dynamics in networks with heterogeneous agents. Int. J. of Engineering Business Management. Vol. 9, pp. 1-17.

Journals in Game Theory

INTERNATIONAL GAME THEORY REVIEW

Editors

David W.K. Yeung Leon A. Petrosyan Hans Peters

WORLD SCIENTIFIC

Game-theoretic methods for community detection in networks

Vladimir Mazalov1 and Natalia Nikitina2

1,2Karelian Research Center of the RAS, Russia [email protected] [email protected]

Keywords: network partitioning, community detection, cooperative game, hedonic game, maximum likelihood method, Gibbs sampling

Traditional methods for detecting communities structure are based on selecting denser subgraphs inside the network. Here we propose to use the methods of cooperative game theory that emphasize the mechanisms of cluster formation. A cooperative game related with network structure is formed and then we find the stable partition of the network into coalitions of the nodes. This method is related with maximization problem of the potential function and is based on the theory of hedonic games. This approach allows to detect clusters with various resolution. Here we propose the maximum likelihood method for tuning the resolution parameter in the hedonic game. The results of computer simulations for different communication networks are presented. This research is supported by the Russian Fund for Basic Research (projects 16- 51-5506, 16-01-00183)

Journals in Game Theory

INTERNATIONAL JOURNAL OF GAME THEORY

Editor-in-Chief Shmuel Zamir

SPRINGER

100

Oligopolistic revenue management with myopic buyers

Vincent Meisner1

1TU Berlin, Germany [email protected]

Keywords: revenue management, myopic buyers, oligopoly

Revenue management (RM) is concerned with the dynamic pricing of goods that are available in fixed capacities and that have to be sold before a deadline. Classical applications are industries such as airlines, hotel or rental cars. Although none of the industries mentioned above is truly monopolistic, the literature on RM in oligopolistic settings is scant. In this paper, I investigate the market interaction between n perfectly myopic customers and multiple sellers selling K

101 the one of the case of letting a competitor sell and the one of the case of selling herself. The reservation price of a smaller (in terms of remaining capacity) seller is less than the reservation price of a larger competitor and, hence, the smaller seller is willing to undercut further. As a result, the largest seller depletes her entire capacity last, while smaller sellers sell at her reservation price before. In spirit, such pricing dynamics are similar to martinez2011 who generalize the dynamic Bertrand-Edgeworth model of dudey1992. In martinez2011, negative prices are an odd but unavoidable feature which can occur in my setting as well. The reason is that, if two sellers are simultaneously the largest, each of them is willing to pay to become a smaller seller who then sells her entire stock first. While single customers with common known valuation arrive sequentially over time in the dynamic Bertrand-Edgeworth model, customers with heterogeneous valuations are already present in my setting. I can circumvent the negative prices by imposing that prices are sticky in the sense that a seller cannot immediately alter prices after a sale. In this case, a seller charging negative prices for the reason above would immediately sell out at the negative price, which results in a loss. Overall, however, sticky prices tend to be lower because the continuation payoffs of smaller sellers decrease which lowers theses sellers’ incentive to charge high prices. For applications such as airline ticket pricing, I extend my model to sequentially (and stochastically) arriving customers. If it is commonly known that additional customers, perhaps with stochastically higher values, may arrive after some point in time, a monopolist prefers to wait for them while under oligopoly the price competition to some extend unravels to the time before the additional buyers arrive. If sellers discount revenues after the arrival of the additional customers with rate δ, a monopolist would only sell to * customers with valuations above a threshold x (= v if δ=1). In contrast, competitive pressure leads sellers to serve lower valuation customers already in the first period. That is, competition raises an ex-post allocation inefficiency following the fact that higher- value customers arriving later might be rationed in favor of lower-valuation customers already present.

102

Numerical Construction of the Information Sets and Optimization of Detection Time in the Simple Search Game with a Team of Pursuers on the Plane

Semyon Mestnikov1 and Nikolay Petrov2

1Yakutsk State University, Russia [email protected] 2North-Eastern Federal University, Russia [email protected]

Keywords: Differential search game, Information set, Mixed strategies, Detection probability

The zero-sum differential search game between pursuer P and evader E in the class of mixed strategies with finite spectrum is considered([1]; [2]). The dynamic of the game is described by the following differential equation

where r>0 is the radius of the uncertainty disk of the initial disposition of Player E and the quantities r, l, α, and β are parameters of the game. The detection set S(x) of player P is the disk of radius l centered at the position of the pursuer. We consider an auxiliary game between evader E and some team 푃̅ = {푃1, … , 푃푘} of similar pursuers acting as one. To optimize the detection time of evader E by the team of pursuers 푃̅ =

{푃1, … , 푃푘}, the necessary conditions on parameters of the game were found in which pursuers detects of evader. In this article, we consider a simple search game, when pursuers move only on polyline and we do not impose any constraints on the movements of the evader. In simple case, where one pursuer participates in the search game, he chooses the moment of time t* and makes a turn. In this case, the trajectory of player P on the plane will be represented as a broken line consisting of three vertices A, B, C. Where the first vertex A is initial state of the pursuer P, the second vertex B is point that the player reaches at the time t*, the third C - the point at which full investigation of the reachability domain of the player E occurs.It is noted that when the player reaches for the point C, the information set - uncertainty disk of the player E becomes empty. Thus, by minimizing

103 the sum of the lengths of the segments AB + BC by the moment of the time t*, we can optimize the total time of the guaranteed detection of the evader E. Also some cases in which two players P and P are involved and the initial 1 2 position of the players P and P can be the same or different are considered. In this case, 1 2 the trajectory of each player with one turn is searched, where pursuers acting together and are guaranteed to detect the player E. For visual example,we numerically construct the approximation of the information sets for cases in games with one and two pursuers. At the same time, optimization of time detection is taken place programmatically, depending on the numerically game parameters given.

References

[1]. Petrosjan L. A., Garnaev A.Yu. (1992) Search games.(Russian) St.-Peterbg. Gos. Univ., St. Petersburg. [2]. Zenkevich N.A., Mestnikov S.V. (1991) Dynamical search of a moving object in conflict condition. (Russian) Leningrad Univ., Voprosy Mekh. Protsess. Upravl., No. 14, 68–76 [3]. Mestnikov S.V. (2002) Approximation of the information set in a differential search game with a team of pursuers. (English) Petrosjan, L. A. (ed.) et al., 10th international symposium on dynamic games and applications. St. Petersburg, Russia. In 2 vol. St. Petersburg: International Society of Dynamic Games, St. Petersburg State Univ.. 630- 631 [4]. Mestnikov S.V., Petrov N.V., Everstova G.V. (2014) Numerical Construction of the Information Sets in the Simple Search Game with a Team of Pursuers and Estimates for a Detection Probability.(English) GAME THEORY AND MANAGEMENT. Collected abstracts of papers presented on the Eighth International Conference Game Theory and Management (GTM2014) / Editors Leon A. Petrosyan and Nikolay A. Zenkevich. – SPb.: Graduate School of Management SPbU, 2014, 207–208

104

Market Manipulation: Game-theoretic Research Agenda

Victor Naroditskiy1

1OneMarketData LLC / BigDataSolutions, Russia [email protected]

Keywords: market manipulation, trade surveillance, mechanism design, manipulative bidding, financial markets

Auctions are one of the most prominent applications of game theory. Auctions have been analyzed both theoretically and empirically for equilibrium strategies and for auction rules that lead to desirable equilibrium strategies. By definition, a game is a strategic situation where strategizing (i.e., taking the effect of own actions on the actions of other participants into account) is a rational thing to do. In contrast, strategizing, or manipulation, is illegal when it comes to trading in financial markets. In particular, trading on exchanges (i.e., trading in continuous double auctions) is subject to regulation that can be summarized as "thou shall not manipulate". For example, European "Market Abuse Regulation" (MAR) that came into force July 3, 2016 reads (MiFID, 2014): ... market manipulation shall comprise the following activities: (a) entering into a transaction, placing an order to trade or any other behaviour which: (i) gives, or is likely to give, false or misleading signals as to the supply of, demand for, or price of, a financial instrument, a related spot commodity contract or an auctioned product based on emission allowances; ... Making manipulative behavior illegal requires the ability to detect it. This is where game theory comes in. Can we propose ways of detecting manipulative behavior by looking at the actions of all players? Manipulative behavior here can be defined as any strategy aimed at enticing other players into taking actions that are desirable for the manipulator. An example of behavior that regulators go after is below (SEC, 2017): As Avalon further anticipated, planned and intended, in response to Avalon's large options purchases, (i) market makers for those options would typically hedge their risk in response to Avalon's large options trades by trading in the corresponding stock, and (ii) this stock trading impacted the stock price and thus resulted in pushing the price of the options further in a favorable direction for Avalon.

105

Three reasons add to the appeal of studying market manipulation. Firstly, there is a wealth of data available for analysis. All of the bids (a.k.a. orders) as well as all of the trades (i.e., bids that were executed) are recorded by exchanges and are available as market data from exchanges and data distributors. We are talking about decades of data with billions of data points per day. The data may be provided for free for academic use. Secondly, market manipulation is of high practical importance. Regulators impose billion dollar fines. Interestingly, the responsibility for detecting manipulation lies with those who trade. All firms that engage in trading are required to have their own monitoring in place to check their own bidding for signs of market manipulation. This gave rise to a sizable (~.5 billion) and fast growing "trade surveillance" industry. Thirdly, there are interesting mechanism design questions. Exchanges and trading venues are keen to be able to detect manipulation and design mechanisms that prevent manipulation (Dynamic Auction / Introducing innovation for Borsa Italiana cash markets). Research questions include: 1. Finding signs of market manipulation in market data. E.g.:  Detecting spoofing/ momentum ignition / fix manipulation by analyzing market data 2. Modeling specific manipulative practices such as insider trading, spoofing, momentum ignition. E.g.:  Market conditions conditions when spoofing is likely to occur 3. Game-theoretic models that characterize manipulative behavior in continuous double auctions E.g.:  Identify characteristics of manipulative bidding  Find effects of market manipulation (Aggarwal & Wu, 2006)  Characterize conditions when manipulation is profitable (Allen & Gale, 1992) 4. Mechanism design to find rules that make financial markets less prone to manipulation References

[1]. Aggarwal, R. K., & Wu, G. (2006). Stock market manipulations. The Journal of Business, 1915-1953. [2]. Allen, F., & Gale, D. (1992). Stock-Price Manipulation. Review of Financial Studies, 502-529. [3]. Dynamic Auction / Introducing innovation for Borsa Italiana cash markets. (n.d.). Retrieved from Borsa Italiana: http://www.borsaitaliana.it/borsaitaliana/pubblicazioni/pubblicazioni/anti- spoofing.en.pdf

106

[4]. MiFID. (2014, April 16). Regulation (EU) No 596/2014 of the European Parliament and of the Council. Retrieved from https://eur-lex.europa.eu/legal- content/EN/TXT/?uri=CELEX%3A02014R0596-20160703 [5]. SEC. (2017, March 10). SEC Avalon Complaint. Retrieved from SEC: https://www.sec.gov/litigation/complaints/2017/comp-pr2017-63.pdf

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

107

Allocation problems for generalized games with restricted cooperation

Natalia Naumova1 and Elena Chernysheva2

1Saint-Petersburg State University, Russia [email protected] 2LLC SKYROS-TELECOM, Russia [email protected]

Keywords: generalized nucleolus, generalized antinucleolus, weakly mixed collections of coalitions

Let N={1,…,n} be a set of players, A be a collection of some subsets of N, c>0. Let be the set of allocations of the resourse c between n players, i.e., X is the set of imputations. Denote x ={x } , . For S∈A, let S i i∈S X ={x :x∈X}, G be a continuous strictly increasing in each variable function defined on S S S X . We suppose that each x can be used by each S∈A such that i∈S, and G (x ) is either S i S S the gain of S at the allocation x or the loss of S at x. Let (N,v) be a TU-cooperative game with v(N)>0. Then (N,v) generates c=v(N) and G1(x )=x(S)−v(S). If moreover v(S)>0 for each S∈A, then (N,v) also generates S

G2(x )=x(S)/v(S). S We consider the following solutions. An imputation x belongs to weakly {G } S S∈A - equal sacrifice solution if for each P,Q∈A with P∩Q=∅, x(P)>0 implies G (x )≤G (x ) P P Q Q For G =G2,S we obtain weakly proportional solution of (N,A,c,v) and for G =G1,S S S we obtain weakly uniform losses solution of (N,A,c,v) defined in [2]. A generalization of these solutions (weakly U-equal sacrifice solution) was introduced in [4]. The condition on A that guarantees existence of weakly {G } - equal sacrifice imputations for all S S∈A continuous strictly increasing in each variable functions {G } is the same as the S S∈A condition for weakly proportional solution. For concave G , we define lexicographically maxmin solution that coincides for S G (x )=U(x(S))+c with U-nucleolus defined in [4]. It concerns the case when G (x ) is S S S S S the gain of S. If G (x ) is the loss of S, for convex G , lexicographically minmax solution S S S

108 is defined. If and G =G2,S then this solution coincides with S antinucleolus that was introduced in [5]. For i∈N, denote A ={T∈A:i∈T}. i A is a weakly positive mixed collection of coalitions if for each i∈N, Q∈A , S∈A, i with Q∩S=∅, there exists j∈N such that A ⊃A ∪{S}∖{Q}. j i A is a weakly negative mixed collection of coalitions if for each i∈N, Q∈A , S∈A, i with Q∩S=∅, there exists j∈N such that A ⊂A ∪{S}∖{Q}. Let G (x )=g (x(S))S, where j i S S S g are strictly increasing continuous functions. If A is a weakly positive mixed collection S of coalitions then the lexicographically maxmin solution is contained in the weakly {G } - equal sacrifice solution. For G =G1,S it was proved in [3]. If A is a weakly S S∈A S negative mixed collection of coalitions then the lexicographically minmax solution is contained in the weakly {G } - equal sacrifice solution. If G =G1,S then the S S∈A S lexicographically minmax solution is contained in the weakly proportional solution at each (N,v) iff A is a weakly negative mixed collection of coalitions. Let G (x )=k x(S)+c with S S S S k >0 and either A be weakly positive mixed or A be negative mixed. Then we describe S iteration procedures that converge to points in the weakly {G } - equal sacrifice S S∈A solution. These iteration procedures are similar to the iteration procedure in [1].

References

[1]. Maschler M., Peleg B. (1976) Stable sets and stable points of set-valued dynamic systems with applications to game theory, SIAM Journal on Control and Optimization, 14, 985-995 [2]. Naumova N. (2011) Claim problems with coalition demands. In: Contributions to Game Theory and Management GTM2010 Vol. 4 Collected Papers. Ed. by L.A.Petrosjan and N.A.Zenkevich. Graduate School of Management St.Petersburg University, St. Petersburg, 311-326 [3]. Naumova N. (2012) Generalized proportional solutions to games with restricted cooperation. In: Contributions to Game Theory and Management GTM2011 Vol. 5 Collected Papers. Ed. by L.A.Petrosjan and N.A.Zenkevich. Graduate School of Management St.Petersburg University, St. Petersburg, 230-242 [4]. Naumova N.(2013) Solidary solutions to games with restricted cooperation. In: Contributions to Game Theory and Management GTM2012 Vol. 6 Collected Papers. Ed. by L.A.Petrosjan and N.A.Zenkevich. Graduate School of Management St.Petersburg University, St. Petersburg, 316-337 [5]. Sudholter P., Peleg B.(1998) Nucleoli as maximizers of collective satisfaction functions. Social Choice and Welfare, 15, 383-411.

109

Channel Coordination with Sales Rebate Contract in Cooperative Supply Networks

Natalia Nikolchenko1 and Nikolay Zenkevich2

1,2Graduate School of Management, St. Petersburg State University, Russia [email protected] [email protected]

Keywords: coordinating contracts, sales rebate contract, channel coordination, bargaining power in contract decision making, supply network

Channel coordination is an effective way to improve supply chain performance. Supply chain coordination with contracts is generally used for removing inefficiency along the supply chain (SC) and aligning supply chain members' objectives (Cachon, 2001; Chen, 2011). The sales rebate contract (SRC) is one of the contracts that coordinate the supply chain with one compliance regime, and channel rebates are widely adopted in the hardware, software and auto industries (Taylor, 2002). The SRC provides a direct incentive for retailers to increase sales because the rebate only applies to items that are sold to end-users. Thus, the SRC motivates retailers to sell goods at a lower price to increase sales (Wong et al., 2009). The target rebate offers an advantage to the manufacturer. By setting the target properly, the manufacturer can induce the distributor or retailer to behave in a way that reflects the marginal revenue of the rebate while shielding the manufacturer from the full cost of doing so. A sales rebate contract is difficult to implement in a traditional supply chain, mainly because a traditional supply chain does not have a mechanism to facilitate continuous information exchange between chain members. The supplier needs to know the exact quantity sold by the retailer in order to pay the rebate, but difficulties arise when the supplier cannot acquire the retailer’s sales data directly. On the other hand, the data obtained from the retailer may not be authentic as the retailer may claim more rebates than what the actual sales allow. (Taylor, 2002).. The sales rebate contract coordinates the supply chain in the setting in which the distributor or retailer determine the optimal order quantity to optimize the overall supply chain profit. The SRC is a more efficient mechanism to increase sales than the quantity discount contract, it relies on a complex administration procedure as manufacturer or distributor need to know the exact quantity sold. (Taylor, 2002; Wong et. al., 2009).

110

The research is based on the existing literature in supply chain coordination contracts. The purpose of the study is to show how sales rebate contract helps achieve supply chain coordination. The cooperative multi-players game involving single manufacturer, single distributor and multi retailers is studied. The game solution is the optimal parameters of the contract in terms of coordination. The three-echelon supply chain model including single manufacturer, single distributor and multiple retailers is proposed. The last part of the study addresses the case of simple two echelon supply chain including manufacturer and distributor related to the pharmaceutical industry. The relationships based on sales rebate contract are considered. The obtained results are presented in the study.

References

[1]. Cachon, G., 2001. Supply chain coordination with contracts. In: Graves, S., Kok, T.D. (Eds.), Handbooks in Operations Research and Management Science: Supply Chain Management. North-Holland, Philadelphia PA, pp. 1-95. [2]. Chen, J.: “Returns with wholesale-price-discount contract in a newsvendor problem”. Int. J. Prod. Econ. 130 (1), 2009, pp.104-111. [3]. Taylor, T.: “Coordination under channel rebates with sales effort effect”. Management Science 48 (8), 2002, pp. 992–1007. [4]. Wong, W., Qi, J., Leung, S.: “Coordinating supply chains with sales rebate contracts and vendor-managed inventory”. Int. J. Prod. Econ. 120 (1), 2009, pp.151-161.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. МЕНЕДЖМЕНТ

Главный редактор Н. П. Дроздова

Издательство Санкт-Петербургского государственного университета

111

Stackelberg Equilibrium in a Dynamic Incentive Model With Full Information

Guennady Ougolnitsky1 and Dmitry Rokhlin2

1,2Southern Federal University, Russia [email protected] [email protected]

Keywords: dynamic incentive model, discrete time, Stackelberg game

Consider a discrete time controlled system with a compact metric state space X, endowed with the Borel σ-algebra B(X). Assume that the set Γ(x) of admissible actions is a non-empty closed subset of the interval [0,1]. The system dynamics is described by the transition kernel q(B|x,a). That is, the probability of its transition from a state x∈X to a set B∈B(X) under an action a∈Γ(x) equals to q(B|x,a). Furthermore, assume that a is an effort of the performer, aimed at the management of the system and leading to the cost g(x,a). Simultaneously, the regulator receives the gain f(x,a). It is supposed that the functions 푓, 푔: 퓧 × [0,1] → 푅+ are continuous, f(x,0)≤0, g(x,0)=0 and 푚푎푥푎∈[0,1] 푓(푥, 푎) − 푔(푥, 푎)) > 0 The regulator does not directly control the system but communicates the performer an incentive function c from the family L of upper semicontinuous functions

푐: 퓧 × [0,1] → 푅+ The performer uses feedback controls u from the family U of Borel functions u:X↦[0,1], satisfying the condition u(x)∈Γ(x). By fixing such function and an ∞ initial staty x, we get the probability measure 푃푥,푢 on the space (A×X) of trajectories, which can be formally written as follows (see, e.g., [5, Appendix C]):

where δ is the Dirac measure, concentrated at y. The expectation with respect to y

푃푥,푢 we denote by 퐸푥,푢. If the regulator chooses an incentive function c∈L, and the performer chooses a control u∈U, then the discounted gains of the regulator and the performer are defined by the formulas

112

where β∈[0,1) is the discount factor. Denote by T(c) the set of optimal strategies of the performer. Consider the Stackelberg game, where the regulator is pessimistic and assumes that the performer takes a worst optimal strategy:

Hence, the optimal gain of the performer equals to Let us call

V the game value (for the regulator). For ε>0 a function c ∈L is called ε Stackelberg 1 ε strategy (1, Definition 4.7]), if A version of Stackelberg game, adapted to the stimulation problem, is called the reverse (or inverse) Stackelberg game (following [3]). In such a game the regulator takes an incentive function, depending on the performer actions, and communicates it to the performer. The concrete problem, formulated above conceptually similar to that considered in [4]. The obtained results can be considered as a generalization of Theorems 1 and 2 from the mentioned paper to the model with Markovian dynamics, infinite horizon and discounted optimality criteria.

Theorem 1 Assume that there exists a continuous optimal solution v ∈U of the problem (1)

and some technical conditions holds true. Then value V of the Stackelberg game, 1 introduced above, coincides with V and the upper semicontinuous function

(2)

is an ε Stackelberg strategy. Moreover, T(c )={ v }. ε This result shows that the regulator should imagine that he chooses the actions by himself and his costs coincide with the costs of the performer. After extracting of an optimal solution v from the control problem (1), the regulator should announce the incentive function c . If the performer acts optimally from his point of view, then the ε

113 regulator gain 퐽1(푥, 푐̅푒, 푣̅) = 푉(푥) − 휀 differs by ε from his optimal gain, and the performer gain equals to ε. Note, that the ε Stackelberg strategy (2) is not unique.

For a finite X the continuity condition of v automatically holds true. In the general case, under some technical conditions, the continuity of an optimal solution v follows from its uniqueness. We also obtained a version of Theorem 1 without such continuity assumption. In this version we consider (ε,η) Stackelberg solution in the sense of [6, Definition 4.1] for a Stackelberg game over the classes of universally measurable functions instead of L, U.

References

[1]. Basar T., Olsder G.J. Dynamic noncooperative game theory. Philadelphia: SIAM, 1999. [2]. Dockner E., Jørgensen S., Van Long N., Sorger G. Differential games in economics and management science. Cambridge: Cambridge University Press, 2000. [3]. Ho Y.-C., Luh P., Muralidharan R. Information Structure, Stackelberg Games, and Incentive Controllability // IEEE Trans. Automat. Control. 1981. V. 26. No. 2. P. 454– 460. [4]. Novikov D.A., Shokhina T.E. Incentive Mechanisms in Dynamic Active Systems // Autom. Remote Control. 2003. V. 64. No. 12. P. 1912–1921. [5]. Hernández-Lerma O., Lasserre J.B. Discrete-time Markov control processes: basic optimality criteria. N.-Y.: Springer, 1996. [6]. Morgan J. Constrained well-posed two-level optimization problems / Clarke F.H., Dem’yanov V.F., Giannessi F., editors. Nonsmooth optimization and related topics. Boston: Springer, 1989, P. 307–325.

114

Outsource to or Compete with a More Efficient Input Supplier? Strategic Outsourcing versus Strategic Competition

Konstantinos Papadopoulos1 and Chariklia Dermentzi2

1,2Aristotle University of Thessaloniki, Greece 1 [email protected] [email protected]

Keywords: Make-Buy, strategic outsourcing, common supplier

Firms outsource production of inputs to other parties for cost reduction or strategic purposes. The literature on strategic outsourcing has highlighted many key factors that affect a firm's buy or make decision (See for instance Arya et al (2008b), Buehler and Haucap (2006), Chen (2011), Chen et al. (2010), Shy and Stenbacka (2003)). In this article, we focus on a particular market structure where downstream firms outsource the production of an essential input to a common supplier. In this context it has been shown by Arya et al. (2008), that a downstream firm, which has in-house production capability, may accept to buy input from a supplier at a unit price which is higher than its own in-house unit production cost. This result, which challenges conventional wisdom, relies on strategic considerations. If the downstream firm decides not to outsource, it will be worse off at equilibrium, because the supplier will provide better contract terms to the rival and downstream competition will be adversely affected. Therefore, a firm may be willing to pay a premium to outsource to its rival's supplier in order to increase rival's cost. In this work, we show that the above result relies crucially on the assumption that there can be no outsourcing among downstream rivals. We may observe that in the model of Arya et. al (2008) at a Nash equilibrium the supplier sets wholesale prices for both downstream firms that are higher than the unit production cost of the firm with input production capability. The above observation suggests that the latter firm may contemplate higher profits by making the input internally and selling part of it to its rival, instead of a paying a premium to increase rival's cost. Thus, the firm with input production capability may benefit from becoming a wholesale competitor towards the supplier. The supplier anticipating the eventual price competition with one of the downstream firms at a later stage of the game, will charge a wholesale price equal to the marginal cost of the downstream firm. So allowing for the possibility of outsourcing among rivals, we show

115 that outsourcing among rivals does not actually occur at equlibrium, the wholesale equilibrium price is uniform and all downstream firms will buy input from the common supplier which is the least cost producer of the input. This result is in line with the conventional wisdom that a firm will not accept to buy input at a price higher that its internal production cost. Furhermore we show that when outsourcing to rival is allowed, consumer prices are strictly lower compared to the case where it is not. We wish to extend the model to bargaining, two part tariffs, entry deterrence and various cost specifications.

Periodicals in Game Theory

CONTRIBUTIONS TO GAME THEORY AND MANAGEMENT

Volumes 1–11

Edited by Leon A. Petrosyan Nikolay A. Zenkevich

St. Petersburg State University

116

Dynamic Game Model of Executors Incentives in Projects for the Development of New Production

Oleg Pavlov1

1Samara National Reseach University, Russia [email protected]

Keywords: hierarchical dynamic game, learning curve effect, new products development project

The incentive problem of executors of the new products development project at the industrial enterprise is considered in this article. Mastering of a new product leads to the learning curve effect, which implies reduction of time spent (labor intensity) on performing repetitive tasks by workers. The project of the new products development is considered as a managed hierarchical dynamic system, consisting of a project management board (principal) and executors (agents). In this dynamic game model there are dynamics of decision making and dynamics of the managed system. The inequality of participants is fixed by the moves order, the first move is made by the principal. With the exact volume of work selected by the agent u(t), the principal chooses the incentive function. It is assumed that agents are not linked to each other and perform independently. The dynamics of the new product production is described by a differential equation.

The incentive problem is formalized as a Hermeyer differential game of Г 2t type for two players with feedback on management:

T T t t J p   g p ( (u(t)),u(t))e dt  extr, Ja  { (u(t))  С(x(t),u(t))}e dt  max, 0 0 dx(t)  u(t), 0  u(t)  x  R  x(t), t  0,T, dt 0

x(0)  x0,

x(T)  x0  R.

where g p ((u(t)),u(t)) is a goal function of the principal,  is principal’s discount rate,  (u(t)) is incentive function of the principal, u(t) - production volume of the agent at time t, x(t) is cumulative production volume, T is project time,

117

С(x(t),u(t)) - agent's costs in monetary terms, x0 - product volume produced by the agent prior to the start of the project, R stands for product volume to be produced by the time T. Two options are considered as a goal function of the principal. 1. Maximization of discounted profit of the principal: , where p is a price of a product. g p ((u(t)),u(t))  pu(t) (u(t))  max 2. Minimization of the discounted costs of agent incentives: g ((u(t)),u(t)) (u(t))  min . p The agent’s costs in monetary terms С(x(t),u(t)) are defined as the multiplication of labor intensity с(x(t)) , the cost of the norm-hour at the enterprise s and the production volume u(t): С(x(t),u(t))  sс(x(t))u(t). The dynamics of the change in labor intensity from the cumulative production volume is described by different learning curve models. There are power, exponential and logistic models considered in this paper.

The power model of learning curve has the following form: c(x(t))  ax(t)b , where а is agent’s time costs for the first product, b – learning index. The learning index characterizes the decrease rate in labor intensity with an increase in the cumulative production volume. x(t) Exponential model of learning curve: c(x(t))  k  e . where  is learning index, k ,  are exponential model parameters.

 1  Logistic model: , c(x(t))  cmin  (cmax  cmin ) x(t)  1 e  c where cm i n , m a x - minimum and maximum values of labor intensities,  - learning index,  - exponential model parameters. The algorithm for solving the problem is as follows. 1. The principal chooses the incentive function which consists in compensating

R the costs of the agent if he chooses the optimal planned trajectory of the principal u (t) and in the absence of material payments, otherwise.

118

 R  (u(t))  C(x(t),u(t)), if u(t)  u (t), дляt [0,T], 0, otherwise.  R 2. The optimal planned trajectory of the principal u (t) is determined from the solution of the optimal control problem:

T t dx(t) J p  e g p (С(x(t),u(t)),u(t))dt  extr,  u(t), 0 dt 0  u(t)  x  R  x(t), x(0)  x , x(T)  x  R 0 t  0,T, 0 0 . Target functions of the principal taking into account the chosen incentive function will take the following form. 1. Maximization of discounted profit of the principal: g (C(x(t),u(t)),u(t))  pu(t) C(x(t),u(t))  max p . 2. Minimization of the discounted costs of agent incentives: g (C(x(t),u(t)),u(t))  C(x(t),u(t))  min p . To solve the stated problem of optimal control, we used the Pontryagin maximum principle. The direct application of the Pontryagin maximum principle to the formulated problem is impossible, since in this case there is a special control. A transition is made to an equivalent problem in which there is a logarithmic form of the goal function of the principal:

T t J p   ln{g p (С(x(t),u(t)),u(t))}e dt  extr. 0 Analytical solutions of problems for various models of learning curve and various target functions of the center are obtained. The reported study was funded by RFBR and Samara region according to the research project № 17-46-630606.

119

Existence of IDP-core in Cooperative Differential Games

Ovanes Petrosian 1 and Victor Zakharov2

1,2Saint-Petersburg State University, Russia [email protected] [email protected]

Keywords: Differential game, Cooperative Differential Game, IDP-core, Linear Programming

In this paper, we consider cooperative differential game with non-transferable payoffs and study non-emptiness conditions of IDP-core presented in [1]. In order to do that we use the approach proposed in [2] and used for Core and SC-core in static TU- cooperative games. Corresponding theorem for n-player differential game is proved and special conditions for a case of 3-players differential game are presented. V. Zakharov in [2] proposed necessary and sufficient conditions for non- emptiness of Core, which simplify test whether single-point solutions, such as Shapley’s vector, Banzaf’s influence index and others, belong to the Core. In [3] and [4] on the basis of the proposed approach, geometric properties of number of selectors were investigated. The approach implies that non-emptiness property of Core can be verified by solving the linear programming problem constructed using a system of constrains from it’s definition. Notion of IDP-core was introduced in the paper [1]. There authors constructed a new cooperative solution and proved that it is a subset of Core and possesses the property of strong time consistency. Verification of time consistency of cooperative solution, together with construction of cooperative trajectory, method of allocating total payoff among players, are the main problems in the theory of cooperative differential games. Concept of time consistency of cooperative solutions was mathematically formalized by L. Petrosyan [5]. In [7], he introduced the notion of imputation distribution procedure (IDP), which is used for constructing time consistent solutions, and in paper [6] he defined the concept of strong time consistency. IDP-core is constructed using a system of linear constraints for imputation distribution procedures. These conditions are defined for each time instant of differential game. From non-emptiness of the set described by these constraints, i.e. non-emptiness of corresponding set of IDPs at each time instant, it follows that IDP-core is not empty. We apply the technique proposed in [2] to study non-emptiness of IDP-core at each time

120 instant and if it is so, then we conclude that IDP-core is non-empty. Obtained results can be used for numerical construction of IDP-core and verification of its non-emptiness. Also a special case of this approach for 3-player differential game is presented. It is shown that it is possible to construct analytically conditions for non-emptiness of IDP- core depending on the characteristic function. Consequently it is possible to define analytical formula for selectors of the IDP-core, or more accurately for the imputation distribution procedures of the IDP-core selectors. According to the analytical form of the conditions for non-emptiness of IDP-core it is possible to define the exact location of solution of linear programming problem described above in set of admissible solutions.

References

[1]. O. Petrosian, E. Gromova, S. Pogozhev. Strong Time-consistent Subset of Core in Cooperative Differential Games with Finite Time Horizon // Mathematical Game Theory and its Applications. 2016. V. 8. N. 4. P. 79–106. [2]. V. Zakharov, O-Hun Kwon. Linear programming approach in cooperative games // J. Korean Math. Soc. 1997. V. 34. N 2. P. 423–435. [3]. V. Zakharov, A. Akimova. Nucleous as a selector of the subcore // Proc. of 11th IFAC Workshop Control Applications of Optimization, St. Petersburg, Russia, July, 2000. V. 33. N. 16. P. 675–680. [4]. V. Zakharov, A. Akimova. Geometric Properties of the Core, Subcore, Nucleolus, Volume VIII // Nova Science Publishers, 2002. P. 281–289. [5]. L. Petrosyan Time-consistency of solutions in multi-player differential games // Vestnik Leningrad State University. 1977. V. 4. N. 19. P. 46–52. [6]. L. Petrosyan Strongly time-consistent differential optimality principles // Vestnik St. Petersburg Univ. Math. 1993. V. 4. P. 35–40. [7]. L. Petrosyan, N. Danilov Stability of solutions in non-zero sum differential games with transferable payoffs // Vestnik Leningrad State University. 1979. V. 1. P. 52–79.

VESTNIK OF SAINT PETERSBURG UNIVERSITY. MANAGEMENT

Editor-in Chief Natalia P. Drozdova

St. Petersburg University Press

121

Increasing MANET performance by using the game theory principles and techniques

Taisiia Plekhanova1, Stewart Blakeway2, Ekaterina Gromova3 and Anna Kirpichnikova4

1,3Saint-Petersburg State University, Russia [email protected] [email protected] 2,4Liverpool Hope University, United Kingdom [email protected] [email protected]

Keywords: Increasing MANET performance, Multistage games, Nash equilibria, Drone placement, Graphs

In this paper, we describe a novel game-theoretic formulation of the optimal mobile agents’ placement problem which arises in the context of Mobile Ad-hoc Networks (MANETs). This problem is modelled as a sequential multistage game. Various types of topologies are considered which are represented as square, hexagonal and triangular grids. Algorithms were coded in JavaScript which tested the theory of optimising mobile agent placement and resulted in achieving Nash Equilibria. To establish if the proposed algorithm increased network performance the novel algorithm was implemented in C++ and tested using Network Simulator 3. The analysis of results derived from the simulation runs demonstrate that the proposed novel algorithm increases network performance (throughput and end-to-end delay) by using the game theory principles and techniques.

References

[1]. Gromova E., Gromov D., Timonin N., Kirpichnikova A., Blakeway S. A dynamic game of mobile agents placement on MANET. Proc. of the IEEE conference SIMS 2016. [2]. Yu F. R. Cognitive Radio Mobile Ad Hoc Networks. Springer, 2011. [3]. Z. Han, D. Niyato, W. Saad, T. Basar, A. Hjorungnes. Game Theory in Wireless and Communication Networks. Theory, Models, and Applications. Cambridge University Press, 2012. [4]. Dijkstra E. W. A note on two problems in connection with graphs. Numerische Matematik, 1959. С. 269-271. [5]. Wilson R. J. Introduction to Graph Theory. Edinburgh: Oliver and Boyd, 1972. [6]. Plekhanova T., Gromova E., Gromov D., Kirpichnikova A., Blakeway S. The Strategic Placement of Mobile Agents on a Hexagonal Graph using Game Theory. Proc. of the IEEE conference ICAT 2017."

122

One Way Flow Dynamic Network Games With Shock And Network Formation Cost

Sun Qiushi1

1Saint-Petersburg State University, China [email protected]

Keywords: network formation cost, imputation, subgame consistency

In this paper, one way flow dynamic network games with shock and net work formation cost are considered. After the first network formation stage, a particular player with a positive probability will become in active in the game by breaking all the links he formed or accepted and obtaining zero payoffs. This influence is called “shock.” And shock can occur only once, and the stage number, between which two stage shock occurs, is chosen in random. In the cooperative case of the game, subgame consistency of the Shapley value, based on a characteristic function, which is constructed in a special way, is investigated. To prevent players from breaking the cooperative agreement, another method of stage payments, imputation distribution procedure is introduced.

ВЕСТНИК САНКТ–ПЕТЕРБУРГСКОГО УНИВЕРСИТЕТА. ПРИКЛАДНАЯ МАТЕМАТИКА, ИНФОРМАТИКА, ПРОЦЕССЫ УПРАВЛЕНИЯ

Главный редактор Л.А. Петросян

Издательство Санкт-Петербургского государственного университета

123

A Note on Four-Players Triple Game

Enkhbat Rentsen1, Batbileg Sukhee2, Tungalag Natsagdorj3, Alexander Gornov4 and Anton Anikin5

1,2,3National University of Mongolia, Mongolia [email protected] [email protected] [email protected] 4,5Matrosov Institute of System Dynamics and Control Theory, SB RAS, Russia [email protected] [email protected]

Keywords: Nash equilibrium, triple game, local search, global search

We consider the four-person matrix game where each of them plays with other three players. We call such game four-players triple game. In this game we introduce a definition of Nash equilibrium similarly to [1]. The game reduces to a nonconvex optimization problem. For solving the optimization problem, we propose a global optimization method that combines the ideas of the classical multistart and local search methods. References

[1]. Strekalovsky, A. S. and Enkhbat, R. (2014) Polymatrix games and optimization problems. Automation and Remote Control 75(4), 632--645.

VESTNIK OF SAINT PETERSBURG UNIVERSITY. MANAGEMENT

Editor-in-Chief Natalia P. Drozdova

St. Petersburg University Press

124

Cooperation in the Bioresource Management Problems

Anna N. Rettieva1

1Institute of Applied Mathematical Research Karelian Research Center of RAS, Russia [email protected]

Keywords: dynamic games, bioresource management problems, cooperative behavior, incentive equilibrium, asymmetric players

The presented talk is dedicated to overview the results of rational behavior analysis in a dynamic bioresource management problems. The primary aim of rational nature exploitation lies in sustainable development of a population. Therefore, studying the difference between cooperative and egoistic (individual) behavior in optimal bioresource management problems represents an important issue. As is well-known, cooperation brings to a sparing mode of bioresource exploitation. There are several methodological schemes to maintain the cooperation. Here we focus on two of them: incentive equilibrium and time-consistent imputation distribution procedure. The concept of cooperative incentive equilibrium was introduced in [1]. In our papers [2], [4] we presented the new scheme where the center controls the cooperation agreement by changing the harvesting territory. The concept of time-consistency (dynamic stability) was introduced by Petrosyan L.A. [6]. The applications of these concepts to bioresource management problems are presented and, furthermore, the new condition for rational behavior is given [3]. Traditionally, cooperative behavior analysis in bioresource management problems bases on the assumption of identical discount factors for all players. If these factors differ (players are asymmetric), standard techniques do not assist in evaluating players’ payoffs under their cooperation. Consequently, a substantial role in cooperative behavior analysis of bioresource management problems belongs to seeking an optimal compromise in the case of heterogeneous goals pursued by players (different discount factors and fishing costs). Our paper [7] suggests to design and stimulate cooperative behavior using the Nash bargaining solution. Cooperative behavior analysis in bioresource management problems with different planning horizons is an important problem, both theoretically and practically.

125

Paper [5] introduced the Nash bargaining solution to construct cooperative strategies in the case of different planning horizons. The model with random planning horizons in the bioresource exploitation process is most adequate to the reality: external random factors can cause cooperative agreement breach and the participants know nothing about them a priori. To define cooperative behavior in this case, we employ the Nash bargaining solution; the role of the status quo point belongs to the noncooperative payoffs of the players [8]. According to the aforesaid, cooperative behavior design in very important. Here we present our results in this direction. Almost all the results are derived analytically, which allows their direct application to concrete biological populations with appropriate parameters. The research was supported by the Russian Science Foundation, project no. 17- 11-01079. References

[1]. Ehtamo H., Hamalainen R.P. A cooperative incentive equilibrium for a resource management problem // J. of Economic Dynamics and Control. 1993. V. 17. P. 659–678. [2]. Mazalov V.V., Rettieva A.N. Incentive equilibrium in discrete-time bioresource sharing model // Doklady Mathematics. 2008. V. 78(3). P. 953–955. [3]. Mazalov V.V., Rettieva A.N. Incentive conditions for rational behavior in discrete-time bioresource management problem // Doklady Mathematics. 2010. V. 81(3). P. 399–402. [4]. Mazalov V.V., Rettieva A.N. Fish wars and cooperation maintenance // Ecological Modelling. 2010. V. 221. P. 1545–1553. [5]. Mazalov V.V., Rettieva A.N. Asymmetry in a cooperative bioresource management problem // Game-Theoretic Models in Mathematical Ecology. 2015. P. 113–152. [6]. Petrosyan L.A. Stability of differential games with many players’ solutions. Vestnik of Leningrad university. 1977. V. 19. P. 46–52.

126

Acceptable points in games with ordered outcomes

Victor Rozen1

1Regional Akademy, Russia [email protected]

Keywords: game with ordered outcomes, acceptable point, antagonistic game with ordered outcomes

Acceptable point concept is some generalization of equilibrium point concept.

An important advantage of the acceptability concept is the fact that every finite game G with ordered outcomes has an acceptable outcome in pure strategies. Recall that an outcome is called acceptable in game if no player having an objection in the form of a strategy to this outcome. We consider some problems concerning the existence and description of the set of acceptable outcomes in games with ordered outcomes. For antagonistic games with ordered outcomes, the complete description of acceptable points structure is given.

References

[1]. Aumann R. J. (1959). Acceptable points in general cooperative n -person games. Contribuutions to the theory of games, vol.4. Annals of Math. Studies, 40, Princeton, Princeton University Press, p. 287-324.

[2]. Rozen, V.V. (2013). Decision making under qualitative criteria. Mathematical models. (in Russian). Palmarium Academic Publishing. Saarbrucken, Deutschland/Германия.

127

Opinion dynamics game in a social network with two influence nodes

Artem Sedakov1 and Mengke Zhen2

1,2Saint Petersburg State University, Russia [email protected] [email protected]

Keywords: network, influence, opinion dynamics, equilibrium

We consider an opinion dynamics game in a social network with two influence nodes. Pursuing certain goals, the influence nodes affect other members of the network by the selection of their levels of influence. Considering this model as a 2-person noncooperative dynamic game and choosing Nash equilibrium as its solution, we find the equilibrium levels of influence for both influence nodes at any game stage. We also perform numerical simulation for both low and high levels of players’ influence on agents. The first author acknowledges the Russian Science Foundation (grant No. 17-11- 01079).

Journals in Game Theory

INTERNATIONAL JOURNAL OF GAME THEORY

Editor Shmuel Zamir

SPRINGER

128

Looking Forward Approach for dynamiccooperative advertising game model

Lihong Shi1 and Ovanes Petrosian2

1Saint Petersburg State University, China 2Saint Petersburg State University, Russia [email protected] [email protected]

Keywords: Advertising Oligopoly, Dynamic Game, Cooperative Game, Looking Forward Approach, Time Consistency

In this paper we examine a dynamic infinite cooperative advertising game model of Naik, Prasad and Sethi[9] where each firm’s market share depends on its own and its competitors’ advertising decisions. We apply Looking Forward Approach in order to model behavior of players when information about the process updates dynamically. We refrain from reviewing this literature, given that there are recent surveys. Huang, Leng, and Liang[4] give an extensive and comprehensive survey of dynamic models of advertising competition since 1994, their coverage starting where the previous survey in Feichtinger, Hartl, and Sethi[2] stopped. Jorgensen and Zaccour[5] cover advertising models in oligopolies (horizontal strategic interactions) and in marketing channels (vertical strategic interactions), whereas He, Prasad, Gutierrez, and Sethi[3] focus on Stackelberg differential game models in supply chains and marketing channels that include advertising. Finally, Aust and Buscher[1] and Jorgensen and Zaccour[6] concentrate on cooperative advertising in marketing channels. Looking Forward Approach is used for constructing game theoretical models and defining solutions for conflict-controlled processes where information about the process updates dynamically. Existing dynamic games often rely on the assumption of time invariant game structures for the derivation of equilibrium solutions. However, many events in the considerably far future are intrinsically unknown. It is supposed that players lack certain information about the dynamical system and payoff function on the whole time interval on which the game is played. At each time instant information about the game structure updates and players receive new updated information about dynamical system and payoff functions. This new approach for the analysis of dynamic games via information updating provides a more realistic and practical alternative to the study of infinite horizon dynamic games. Optimal strategies in the game models with Looking

129

Forward Approach depend mainly on the length of information horizon and time between instants when information updates. It brings up the question of constructing new or transform existing Bellman equations such that the solution depends upon this parameters. Construction of Bellman equations where information horizon is defined as parameters will give researchers the ability to study behavior of players as a function of the value of information horizon. In the paper we construct corresponding Bellman equation for dynamic game model. The concept of Looking Forward Approach[7,8,10] is new in game theory especially in cooperative differential games. In this paper we consider oligopoly advertising model of Naik, Prasad and Sethi [9], transferring it into discrete case. Then we consider the cooperative case of the initial game and apply the Looking Forward Approach to it, then provide results about the time consistency of the resulting solution and results related to connection of solutions chosen by the players in each truncated subgame and in the overall game. Results of numerical simulation in Matlab are presented.

References

[1]. Gerhard Aust and Udo Buscher. Cooperative advertising models in supply chain management: A review. European Journal of Operational Research,234(1):1–14, 2014. [2]. Gustav Feichtinger, Richard F Hartl, and Suresh P Sethi .Dynamic optimal control models in advertising: recent developments. Management Science,40(2):195–226, 1994. [3]. Xiuli He, Ashutosh Prasad, Suresh P Sethi, and Genaro J Gutierrez. A survey of stackelberg differential game models in supply and marketing channels, Journal of Systems Science and Systems Engineering, 16(4):385–413, 2007. [4]. Jian Huang, Mingming Leng, and Liping Liang. Recent developments in dynamic advertising research, European Journal of Operational Research,220(3):591–609, 2012. [5]. Steffen Jørgensen and Georges Zaccour. Differential games in marketing. international series in quantitative marketing, 2004. [6]. Steffen Jørgensen and Georges Zaccour. A survey of game-theoretic models of cooperative advertising, European Journal of Operational Research,237(1):1–14, 2014. [7]. Ovanes Petrosian and Andrey Barabanov. Looking forward approach in cooperative differential games with uncertain stochastic dynamics, Journal of Optimization Theory and Applications, 172(1):328–347, 2017. [8]. Ovanes Petrosian, Maria Nastych, and Dmitrii Volf. Differential game of oil market with moving informational horizon and non-transferable utility, In Constructive Nonsmooth Analysis and Related Topics (dedicated to thememory of VF Demyanov)(CNSA), 2017, pages 1–4. IEEE, 2017. [9]. Prasad, A., Sethi, S. P., Naik, P. A. (2009). Optimal control of an oligopoly model of advertising. INCOM 2009,Moscow,Russia. [10]. Yeung D.W.K., Petrosian O. (2017) Infinite horizon Dynamic Games: a New Approach via Information Updating, International Game Theory Review.

130

Apology of Schumpeter: Positive Social Welfare effect of Oligopolization

Alexander Sidorov1

1Sobolev Institute of Mathematics, Russia [email protected]

Keywords: Bertrand competition, monopolistic competition, additive preferences, Ford effect, Schumpeter hypothesis

The economy involves two sector with different competitions regimes supplying a horizontally differentiated good and one production factor - labor. There is a continuum [0,1] of identical consumers endowed with one unit of labor. The labor market is perfectly competitive and labor is chosen as the numéraire. There are two types of firms: a finite number n≥2 of the “big” oligopolistic firms and a continuum mass M of the “small” monopolistic competitive firms. Each variety is produced by a single firm and each firm produces a single variety, thus the horizontally differentiated good 푥 = {푥푘 ≥ 0|푘 ∈ {1, … , 푛} ∪ 푀} consists of two parts – the finite dimensional oligopolistic part (푥 , … , 푥 ) ∈ 푅푛, and MC-produced bundle of varieties x ≥0 | j∈M . The relative share 1 푛 + { j } of labor hired by one oligopolistic firm is denoted as s, thus the total amount of oligopolistic employee is n⋅s, while the monopolistic competitive share in labor market is 1−ns. The cost of producing q units of variety 푘 ∈ {1, … , 푛} ∪ 푀 is linear f+c⋅q . k i 푛 Consumers share the same additive preferences given by 푈(풙) = ∑푖=1 푢(푥푖) +

∫푀 푢(푥푗)푑푗, where u(⋅) is thrice continuously differentiable, strictly increasing, strictly 푛 concave, and such that u(0)=0. A consumer’s budget constraint is ∑푖=1 푝푖푥푖 +

∫푀 푝푗푥푗푑푗 = 푦, where the income y is equal to her wage plus her share in total profits 푦 = 1 + ∑푛 ∏ + ∫ ∏ 푑푗 ≥ 1. In turn, the profit made by the firm selling amount q of 푖=1 푖 푀 푖 k variety 푘 ∈ {1, … , 푛} at price 푝푘 is given by 훱푘 = (푝푘 − 푐)푞푘 − 푓. The main difference between “big” and “small” firms in this model is that “small” firms are free to enter into industry, while the number n and size s of oligopolies are typically subjected to more sophisticated laws, e.g., to some kind of antitrust legislation. This feature discriminate the presented model from the previous one from (Parenti et al., 2017), where oligopolies were also free to entry until the zero-profit cut-off.

131

The market equilibrium is defined in standard way, i.e., product markets clear for all varieties, as well as labor market, while consumes and firms maximize their objective functions (utility and profits). Due to symmetry of model setup, we shall focus only on symmetric equilibria. By definition, the income level influences firms’ demands, whence their profits. As a result, firms must anticipate accurately what the total income will be. In addition, firms should be aware that they can manipulate the income level, whence their “true” demands, through their own strategies with the aim of maximizing profits. This feedback effect is known as the Ford effect (d’Aspremont et al., 1996). Note that these considerations may concern only the “big” oligopolistic firms 푖 ∈ {1, … , 푛}. The “non- atomic” monopolistic competitive firms j∈M have negligible effect on market statistics, 푑푦 such that consumer’s income y(p), in other words, we obtain that = 0, 푗 ∈ 푀 while for 푑푝푗 oligopolies these derivatives are non-zero. This difference between “big” and “small” imply the difference in equilibrium prices. In what follows, we consider the relative

푝̅−푐 푝̂−푐 markups 푚 ≡ , 휇 ≡ , instead of equilibrium prices ̄p, ̂p for oligopolistic and 푝̅ 푝̂ monopolistic competitive sub-sectors, respectively. Following (Zhelobodko et al., 2012) 푥푢′′(푥) we define the relative lave for variety (RLV) as follows 푟 (푥) ≡ − . 푢 푢′(푥) Proposition 1. The symmetric equilibrium markups satisfy the following equation

For sufficiently small f/c, n and s these equations have the unique solution. Now consider the Social Welfare function V≡nu(̄x)+Mu(̂x), where ̄x, ̂x are equilibrium demands in oligopolistic and monopolistic competitive sub-sectors, respectively, and M is an equilibrium mass of monopolistic competitors. These values are determined by equilibrium markups μ and m, which depend, in particular, on s and n. Our research question is what is an “optimum”structure of industry? In other words, was J.Schumpeter right, assuming that existing of oligopolies may be advantageous for consumers or is it always harmful for them?

Proposition 2. There exists the “optimum size” s* of oligopolistic firm, which does not depend on n.

132

Now assume that size of oligopolies s is already chosen and some Social Planner, e.g., government, decides which number n* of “big” firms provide the maximum of Social Welfare. First apply this question to CES model.

Theorem 1. Let 푢(푥) = 푥푝, 0 < 푝 < 1, then for all s≠s* the optimum industry structure is pure monopolistic competition, i.e., n*=0, while at s=s* consumers’ welfare does not depend on n. In what follows we assume that the elementary utility function u(x) satisfies the following additional

Assumption Derivatives u(k)(x), k=1,2,3 are bounded in neighborhood of zero. It is obvious, that CES utility contradicts with this assumption, while many other classes of utility fumctions, e.g., quadratic, CARA, HARA, etc., satisfy it. Theorem 2. For all sufficiently small f/c there exists non-zero neighborhood of optimum size interval s

References

[1]. d’Aspremont, C., Dos Santos Ferreira, R. and Gérard-Varet, L. (1996) On the Dixit- Stiglitz model of monopolistic competition. American Economic Review 86, 623-629. [2]. Parenti, M., A.V. Sidorov, J.-F. Thisse, and E.V. Zhelobodko (2017) Cournot, Bertrand or Chamberlin: Toward a reconciliation, International Journal of Economic Theory, 13(1), 29–45. [3]. Zhelobodko, E., S. Kokovin, M. Parenti and J.-F. Thisse (2012) Monopolistic competition in general equilibrium: Beyond the constant elasticity of substitution. Econometrica 80, 2765-2784

133

A Modified Multi-Armed Bandit Problem and its Application to Internet Advertising

Dmitriy Smirnov1 and Ekaterina Gromova2

1,2Saint Petersburg State University, Russia [email protected] [email protected]

Keywords: decision making under uncertainty, Nature player, MAB, games of incomplete information

The multi-armed bandit problem is a well-known model for studying the exploration vs. exploitation trade-off in sequential decision problems which was originally described by Robbins [10] in 1985. There are many versions and generalizations of the multi-armed bandit problem in the literature. In our paper we will consider a basic version of this problem: the stochastic multi-armed bandit problem. Note that the multi-armed bandit problem can be considered as a game in which one player is the nature (Nature player) and that is why are an integral part of games of incomplete information. Sometimes the formulation of the multi-armed bandit problem may include some additional information that can be used to make a decision. For example, in the field of the internet marketing, the individual information about the visitor of the web resource can be used to improve the decision. In our paper we consider a special type of additional information containing predictions about the payoff when player makes a decision. This kind of information we call expert’s predictions. To consider expert’s predictions in the multi-armed bandit problem we proposed a new formulation of the multi-armed bandit problem and also modified a known algorithm to solve it. In our formulation of the multi-armed bandit problem we consider that at each round t the expert offers a vector (푏 (푡), … 푏 (푡)), where b (t) is the prediction about the 1 푛 i value of the payoff as the result of the choice of action (alternative) a at round t. The i vector (푏1(푡), … 푏푛(푡)) we call a the prediction of the expert at time t. In our work we consider the multi-armed bandit problem with one and several experts (in this case we have a matrix of predictions). In order to solve the multi-armed bandit problem with experts we modified the algorithm UCB1 that was proposed by Auer et al. in [1]. In the modified version the

134 dependence of the chosen alternative on the value and the accuracy of expert’s predictions was added. In the case of the multi-armed bandit problem with several experts at the time t we extract from the matrix of experts predictions one vector of predictions and use previously modified algorithm UCB1. We have proposed three methods of extraction one prediction from the matrix of experts predictions. To show how the reward depends on predictions of expert we demonstrate the results of numerical experiments which are based on the software implementation of the multi-armed bandit problem simulation. We show that the proposed modified multi-armed bandit problem can be applied to the testing of web-pages in order to increase the conversion and therefore sales.

References

[1]. Auer P., Cesa-Bianchi N., Fischer P. Finite-time Analysis of the Multiarmed Bandit Problem // Machine Learning. 2002. Vol. 47, No 2-3. P. 235–256. [2]. Bather J.A. The Minimax Risk for the Two-Armed Bandit Problem // Mathematical Learning Models - Theory and Algorithms. Lecture Notes in Statistics. V 20, P. 1–11. Springer-Verlag, New York Inc. 1983. [3]. Borovkov A. A. Mathematical statistics. Additional chapters. // Moscow, Nauka, 1984. 143 p. [4]. Bure V. M., Parilina E. M. Theory of Probability and Mathematical Statistics // St. Petersburg, "Lan" Publ. 2013. [5]. Ferguson T. S. Mathematical statistics: A decision theoretic approach. Academic press, 2014. Ò. 1. [6]. Lai T.Z. Adaptive treatment allocation and the multi-armed bandit problem // The annals of statistics, 1987, v. 15, P. 1091–1114. [7]. Lindley D. V. Making decisions. 1991. [8]. McKinsey J. C. C. Introduction to the Theory of Games. Courier Corporation, 2003. [9]. Petrosyan L. A., Zenkevich N. A., Shevkoplyas E. V. Teoriya igr (Game Theory), St //Petersburg: BKhV-Peterburg 2012. [10]. Robbins H. Some aspects of the sequential design of experiments // Herbert Robbins Selected Papers. Springer New York, 1985. Ñ. 169–177. [11]. Smirnov D.:S. The multi-armed bandit problem with an expert. Control processes and stability. 2017. Ò. 4(20). No 1. P. 681–685.

135

The One-Shot Crowdfunding Game

Rann Smorodinsky1, Moran Koren2 and Itai Arieli2

1 ,2,3Technion, Israel [email protected] [email protected] [email protected]

Keywords: Crowdfunding, Equilibrium, Information aggregation, Efficiency

The evolution of the `sharing economy' has made it possible for the general public to invest in early-stage innovative and economically risky projects and products. These funding schemes, dubbed `crowdfunding', have been gaining popularity among entrepreneurs and it is reported that crowdfunding for supporting new and innovative products has been overwhelming with over 34 Billion Dollars raised in 2015. In addition to serving as an alternative to venture capital funds as a source for fund raising for nascent stage products, the crowdfunding option also serves as a means to gauge market traction for new products. It is implicitly assumed that a successful crowdfunding campaign suggests a high market demand for the new offering. From the contributor's perspective, the investment in a crowdfunding campaign has two risky aspects. First, the risk of whether the firm will have enough funds to produce and deliver the product; and second, the quality and value of the product is unknown at the time of the campaign and could possibly be disappointing even if eventually delivered. In many on-line crowd-funding platforms such as ``Kickstarter" and ``Indiegogo" a typical campaign format has two critical components. First, it sets a price for the future product and second it sets a threshold. Contributions are collected only if in total they exceed this threshold. Both values are determined by the fund raising firm. This format is designed to mitigate the aforementioned risks. If the threshold is set high enough then contributions are collected only when the company has enough funds on the one hand, and the `wisdom-of-the-crowd' points to a high valued product. We model a crowdfunding campaign as a game of incomplete information played by a set of N contributors. Potential contributors receive an IID signal about the common value of the product and must decide whether to participate or not. We propose two performance measures for a crowdfunding campaign: 1) The correctness index of a game is defined as the probability that the game ends up with a the correct decision. That is, the probability the product is funded when its

136 value is high or the probability that the product is rejected when its value is low. The correctness index measures how well the crowdfunding aggregates the private information from the buyers in order to make sure the firm pursues the product only when it is viable. 2) The market penetration index is the expected number of buyers provided that the product is supplied, i.e, the threshold is surpassed. This number serves as a proxy for success of the campaign as a means to attract further investments. Our theoretical results provide limits on the success, in both aspects, of large crowdfunding games. We state and prove three results: 1) We provide a constructive proof for the existence of a symmetric, non-trivial equilibrium and we show it is unique. In every such equilibrium players with a high signal surely contribute while those with a low signal either decline or take a mixed strategy whereby they contribute at a positive probability, strictly less than one. 2) In large games, we provide a tight bound on the correctness index which is strictly less than one. Thus, no matter how the campaign goal is set, full information aggregation cannot be guaranteed. We compare this with the efficiency guarantees of majority voting implied by Condorcet Jury Theorem. 3) In large games, we provide a bound on the penetration index and we show that by setting the campaign goal optimally the resulting market penetration is higher than the benchmark case where the campaign goal is set to a single buyer (B=1). Calculations, provided in the paper, demonstrate that the asymptotic results approximately hold for small populations of potential

137

The Formation of Social Groups under Status Concern

Manuel Staab1

1London School of Economics, United Kingdom [email protected]

Keywords: Bertrand competition, monopolistic competition, additive preferences, Ford effect, Schumpeter hypothesis

When people interact in a social environment, whether it is at work or school, in clubs or in their neighborhood, social spillovers tend to play an important role. At work, cooperation with colleagues might be essential, at school and university, studying with peers can promote understanding and enhance the learning experience. For any team sport, other players are a pre-requisite. In many of these situations, we would like to be surrounded by ‘strong’ peers as their ability influences the benefit we gain from the interaction. At the same time, we might want to be someone with a relatively high standing in the group. This presents a clear tension: the stronger the peers, the lower one’s own standing. This paper develops a model to explore the importance of this (potential) tension in the formation of social groups very much in the spirit of frank85. In a non-cooperative setting, it addresses the questions what groups can be formed and what groups might be offered by a social planner, monopolist, or competitive firm when agents care about both the quality of peers, as well as their standing within their group. The focus lies on two key aspects: segregation and social exclusion. It is explored how status concern affects the segregation of agents i.e. how fine agents can be sorted into groups. And it is examined what status concern implies about social exclusion, addressing the question how many agents might not be offered any social group. It is shown that status concern reduces the possibility of, as well as the benefit from segregation. More precisely, ‘splitting’ a population into several, separate groups is less beneficial under status concern - both in terms of aggregate welfare and, under some restrictions, in terms of revenue. This means any provider gains less from posting prices that allow for finer sorting. Status concern is a force for homogeneity across groups as it limits the degree to which groups can differ in their composition of types. For example, if two groups are priced equally, then they have to be identical in their probability distribution over types, not just their quality. Additionally, there might be no prices that make a given group structure incentive compatible even though such prices exist if agents

138 only care about quality. No matter the objective of the group provider, status concern leaves less room for manoeuvre. Sorting cannot be arbitrarily fine as the groups take the form of non-overlapping intervals and the number of such intervals in equilibrium is necessarily finite. If status concern is relatively more important, less segregation can be achieved. In the extreme case where agents have preferences only over their rank, no segregation is possible and all agents that join a group, pay the same price. In contrast, board09 finds in a closely related setting without status concern that for sufficiently convex quality functions, full separation can indeed be both a welfare and profit maximizing equilibrium.

References

[1]. Simon Board2009board09 Board, Simon. 2009. “Monopolistic group design with peer effects.” Theoretical Economics, 4(1): 89 – 125. [2]. Robert H. Frank1985frank85 Frank, Robert H. 1985. Choosing the Right Pond. Oxford University Press. [3]. Ori Heffetz Robert H. Frank2011heffetz11 Heffetz, Ori, and Robert H. Frank. 2011. “Chapter 3 - Preferences for Status: Evidence and Economic Implications.” In . Vol. 1 of Handbook of Social Economics, , ed. Jess Benhabib, Alberto Bisin and Matthew O. Jackson, 69 – 91. [4]. Luis Rayo2013rayo13 Rayo, Luis. 2013. “Monopolistic Signal Provision.” B.E. Journal of Theoretical Economics, 13(1): 27 – 58.

МАТЕМАТИЧЕСКАЯ ТЕОРИЯ ИГР И ЕЕ ПРИЛОЖЕНИЯ

139

Cluster analysis as a cooperative game with coalition structure

Kseniya Staroverova1 and Vladimir Bure2

1,2Saint Petersburg State University, Russia 1 [email protected] 2 [email protected]

Keywords: clustering, coalition structure, stability

Consider a cooperative game where players are some objects which are supposed to be clustered. We can define a characteristic function in such a way that it is high for such coalitions which consist of more similar objects than objects from other coalitions. Therefore, we have a natural transformation from a clustering problem to a cooperative game. Methods of the cooperative game theory were already used for clustering analysis, but a characteristic function was built another way. The main difference is that we consider the function which might not be superadditive and monotone. Therefore, in general, not only the grand coalition but smaller ones can be formed, that is a more desirable outcome for the clustering problem. Several single-valued cooperative solution concepts are considered with this type of a characteristic function and stability conditions are found. Clustering is the unsupervised assignment of data objects to subsets such that objects within a subset are more similar to each other than objectss from other subsets. The conventional theory uses some objective function and optimization techniques to find structure in the data. The idea of applying methods of the cooperative game theory was already used for clustering analysis in works of ([1]) and ([2]) where a total worth of a coalition was computed as the sum of pairwise similarities between the points. In this case, game is convex, so players always form the unique grand coalition. That is not natural for clustering. The clustering algorithm is based on the fact that in such a game Shapley values of close data objects are almost equal. In this work, we consider clustering as a cooperative game with coalitional structure. We suggest that not only grand coalition but smaller ones may be formed. The idea is to find such coalitional structure that will be profitable for all players. The existence of stable coalitional structure means the presence of crisp clustering structure with regard to the characteristic function and some single-valued cooperative solution concept.

140

References

[1]. Dhamal, S., Bhat, S., Anoop, K. R. and Embar, V. R. (2012). Pattern Clustering using Cooperative Game Theory. CoRR, abs/1201.0461, 1–6. [2]. Garg, V. K., Narahari, Y. and Murty, M. N (2013). Novel Biobjective Clustering (BiGC) Based on Cooperative Game Theory. IEEE Trans. Knowl. Data. Eng. 25(5), 1070–1082. [3]. Sedakov, A., Parilina, E., Volobuev, Y. and Klimuk, D. (2013). Existence of Stable Coalition Structures in Three-person Games. Contribu. Game Theor. Manag. 6, 407– 422. [4]. Parilina, E. and Sedakov, A. (2014). Stable bank cooperation for cost reduction problem. Czech Econ. Rev. 8(1), 7–25.

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

141

Optimal Synthesis in Mathematical Models with Piecewise Monotone Dynamics

Nina Subbotina1 and Natalia Novoselova2

1,2Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Russia [email protected] [email protected]

Keywords: optimal control, Rankine–Hugoniot line, Hamilton–Jacobi–Bellman equation, Cauchy method of characteristics.

This work is a development of researchers in the model of chemotherapy of a malignant tumor in [1]. We present numerical-analytical methods to construct optimal control synthesis (optimal feedbacks) with reference to the piecewise monotone mathematical model. We consider the case when the therapy function, which describes the effect of the drug on the cell growth rate, has two maxima. The case of a single maximum of the therapy function was investigated in [2]. We consider the following interaction process:

(1) here m is quantity of malignant cells, h is quantity of drug, f(h) is a therapy function that describes the effect of medicine on tumor cells, the function g(m) describes growth of tumor cells, u(t) is a restricted control: u(t)∈[0,Q], 푡 ∈ [푡0, 푇] ⊂ [0, 푇].

For any initial state (푡0, ℎ0, 푚0), the aim of the optimal control problem is minimization of the terminal cost function:

(2) We construct optimal feedbacks based on the value function [3, 4, 5] and use the fact that the value function is the unique minimax (viscosity) solution of the Cauchy problem for the basic Hamilton–Jacobi–Bellman equation (HJB) [6, 7]. The case when g(m)=0 has already been considered earlier in paper [1]. We have obtained a construction of the value function.

142

Using the classical method of Cauchy characteristics for the linear first-order partial differential auxiliary equations we constructed a finite number of smooth functions. Gluing together this functions we have got a continuous function ϕ and checked the coincidence of the constructed function ϕ with the minimax solution of the Cauchy problem for the basic HJB equation, and, consequently, with the value function of the optimal control problem under consideration. A new element of the construction is the line of nonsmooth gluing with the use of the Rankin–Hugoniot conditions [8, 9]. This line plays a key role for the optimal feedback strategy, because it determines its discontinuity line. In this paper, we investigate the construction of the value function and optimal feedbacks in the case, when g(m)≠0. We investigate whether the method used in the paper [1] can apply to the problem (1),(2). Note that approach is related to the construction of an optimal synthesis of control for some economic mathematical models, as well as models of environmental management.

References

[1]. N.N. Subbotina, N.G.Novoselova. Optimal result in a control problem with piecewise monotone dynamics. Journal "Proceedings of the Institute of Mathematics and Mechanics" 2017; 23(4): 265-280. [2]. Bratus, A. S., Chumerina, E. S. Optimal control in therapy of solid tumor growth. Computational Mathematics and Mathematical Physics 2008; 48(6):892-911. [3]. Pontryagin, L. S., Boltyansky, V. G., Gamkrelidze, R. V., Mishchenko, E. F. The Mathematical Theory of Optimal Processes. Moscow: Nauka, 1961. [4]. Krasovskii N.N. Motion Control Theory. Moscow: Nauka, 1968. [5]. N.N. Krasovskii, A.I.Subbotin Game-Theoretical Control Problems. Springer-Verlag, New York, 1988. [6]. Subbotin, A. I. Generalized Solutions of First-Order PDEs: The Dynamical Optimization Perspective. Birkhauser: Boston, 1995. [7]. Crandall, M.G., Evans, L.C., Lions, P.-L.: Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., vol. 282., no 2, pp. 487-502, 1984. [8]. Goritskiy A.YU., Kruzhkov S.N., Chechkin G.A. First-Order Partial Differential Equations (Study Guide). Moscow: Lomonosov Moscow State University, Faculty of Mechanics and Mathematics, 1999. [9]. Subbotina, N. N., Kolpakova, E. A., Tokmantsev, Shagalova L.G. The Method of Characteristics for Hamilton–Jacobi–Bellman equations. RIO UrO RAN, Yekaterinburg, 2013.

143

On Gamesmen and Fair men: Explaining Fairness in Noncooperative Bargaining Games

Ramzi Suleiman1

1Triangle Center for Research & Development (TCRD), & University of Haifa, Israel [email protected]

Keywords: Bargaining, Alternating offers, Ultimatum game, Subgame perfect equilibrium, Golden Ratio

Experiments on bargaining games have repeatedly shown that subjects fail to use backward induction, and that they only rarely make demands in accordance with the subgame perfect equilibrium (SPE). In a recent article, we proposed an alternative model, termed "economic harmony" in which we modified the individual's utility by defining it as a function of the ratio between the actual and aspired payoffs. We also abandoned the notion of equilibrium, in favor of a new notion of "harmony", defined as the intersection of strategies, at which all players are equally satisfied. We showed that the proposed model yields excellent predictions of offers in the ultimatum game, and requests in the sequential CPR dilemma game. Strikingly, the predicted demand in the UG is equal to the famous Golden Ratio (≈ 0.62 of the entire pie). The same prediction was recently derived independently by Schuster (Scientific reports, 2017). In the present paper, we extend the solution to bargaining games with alternating offers. We show that the derived solution predicts the opening demands reported in several experiments, on games with equal and unequal discount factors, and game horizons. Our solution also predicts several unexplained findings, including the puzzling "disadvantageous counteroffers", and the insensitivity of opening demands to variations in the players' discount factors, and game horizon. Strikingly, we find that the predicted opening demand in the alternating offers game, is also equal to the Golden Ratio.

144

On Construction of Optimal Strategies for Games with Dynamic Updating

Denis Tikhomirov1 and Ovanes Petrosian2

1National Research University Higher School of Economics, Russia [email protected] 2Saint-Petersburg State University, Russia [email protected]

Keywords: Differential Games, Optimal Control, Looking Forward Approach

Differential games with dynamic updating allow to construct mathematical models for conflict-controlled processes, when information updates dynamically and players choose strategy according to the information [1,2,3]. ̅̅̅̅ Information about the game updates at fixed time instants 푡 = 푡0 + 푗훥푡, 푗 = 0, 푙, 푇 where 푙 = − 1. On the time interval [푡 + 푗훥푡, 푡 + (푗 + 1)훥푡] players have full 훥푡 0 0 information about the game structure on the interval [푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅] with length 푇̅, which is called an information horizon. Currently this class of game is modelled by a series of truncated subgames 훤̅푗(푥푗,0, 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅), each one of them describing players behavior on the interval 훥푡 between the instants when information updates,

푗 푗 particularly defining feedback optimal control 푢푗(푡, 푥) = (푢1(푡, 푥), … , 푢푛(푡, 푥)) (as a ̅̅̅̅ closed-loop control) and corresponding state function 푥푗(푡), 푗 = 0, 푙. When considering a cooperative differential game on the first step it is necessary to solve an optimal control problem, corresponding to maximization of joint payoff of all players. Therefore, for every truncated subgame 훤̅푗(푥푗,0, 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅) we are to solve the following optimization problem:

푑푥 푡 +푗훥푡+푇̅ = 푔(푡, 푥, 푢) 푖 0 푑푡 ∑푖 퐾푗 = ∑푖 ∫ ℎ푖(휏, 푥, 푢) 푑휏 → max, subject to { , (*) 푡0+푗훥푡 푢∈푈 푥(푡0 + 푗훥푡) = 푥0푗

푛 where ℎ푖(푡, 푥, 푢) is an instantaneous payoff of player 푖, 푥: ℝ ↦ ℝ is a state 푛 푛 function, 푢: ℝ ↦ ℝ is a control (strategy), 푥0푗 ∈ ℝ is an initial state of truncated subgame with number 푗. The existing approach requires significant amount of computation resources and does not allow to derive an analytical solution defined on the time interval [푡0, 푇].

145

On the first step in the paper we present a technique of deriving an analytical

푗 푗 form of a closed-loop control 푢푗(푡, 푥) = (푢1(푡, 푥), … , 푢푛(푡, 푥)) for an arbitrary truncated subgame 훤̅푗(푥푗,0, 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅), i.e. for an arbitrary values of parameters 푡0, 푗, 푡−푡 −푗훥푡 훥푡, 푇̅. The technique is based on using the substitution 푧 = 0 , which transforms an 푗 푇̅ optimal control problem (*) defined on the time interval [푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅] to the optimal control problem on the time interval [0,1].

As a result 푡0, 푗, 훥푡, 푇̅ become model parameters and controls on every interval [푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅] will also be a function of 푡0, 푗, 훥푡, 푇̅:

푢푗(푡, 푥) = 푢푗(푇̅푧푗 + 푡0 + 푗훥푡, 푥) = 푢(푇̅푧푗 + 푡0 + 푗훥푡, 푥; 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅), ̅̅̅̅ where 푧푗 ∈ [0,1], 푗 = 0, 푙. Control function 푢(푇̅푧푗 + 푡0 + 푗훥푡, 푥; 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅) does not depend upon the truncated subgame, only upon the parameters of truncated subgame, i.e. 푡0, 푗, 훥푡, 푇̅. Therefore, the optimal control on whole interval can be constructed: ̅̅̅̅ 푢̅(푡, 푥) = 푢(푡, 푥; 푡0 + 푗훥푡, 푡0 + 푗훥푡 + 푇̅), 푡 ∈ [푡0 + 푗훥푡; 푡0 + 푗훥푡 + 푇̅], 푗 = 0, 푙. On the second step we present a technique for construction an optimal control

푢̅(푥, 푡), 푡 ∈ [푡0, 푇] when 훥푡 → 0, i.e. in case when information about the game is continuously updated. As a result, the optimal control will have a form: 푢̅(푡, 푥) = 푢̅(푡, 푥; 푡, 푡 + 푇̅), where 푡 (first argument) is the current time instant, 푡 (third argument) is the initial time instant of truncated subgame defined on the interval [푡, 푡 + 푇̅]. The way of constructing optimal control 푢̅(푡, 푥) is demonstrated on the differential game model of non-renewable resource extraction with symmetrical players [5,6].

References:

[1]. Gromova E., Petrosian O. Control of information horizon for cooperative differential game of pollution control // 2016 International Conference Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference). 2016 [2]. Barabanov A. Petrosian O. Looking Forward Approach in Cooperative Differential Games with Uncertain Stochastic Dynamics // Journal of Optimization Theory and Applications. 2017. Vol. 172. No. 1. P. 328-347 [3]. Petrosian O. Looking Forward Approach in Cooperative Differential Games // International Game Theory Review. 2016. Vol. 18. No. 2. P. 1-14 [4]. Bellman R. Dynamic Programming and a New Formalism in the Calculus of Variations Proc Natl Acad Sci U S A. 1954 Apr; 40(4): 231–235. [5]. Dockner E.J., Jorgensen S., Van Long N., Sorger G. Differential Games in Economics and Management Science. – Cambridge Univ. Press, 2000. [6]. Shevkoplyas E. V. Optimal Solutions in Differential Games with Random Duration // Journal of Mathematical Sciences, 2014, 199:6, 715–722

146

Cooperation in Multistage and Repeated Hierarchical Games

Puneet Tomar1

1Saint Petersburg State University, Russia [email protected]

Keywords: Repeated Games, Cooperated Games, Hierarchical Games, multistage games

The paper is focused on dynamic aspects of cooperation in n-person multistage and repeated hierarchical games. I will not restrict myself to the games with perfect information when I ∈ N use pure strategies 푢푖(∙). There exist an important subclass of multistage nonzero-sum games, referred to as hierarchical games. Hierarchical games model conflict controlled system with a hierarchical structure. This structure is determined by a sequence of control levels ranking in a particular priority order. Mathematically, it is convenient to classify hierarchical games according to the number of levels and the nature of the vertical relation. The simplest form of them is a two-level system. The functioning of a two-level conflict-controlled system is as follows. The control or coordinating center

A0 is at the first level of the hierarchy, select a vector u = (푢1, …...푢푛) from a given control set U, where ui is a control influence of the center on its subordinate divisions Bi, standing at the second level of the hierarchy. Bi, I = 1,…..,n in its turn, select controls 푣푖 ∈ 푉푖(푢푖), where Vi(ui) is the set of controls for division Bi predetermined by the control u of center A0. Now, the control centre has the priority right to make the first move and may restrict the possibilities of its subordinate divisions by channeling their actions as desired. The aim of center A0 is to maximize the functional K0(u, v1, … , vn) over u, whereas divisions Bi, I = 1,…..,n, which have their own goals to pursue, seek to maximize the functionals Ki(ui, vi) over vi. In this paper we prove the cooperation between multi-stage and repeated hierarchical games as a cooperative (n+)-person game in normal form. I shall formalize this problem as a cooperative (n+1)-person game G in normal form.

147

Self-associated consistency and the Shapley value

Genjiu Xu1, Wenzhong Li2, Jun Su3 and Wenna Wang4

1,2,4Northwestern Polytechnical University, China [email protected] [email protected] 4 [email protected] 3Xi'an University of Science and Technology, China [email protected]

Keywords: TU games, self-associated consistency, Shapley value, matrix approach

In this paper, a so-called self-associated game is introduced for a solution of TU games. Every coalition can be viewed as a unied player and every coalition revalues its worth in terms of the marginal contribution of the unied player in the corresponding to coalitioncontracted game. It generates the characteristic function of the self-associated game. A solution is self-associated consistent when it allocates to every player invariably in a game and its self-associated game. We show that the Shapley value is self-associated consistent and is also characterized as the unique solution for TU games satisfying the inessential game property, continuity and self-associated consistency. The characterization is obtained by applying the matrix approach as the pivotal technique for characterizing linear transformations on game space.

Periodicals in Game Theory

ANNALS OF THE INTERNATIONAL SOCIETY OF DYNAMIC GAMES

Volumes 1–14

Series Editor Tamer Basar

Birkhäuser, Boston

148

The allocation of marginal surplus in cooperative situations

Genjiu Xu1, Wenzhong Li2, Rong Zou3 and Dongshuang Hou4

1,2,3,4 Northwestern Polytechnical University, China [email protected] [email protected] 3 [email protected] [email protected]

Keywords: game theory, average-surplus value, Shapley value, Solidarity value, mechanism design

Marginal contributions and marginal surplus are two signicant indexes to measure one player's important degree. According to marginal contributions, the Shapley value and the Solidarity value are introduced as two classical distribution schemes in cooperative situations, which assign the expectation of marginal contributions and average marginal contributions to every player respectively. Diering from the Shapley value and the Solidarity value, in this paper, we propose a new allocation from the perspective of marginal surplus, called the average-surplus value. We rst give out the procedural denition of the value. According to the allocation scheme, every player obtains the sum of his individual worth and the expectation of average marginal surplus. Then we characterize the average-surplus value by three classical methods of cooperative game theory, which are axiomatization, potential function and mechanism design. Especially, the average-surplus value is characterized by two sets of dierent axioms.

149

Solutions of Cooperative Dynamic Games with Control Lags

David Yeung1 and Leon Petrosyan2

1Hong Kong Shue Yan University, China [email protected] 2Saint-Petersburg State University, Russia [email protected]

Keywords: Cooperative dynamic game, control lag, subgame consistent solution

Controls with lags are control strategies which effects last for more than one stage of the game after the controls had been executed. Lags in control are not uncommon and they often make the adverse effects in a non-cooperative game equilibrium more accentuate and prolonged. Cooperation provides an effective mean to alleviate the problem and obtains an optimal solution. This presentation is about new class of cooperative dynamic games in which control lags exist. New dynamic optimization technique in dynamic games with control lags is developed to derive the Pareto optimal cooperative controls. Subgame consistent solutions and corresponding imputation distribution procedure are provided to formulate a dynamically stable cooperative scheme under control lags.

Periodicals in Game Theory

GAME THEORY AND APPLICATIONS Volumes 1–17

Edited by Leon A. Petrosyan & Vladimir V. Mazalov

NOVA SCIENCE

150

Strategic Cautiousness as an Expression of Robustness to Ambiguity

Gabriel Ziegler1 and Peio Zuazo-Garin2

1Northwestern University, United States [email protected] 2University of the Basque Country, Spain [email protected]

Keywords: Game theory, decision theory, ambiguity, Knightian uncertainty, incomplete preferences, Bayesian rationality, cautiousness, iterated admissibility

This paper proposes a new take in this longstanding problem by suggesting novel theoretical foundation for the interplay between strategic reasoning and cautiousness based on robustness to ambiguity. Robustness to ambiguity arises as a natural benchmark for rational players who face what is now known as Knightian Uncertainty. Specifically, we operate on the decision-theoretic formalization of each player by allowing for incomplete preferences via a model à la Bewley (2002) where: (i) each player’s strategic uncertainty is represented by a possibly non-singleton set of beliefs that reflects the player’s ambiguity, and (ii) a rational player best-responds to every belief in her set, so that the resulting choice is robust to the possible ambiguity faced by the player. Under this set-up, we say that a player believes in certain behavior by her opponents’ if at least one of the beliefs in her set assigns full probability to the collection of states representing such behavior. Similarly, we say that a player assumes certain behavior by her opponents’ if at least one of the beliefs in her set has full-support on the collection of states representing such behavior. Consequently, the introduction of ambiguity and the requirement of robustness give great flexibility: it is possible for a player to assume certain behavior and, simultaneously, believe in certain more restrictive behavior, and the player’s rational response needs to be a best-reply to both beliefs imposing these restrictions. Hence, in particular, the tension between strategic reasoning and cautiousness is solved: a player can be strategically sophisticated by having one belief that assigns zero probability to her opponents playing strictly dominated strategies and, at the same time, cautious, by having another different belief that has full-support on her opponents’ set of strategies. We exploit the potential for reconciliation of the aforementioned approach to construct a unifying framework that provides reasoning-based foundations to iterated strategy elimination procedures that may display cautiousness. First, we show in Theorem

151

1 that rationalizability characterizes the behavioral implications of rationality and common belief in rationality. Next, we introduce two different notions of cautiousness: weak and strong cautiousness. A player is said to be weakly cautious when at least one of her beliefs has full-support on the set of states of the world (i.e., those that not only represent strategic uncertainty, but also higher-order beliefs thereof). More restrictively, a player is said to be strongly cautious when her sets of beliefs has non-empty topological interior, this feature representing that the player takes into account any possible small enough perturbation on her beliefs. Based on these two notions of cautiousness we show in Theorems 2 that the Dekel-Fudenberg procedure characterizes the behavioral implications of rationality, weak cautiousness and common belief thereof, and in Theorem 3, that strict rationalizability characterizes the behavioral implications of rationality, strong cautiousness and common belief thereof. It is easy to see that the foundations of both the Dekel-Fudenberg procedure and strict rationalizability necessarily requires the presence of ambiguity whenever strategic reasoning has any bite.* The Dekel-Fudenberg requires the set of ambiguous belief to include a belief with full-support in the set of opponents’ strategies (due to weak cautiousness) and another belief that assigns full-probability only to the opponents’ strategies that survive the first elimination step. Strict rationalizability is founded in terms of strong cautiousness, which always requires the presence of ambiguity as a singleton set of beliefs cannot have non-empty topological interior. Finally, we also provide foundations for iterated admissibility—the iterated elimination of weakly dominated strategies, by showing in Theorem 4 that it characterizes the behavioral implications of rationality, weak cautiousness and common assumption thereof. Again, this characterization requires the presence of ambiguity as, in case the elimination procedure consists on multiple rounds, the set of ambiguous beliefs need to contain a specific belief with full-support on the set of opponents’ strategies that survive each of the rounds. Despite the main approach in the paper is conceptual and focused on the link between cautiousness in reasoning-based processes and robustness to ambiguity, the results provide some purely methodological contribution w.r.t. the use of incomplete preferences in game theory, which is a subject of interest itself besides its interpretation as a reflection of ambiguity. Our approach can be regarded as complementary to the lexicographic probability system one. However, apart from the transparent link between cautiousness and robustness

* I.e., the elimination procedures requires more than a single round. 152 to ambiguity the framework allows for, the nice structure of the sets of ambiguous beliefs representing incomplete preferences has some additional advantages. First, it is easy to show that rationality, weak cautiousness and common assumption thereof is a non-empty event and thus, that iterated admissibility is properly founded for all games. Second, the definitions and formalism involved do not require departures from the canonical definition of the objects involved: (i) the modeling of higher-order beliefs (i.e., the type structures employed), including the definition of coherence, and the version of assumption we rely on are natural extensions of their counterparts in the realm of standard Bayesian preferences, and (ii) the notions of cautiousness our theorems invoke do not necessarily restrict to environments where the sets of states have a specific structure (e.g., games). Finally, the presence of ambiguity via incompleteness preferences has been shown to be empirically testable by recent work by Cettolin and Riedl (2016).

Journals in Game Theory

DYNAMIC GAMES AND APPLICATIONS

Editor-in-Chief Georges Zaccour

Birkhäuser, Boston

153

Index

Ageev ...... 5 Hanke ...... 57 Aizenberg ...... 7, 10 Haufe ...... 35 Anikin ...... 124 Herings ...... 18 Arieli ...... 136 Heее ...... 15 Artemov ...... 15 Hohmann ...... 43 Askoura ...... 13 Hou гг ...... 149 Barbieri ...... 16 Ibragimov ...... 60 Bartel ...... 83 Ivakina ...... 61 Bayer ...... 18, 71 Jieе… ...... 63 Belyavsky...... 21 Khmelnitskaya ...... 64, 65 Blakeway ...... 122 Kirpichnikova ...... 122 Bure ...... 140 Kleimenov ...... 66 Bykadorov ...... 10, 24 Kleshnina ...... 68 Cardona ...... 26 Kokovin ...... 10 Carroni ...... 30 Koren ...... 136 Chang ...... 27 Korolev ...... 97 Chernysheva ...... 108 Kovac ...... 69 Cheее ...... 15 Kozics ...... 71 Colombo...... 30 Krasovskii ...... 73 Danilova ...... 21 Kreiss ...... 35 Del Pozo ...... 94 Kreps ...... 47, 76 Dermentzi...... 115 Kumacheva ...... 79 Dietzenbacher ...... 33 Kuznetsova ...... 50, 81 Dodonova ...... 81 Kuzyutin...... 83 Dotsenko ...... 34 Liii ...... 85 Ehrhart ...... 35, 57 Liгг...... 148, 149 Ejov ...... 68 Lonyagina ...... 86 Fedyanin...... 40 Lozovanu ...... 88 Fernández ...... 38 Ludwig ...... 43 Filar ...... 68 Ma ...... 91 Filippini ...... 30 Manuel ...... 93, 94 Flad ...... 43 Marova ...... 95 Flesch ...... 44 Martin ...... 93 Freitas ...... 26 Matveenko ...... 97 G.-Tóth ...... 38 Mazalov ...... 100 Garcia-Meza ...... 12 McKerral ...... 68 Garmash ...... 97 Meisner ...... 101 Gavrilovich ...... 47 Mestnikov ...... 103 Gladkova ...... 86 Naroditskiy...... 105 Glukhov ...... 50 Natsagdorj ...... 124 Gornov ...... 124 Naumova ...... 108 Gromova ...... 52, 95, 122, 134 Nikitina ...... 100 Gubar ...... 79 Nikolchenko ...... 86, 110 Gudmundsson ...... 54 Novoselova ...... 142 Gusev ...... 55 Ortega ...... 94 Habis ...... 54 Ott ...... 57

154

Ougolnitsky ...... 21, 112 Su гггг ...... 148 Pankratova ...... 5 Subbotina ...... 142 Papadopoulos ...... 115 Sukhee ...... 124 Parilina ...... 64 Suleiman ...... 144 Pavlov ...... 117 Talman ...... 65 Peeters ...... 18 Tarashnina ...... 5 Peters ...... 33 Tarasyev ...... 73 Petrosian...... 120, 129, 145 Thuijsman ...... 18 Petrosyan ...... 63, 150 Tikhomirov ...... 145 Petrov ...... 103 Tomar ...... 85, 147 Pickl ...... 88 Tomilina ...... 79 Plekhanova ...... 122 Tur ...... 52 Prokhorov...... 60 Venable ...... 16 Qiushi ...... 123 Vermeulen ...... 44 Rentsen...... 124 Vikulova...... 52 Rettieva ...... 125 Walden ...... 60 Rokhlin ...... 112 Wang гг ...... 148 Rozen ...... 127 Xuгг ...... 148, 149 Rubi-Barcelo ...... 26 Yeung ...... 150 Salhi ...... 38 Zakharov ...... 120 Schmidt ...... 69 Zenkevich...... 61, 110 Sedakov ...... 64, 128 Zhang ...... 65 Shi ...... 129 Zhen ...... 128 Sidorov ...... 131 Zheng ...... 16 Smirnov ...... 134 Zhitkova ...... 79 Smirnova ...... 83 Ziegler ...... 151 Smorodinsky ...... 136 Zou гг ...... 149 Sperisen ...... 16 Zseleva ...... 44 Staab ...... 138 Zuazo-Garin ...... 151 Staroverova ...... 140

155

Game Theory and Management

Editors: Leon A. Petrosyan, Nikolay A. Zenkevich

Abstracts

Authorized: 18.06.2013. Author’s sheets: 13 150 copies Publishing House of the Graduate School of Management, SPbU Volkhovsky per. 3, St. Petersburg, 199004, Russia tel. +7 (812) 323 84 60 www.gsom.spbu.ru