DOCSLIB.ORG

Prophylaxy Copie.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Prophylaxy Copie.Pdf Social interactions and the prophylaxis of SI epidemics on networks Géraldine Bouveret, Antoine Mandel To cite this version: Géraldine Bouveret, Antoine Mandel. Social interactions and the prophylaxis of SI epi- demics on networks. Journal of Mathematical Economics, Elsevier, 2021, 93, pp.102486. 10.1016/j.jmateco.2021.102486. halshs-03165772 HAL Id: halshs-03165772 https://halshs.archives-ouvertes.fr/halshs-03165772 Submitted on 17 Mar 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Social interactions and the prophylaxis of SI epidemics on networkssa G´eraldineBouveretb Antoine Mandel c March 17, 2021 Abstract We investigate the containment of epidemic spreading in networks from a nor- mative point of view. We consider a susceptible/infected model in which agents can invest in order to reduce the contagiousness of network links. In this setting, we study the relationships between social efficiency, individual behaviours and network structure. First, we characterise individual and socially efficient behaviour using the notions of communicability and exponential centrality. Second, we show, by computing the Price of Anarchy, that the level of inefficiency can scale up to lin- early with the number of agents. Third, we prove that policies of uniform reduction of interactions satisfy some optimality conditions in a vast range of networks. In setting where no central authority can enforce such stringent policies, we consider as a type of second-best policy the implementation of cooperation frameworks that allow agents to subsidise prophylactic investments in the global rather than in the local network. We then characterise the scope for Pareto improvement opened by such policies through a notion of Price of Autarky, measuring the ratio between social welfare at a global and a local equilibrium. Overall, our results show that individual behaviours can be extremely inefficient in the face of epidemic propaga- tion but that policy can take advantage of the network structure to design welfare improving containment policies. Keywords: Network, Epidemic Spreading, Public Good, Price of Anarchy JEL codes: D85, D62, I18 aAntoine Mandel acknowledges support from ITN ExSIDE European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 721846. Declarations of interest: none bNanyang Technological University, School of Physical and Mathematical Sciences, 21 Nanyang Link, 637371 Singapore, Email: [email protected] (corresponding author) cParis School of Economics-Universit´eParis 1 Panth´eon-Sorbonne, Centre d'Economie´ de la Sor- bonne, 106 Boulevard de l'hopital, 75013, Paris, France, Email: [email protected] 1 1 Introduction In the context of the 2020 COVID-19 pandemic, very strong policy measures have been implemented to contain epidemic diffusion. State of emergency has been declared in certain countries and certain civil liberties (e.g. freedom of assembly) have been suspended. The implementation of such stringent policies, labelled as social distancing measures, has been justified by the role of social interactions in epidemic diffusion. In economic terms, the premise is that individual behaviour is extremely inefficient in the presence of disease/network externalities. Yet, there is, to our knowledge, no normative analysis of the challenges posed by the containment of epidemic spreading in a network. This is the issue we address in this paper. The containment of epidemic processes defines a specific class of externality prob- lems: through prophylactic investment, agents can reduce their own contamination risk but also reduce the risk of contagion of their peers in the network. The external effect hence created has certain features of a public good as the investment of each agent benefits to all the agents to whom it is connected. However, the magnitude of the effect depends on the specific connectivity between each pair of agents and thus on the structure of the network. In this setting, our first aim is to characterise, as a function of the network structure, individual and socially efficient behaviours. Second, we measure, using the notion of Price of Anarchy (PoA), the inefficiency induced by individual behaviours. Third, we investigate policy measures that can be implemented to overcome these inefficiencies. We place ourselves in a setting where the network structure is given, each agent can be initially contaminated with a certain probability, and contagion spreads through network links proportionally to their contagiousness. Once infected, agents remain so permanently, i.e. we consider a susceptible/infected type of model according to the epidemiological terminology. In this context, agents aim at minimising their probability of contagion before a given date. In a narrow interpretation, this date can be seen as the expected date at which a treatment will be available. In a broader sense, the objective of each individual is to reduce the speed of incoming epidemic propagation. We assume that agents can invest in the network to reduce the speed of contagion. More precisely, they can decrease the contagiousness of links at a fixed linear cost. As the impact of individual investments depends on global contagiousness, and hence on the investment 2 of other players, the situation defines a non-cooperative game. We consider two variants of the game. The local game in which an agent can only invest in the links through which it is connected. The global game in which an agent can invest in each link of the network. The local game naturally applies to settings where agents are individuals that can take individual and costly measures to limit their social interactions. The global game corresponds to a more complex setting where agents are usually organisations (regions, countries) that are involved in a scheme that allows one agent to subsidise, directly or indirectly, the investment of other agents in the reduction of contagiousness. Our main results characterise the relationships between social efficiency, individual behaviours and network structure. First, we show that individually rational and so- cially efficient behaviours can be characterised using the notions of communicability and exponential centrality (Estrada and Hatano, 2008; Estrada and Higham, 2010). It is individually rational to invest in a link proportionally to the communicability be- tween the investor and the edges of the link while it is socially efficient to invest in a link proportionally to the total communicability/exponential centrality of its edges. Second, we derive a quantitative measure of the inefficiency induced by individual be- haviours using the notion of PoA. We show that in worst cases the level of inefficiency can scale up to linearly with the number of agents. This strongly calls for public policy interventions. In this respect, we show the "-optimality of a policy of uniform reduction of interactions in a wide range of networks. This latter result provides normative foun- dations for the social distancing policies implemented during the COVID-19 pandemic. The implementation of such policies nevertheless requires the existence of an authority with sufficient legitimacy to implement such coercive measures. It can be thus imple- mented in a domestic context but is much harder to implement at the global scale, unless all agents/countries have individual incentives to do so. If this is not the case, we regard the shift from a local to a global game as a type of second-best policy. In the latter game, agents can subsidise investments towards contagiousness reduction in the global rather than in the local network. The scope for Pareto improvement gener- ated by such policies is then characterised through a notion of Price of Autarky (PoK), which assesses the ratio between social welfare at a global and a local equilibrium. We derive a lower bound on this PoK, as a function of the network structure and thus give sufficient conditions under which a shift to the global game actually improves social welfare. Overall, our results underline the possible extreme inefficiency of individual 3 behaviours to limit epidemic propagation, but also the possibility to design efficient containment policies taking into account the network structure. The remaining of this paper is organised as follows. Section2 reviews the related literature. Section3 introduces epidemic dynamics as well as our behavioural model of the containment of epidemic spreading. Section4 provides our main results on the relationship between individual behaviours, social efficiency and network structure. Sec- tion5 investigates the social efficiency of policy measures aiming at reducing epidemic diffusion. Section6 concludes. All proofs are given in the appendix. 2 Related literature The paper builds on the very large literature on the optimal design and defense of net- works (see, e.g., Bravard et al.(2017)) and on epidemic spreading in networks. The latter literature has been extensively reviewed in Pastor-Satorras et al.(2015) and gen- erally combines an epidemiological
Load more

Recommended publications
  • Potential Games. Congestion Games. Price of Anarchy and Price of Stability
    8803 Connections between Learning, Game Theory, and Optimization Maria-Florina Balcan Lecture 13: October 5, 2010 Reading: Algorithmic Game Theory book, Chapters 17, 18 and 19. Price of Anarchy and Price of Staility We assume a (finite) game with n players, where player i's set of possible strategies is Si. We let s = (s1; : : : ; sn) denote the (joint) vector of strategies selected by players in the space S = S1 × · · · × Sn of joint actions. The game assigns utilities ui : S ! R or costs ui : S ! R to any player i at any joint action s 2 S: any player maximizes his utility ui(s) or minimizes his cost ci(s). As we recall from the introductory lectures, any finite game has a mixed Nash equilibrium (NE), but a finite game may or may not have pure Nash equilibria. Today we focus on games with pure NE. Some NE are \better" than others, which we formalize via a social objective function f : S ! R. Two classic social objectives are: P sum social welfare f(s) = i ui(s) measures social welfare { we make sure that the av- erage satisfaction of the population is high maxmin social utility f(s) = mini ui(s) measures the satisfaction of the most unsatisfied player A social objective function quantifies the efficiency of each strategy profile. We can now measure how efficient a Nash equilibrium is in a specific game. Since a game may have many NE we have at least two natural measures, corresponding to the best and the worst NE. We first define the best possible solution in a game Definition 1.
    [Show full text]
  • Price of Competition and Dueling Games
    Price of Competition and Dueling Games Sina Dehghani ∗† MohammadTaghi HajiAghayi ∗† Hamid Mahini ∗† Saeed Seddighin ∗† Abstract We study competition in a general framework introduced by Immorlica, Kalai, Lucier, Moitra, Postlewaite, and Tennenholtz [19] and answer their main open question. Immorlica et al. [19] considered classic optimization problems in terms of competition and introduced a general class of games called dueling games. They model this competition as a zero-sum game, where two players are competing for a user’s satisfaction. In their main and most natural game, the ranking duel, a user requests a webpage by submitting a query and players output an or- dering over all possible webpages based on the submitted query. The user tends to choose the ordering which displays her requested webpage in a higher rank. The goal of both players is to maximize the probability that her ordering beats that of her opponent and gets the user’s at- tention. Immorlica et al. [19] show this game directs both players to provide suboptimal search results. However, they leave the following as their main open question: “does competition be- tween algorithms improve or degrade expected performance?” (see the introduction for more quotes) In this paper, we resolve this question for the ranking duel and a more general class of dueling games. More precisely, we study the quality of orderings in a competition between two players. This game is a zero-sum game, and thus any Nash equilibrium of the game can be described by minimax strategies. Let the value of the user for an ordering be a function of the position of her requested item in the corresponding ordering, and the social welfare for an ordering be the expected value of the corresponding ordering for the user.
    [Show full text]
  • Strategic Behavior in Queues
    Strategic behavior in queues Lecturer: Moshe Haviv1 Dates: 31 January – 4 February 2011 Abstract: The course will first introduce some concepts borrowed from non-cooperative game theory to the analysis of strategic behavior in queues. Among them: Nash equilibrium, socially optimal strategies, price of anarchy, evolutionarily stable strategies, avoid the crowd and follow the crowd. Various decision models will be considered. Among them: to join or not to join an M/M/1 or an M/G/1 queue, when to abandon the queue, when to arrive to a queue, and from which server to seek service (if at all). We will also look at the application of cooperative game theory concepts to queues. Among them: how to split the cost of waiting among customers and how to split the reward gained when servers pooled their resources. Program: 1. Basic concepts in strategic behavior in queues: Unobservable and observable queueing models, strategy profiles, to avoid or to follow the crowd, Nash equilibrium, evolutionarily stable strategy, social optimization, the price of anarchy. 2. Examples: to queue or not to queue, priority purchasing, retrials and abandonment, server selection. 3. Competition between servers. Examples: price war, capacity competition, discipline competition. 4. When to arrive to a queue so as to minimize waiting and tardiness costs? Examples: Poisson number of arrivals, fluid approximation. 5. Basic concepts in cooperative game theory: The Shapley value, the core, the Aumann-Shapley prices. Ex- amples: Cooperation among servers, charging customers based on the externalities they inflict on others. Bibliography: [1] M. Armony and M. Haviv, Price and delay competition between two service providers, European Journal of Operational Research 147 (2003) 32–50.
    [Show full text]
  • Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction
    Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction Renato Paes Leme Eva´ Tardos Department of Computer Science Department of Computer Science Cornell University, Ithaca, NY Cornell University, Ithaca, NY [email protected] [email protected] Abstract—The Generalized Second Price Auction has for advertisements and slots higher on the page are been the main mechanism used by search companies more valuable (clicked on by more users). The bids to auction positions for advertisements on search pages. are used to determine both the assignment of bidders In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full to slots, and the fees charged. In the simplest model, information setting, socially optimal Nash equilibria are the bidders are assigned to slots in order of bids, and known to exist (i.e., the Price of Stability is 1). This paper the fee for each click is the bid occupying the next is the first to prove bounds on the price of anarchy, and slot. This auction is called the Generalized Second Price to give any bounds in the Bayesian setting. Auction (GSP). More generally, positions and payments Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated in the Generalized Second Price Auction depend also on strategies. In the full information setting we prove a bound the click-through rates associated with the bidders, the of 1.618 for the price of anarchy for pure Nash equilibria, probability that the advertisement will get clicked on by and a bound of 4 for mixed Nash equilibria.
    [Show full text]
  • Statistical GGP Game Decomposition Aline Hufschmitt, Jean-Noël Vittaut, Nicolas Jouandeau
    Statistical GGP Game Decomposition Aline Hufschmitt, Jean-Noël Vittaut, Nicolas Jouandeau To cite this version: Aline Hufschmitt, Jean-Noël Vittaut, Nicolas Jouandeau. Statistical GGP Game Decomposition. Computer Games - 7th Workshop, CGW 2018, Held in Conjunction with the 27th International Con- ference on Artificial Intelligence, IJCAI 2018, Jul 2018, Stockholm, Sweden. pp.79-97, 10.1007/978- 3-030-24337-1_4. hal-02182443 HAL Id: hal-02182443 https://hal.archives-ouvertes.fr/hal-02182443 Submitted on 15 Oct 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. In Proceedings of the IJCAI-18 Workshop on Computer Games (CGW 2018) Statistical GGP Game Decomposition Aline Hufschmitt, Jean-No¨elVittaut, and Nicolas Jouandeau LIASD - University of Paris 8, France falinehuf,jnv,[email protected] Abstract. This paper presents a statistical approach for the decompo- sition of games in the General Game Playing framework. General game players can drastically decrease game search cost if they hold a decom- posed version of the game. Previous works on decomposition rely on syn- tactical structures, which can be missing from the game description, or on the disjunctive normal form of the rules, which is very costly to compute.
    [Show full text]
  • Common Belief Foundations of Global Games∗
    Common Belief Foundations of Global Games Stephen Morris Hyun Song Shin Muhamet Yildiz Princeton University Bank of International Settlements M.I.T. January 2015 Abstract Complete information games– i.e., games where there is common certainty of payoffs– often have multiple rationalizable actions. What happens if the common certainty of payoffs assumption is relaxed? Consider a two player game where a player’spayoff depends on a payoff parameter. Define a player’s"rank belief" as the probability he assigns to his payoff parameter being higher than his opponent’s. We show that there is a unique rationalizable action played when there is common certainty of rank beliefs, and we argue this is the driving force behind selection results in the global games literature. We consider what happens if players’payoff parameters are derived from common and idiosyncratic terms with fat tailed distributions, and we approach complete information by shrinking the distribution of both terms. There are two cases. If the common component has a thicker tails, then there is common certainty of rank beliefs, and thus uniqueness and equilibrium selection. If the idiosyncratic terms has thicker tails, then there is approximate common certainty of payoffs, and thus multiple rationalizable actions. 1 Introduction Complete information games often have many equilibria. Even when they have a single equilibrium, they often have many actions that are rationalizable, and are therefore consistent with common This paper incorporates material from a working paper of the same title circulated in 2009 (Morris and Shin (2009)). We are grateful for comments from seminar participants at Northwestern on this iteration of the project.
    [Show full text]
  • Essays on Supermodular Games, Design, and Political Theory
    Selection, Learning, and Nomination: Essays on Supermodular Games, Design, and Political Theory Thesis by Laurent Mathevet In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2008 (Defended May 15th, 2008) ii c 2008 Laurent Mathevet All Rights Reserved iii For my parents, Denise and Ren´eMathevet iv Acknowledgements Many people have helped me along the way. Christelle has been a source of happiness, and without her uncompromising support and faith in me, I would have been miserable. My parents, to whom this dissertation is dedicated, have shown continual and unconditional support despite the distance. They are a constant source of inspiration. I also wish to thank my advisors, Federico Echenique and Matthew Jackson, for their help and encouragement. Federico has spent numerous hours advising me, meeting with me nearly every week since my second year as a graduate student. He has endeavored to extract the most out of me, discarding unpromising thoughts and results, reading my drafts countless times and demanding (countless!) rewrites. Most importantly, he found the right words in those difficult times of a PhD graduate student. Matt has also been a wonderful advisor, available, demanding, and encouraging. I have benefited greatly from his comments, which are the kind that shapes the core of a research project. Matt has funded me for several years, which allowed me to concentrate on research. I also thank him for his invitation to spend a term at Stanford to further my research. I am indebted to Preston McAfee, whose personality, charisma, and sharpness have made meetings with him among the most fruitful and enjoyable.
    [Show full text]
  • Game Theory for Signal Processing in Networks
    Game Theory for Signal Processing in Networks Giacomo Bacci, Samson Lasaulce, Walid Saad, and Luca Sanguinetti Abstract In this tutorial, the basics of game theory are introduced along with an overview of its most recent and emerging applications in signal processing. One of the main features of this contribution is to gather in a single paper some fundamental game-theoretic notions and tools which, over the past few years, have become widely spread over a large number of papers. In particular, both strategic-form and coalition-form games are described in details while the key connections and differences betweenthemareoutlined.Moreover,aparticularattention is also devoted to clarify the connections between strategic-form games and distributed optimization and learning algorithms. Beyond an introduction to the basic concepts andmainsolutionapproaches,severalcarefullydesigned examples are provided to allow a better understanding of how to apply the described tools. I. INTRODUCTION game theory (GT) is a branch of mathematics that enables the modeling and analysis of the interactions between several decision-makers (called players) who can have conflicting or common objectives. A game is a situation in which the benefit or cost achieved by each player from an interactive situation depends, not only on its own decisions, but also on those taken by the other players. For example, the time a car driver needs to get back home generally depends not only on the route he/she chooses but also on the decisions taken by the other drivers. Therefore, in a game, the actions and objectives of the players are tightly coupled.Untilveryrecently,GThas been used only marginally in signal processing (SP), with notable examples being some applications in robust detection and estimation [1] as well as watermarking [2] (in which the watermarking problem is seen as a game between the data embedder and the attacker).
    [Show full text]
  • Convergence to Approximate Nash Equilibria in Congestion Games†
    Convergence to Approximate Nash Equilibria in Congestion Games† Steve Chien‡ Alistair Sinclair§ Abstract We study the ability of decentralized, local dynamics in non-cooperative games to rapidly reach an approximate Nash equilibrium. For symmetric congestion games in which the edge delays satisfy a “bounded jump” condition, we show that convergence to an ε-Nash equilibrium occurs within a number of steps that is polynomial in the number of players and ε−1. This appears to be the first such result for a class of games that includes examples for which finding an exact Nash equilibrium is PLS-complete, and in which shortest paths to an exact equilibrium are exponentially long. We show moreover that rapid convergence holds even under only the apparently minimal assumption that no player is excluded from moving for arbitrarily many steps. We also prove that, in a generalized setting where players have different “tolerances” εi that specify their thresholds in the approximate Nash equilibrium, the number of moves made by a player before equilibrium is reached depends only on his associated εi, and not on those of the other players. Finally, we show that polynomial time convergence still holds even when a bounded number of edges are allowed to have arbitrary delay functions. 1 Introduction The emerging field of algorithmic game theory has led to a fundamental re-examination, from a computational perspective, of the classical concept of Nash equilibrium [20]. Much of this activity has focused on understanding the structure of Nash equilibria (as expressed, notably, in the “price of anarchy,” see e.g. [22, 25, 24]) and the computational complexity of finding them (see, e.g., [10, 7, 4]).
    [Show full text]
  • Bounded Rationality in Keynesian Beauty Contests: a Lesson for Central Bankers?
    Vol. 14, 2020-16 | June 04, 2020 | http://dx.doi.org/10.5018/economics-ejournal.ja.2020-16 Bounded rationality in Keynesian beauty contests: a lesson for central bankers? Felix Mauersberger, Rosemarie Nagel, and Christoph Bühren Abstract The great recession (2008) triggered an apparent discrepancy between empirical findings and macroeconomic models based on rational expectations alone. This gap led to a series of recent developments of a behavioral microfoundation of macroeconomics combined with the underlying experimental and behavioral Beauty Contest (BC) literature, which the authors review in this paper. They introduce the reader to variations of the Keynesian Beauty Contest (Keynes, The general theory of employment, interest, and money, 1936), theoretically and experimentally, demonstrating systematic patterns of out-of-equilibrium behavior. This divergence of (benchmark) solutions and bounded rationality observed in human behavior has been resolved through stepwise reasoning, the so-called level k, or cognitive hierarchy models. Furthermore, the authors show how the generalized BC function with limited parameter specifications encompasses relevant micro and macro models. Therefore, the stepwise reasoning models emerge naturally as building blocks for new behavioral macroeconomic theories to understand puzzles like the lacking rise of inflation after the financial crisis, the efficacy of quantitative easing, the forward guidance puzzle, and the effectiveness of temporary fiscal expansion. (Published in Special Issue Bio-psycho-social foundations of macroeconomics) JEL E12 E13 E7 D80 D9 C91 Keywords Beauty Contest game; expectation formation; equilibration; level k reasoning; behavioral macroeconomics; game theory; experimental economics Authors Felix Mauersberger, University of Bonn, Germany Rosemarie Nagel, ICREA-UPF Barcelona GSE, Spain, [email protected] Christoph Bühren, Clausthal University of Technology, Germany Citation Felix Mauersberger, Rosemarie Nagel, and Christoph Bühren (2020).
    [Show full text]
  • Efficiency of Mechanisms in Complex Markets
    EFFICIENCY OF MECHANISMS IN COMPLEX MARKETS A Dissertation Presented to the Faculty of the Graduate School of Cornell University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Vasileios Syrgkanis August 2014 ⃝c 2014 Vasileios Syrgkanis ALL RIGHTS RESERVED EFFICIENCY OF MECHANISMS IN COMPLEX MARKETS Vasileios Syrgkanis, Ph.D. Cornell University 2014 We provide a unifying theory for the analysis and design of efficient simple mechanisms for allocating resources to strategic players, with guaranteed good properties even when players participate in many mechanisms simultaneously or sequentially and even when they use learning algorithms to identify how to play and have incomplete information about the parameters of the game. These properties are essential in large scale markets, such as electronic marketplaces, where mechanisms rarely run in isolation and the environment is too complex to assume that the market will always converge to the classic economic equilib- rium or that the participants will have full knowledge of the competition. We propose the notion of a smooth mechanism, and show that smooth mech- anisms possess all the aforementioned desiderata in large scale markets. We fur- ther give guarantees for smooth mechanisms even when players have budget constraints on their payments. We provide several examples of smooth mech- anisms and show that many simple mechanisms used in practice are smooth (such as formats of position auctions, uniform price auctions, proportional bandwidth allocation mechanisms, greedy combinatorial auctions). We give algorithmic characterizations of which resource allocation algorithms lead to smooth mechanisms when accompanied by appropriate payment schemes and show a strong connection with greedy algorithms on matroids.
    [Show full text]
  • Intrinsic Robustness of the Price of Anarchy
    Intrinsic Robustness of the Price of Anarchy Tim Roughgarden Department of Computer Science Stanford University 353 Serra Mall, Stanford, CA 94305 [email protected] ABSTRACT ble to eliminate in many situations, with large networks be- The price of anarchy, defined as the ratio of the worst-case ing one obvious example. The past ten years have provided objective function value of a Nash equilibrium of a game and an encouraging counterpoint to this widespread equilibrium that of an optimal outcome, quantifies the inefficiency of self- inefficiency: in a number of interesting application domains, ish behavior. Remarkably good bounds on this measure have decentralized optimization by competing individuals prov- been proved in a wide range of application domains. How- ably approximates the optimal outcome. ever, such bounds are meaningful only if a game’s partici- A rigorous guarantee of this type requires a formal behav- pants successfully reach a Nash equilibrium. This drawback ioral model, in order to define“the outcome of self-interested motivates inefficiency bounds that apply more generally to behavior”. The majority of previous research studies pure- weaker notions of equilibria, such as mixed Nash equilibria, strategy Nash equilibria, defined as follows. Each player i correlated equilibria, or to sequences of outcomes generated selects a strategy si from a set Si (like a path in a network). by natural experimentation strategies, such as simultaneous The cost Ci(s) incurred by a player i in a game is a function regret-minimization. of the entire vector s of players’ chosen strategies, which We prove a general and fundamental connection between is called a strategy profile or an outcome.
    [Show full text]