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Metastability of Logit Dynamics for Coordination Games
, 2012
"... Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a 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. ..."
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Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a 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 longterm 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. In several cases it happens that on a timescale shorter than mixing time the chain is “quasistationary”, meaning that it stays close to some small set of the state space, while in a timescale multiple of the mixing time it jumps from one quasistationaryconfiguration to another; this phenomenon is usually called “metastability”. 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. In particular, we study noriskdominant coordination games on the clique (which is equivalent to the wellknown Glauber dynamics for the Ising model) and coordination games on a ring (both the riskdominant and noriskdominant case). We also describe a simple “artificial” game that highlights the distinctive features of our metastability notion based on distributions.
Decentralized Dynamics for Finite Opinion Games
, 2012
"... Game theory studies situations in which strategic players can modify the state of a given system, due to the absence of a central authority. Solution concepts, such as Nash equilibrium, are defined to predict the outcome of such situations. In the spirit of the field, we study the computation of sol ..."
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Cited by 2 (2 self)
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Game theory studies situations in which strategic players can modify the state of a given system, due to the absence of a central authority. Solution concepts, such as Nash equilibrium, are defined to predict the outcome of such situations. In the spirit of the field, we study the computation of solution concepts by means of decentralized dynamics. These are algorithms in which players move in turns to improve their own utility and the hope is that the system reaches an “equilibrium” quickly. We study these dynamics for the class of opinion games, recently introduced by [1]. These are games, important in economics and sociology, that model the formation of an opinion in a social network. We study bestresponse dynamics and show that the convergence to Nash equilibria is polynomial in the number of players. We also study a noisy version of bestresponse 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.
Logit Dynamics with Concurrent Updates for Local Interaction Games?
"... Abstract. Logit dynamics are a family of randomized best response dynamics based on the logit choice function [21] that is used to model players with limited rationality and knowledge. In this paper we study the alllogit dynamics, where at each time step all players concurrently update their strate ..."
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Cited by 1 (1 self)
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Abstract. Logit dynamics are a family of randomized best response dynamics based on the logit choice function [21] that is used to model players with limited rationality and knowledge. In this paper we study the alllogit dynamics, where at each time step all players concurrently update their strategies according to the logit choice function. In the well studied onelogit dynamics [7] instead at each step only one randomly chosen player is allowed to update. We study properties of the alllogit dynamics in the context of local interaction games, a class of games that has been used to model complex social phenomena [7, 23, 26] and physical systems [19]. In a local interaction game, players are the vertices of a social graph whose edges are twoplayer potential games. Each player picks one strategy to be played for all the games she is involved in and the payoff of the player is the (weighted) sum of the payoffs from each of the games. We prove that local interaction games characterize the class of games for which the alllogit dynamics are reversible. We then compare the stationary behavior of onelogit and alllogit dynamics. Specifically, we look at the expected value of a notable class of observables, that we call decomposable observables. 1
Logit Dynamics: A Model for Bounded Rationality DIODATO FERRAIOLI
"... We describe logit dynamics, which are used to model bounded rationality in games, and their related equilibrium concept, the logit equilibrium. We also present some results about the convergence time of these dynamics and introduce a suitable approximation of the logit equilibrium. We conclude by de ..."
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We describe logit dynamics, which are used to model bounded rationality in games, and their related equilibrium concept, the logit equilibrium. We also present some results about the convergence time of these dynamics and introduce a suitable approximation of the logit equilibrium. We conclude by describing some interesting future extensions to logit dynamics.
Imperfect bestresponse mechanisms ∗
, 1208
"... Bestresponse mechanisms (Nisan, Schapira, Valiant, Zohar, 2011) provide a unifying framework for studying various distributed protocols in which the participants are instructed to repeatedly best respond to each others ’ strategies. Two fundamental features of these mechanisms are convergence and i ..."
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Bestresponse mechanisms (Nisan, Schapira, Valiant, Zohar, 2011) provide a unifying framework for studying various distributed protocols in which the participants are instructed to repeatedly best respond to each others ’ strategies. Two fundamental features of these mechanisms are convergence and incentive compatibility. This work investigates convergence and incentive compatibility conditions of such mechanisms when players are not guaranteed to always best respond but they rather play an imperfect bestresponse strategy. That is, at every time step every player deviates from the prescribed bestresponse strategy according to some probability parameter. The results explain to what extent convergence and incentive compatibility depend on the assumption that players never make mistakes, and how robust such protocols are to “noise ” or “mistakes”. 1
GRAPHICAL POTENTIAL GAMES
"... Abstract. We study the class of potential games that are also graphical games with respect to a given graph G of connections between the players. We show that, up to strategic equivalence, this class of games can be identified with the set of Markov random fields on G. From this characterization, an ..."
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Abstract. We study the class of potential games that are also graphical games with respect to a given graph G of connections between the players. We show that, up to strategic equivalence, this class of games can be identified with the set of Markov random fields on G. From this characterization, and from the HammersleyClifford theorem, it follows that the potentials of such games can be decomposed to local potentials. We use this decomposition to strongly bound the lengths of strict betterresponse paths. This result extends to generalized graphical potential games, which are played
Metastability of Logit Dynamics for Coordination
"... Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a 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. ..."
Abstract
 Add to MetaCart
(Show Context)
Logit Dynamics [Blume, Games and Economic Behavior, 1993] is a 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 longterm 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. In several cases it happens that on a timescale shorter than mixing time the chain is “quasistationary”, meaning that it stays close to some small set of the state space, while in a timescale multiple of the mixing time it jumps from one quasistationary configuration to another; this phenomenon is usually called “metastability”. 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. In particular, we study noriskdominant coordination games on the clique (which is equivalent to the wellknown Glauber dynamics for the Ising model) and coordination games on a ring (both the riskdominant and noriskdominant case). We also describe a simple “artificial ” game that highlights the distinctive features of our metastability notion based on distributions. 1 1
Metastability of Asymptotically WellBehaved 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, ..."
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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. [4]. These are distributions that remain stable (i.e., players do not go too far from it) for a superpolynomial 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 wellbehaved, 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.