DOCSLIB.ORG

The Price of Stability for Network Design with Fair Cost Allocation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Price of Stability for Network Design with Fair Cost Allocation The Price of Stability for Network Design with Fair Cost Allocation Elliot Anshelevich∗ Anirban Dasguptay Jon Kleinberg z Ev´ a Tardosx Tom Wexler{ Tim Roughgardenk Abstract scheme can be derived from the Shapley value, and has a number of basic economic motivations. We Network design is a fundamental problem for show that the price of stability for network design which it is important to understand the effects with respect to this fair cost allocation is O(log k), of strategic behavior. Given a collection of self- where k is the number of users, and that a good interested agents who want to form a network con- Nash equilibrium can be achieved via best-response necting certain endpoints, the set of stable solutions dynamics in which users iteratively defect from a | the Nash equilibria | may look quite different starting solution. This establishes that the fair cost from the centrally enforced optimum. We study the allocation protocol is in fact a useful mechanism quality of the best Nash equilibrium, and refer to for inducing strategic behavior to form near-optimal the ratio of its cost to the optimum network cost equilibria. We discuss connections to the class of as the price of stability. The best Nash equilibrium potential games defined by Monderer and Shapley, solution has a natural meaning of stability in this and extend our results to cases in which users are context | it is the optimal solution that can be pro- seeking to balance network design costs with laten- posed from which no user will \defect". cies in the constructed network, with stronger results We consider the price of stability for network de- when the network has only delays and no construc- sign with respect to one of the most widely-studied tion costs. We also present bounds on the conver- protocols for network cost allocation, in which the gence time of best-response dynamics, and discuss cost of each edge is divided equally between users extensions to a weighted game. whose connections make use of it; this fair-division ∗Cornell University, Department of Computer Science, 1. Introduction Upson Hall, Ithaca, NY 14853. Supported by ITR grant 0311333. Email: [email protected]. yCornell University, Department of Computer Science, In many network settings, the system behavior Upson Hall, Ithaca, NY 14853. Supported by the Depart- arises from the actions of a large number of inde- ment of Computer Science. Email: [email protected]. pendent agents, each motivated by self-interest and zCornell University, Department of Computer Sci- optimizing an individual objective function. As a ence, Upson Hall, Ithaca, NY 14853. Email: klein- [email protected]. Supported in part by a David and Lucile result, the global performance of the system may Packard Foundation Fellowship and NSF grants 0081334 and not be as good as in a case where a central author- 0311333. ity can simply dictate a solution; rather, we need xCornell University, Department of Computer Science, to understand the quality of solutions that are con- Upson Hall, Ithaca, NY 14853. Supported in part by NSF grant CCR-032553, ITR grant 0311333, and ONR grant sistent with self-interested behavior. Recent theo- N00014-98-1-0589. Email: [email protected]. retical work has framed this type of question in the {Cornell University, Department of Computer Science, following general form: how much worse is the so- Upson Hall, Ithaca, NY 14853. Supported by ITR grant lution quality at a Nash equilibrium1, relative to 0311333. Email: [email protected]. kUC Berkeley, Computer Science Division, Soda Hall, 1Recall that a Nash equilibrium is a state of the system Berkeley, CA 94720. Supported by an NSF Postdoctoral Fel- in which no agent has an interest in unilaterally changing its lowship. Email: [email protected]. own behavior. the quality at a centrally enforced optimum? Ques- (Note that this cost can depend on the choices of tions of this genre have received considerable atten- the other users as well.) Although there are in tion in recent years, for problems including routing principle many possible cost-sharing mechanisms, [24, 25, 4], load balancing [5, 6, 16, 23], and facility research in this area has converged on a few mech- location [26]. anisms with good theoretical and empirical behav- An important issue to explore in this area is the ior; here we focus on the following particularly nat- middle ground between centrally enforced solutions ural one: the cost of each edge is shared equally and completely unregulated anarchy. In most net- by the set of all users whose paths contain it, so ce working applications, it is not the case that agents that Ci(S1; S2; : : : ; Sk) = . This X jfj : e 2 Sj gj are completely unrestricted; rather, they interact e2Si with an underlying protocol that essentially pro- equal-division mechanism has a number of basic eco- poses a collective solution to all participants, who nomic motivations; it can be derived from the Shap- can each either accept it or defect from it. As a re- ley value [20], and it can be shown to be the unique sult, it is in the interest of the protocol designer to cost-sharing scheme satisfying a number of differ- seek the best Nash equilibrium; this can naturally ent sets of axioms [9, 11, 20]. For the former rea- be viewed as the optimum subject to the constraint son, we will refer to it as the Shapley cost-sharing that the solution be stable, with no agent having mechanism. Note that the total edge cost of the de- an incentive to unilaterally defect from it once it signed network is equal to the sum of the costs in the is offered. Hence, one can view the ratio of the union of all Si, and the costs allocated to users in solution quality at the best Nash equilibrium rela- the Shapley mechanism completely pay for this total n tive to the global optimum as a price of stability, edge cost: Ci(S1; S2; : : : ; Sk) = 2[ ce. Pi=1 Pe iSi since it captures the problem of optimization sub- Now, the general question is to determine how ject to this constraint. Some recent work [1, 4] has this basic cost-sharing mechanism serves to influ- considered this definition (termed the \optimistic ence the strategic behavior of the users, and what price of anarchy" in [1]); it stands in contrast to the effect this has on the structure and overall cost larger line of work in algorithmic game theory on of the network one obtains. Given a solution to the price of anarchy [21] | the ratio of the worst the network design problem consisting of a vec- Nash equilibrium to the optimum | which is more tor of paths (S1; : : : ; Sk) for the n users, user i suited to worst-case analysis of situations with es- would be interested in deviating from this solu- sentially no protocol mediating interactions among 0 tion if there were an alternate si-ti path Si so that the agents. Indeed, one can view the activity of a 0 changing to Si would lower its cost under the result- protocol designer seeking a good Nash equilibrium 0 ing allocation: Ci(S1; : : : ; Si−1; S ; Si+1; : : : ; Sk) < as being aligned with the general goals of mecha- i Ci(S1; : : : ; Si−1; Si; Si+1; : : : ; Sk): We say that a set nism design | producing a game that yields good of paths is a Nash equilibrium if no user has an inter- outcomes when players act in their own self-interest. est in deviating. As we will see below, there exists a set of paths in Nash equilibrium for every instance Network Design Games. Network design is a of this network design game. (In this paper, we natural area in which to explore the price of stabil- will only be concerned with pure Nash equilibrium; ity, given the large body of work in the networking i.e. with equilibria where each user deterministically literature on methods for sharing the cost of a de- chooses a single path.) signed network | often a virtual overlay, multicast The goal of a network design protocol is to sug- tree, or other sub-network of the Internet | among gest for each user i a path Si so that the resulting a collection of participants. (See e.g. [9, 11] for set of paths is in Nash equilibrium and its total cost overviews of work in this area). exceeds that of an optimal set of paths by as small a A cost-sharing mechanism can be viewed as the factor as possible; this factor is the price of stability underlying protocol that determines how much a of the instance. It is useful at this point to consider network serving several participants will cost to a simple example that illustrates how the price of each of them. Specifically, say that each user i has stability can grow to a super-constant value (with a pair of nodes (si; ti) that it wishes to connect; it k). Suppose k players wish to connect the common chooses an si-ti path Si; and the cost-sharing mech- source s to their terminal ti, assume player i has anism then charges user i a cost of Ci(S1; : : : ; Sk). its own path of cost 1=i, and all players can share a t 1 1 1 1 1 2 3 k-1 k ... s s1 s2 s3 k-1 sk 1+ε 0 0 0 0 0 v Figure 1. An instance in which the price of stability converges to H(k) = Θ(log k) as " ! 0.
Load more

Recommended publications
  • CSC304 Lecture 4 Game Theory
    CSC304 Lecture 4 Game Theory (Cost sharing & congestion games, Potential function, Braess’ paradox) CSC304 - Nisarg Shah 1 Recap • Nash equilibria (NE) ➢ No agent wants to change their strategy ➢ Guaranteed to exist if mixed strategies are allowed ➢ Could be multiple • Pure NE through best-response diagrams • Mixed NE through the indifference principle CSC304 - Nisarg Shah 2 Worst and Best Nash Equilibria • What can we say after we identify all Nash equilibria? ➢ Compute how “good” they are in the best/worst case • How do we measure “social good”? ➢ Game with only rewards? Higher total reward of players = more social good ➢ Game with only penalties? Lower total penalty to players = more social good ➢ Game with rewards and penalties? No clear consensus… CSC304 - Nisarg Shah 3 Price of Anarchy and Stability • Price of Anarchy (PoA) • Price of Stability (PoS) “Worst NE vs optimum” “Best NE vs optimum” Max total reward Max total reward Min total reward in any NE Max total reward in any NE or or Max total cost in any NE Min total cost in any NE Min total cost Min total cost PoA ≥ PoS ≥ 1 CSC304 - Nisarg Shah 4 Revisiting Stag-Hunt Hunter 2 Stag Hare Hunter 1 Stag (4 , 4) (0 , 2) Hare (2 , 0) (1 , 1) • Max total reward = 4 + 4 = 8 • Three equilibria ➢ (Stag, Stag) : Total reward = 8 ➢ (Hare, Hare) : Total reward = 2 1 2 1 2 ➢ ( Τ3 Stag – Τ3 Hare, Τ3 Stag – Τ3 Hare) 1 1 1 1 o Total reward = ∗ ∗ 8 + 1 − ∗ ∗ 2 ∈ (2,8) 3 3 3 3 • Price of stability? Price of anarchy? CSC304 - Nisarg Shah 5 Revisiting Prisoner’s Dilemma John Stay Silent Betray Sam
    [Show full text]
  • 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]
  • Lecture Notes
    GRADUATE GAME THEORY LECTURE NOTES BY OMER TAMUZ California Institute of Technology 2018 Acknowledgments These lecture notes are partially adapted from Osborne and Rubinstein [29], Maschler, Solan and Zamir [23], lecture notes by Federico Echenique, and slides by Daron Acemoglu and Asu Ozdaglar. I am indebted to Seo Young (Silvia) Kim and Zhuofang Li for their help in finding and correcting many errors. Any comments or suggestions are welcome. 2 Contents 1 Extensive form games with perfect information 7 1.1 Tic-Tac-Toe ........................................ 7 1.2 The Sweet Fifteen Game ................................ 7 1.3 Chess ............................................ 7 1.4 Definition of extensive form games with perfect information ........... 10 1.5 The ultimatum game .................................. 10 1.6 Equilibria ......................................... 11 1.7 The centipede game ................................... 11 1.8 Subgames and subgame perfect equilibria ...................... 13 1.9 The dollar auction .................................... 14 1.10 Backward induction, Kuhn’s Theorem and a proof of Zermelo’s Theorem ... 15 2 Strategic form games 17 2.1 Definition ......................................... 17 2.2 Nash equilibria ...................................... 17 2.3 Classical examples .................................... 17 2.4 Dominated strategies .................................. 22 2.5 Repeated elimination of dominated strategies ................... 22 2.6 Dominant strategies ..................................
    [Show full text]
  • Potential Games and Competition in the Supply of Natural Resources By
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by ASU Digital Repository Potential Games and Competition in the Supply of Natural Resources by Robert H. Mamada A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved March 2017 by the Graduate Supervisory Committee: Carlos Castillo-Chavez, Co-Chair Charles Perrings, Co-Chair Adam Lampert ARIZONA STATE UNIVERSITY May 2017 ABSTRACT This dissertation discusses the Cournot competition and competitions in the exploita- tion of common pool resources and its extension to the tragedy of the commons. I address these models by using potential games and inquire how these models reflect the real competitions for provisions of environmental resources. The Cournot models are dependent upon how many firms there are so that the resultant Cournot-Nash equilibrium is dependent upon the number of firms in oligopoly. But many studies do not take into account how the resultant Cournot-Nash equilibrium is sensitive to the change of the number of firms. Potential games can find out the outcome when the number of firms changes in addition to providing the \traditional" Cournot-Nash equilibrium when the number of firms is fixed. Hence, I use potential games to fill the gaps that exist in the studies of competitions in oligopoly and common pool resources and extend our knowledge in these topics. In specific, one of the rational conclusions from the Cournot model is that a firm’s best policy is to split into separate firms. In real life, we usually witness the other way around; i.e., several firms attempt to merge and enjoy the monopoly profit by restricting the amount of output and raising the price.
    [Show full text]
  • Online Learning and Game Theory
    Online Learning and Game Theory Your guide: Avrim Blum Carnegie Mellon University [Algorithmic Economics Summer School 2012] Itinerary • Stop 1: Minimizing regret and combining advice. – Randomized Wtd Majority / Multiplicative Weights alg – Connections to minimax-optimality in zero-sum games • Stop 2: Bandits and Pricing – Online learning from limited feedback (bandit algs) – Use for online pricing/auction problems • Stop 3: Internal regret & Correlated equil – Internal regret minimization – Connections to correlated equilib. in general-sum games • Stop 4: Guiding dynamics to higher-quality equilibria – Helpful “nudging” of simple dynamics in potential games Stop 1: Minimizing regret and combining expert advice Consider the following setting… Each morning, you need to pick one of N possible routes to drive to work. Robots But traffic is different each day. R Us 32 min Not clear a priori which will be best. When you get there you find out how long your route took. (And maybe others too or maybe not.) Is there a strategy for picking routes so that in the long run, whatever the sequence of traffic patterns has been, you’ve done nearly as well as the best fixed route in hindsight? (In expectation, over internal randomness in the algorithm) Yes. “No-regret” algorithms for repeated decisions A bit more generally: Algorithm has N options. World chooses cost vector. Can view as matrix like this (maybe infinite # cols) World – life - fate Algorithm At each time step, algorithm picks row, life picks column. Alg pays cost for action chosen. Alg gets column as feedback (or just its own cost in the “bandit” model). Need to assume some bound on max cost.
    [Show full text]
  • Potential Games with Continuous Player Sets
    Potential Games with Continuous Player Sets William H. Sandholm* Department of Economics University of Wisconsin 1180 Observatory Drive Madison, WI 53706 [email protected] www.ssc.wisc.edu/~whs First Version: April 30, 1997 This Version: May 20, 2000 * This paper is based on Chapter 4 of my doctoral dissertation (Sandholm [29]). I thank an anonymous referee and Associate Editor, Josef Hofbauer, and seminar audiences at Caltech, Chicago, Columbia, Harvard (Kennedy), Michigan, Northwestern, Rochester, Washington University, and Wisconsin for their comments. I especially thank Eddie Dekel and Jeroen Swinkels for their advice and encouragement. Financial support from a State Farm Companies Foundation Dissertation Fellowship is gratefully acknowledged. Abstract We study potential games with continuous player sets, a class of games characterized by an externality symmetry condition. Examples of these games include random matching games with common payoffs and congestion games. We offer a simple description of equilibria which are locally stable under a broad class of evolutionary dynamics, and prove that behavior converges to Nash equilibrium from all initial conditions. W e consider a subclass of potential games in which evolution leads to efficient play. Finally, we show that the games studied here are the limits of convergent sequences of the finite player potential games studied by Monderer and Shapley [22]. JEL Classification Numbers: C72, C73, D62, R41. 1. Introduction Nash equilibrium is the cornerstone of non-cooperative game theory, providing a necessary condition for stable behavior among rational agents. Still, to justify the prediction of Nash equilibrium play, one must explain how players arrive at a Nash equilibrium; if equilibrium is not reached, the fact that it is self-sustaining becomes moot.
    [Show full text]
  • Noisy Directional Learning and the Logit Equilibrium*
    CSE: AR SJOE 011 Scand. J. of Economics 106(*), *–*, 2004 DOI: 10.1111/j.1467-9442.2004.000376.x Noisy Directional Learning and the Logit Equilibrium* Simon P. Anderson University of Virginia, Charlottesville, VA 22903-3328, USA [email protected] Jacob K. Goeree California Institute of Technology, Pasadena, CA 91125, USA [email protected] Charles A. Holt University of Virginia, Charlottesville, VA 22903-3328, USA [email protected] PROOF Abstract We specify a dynamic model in which agents adjust their decisions toward higher payoffs, subject to normal error. This process generates a probability distribution of players’ decisions that evolves over time according to the Fokker–Planck equation. The dynamic process is stable for all potential games, a class of payoff structures that includes several widely studied games. In equilibrium, the distributions that determine expected payoffs correspond to the distributions that arise from the logit function applied to those expected payoffs. This ‘‘logit equilibrium’’ forms a stochastic generalization of the Nash equilibrium and provides a possible explanation of anomalous laboratory data. ECTED Keywords: Bounded rationality; noisy directional learning; Fokker–Planck equation; potential games; logit equilibrium JEL classification: C62; C73 I. Introduction Small errors and shocks may have offsetting effects in some economic con- texts, in which case there is not much to be gained from an explicit analysis of stochasticNCORR elements. In other contexts, a small amount of randomness can * We gratefully acknowledge financial support from the National Science Foundation (SBR- 9818683U and SBR-0094800), the Alfred P. Sloan Foundation and the Dutch National Science Foundation (NWO-VICI 453.03.606).
    [Show full text]
  • Strong Nash Equilibria in Games with the Lexicographical Improvement
    Strong Nash Equilibria in Games with the Lexicographical Improvement Property Tobias Harks, Max Klimm, and Rolf H. M¨ohring Preprint 15/07/2009 Institut f¨ur Mathematik, Technische Universit¨at Berlin harks,klimm,moehring @math.tu-berlin.de { } Abstract. We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bot- tleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium. 1 Introduction The theory of non-cooperative games is used to study situations that involve rational and selfish agents who are motivated by optimizing their own utilities rather than reaching some social optimum.
    [Show full text]
  • Dynamics of Profit-Sharing Games
    Proceedings of the Twenty-Second International Joint Conference on Artificial Intelligence Dynamics of Profit-Sharing Games John Augustine, Ning Chen, Edith Elkind, Angelo Fanelli, Nick Gravin, Dmitry Shiryaev School of Physical and Mathematical Sciences Nanyang Technological University, Singapore Abstract information about other players. Such agents may simply re- spond to their current environment without worrying about An important task in the analysis of multiagent sys- the subsequent reaction of other agents; such behavior is said tems is to understand how groups of selfish players to be myopic. Now, coalition formation by computationally can form coalitions, i.e., work together in teams. In limited agents has been studied by a number of researchers in this paper, we study the dynamics of coalition for- multi-agent systems, starting with the work of [Shehory and mation under bounded rationality. We consider set- Kraus, 1999] and [Sandholm and Lesser, 1997]. However, tings where each team’s profit is given by a concave myopic behavior in coalition formation received relatively lit- function, and propose three profit-sharing schemes, tle attention in the literature (for some exceptions, see [Dieck- each of which is based on the concept of marginal mann and Schwalbe, 2002; Chalkiadakis and Boutilier, 2004; utility. The agents are assumed to be myopic, i.e., Airiau and Sen, 2009]). In contrast, myopic dynamics of they keep changing teams as long as they can in- non-cooperative games is the subject of a growing body of crease their payoff by doing so. We study the prop- research (see, e.g. [Fabrikant et al., 2004; Awerbuch et al., erties (such as closeness to Nash equilibrium or to- 2008; Fanelli et al., 2008]).
    [Show full text]
  • Informational Incentives for Congestion Games
    Informational Incentives for Congestion Games Hamidreza Tavafoghi and Demosthenis Teneketzis network, leading to a decrease in social welfare. In addition Abstract— We investigate the problems of designing public to the above-mentioned theoretical and experimental works, and private information disclosure mechanisms by a principal there are empirical evidences that identify negative impacts in a transportation network so as to improve the overall conges- tion. We show that perfect disclosure of information about the of information provision on the network’s congestion [16], routes’ conditions is not optimal. The principal can improve the [26], [28], [15], [10]. Therefore, it is important to investigate congestion (i.e. social welfare) by providing coordinated routing how to design appropriate information provision mechanisms recommendation to drivers based on the routes’ conditions. in a manner that is socially beneficial and leads to a reduction When the uncertainty about the routes’ conditions is high in the overall congestion in the transportation network. relative to the ex-ante difference in the routes’ conditions (the value of information is high), we show that the socially efficient In this paper, we study the problem of designing socially routing outcome is achievable using a private information optimal information disclosure mechanisms. We consider a disclosure mechanism. Furthermore, we study the problem congestion game [25], [29] in a parallel two-link network. of optimal dynamic private information disclosure mechanism We consider an information provider (principal) who wants design in a dynamic two-time step setting. We consider different to disclose information about the condition of the network pieces of information that drivers may observe and learn from at t = 1 and investigate qualitative properties of an optimal to a fixed population of drivers (agents).
    [Show full text]
  • The Price of Stability for Network Design with Fair Cost Allocation∗
    THE PRICE OF STABILITY FOR NETWORK DESIGN WITH FAIR COST ALLOCATION¤ ELLIOT ANSHELEVICHy , ANIRBAN DASGUPTAz , JON KLEINBERG x , EVA¶ TARDOS{, TOM WEXLERk, AND TIM ROUGHGARDEN¤¤ Abstract. Network design is a fundamental problem for which it is important to understand the e®ects of strategic behavior. Given a collection of self-interested agents who want to form a network connecting certain endpoints, the set of stable solutions | the Nash equilibria | may look quite di®erent from the centrally enforced optimum. We study the quality of the best Nash equilibrium, and refer to the ratio of its cost to the optimum network cost as the price of stability. The best Nash equilibrium solution has a natural meaning of stability in this context | it is the optimal solution that can be proposed from which no user will defect. We consider the price of stability for network design with respect to one of the most widely-studied protocols for network cost allocation, in which the cost of each edge is divided equally between users whose connections make use of it; this fair-division scheme can be derived from the Shapley value, and has a number of basic economic motivations. We show that the price of stability for network design with respect to this fair cost allocation is O(log k), where k is the number of users, and that a good Nash equilibrium can be achieved via best-response dynamics in which users iteratively defect from a starting solution. This establishes that the fair cost allocation protocol is in fact a useful mechanism for inducing strategic behavior to form near-optimal equilibria.
    [Show full text]
  • Algorithms for Generalized Potential Games with Mixed-Integer Variables
    Noname manuscript No. (will be inserted by the editor) Algorithms for generalized potential games with mixed-integer variables Simone Sagratella Received: date / Accepted: date Abstract We consider generalized potential games, that constitute a funda- mental subclass of generalized Nash equilibrium problems. We propose differ- ent methods to compute solutions of generalized potential games with mixed- integer variables, i.e., games in which some variable are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how dif- ferent potential functions can be used to select equilibria. We propose a new Gauss-Southwell algorithm to compute approximate equilibria of any general- ized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches. Keywords Generalized Nash equilibrium problem · Generalized potential game · Mixed-integer nonlinear problem · Parametric optimization 1 Introduction Generalized Nash equilibrium problems (GNEPs) are widely used tools to model multi agent systems in many fields [16]. In the past two decades, sev- The work of the author has been partially supported by Avvio alla Ricerca 2015 Sapienza University of Rome, under grant 488.
    [Show full text]