correlated equilibrium

In this paper, we formulate an evolutionary multiple access control game with continuousvariable actions and coupled constraints. We characterize equilibria of the game and show that the pure equilibria are Pareto optimal and also resilient to deviations by coalitions of any size, i.e., they are strong equilibria. We use the concepts of price of anarchy and strong price of anarchy to study the performance of the system. The paper also addresses how to select one specific equilibrium solution using the concepts of normalized equilibrium and evolutionarily stable strategies. We examine the long-run behavior of these strategies under several classes of evolutionary game dynamics, such as Brown-von Neumann-Nash dynamics, Smith dynamics and replicator dynamics. In addition, we examine correlated equilibrium for the single-receiver model. Correlated strategies are based on signaling structures before making decisions on rates. We then focus on evolutionary games for hybrid additive white Gaussian noise multiple access channel with multiple users and multiple receivers, where each user chooses a rate and splits it over the receivers. Users have coupled constraints determined by the capacity regions. Building upon the static game, we formulate a system of hybrid evolutionary game dynamics using Gfunction dynamics and Smith dynamics on rate control and channel selection, respectively. We show that the evolving game has an equilibrium and illustrate these dynamics with numerical examples. * The material in this paper was partially presented in [9] and [10].

We consider the class of symmetric two-player games that have the property that for any mixed strategy of the opponent, a player's best responses are included in the support of this mixed strategy-the total bandwagon or coordination property (CP). We show that for any number of pure strategies n, a symmetric two-player game has CP if and only if the game has 2 n -1 symmetric Nash equilibria. In view of the importance of 1/2-dominance as an equilibrium selection criterion, we show that if in addition to CP a game is supermodular, it will always have a 1/2-dominant equilibrium, and the 1/2-dominant equilibrium will be either the lowest or the highest strategy profile. Furthermore we show that if a game with CP has a unique potential maximizer, it will be equivalent to a 1/2-dominant equilibrium. As a specific application, we consider the minimum-effort game and reexamine the experimental equilibrium selection result of to compare it with the theoretical prediction. Our results allow us to give a new insight into an aspect of the experiment that so far could not be explained.

In this work, we first deal with the modeling of game situations that reach one of possibly many Nash equilibria. Before an instance of such a game starts, an external observer does not know, a priori, what is the exact profile of actions -- constituting an equilibrium -- that will occur; thus, he assigns subjective probabilities to players' actions. Such scenario is formalized as an observable game, which is a newly introduced structure for that purpose. Then, we study the decision problem of determining if a given set of probabilistic constraints assigned a priori by the observer to a given game is coherent, called the PCE-Coherence problem. We show several results concerning algorithms and complexity for PCE-Coherence when pure Nash equilibria and specific classes of games, called GNP-classes, are considered. In this context, we also study the computation of maximal and minimal probabilistic constraints on actions that preserves coherence. Finally, we study these problems whe...

In all social and economic interactions, individuals or coalitions choose not only with whom to interact but how to interact, and over time both the structure (the "with whom") and the strategy ("the how") of interactions change. Our objectives here are to model the structure and strategy of interactions prevailing at any point in time as a directed network and to address the following open question in the theory of social and economic network formation: given the rules of network and coalition formation, the preferences of individuals over networks, the strategic behavior of coalitions in forming networks, and the trembles of nature, what network and coalitional dynamics are likely to emergence and persist. Our main contributions are (i) to formulate the problem of network and coalition formation as a dynamic, stochastic game, (ii) to show that this game possesses a stationary correlated equilibrium (in network and coalition formation strategies), (iii) to show that, together with the trembles of nature, this stationary correlated equilibrium determines an equilibrium Markov process of network and coalition formation which respects the rules of network and coalition formation and the preferences of individuals, and (iv) to show that, although uncountably many networks may form, this endogenous process of network and coalition formation generates a unique, finite, disjoint collection of nonempty subsets of networks and coalitions, each constituting a basin of attraction, and possesses a unique, finite, nonempty set of ergodic measures. Moreover, we extend to the setting of endogenous Markov dynamics the notions of pairwise stability , strong stability (Jackson-van den Nouweland, 2005), and Nash stability , and we show that in order for any network-coalition pair to be stable (pairwise, strong, or Nash) it is necessary and sufficient that the pair reside in one of finitely many basins of attraction -and hence reside in the support of an ergodic measure. The results we obtain here for endogenous network dynamics and stochastic basins of attraction are the dynamic analogs of our earlier results on endogenous network formation and strategic basins of attraction in static, abstract games of network formation (

Modeling club structures as bipartite directed networks, we formulate the problem of club formation as a noncooperative game of network formation and identify conditions on network formation rules and players' network payoffs sufficient to guarantee that the game has a potential function. Our sufficient conditions on network formation rules require that each player be choose freely and unilaterally those clubs he joins and also his activities within these clubs (subject to his set of feasible actions). We refer to our conditions on rules as noncooperative free mobility. We also require that players' payoffs be additively separable in player-specific payoffs and externalities (additive separability) and that payoff externalities -a function of club membership, club activities, and crowdingbe identical across players (externality homogeneity). We then show that under these conditions, the noncooperative game of club network formation is a potential game over directed club networks and we discuss the implications of this result.

We model an oligopolistic industry where a number of firms that are asymmetrically informed about the environment compete via quantities, and we study how the information available to a firm affects its equilibrium profits. Indeed we find that if all firms have access to the same constant returns to scale technology, in any Bayesian equilibrium the information advantage of a firm is rewarded.

This paper studies extensive form games with public information where all players have the same information at each point in time. We prove that when there are at least three players, all communication equilibrium payos can be obtained by unmediated cheap-talk procedures. The result encompasses repeated games and stochastic games.

In many situations, such as trade in stock exchanges, agents have many opportunities to act within a short interval of time. The agents in such situations can often coordinate their actions in advance, but coordination during the game consumes too much time. An equilibrium in such situations has to be sequential in order to handle mistakes made by players. In this paper, we present a new solution concept for infinite-horizon dynamic games, which is appropriate for such situations: a sequential normal-form correlated approximate equilibrium. Under additional assumptions, we show that every such game admits this kind of equilibrium.

We explore the potential for correlated equilibrium to capture conformity to norms and the coordination of behavior within social groups. Given a partition of players into social groups we propose three properties one may expect of a correlated equilibrium: within-group anonymity, group independence and stereotyped beliefs. Within-group anonymity requires that players within the same social group have equal opportunities and equal payoffs. Group independence requires that there be no correlation of behavior between groups. If beliefs are stereotyped then any two members of a social group are expected to behave identically. * This paper is a revision of Cartwright and Wooders (2003a). The main results of that paper were presented at the the 2002 General Equilibrium Conference held in Athens in May 2002, at Northwestern University in August 2002, at the Purdue Economic Theory conference in honor of C.D. Aliprantis and at Hebrew University in 2006. We thank the participants for their interest and comments. We demonstrate that there are subjective correlated equilibrium satisfying within-group anonymity, group independence and stereotyping. These results apply when players within social groups are 'similar', and not necessarily identical. A number of related issues, such as fairness, are also discussed.

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: in the long-run, the player is guaranteed an average payoff as large as the best-reply payoff to the empirical distribution of play of the other players; i.e., there is no``regret.

Is conformity amongst similar individuals consistent with self-interested behavior? We consider a model of incomplete information in which each player receives a signal, interpreted as an allocation to a role, and can make his action choice conditional on his role. Our main result demonstrates that 'near to' any correlated equilibrium is an approximate correlated equilibrium 'with conformity' — that is, an equilibrium where all 'similar players' play the same strategy, have the same probability of being allocated to each role, and receive approximately the same payoff; in short, similar players 'behave in an identical way' and are treated nearly equally. To measure 'similarity' amongst players we introduce the notions of approximate substitutes and a (δ, Q)-class games — a game with Q classes of players where all players in the same class are δ-substitutes for each other.

We argue that a social norm and the coordination of behavior within social groups can be expressed by a correlated equilibrium. Given a social group structure (a partition of individuals into social groups), we propose four conditions that one may expect of a correlated equilibrium consistent with social norms. These are: (1) within-group anonymity (conformity within groups), (2) group independence (no conformity between groups), (3) homophily (individuals in the same group have similar attributes), and (4) predictable group behavior (ex post stability). We demonstrate that correlated equilibrium satisfying (1)-(3) exist very generally and equilibrium satisfying (1)-(4) exist in games with many players. We also consider stereotyped beliefs-beliefs that all individuals in a social group can be expected to behave in the same way-and show that stereotyping is not costly to the person who stereotypes but may or may not be beneficial to society.

Coordination between agents can be modelled using correlated equilibrium and a 'device' allocating roles to players. That coordination takes place within a social context suggests properties that one may expect of a correlated equilibrium. In this paper, given a partition of the player set into (social) groups, we demonstrate the existence of an approximate correlated equilibrium satisfying a within group anonymity property and a group independence property. Within group anonymity assures 'fairness' and equality within groups while group independence rules out correlation of actions between groups. * This paper is a major revision of Cartwright and Wooders (2003a). The main results of that paper were presented at the the 2002 General Equilibrium Conference held in Athens in May 2002 and at Northwestern University in August 2002. We thank the participants for their interest and comments.

Today’s wireless devices can be simultaneously connected to multiple communication networks based on different radio access technologies (RATs) such as WiFi, 3G, and LTE. Simultaneous aggregation of each client’s traffic across multiple such RATs or the corresponding base stations (BSs) can substantially improve the quality of experience for each client and improve the overall network utilization and efficiency.Our goal in this paper is to design distributed resource allocation algorithms that can be independently executed by each BS and achieve alpha-fairness among the clients. In particular, we derive a simple water filling based solution and study its theoretical aspects such as convergence, optimality, evolution, and convergence time. We also characterize its performance through a comprehensive multi-RAT simulator, and show that it achieves superior performance to alternative distributed solutions due to its fast and low-overhead operation.

Today's wireless devices can be simultaneously connected to multiple communication networks based on different radio access technologies (RATs) such as WiFi, 3G, and LTE. Simultaneous aggregation of each client's traffic across multiple such RATs or the corresponding base stations (BSs) can substantially improve the quality of experience for each client and improve the overall network utilization and efficiency. Our goal in this paper is to design distributed resource allocation algorithms that can be independently executed by each BS and achieve alpha-fairness among the clients. In particular, we derive a simple water filling based solution and study its theoretical aspects such as convergence, optimality, evolution, and convergence time. We also characterize its performance through a comprehensive multi-RAT simulator, and show that it achieves superior performance to alternative distributed solutions due to its fast and low-overhead operation.

Today’s wireless devices can be simultaneously connected to multiple communication networks based on different radio access technologies (RATs) such as WiFi, 3G, and LTE. Simultaneous aggregation of each client’s traffic across multiple such RATs or the corresponding base stations (BSs) can substantially improve the quality of experience for each client and improve the overall network utilization and efficiency.Our goal in this paper is to design distributed resource allocation algorithms that can be independently executed by each BS and achieve alpha-fairness among the clients. In particular, we derive a simple water filling based solution and study its theoretical aspects such as convergence, optimality, evolution, and convergence time. We also characterize its performance through a comprehensive multi-RAT simulator, and show that it achieves superior performance to alternative distributed solutions due to its fast and low-overhead operation.

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier's archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

We study the fundamental problem of computing an arbitrary Nash equilibrium in bimatrix games. We start by proposing a novel characterization of the set of Nash equilibria, via a bijective map to the solution set of a (parameterized) quadratic program, whose feasible space is the (highly structured) set of correlated equilibria. We then proceed by proposing new subclasses of bimatrix games for which either an exact polynomial-time construction, or at least a FPTAS, is possible. In particular, we introduce the notion of mutual (quasi-) concavity of a bimatrix game, which assures (quasi-) convexity of our quadratic program, for at least one value of the parameter. For mutually concave bimatrix games, we provide a polynomial-time computation of a Nash equilibrium, based on the polynomial tractability of convex quadratic programming. For the mutually quasi-concave games, we provide (to our knowledge) the first FPTAS for the construction of a Nash equilibrium. Of course, for these new polynomially tractable subclasses of bimatrix games to be useful, polynomial-time certificates are also necessary that will allow us to efficiently identify them. Towards this direction, we provide various characterizations of mutual concavity, which allow us to construct such a certificate. Interestingly, these characterizations also shed light to some structural properties of the bimatrix games satisfying mutual concavity. This subclass entirely contains the most popular subclass of polynomial-time solvable bimatrix games, namely, all the constantsum games (rank−0 games). It is though incomparable to the subclass of games with fixed rank [16]: Even rank−1 games may not be mutually concave (eg, Prisoner's dilemma), but on the other hand, there exist mutually concave games of arbitrary (even full) rank. Finally, we prove closeness of mutual concavity under (Nash equilibrium preserving) positive affine transformations of bimatrix games having the same scaling factor for both payoff matrices. For different scaling factors the property is not necessarily preserved.

We focus on the problem of computing an-Nash equilibrium of a bimatrix game, when is an absolute constant. We present a simple algorithm for computing a 3 4-Nash equilibrium for any bimatrix game in strongly polynomial time and we next show how to extend this algorithm so as to obtain a (potentially stronger) parameterized approximation. Namely, we present an algorithm that computes a 2+λ 4-Nash equilibrium, where λ is the minimum, among all Nash equilibria, expected payoff of either player. The suggested algorithm runs in time polynomial in the number of strategies available to the players.

A social choice rule (SCR) F maps preference profiles to lotteries over some finite set of outcomes. F is virtually implementable in (pure and mixed) Nash equilibria provided that for all > 0, there exists a mechanism such that for each preference profile t, its set of Nash equilibrium outcomes at t is-closed to the socially desirable set F (t). Under a domain restriction, we obtain the following result: When there are at least three agents, any F is virtually implementable in Nash equilibrium, as well as in rationalizable strategies, by a bounded mechanism. No "tail-chasing" constructions, common in the constructive proofs of the literature, is used to assure that undesired strategy combinations do not form a Nash equilibrium.

As instructors, we search for alternative methods of explaining and solving problems that are difficult for our students to understand, and for ways to integrate their learning experience across related fields. This is our motivation for using decision trees to teach mixed strategy Nash equilibria. With practice, most students comprehend techniques for finding pure strategy Nash equilibria in normal form games. However, students find it significantly more difficult to find mixed strategy Nash equilibrium solutions to normal form games. The decision tree method we describe in this paper provides instructors an option for explaining mixed strategies to students, and for training students to find mixed strategy Nash equilibrium (heretofore MSNE) solutions to normal form games.

This dataset includes 204,350,0 0 0 games in normal form played by two agents that have the choice between three strategies each, as well as 10 0,0 0 0 games in normal form played by four agents that have the choice between three strategies each. The games are in general position, i.e., there are no ties between the outcomes for each of the agents. These are simple random samples with replacement from the associated populations of strategic games with ordinal semantics. Each game was obtained with random permutations of the payoffs. Their Nash Equilibria as well Perfectly Transparent Equilibria are pre-computed. The existence ratios of the Nash equilibrium, unique Nash equilibrium and Perfectly Transparent Equilibrium were computed from the first sample with two players, and the social utilities from the second sample with four players.

Games may be represented in many different ways, and different representations of games affect the complexity of problems associated with games, such as finding a Nash equilibrium. The traditional method of representing a game is to explicitly list all the payoffs, but this incurs an exponential blowup as the number of agents grows. We study two models of concisely represented games: circuit games, where the payoffs are computed by a given boolean circuit, and graph games, where each agent's payoff is a function of only the strategies played by its neighbors in a given graph. For these two models, we study the complexity of four questions: determining if a given strategy is a Nash equilibrium, finding a Nash equilibrium, determining if there exists a pure Nash equilibrium, and determining if there exists a Nash equilibrium in which the payoffs to a player meet some given guarantees. In many cases, we obtain tight results, showing that the problems are complete for various complexity classes.

We consider repeated games where at any period each player knows only his set of actions and the stream of payoffs that he has received in the past. He knows neither his own payoff function, nor the characteristics of the other players (how many there are, their strategies and payoffs). In this context, we present an adaptive procedure for play-called "modified-regret-matching"-which is interpretable as a stimulus-response or reinforcement procedure, and which has the property that any limit point of the empirical distribution of play is a correlated equilibrium of the stage game. 1 To avoid confusion, we will use the term "action" for the one-shot game, and "strategy" for the multi-stage game. 2 R denotes the real line.

We propose a new and simple adaptive procedure for playing a game: ''regret-matching.'' In this procedure, players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past. It is shown that our adaptive procedure guarantees that, with probability one, the empirical distributions of play converge to the set of correlated equilibria of the game.

Theorem. If player i plays regret-matching (RM), then i is guaranteed to have no regret in the limit (with probability 1) : lim T →∞ R i T (k) = 0 for all k in S i (a.s.) (HC) for any strategies of the other players Condition (HC) = "Hannan Consistency" Hannan 1957: There exists a strategy guaranteeing "no regret"

Trembling hand (TH) equilibria were introduced by Selten in 1975. Intuitively, these are Nash equilibria that remain stable when players assume that there is a small probability that other players will choose off-equilibrium strategies. This concept is useful for equilibrium refinement, i.e., selecting the most plausible Nash equilibria when the set of all Nash equilibria can be very large, as is the case, for instance, for Plurality voting with strategic voters. In this paper, we analyze TH equilibria of Plurality voting. We provide an efficient algorithm for computing a TH best response and establish many useful properties of TH equilibria in Plurality voting games. On the negative side, we provide an example of a Plurality voting game with no TH equilibria, and show that it is NP-hard to check whether a given Plurality voting game admits a TH equilibrium where a specific candidate is among the election winners.

Correlated equilibrium is an established solution concept in game theory describing a situation when players condition their strategies on external signals produced by a correlation device. In recent years, the concept has begun gaining traction also in general artificial intelligence because of its suitability for studying coordinated multi-agent systems. Yet the original formulation of correlated equilibrium assumes entirely rational players and hence fails to capture the subrational behavior of human decision-makers. We investigate the analogue of quantal response for correlated equilibrium, which is among the most commonly used models of bounded rationality. We coin the solution concept the quantal correlated equilibrium and study its relation to quantal response and correlated equilibria. The definition corroborates with prior conception as every quantal response equilibrium is a quantal correlated equilibrium, and correlated equilibrium is its limit as quantal responses approach the best response. We prove the concept remains PPAD-hard but searching for an optimal correlation device is beneficial for the signaler. To this end, we introduce a homotopic algorithm that simultaneously traces the equilibrium and optimizes the signaling distribution. Empirical results on one structured and one random domain show that our approach is sufficiently precise and several orders of magnitude faster than a state-of-the-art non-convex optimization solver. CCS Concepts: • Theory of computation → Algorithmic game theory; Exact and approximate computation of equilibria.

In this paper we introduce the concept of split Nash equilibrium problems associated with two related noncooperative strategic games. Then we apply the Fan-KKM theorem to prove the existence of solutions to split Nash equilibrium problems of related noncooperative strategic games, in which the strategy sets of the players are nonempty closed and convex subsets in Banach spaces. As application of this existence to economics, an example is provided to study the existence of split Nash equilibrium of utilities of two related economies. As applications, we study the existence of split Nash equilibrium in the dual extended Bertrant duopoly model of price competition

A pre-play communication-process is presented which leads to a Nash equilibrium of a strategic form game. In the communication process each player predicts the other players' actions, and he/she communicates privately his/her conjecture through message according to a protocol. All the players receiving the messages learn and revise their conjectures. We show that after a long round of the communication the profile of players' conjectures in the revision process leads a mixed strategy Nash equilibrium of the game.

In this paper, we formulate an evolutionary multiple access control game with continuousvariable actions and coupled constraints. We characterize equilibria of the game and show that the pure equilibria are Pareto optimal and also resilient to deviations by coalitions of any size, i.e., they are strong equilibria. We use the concepts of price of anarchy and strong price of anarchy to study the performance of the system. The paper also addresses how to select one specific equilibrium solution using the concepts of normalized equilibrium and evolutionarily stable strategies. We examine the long-run behavior of these strategies under several classes of evolutionary game dynamics, such as Brown-von Neumann-Nash dynamics, Smith dynamics and replicator dynamics. In addition, we examine correlated equilibrium for the single-receiver model. Correlated strategies are based on signaling structures before making decisions on rates. We then focus on evolutionary games for hybrid additive white Gaussian noise multiple access channel with multiple users and multiple receivers, where each user chooses a rate and splits it over the receivers. Users have coupled constraints determined by the capacity regions. Building upon the static game, we formulate a system of hybrid evolutionary game dynamics using Gfunction dynamics and Smith dynamics on rate control and channel selection, respectively. We show that the evolving game has an equilibrium and illustrate these dynamics with numerical examples.

We consider the class of symmetric two-player games that have the property that for any mixed strategy of the opponent, a player's best responses are included in the support of this mixed strategy-the total bandwagon or coordination property (CP). We show that for any number of pure strategies n, a symmetric two-player game has CP if and only if the game has 2 n − 1 symmetric Nash equilibria. In view of the importance of 1/2-dominance as an equilibrium selection criterion, we show that if in addition to CP a game is supermodular, it will always have a 1/2-dominant equilibrium, and the 1/2-dominant equilibrium will be either the lowest or the highest strategy profile. Furthermore we show that if a game with CP has a unique potential maximizer, it will be equivalent to a 1/2-dominant equilibrium. As a specific application, we consider the minimum-effort game and reexamine the experimental equilibrium selection result of Van Huyck, Battalio and Beil (1990) to compare it with the theoretical prediction. Our results allow us to give a new insight into an aspect of the experiment that so far could not be explained.

Multi-person stopping games with players' priorities are considered. Players observe sequentially offers Y 1 , Y 2 ,. .. at jump times T 1 , T 2 ,. .. of a Poisson process. Y 1 , Y 2 ,. .. are independent identically distributed random variables. Each accepted offer Y n results in a reward G n = Y n r(T n), where r is a non-increasing discount function. If more than one player wants to accept an offer, then the player with the highest priority (the lowest ordering) gets the reward. We construct Nash equilibrium in the multi-person stopping game using the solution of a multiple optimal stopping time problem with structure of rewards {G n }. We compare rewards and stopping times of the players in Nash equilibrium in the game with the optimal rewards and optimal stopping times in the multiple stopping time problem. It is also proved that presented Nash equilibrium is a Pareto optimum of the game. The game is a generalization of the Elfving stopping time problem to multi-person stopping games with priorities. Keywords Stopping game • Nash equilibrium • Pareto-optimality • Multiple stopping Suppose that a company is going to open m new departments that will be ordered (ranked) according to their importance. Rank 1 denotes the most important department, and consequently rank m refers to the least important one. So, the firm offers m secretary positions. The applications will be considered up to a fixed deadline U. There is a selection committee that interviews candidates arriving one by one at jump times of a Poisson process. We assume that candidates' "skills" form a sequence of i.i.d. random variables with known probability distribution. The selection process is organized similarly to the classical secretary problem but taking into account the departments' rankings. So, a candidate who is being

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. This paper gives a counterexample to the main step in Cotter's [K. Cotter, Correlated equilibrium in games with typedependent strategies, J. Econ. Theory 54 (1991) 48-69] argument for correlated equilibrium existence for this class of games, and supplies an alternative proof.

In this work we extend a result of Lehrer characterizing the correlated equilibrium payoffs in undiscounted two player repeated games with partial monitoring to the case in which the signals are permitted to be stochastic. In particular we develop appropriate versions of Lehrer's concepts of "indistinguishable" and "more informative." We also show that any payoff associated with a (correlated) distribution on strategy vectors in the stage game such that neither player can profitably deviate from one of his strategies to another that is indistinguishable and more informative is the payoff of a correlated equilibrium of the supergame.

Correlated equilibrium generalizes Nash equilibrium to allow correlation devices. Cor- related equilibrium captures the idea that in many systems there exists a trusted adminis- trator who can recommend behavior to a set of agents, but can not enforce such behavior. This makes this solution concept most appropriate to the study of multi-agent systems in AI. Aumann showed an example of a game, and of a correlated equilibrium in this game in which the agents' welfare (expected sum of players' utilities) is greater than their welfare in all mixed-strategy equilibria. Following the idea initiated by the price of anarchy literature this suggests the study of two major measures for the value of correlation in a game with nonnegative payofis: 1. The ratio between the maximal welfare obtained in a correlated equilibrium to the maximal welfare obtained in a mixed-strategy equilibrium. We refer to this ratio as the mediation value. 2. The ratio between the maximal welfare to the max...

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: In the long-run, the player is guaranteed an average payo as large as the best-reply payo to the empirical distribution of play of the other players; i.e., there is no regret." Smooth ctitious play Fudenberg and Levine 1995 and regret-matching Hart and Mas-Colell 1998 are particular cases. The motivation and application of this work come from the study of procedures whose empirical distribution of play is, in the long-run, almost a correlated equilibrium. The basic tool for the analysis is a generalization of Blackwell's 1956a approachability strategy for games with vector payo s.

We exhibit and characterize an entire class of simple adaptive strategies, in the repeated play of a game, having the Hannan-consistency property: In the long-run, the player is guaranteed an average payo as large as the best-reply payo to the empirical distribution of play of the other players; i.e., there is no regret." Smooth ctitious play Fudenberg and Levine 1995 and regret-matching Hart and Mas-Colell 1998 are particular cases. The motivation and application of this work come from the study of procedures whose empirical distribution of play is, in the long-run, almost a correlated equilibrium. The basic tool for the analysis is a generalization of Blackwell's 1956a approachability strategy for games with vector payo s.

We present an extension to any finite complete information game with two players. In the extension, players are allowed to communicate directly and, additionally, send private messages to a simple, detail-free mediator, which, in turn, makes public announcements as a deterministic function of the private messages. The extension captures situations in which people engage in face-to-face communication and can observe the opponent's face during the conversation before choosing actions in some underlying game. We prove that the set of Nash equilibrium payoffs of the extended game approximately coincides with the set of correlated equilibrium payoffs of any underlying game.

In this paper, we formulate an evolutionary multiple access control game with continuousvariable actions and coupled constraints. We characterize equilibria of the game and show that the pure equilibria are Pareto optimal and also resilient to deviations by coalitions of any size, i.e., they are strong equilibria. We use the concepts of price of anarchy and strong price of anarchy to study the performance of the system. The paper also addresses how to select one specific equilibrium solution using the concepts of normalized equilibrium and evolutionarily stable strategies. We examine the long-run behavior of these strategies under several classes of evolutionary game dynamics, such as Brown-von Neumann-Nash dynamics, Smith dynamics and replicator dynamics. In addition, we examine correlated equilibrium for the single-receiver model. Correlated strategies are based on signaling structures before making decisions on rates. We then focus on evolutionary games for hybrid additive white Gaussian noise multiple access channel with multiple users and multiple receivers, where each user chooses a rate and splits it over the receivers. Users have coupled constraints determined by the capacity regions. Building upon the static game, we formulate a system of hybrid evolutionary game dynamics using Gfunction dynamics and Smith dynamics on rate control and channel selection, respectively. We show that the evolving game has an equilibrium and illustrate these dynamics with numerical examples.

In this article, we deal with the resolution of a dynamic game of interconnection between mobile network operators (MNOs) sharing their scarce resources and mobile virtual network operators (MVNOs) lacking the infrastructure but trying to bargain to send as much traffic as possible on the MNOs' networks. Besides consumers' perception of the MVNO's brand is based on a word-of-mouth process; the consumers' cheating probability being an exogeneous parameter. The difficulty arises from the fact that the MVNOs ignore the unused bandwidth volumes on both MNOs' networks and how these latter allocate it between the competitive MVNOs. In fact, to increase their revenues, the MVNOs need to learn as fast as possible the hidden information. Five learning strategies have been tested: (i) tit for tat, (ii) fictitious play, (iii) external regret minimization, (iv) repeated explorationexploitation, (v) internal regret minimization. We check numerically that in general, solely the internal regret minimization strategy reaches a correlated equilibrium. Besides MVNOs' power relation and consumers' cheating probability play a central role in the learning rate and the resulting organization. Indeed either one of the MVNOs learns faster, capturing a larger part of the market while the other has no choice but to follow (leader/ follower), or an implicit cooperation occurs i.e., both MVNOs learn and increase their performances simultaneously (peers).

In this paper we investigate complexity issues related to pure Nash equilibria of strategic games. We show that, even in very restrictive settings, determining whether a game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has a strong Nash equilibrium is St-complete. We then study practically relevant restrictions that lower the complexity. In particular, we are interested in quantitative and qualitative restrictions of the way each player's move depends on moves of other players. We say that a game has small neighborhood if the " utility function for each player depends only on (the actions of) a logarithmically small number of other players, The dependency structure of a game G can he expressed by a graph G(G) or by a hypergraph I-I(G). Among other results, we show that if jC has small neighborhood and if I-I(G) has botmdecl hypertree width (or if G(G) has bounded treewidth), then finding pure Nash and Pareto equilibria is feasible in polynomial time. If the game is graphical, then these problems are LOGCFL-complete and thus in the class _NC~ of highly parallelizable problems.

In this paper we show that some decision problems regarding the computation of Nash equilibria are to be considered particularly hard. Most decision problems regarding Nash equilibria have been shown to be NP-complete. While some NP-complete problems can find an alternative to tractability with the tools of Parameterized Complexity Theory, it is also the case that some classes of problems do not seem to have fixed-parameter tractable algorithms. We show that k-Uniform Nash and k-Minimal Nash support are W [2]-hard. Given a game G=(A,B) and a nonnegative integer k, the k-Uniform Nash problem asks whether G has a uniform Nash equilibrium of size k. The k-Minimal Nash support asks whether G has Nash equilibrium such that the support of each player's Nash strategy has size equal to or less than k. First, we show that k-Uniform Nash (with k as the parameter) is W [2]-hard even when we have 2 players, or fewer than 4 different integer values in the matrices. Second, we illustrate that even in zerosum games k-Minimal Nash support is W [2]-hard (a sample Nash equilibrium in a zero-sum 2-player game can be found in polynomial time (von Stengel 2002)). Thus, it must be the case that other more general decision problems are also W [2]-hard. Therefore, the possible parameters for fixed parameter tractability in those decision problems regarding Nash equilibria seem elusive.

This paper introduces a bottleneck game with finite sets of commuters and departing time slots as an extension of congestion games of Konishi et al. (J Econ Theory 72:225–237, 1997a). After characterizing Nash equilibrium of the game, we provide sufficient conditions for which the equivalence between Nash and strong equilibria holds. Somewhat surprisingly, unlike in congestion games, a Nash equilibrium in pure strategies may often fail to exist, even when players are homogeneous. In contrast, when there is a continuum of atomless players, the existence of a Nash equilibrium and the equivalence between the set of Nash and strong equilibria hold as in congestion games (Konishi et al. 1997a).

In a coarse correlated equilibrium (Moulin and Vial 1978), each player finds it optimal to commit ex ante to the future outcome from a probabilistic correlation device instead of playing any strategy of their own. In this paper, we consider a specific two-person game with unique pure Nash and correlated equilbrium and test the concept of coarse correlated equilibrium with a device which is an equally weighted lottery over three symmetric outcomes in the game including the Nash equilibrium, with higher expected payoff than the Nash payoff (as in Moulin and Vial 1978). We also test an individual choice between a lottery over the same payoffs with equal probabilities and the sure payoff as in the Nash equilibrium of the game. Subjects choose the individual lottery, however, they do not commit to the device in the game and instead coordinate to play the Nash equilibrium. We explain this behaviour as an equilibrium in the game.

We introduce a simple network design game that models how independent selfish agents can build or maintain a large network. In our game every agent has a specific connectivity requirement, i.e. each agent has a set of terminals and wants to build a network in which his terminals are connected. Possible edges in the network have costs and each agent's goal is to pay as little as possible. Determining whether or not a Nash equilibrium exists in this game is NP-complete. However, when the goal of each player is to connect a terminal to a common source, we prove that there is a Nash equilibrium as cheap as the optimal network, and give a polynomial time algorithm to find a (1 + ε)-approximate Nash equilibrium that does not cost much more. For the general connection game we prove that there is a 3-approximate Nash equilibrium that is as cheap as the optimal network, and give an algorithm to find a (4.65 + ε)-approximate Nash equilibrium that does not cost much more.

The probability density function (PDF) of a global measure in a large class of highly correlated systems has been suggested to be of the same functional form. Here, we identify the analytical form of the PDF of one such measure, the order parameter in the low temperature phase of the 2D-XY model. We demonstrate that this function describes the fluctuations of global quantities in other correlated, equilibrium and non-equilibrium systems. These include a coupled rotor model, Ising and percolation models, models of forest fires, sand-piles, avalanches and granular media in a self organized critical state. We discuss the relationship with both Gaussian and extremal statistics.

Consider an information network with threats called attackers; each attacker uses a probability distribution to choose a node of the network to damage. Opponent to the attackers is a protector entity called defender; the defender scans and cleans from attacks some part of the network (in particular, a link), which it chooses independently using its own probability distribution. Each attacker wishes to maximize the probability of escaping its cleaning by the defender; towards a conflicting objective, the defender aims at maximizing the expected number of attackers it catches. We model this network security scenario as a non-cooperative strategic game on graphs. We are interested in its associated Nash equilibria, where no network entity can unilaterally increase its local objective. We obtain the following results: • We obtain an algebraic characterization of (mixed) Nash equilibria. • No (non-trivial) instance of the graph-theoretic game has a pure Nash equilibrium. This is an immediate consequence of some covering properties we prove for the supports of the players in all (mixed) Nash equilibria. • We coin a natural subclass of mixed Nash equilibria, which we call Matching Nash equilibria, for this graph-theoretic game. Matching Nash equilibria are defined by enriching the necessary covering properties we proved with some additional conditions involving other structural parameters of graphs, such as Independent Sets.-We derive a characterization of graphs admitting Matching Nash equilibria. All such graphs have an Expanding Independent Set. The characterization enables a non-deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any such graph.-Bipartite graphs are shown to satisfy the characterization. So, using a polynomial time algorithm to compute a Maximum Matching for a bipartite graph, we obtain, as our main result, a deterministic, polynomial time algorithm to compute a Matching Nash equilibrium for any instance of the game with a bipartite graph.