Metastability of Logit Dynamics for Coordination Games

Logit Dynamics [Blume, Games and Economic Behavior, 1993] are randomized best response dynamics for strategic games: at every time step a player is selected uniformly at random and she chooses a new strategy according to a probability distribution biased toward strategies promising higher payoffs. This process defines an ergodic Markov chain, over the set of strategy profiles of the game, whose unique stationary distribution is the long-term equilibrium concept for the game. However, when the mixing time of the chain is large (e.g., exponential in the number of players), the stationary distribution loses its appeal as equilibrium concept, and the transient phase of the Markov chain becomes important. It can happen that the chain is "metastable", i.e., on a time-scale shorter than the mixing time, it stays close to some probability distribution over the state space, while in a time-scale multiple of the mixing time it jumps from one distribution to another. In this paper we give a quantitative definition of "metastable probability distributions" for a Markov chain and we study the metastability of the logit dynamics for some classes of coordination games. We first consider a pure $n$-player coordination game that highlights the distinctive features of our metastability notion based on distributions. Then, we study coordination games on the clique without a risk-dominant strategy (which are equivalent to the well-known Glauber dynamics for the Curie-Weiss model) and coordination games on a ring (both with and without risk-dominant strategy)

Metastability of Asymptotically Well-Behaved Potential Games

One of the main criticisms to game theory concerns the assumption of full rationality. Logit dynamics is a decentralized algorithm in which a level of irrationality (a.k.a. "noise") is introduced in players' behavior. In this context, the solution concept of interest becomes the logit equilibrium, as opposed to Nash equilibria. Logit equilibria are distributions over strategy profiles that possess several nice properties, including existence and uniqueness. However, there are games in which their computation may take time exponential in the number of players. We therefore look at an approximate version of logit equilibria, called metastable distributions, introduced by Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e., players do not go too far from it) for a super-polynomial number of steps (rather than forever, as for logit equilibria). The hope is that these distributions exist and can be reached quickly by logit dynamics. We identify a class of potential games, called asymptotically well-behaved, for which the behavior of the logit dynamics is not chaotic as the number of players increases so to guarantee meaningful asymptotic results. We prove that any such game admits distributions which are metastable no matter the level of noise present in the system, and the starting profile of the dynamics. These distributions can be quickly reached if the rationality level is not too big when compared to the inverse of the maximum difference in potential. Our proofs build on results which may be of independent interest, including some spectral characterizations of the transition matrix defined by logit dynamics for generic games and the relationship of several convergence measures for Markov chains

Metastability of the Logit Dynamics for Asymptotically Well-Behaved Potential Games

Convergence rate and stability of a solution concept are classically measured in terms of “even- tually” and “forever”, respectively. In the wake of recent computational criticisms to this approach, we study whether these time frames can be updated to have states computed “quickly” and stable for “long enough”. Logit dynamics allows irrationality in players’ behavior, and may take time exponential in the number of players n to converge to a stable state (i.e., a certain distribution over pure strategy pro- files). We prove that every potential game, for which the behavior of the logit dynamics is not chaotic as n increases, admits distributions stable for a super-polynomial number of steps in n no matter the players’ irrationality, and the starting profile of the dynamics. The convergence rate to these metastable distributions is polynomial in n when the players are not too rational. Our proofs build upon the new concept of partitioned Markov chains, that might be of indepen- dent interest, and a number of involved technical contributions

Convergence to Equilibrium of Logit Dynamics for Strategic Games

We present the first general bounds on the mixing time of the Markov chain associated to the logit dynamics for wide classes of strategic games. The logit dynamics with inverse noise beta describes the behavior of a complex system whose individual components act selfishly and keep responding according to some partial ("noisy") knowledge of the system, where the capacity of the agent to know the system and compute her best move is measured by the inverse of the parameter beta. In particular, we prove nearly tight bounds for potential games and games with dominant strategies. Our results show that, for potential games, the mixing time is upper and lower bounded by an exponential in the inverse of the noise and in the maximum potential difference. Instead, for games with dominant strategies, the mixing time cannot grow arbitrarily with the inverse of the noise. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, a class of games that have been previously studied in Physics and, more recently, in Computer Science in the context of diffusion of new technologies. We give evidence that the mixing time of the logit dynamics for these games strongly depends on the structure of the underlying graph. We prove that the mixing time of the logit dynamics for these games can be upper bounded by a function that is exponential in the cutwidth of the underlying graph and in the inverse of noise. Moreover, we consider two specific and popular network topologies, the clique and the ring. For games played on a clique we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in the inverse of the noise and in the maximum potential difference, while for games played on a ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter

Logit dynamics for strategic games mixing time and metastability

2010 - 2011A complex system is generally de_ned as a system emerging from the interaction of several and di_erent components, each one with their properties and their goals, usually subject to external inuences. Nowadays, complex systems are ubiquitous and they are found in many research areas: examples can be found in Economy (e.g., markets), Physics (e.g., ideal gases, spin systems), Biology (e.g., evolution of life) and Computer Science (e.g., Internet and social networks). Modeling complex systems, understanding how they evolve and predicting the future status of a complex system are major research endeavors. Historically, physicists, economists, sociologists and biologists have separately studied complex systems, developing their own tools that, however, often are not suitable for being adopted in di_erent areas. Recently, the close relation between phenomena in di_erent research areas has been highlighted. Hence, the aim is to have a powerful tool that is able to give us insight both about Nature and about Society, an universal language spoken both in natural and in social sciences, a modern code of nature. In a recent book [16], Tom Siegfried pointed out game theory as such a powerful tool, able to embrace complex systems in Economics [3, 4, 5], Biology [13], Physics [8], Computer Science [10, 11], Sociology [12] and many other disciplines. Game theory deals with sel_sh agents or players, each with a set of possible actions or strategies. An agent chooses a strategy evaluating her utility or payo_ that does not depend only on agent's own strategy, but also on the strategies played by the other players. The way players update their strategies in response to changes generated by other players de_nes the dynamics of the game and describes how the game evolves. If the game eventually reaches a _xed point, i.e., a state stable under the dynamics considered, then it is said that the game is in an equilibrium, through which we can make predictions about the future status of a game. The classical game theory approach assumes that players have complete knowledge about the game and they are always able to select the strategy that maximizes their utility: in this rational setting, the evolution of a system is modeled by best response dynamics and predictions can be done by looking at well-known Nash equilibrium. Another approach is followed by learning dynamics: here, players are supposed to \learn" how to play in the next rounds by analyzing the history of previous plays. By examining the features and the drawbacks of these dynamics, we can detect the basic requirements to model the evolution of complex systems and to predict their future status. Usually, in these systems, environmental factors can inuence the way each agent selects her own strategy: for example, the temperature and the pressure play a fundamental role in the dynamics of particle systems, whereas the limited computational power is the main inuence in computer and social settings. Moreover, as already pointed by Harsanyi and Selten [9], the complete knowledge assumption can fail due to limited information about external factors that could inuence the game (e.g., if it will rain tomorrow), or about the attitude of other players (if they are risk taking), or about the amount of knowledge available to other players. Equilibria are usually used to make predictions about the future status of a game: for this reason, we like that an equilibrium always exists and that the game converges to it. Moreover, in case that multiple equilibria exist, we like to know which equilibrium will be selected, otherwise we could make wrong predictions. Finally, if the dynamics takes too long time to reach its _xed point, then this equilibrium cannot be taken to describe the state of the players, unless we are willing to wait super-polynomially long transient time. Thus we would like to have dynamics that models bounded rationality and induces an equilibrium that always exists, it is unique and is quickly reached. Logit dynamics, introduced by Blume [6], models a noisy-rational behavior in a clean and tractable way. In the logit dynamics for a game, at each time step, a player is randomly selected for strategy update and the update is performed with respect to an inverse noise parameter _ (that represents the degree of rationality or knowledge) and of the state of the system, that is the strategies currently played by the players. Intuitively, a low value of _ represents the situation where players choose their strategies \nearly at random" because they are subject to strong noise or they have very limited knowledge of the game; instead, an high value of _ represents the situation where players \almost surely" play the best response, that is, they pick the strategies yielding high payo_ with higher probability. This model is similar to the one used by physicists to describe particle systems, where the behavior of each particle is inuenced by temperature: here, low temperature means high rationality and high temperature means low rationality. It is well known [6] that this dynamics de_nes an ergodic _nite Markov chain over the set of strategy pro_les of the game, and thus it is known that a stationary distribution always exists, it is unique and the chain converges to such distribution, independently of the starting pro_le. Since the logit dynamics models bounded rationality in a clean and tractable way, several works have been devoted to this subject. Early works about this dynamics have focused about long-term behavior of the dynamics: Blume [6] showed that, for 2 _ 2 coordination games and potential games, the long-term behavior of the system is concentrated in a speci_c Nash equilibrium; Al_os-Ferrer and Netzer [1] gave a general characterization of long term behavior of logit dynamics for wider classes of games. A lot of works have been devoted to evaluating the time that the dynamics takes to reach speci_c Nash equilibria of a game, called hitting time: Ellison [7] considered logit dynamics for graphical coordination games on cliques and rings; Peyton Young [15] extended this work for more general families of graphs; Montanari and Saberi [14] gave the exact graph theoretic property of the underlying interaction network that characterizes the hitting time in graphical coordination games; Asadpour and Saberi [2] studied the hitting time for a class of congestion games. Our approach is di_erent: indeed, our _rst contribution is to propose the stationary distribution of the logit dynamics Markov chain as a new equilibrium concept in game theory. Our new solution concept, sometimes called logit equilibrium, always exists, it is unique and the game converges to it from any starting point. Instead, previous works only take in account the classical equilibrium concept of Nash equilibrium, that it is known to not satisfying all the requested properties. Moreover, the approach of previous works forces to consider only speci_c values of the rationality parameter, whereas we are interested to analyze the behavior of the system for each value of _. In order to validate the logit equilibrium concept we follow two di_erent lines of research: from one hand we evaluate the performance of a system when it reaches this equilibrium; on the other hand we look for bounds to the time that the dynamics takes to reach this equi- librium, namely the mixing time. This approach is trained on some simple but interesting games, such as 2_2 coordination games, congestion games and two team games (i.e., games where every player has the same utility). Then, we give bounds to the convergence time of the logit dynamics for very interesting classes of games, such as potential games, games with dominant strategies and graphical coordination games. Speci_cally, we prove a twofold behavior of the mixing time: there are games for which it exponentially depends on _, whereas for other games there exists a function independent of _ such that the mixing time is always bounded by this function. Unfortunately, we show also that there are games where the mixing time can be exponential in the number of players. When the mixing is slow, in order to describe the future status of the system through the logit equilibrium, we need to wait a long transient phase. But in this case, it is natural to ask if we can make predictions about the future status of the game even if the equilibrium has not been reached yet. In order to answer this question we introduce the concept of metastable distribution, a probability distribution such that the dynamics quickly reaches it and spends a lot of time therein: we show that there are graphical coordination games where there are some distributions such that for almost every starting pro_le the logit dynamics rapidly converges to one of these distributions and remains close to it for an huge number of steps. In this way, even if the logit equilibrium is no longer a meaningful description of the future status of a game, the metastable distributions resort the predictive power of the logit dynamics. References [1] Carlos Al_os-Ferrer and Nick Netzer. The logit-response dynamics. Games and Economic Behavior, 68(2):413 { 427, 2010. [2] Arash Asadpour and Amin Saberi. On the ine_ciency ratio of stable equilibria in congestion games. In Proc. of the 5th International Workshop on Internet and Network Economics (WINE'09), volume 5929 of Lecture Notes in Computer Science, pages 545{ 552. Springer, 2009. [3] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 1. Elsevier, 1992. [4] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 2. Elsevier, 1994. [5] Robert J. Aumann and S. Hart, editors. Handbook of Game Theory with Economic Applications, volume 3. Elsevier, 2002. [6] Lawrence E. Blume. The statistical mechanics of strategic interaction. Games and Economic Behavior, 5:387{424, 1993. [7] Glenn Ellison. Learning, local interaction, and coordination. Econometrica, 61(5):1047{ 1071, 1993. [8] Serge Galam and Bernard Walliser. Ising model versus normal form game. Physica A: Statistical Mechanics and its Applications, 389(3):481 { 489, 2010. [9] John C. Harsanyi and Reinhard Selten. A General Theory of Equilibrium Selection in Games. MIT Press, 1988. [10] Elias Koutsoupias and Christos H. Papadimitriou. Worst-case equilibria. Computer Science Review, 3(2):65{69, 2009. Preliminary version in STACS 1999. [11] Hagay Levin, Michael Schapira, and Aviv Zohar. Interdomain routing and games. In STOC, pages 57{66, 2008. [12] Jan Lorenz, Heiko Rauhut, Frank Schweitzer, and Dirk Helbing. How social inuence can undermine the wisdom of crowd e_ect. Proceedings of the National Academy of Sciences, 108(22):9020{9025, 2011. [13] John Maynard Smith. Evolution and the theory of games. Cambridge University Press, 1982. [14] Andrea Montanari and Amin Saberi. Convergence to equilibrium in local interaction games. In Proc. of the 50th Annual Symposium on Foundations of Computer Science (FOCS'09). IEEE, 2009. [15] Hobart Peyton Young. The di_usion of innovations in social networks, chapter in \The Economy as a Complex Evolving System", vol. III, Lawrence E. Blume and Steven N. Durlauf, eds. Oxford University Press, 2003. [16] Tom Siegfried. A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. Joseph Henry Press, 1st ed edition, 2006. [edited by author]X n.s

Decentralized Dynamics for Finite Opinion Games

Game theory studies situations in which strategic players can modify the state of a given system, in the absence of a central authority. Solution concepts, such as Nash equilibrium, have been defined in order to predict the outcome of such situations. In multi-player settings, it has been pointed out that to be realistic, a solution concept should be obtainable via processes that are decentralized and reasonably simple. Accordingly we look at the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to decrease their own cost and the hope is that the system reaches an “equilibrium” quickly. We study these dynamics for the class of opinion games, recently introduced by Bindel et al. [FOCS, 2011]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study best-response dynamics and show upper and lower bounds on the convergence to Nash equilibria. We also study a noisy version of best-response dynamics, called logit dynamics, and prove a host of results about its convergence rate as the noise in the system varies. To get these results, we use a variety of techniques developed to bound the mixing time of Markov chains, including coupling, spectral characterizations and bottleneck ratio

A Markov state modelling approach to characterizing the punctuated equilibrium dynamics of stochastic evolutionary games

Stochastic evolutionary games often share a dynamic property called punctuated equilibrium; this means that their sample paths exhibit long periods of stasis near one population state which are infrequently interrupted by switching events after which the sample paths stay close to a different population state, again for a long period of time. This has been described in the literature as a favorable property of stochastic evolutionary games. The methods used so far in stochastic evolutionary game theory, however, do not fully characterize these dynamics. We present an approach that aims at exposing the punctuated equilibrium dynamics by constructing Markov models on a reduced state space which approximate well this dynamic behavior. Besides having good approximation properties, the approach allows a simulation-based algorithm, which is appealing in the case of complex games