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

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)

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

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

Belief-Invariant and Quantum Equilibria in Games of Incomplete Information

Drawing on ideas from game theory and quantum physics, we investigate nonlocal correlations from the point of view of equilibria in games of incomplete information. These equilibria can be classified in decreasing power as general communication equilibria, belief-invariant equilibria and correlated equilibria, all of which contain the familiar Nash equilibria. The notion of belief-invariant equilibrium has appeared in game theory before, in the 1990s. However, the class of non-signalling correlations associated to belief-invariance arose naturally already in the 1980s in the foundations of quantum mechanics. Here, we explain and unify these two origins of the idea and study the above classes of equilibria, and furthermore quantum correlated equilibria, using tools from quantum information but the language of game theory. We present a general framework of belief-invariant communication equilibria, which contains (quantum) correlated equilibria as special cases. It also contains the theory of Bell inequalities, a question of intense interest in quantum mechanics, and quantum games where players have conflicting interests, a recent topic in physics. We then use our framework to show new results related to social welfare. Namely, we exhibit a game where belief-invariance is socially better than correlated equilibria, and one where all non-belief-invariant equilibria are socially suboptimal. Then, we show that in some cases optimal social welfare is achieved by quantum correlations, which do not need an informed mediator to be implemented. Furthermore, we illustrate potential practical applications: for instance, situations where competing companies can correlate without exposing their trade secrets, or where privacy-preserving advice reduces congestion in a network. Along the way, we highlight open questions on the interplay between quantum information, cryptography, and game theory

A Mechanism Design Approach to Measure Awareness

In this paper, we study protocols that allow to discern conscious and unconscious decisions of human beings; i.e., protocols that measure awareness. Consciousness is a central research theme in Neuroscience and AI, which remains, to date, an obscure phenomenon of human brains. Our starting point is a recent experiment, called Post Decision Wagering (PDW) (Persaud, McLeod, and Cowey 2007), that attempts to align experimenters' and subjects' objectives by leveraging financial incentives. We note a similarity with mechanism design, a research area which aims at the design of protocols that reconcile often divergent objectives through incentive-compatibility. We look at the issue of measuring awareness from this perspective. We abstract the setting underlying the PDW experiment and identify three factors that could make it ineffective: rationality, risk attitude and bias of subjects. Using mechanism design tools, we study the barrier between possibility and impossibility of incentive compatibility with respect to the aforementioned characteristics of subjects. We complete this study by showing how to use our mechanisms to potentially get a better understanding of consciousness

Competitive Influence in Social Networks: Convergence, Submodularity, and Competition Effects

In the last 10 years, a vast amount of scientific literature has studied the problem of influence maximization. Yet, only very recently have scientists started considering the more realistic case in which competing entities try to expand their market and maximize their share via viral marketing. Goyal and Kearns [STOC 2012] present a model for the diffusion of two competing alternatives in a social network, which consists of two phases: one for the activation, in which nodes choose whether to adopt any of the two alternatives or none of them, and one for the selection, which is for choosing which of the two alternatives to adopt. In this work we consider this two-phase model, by composing some of the most known dynamics (threshold, voter, and logit models), and we ask the following questions: (1) How is the stationary distribution of the composition of these dynamics related to those of the single composing dynamics? (2) Does the number of adopters of one of the alternatives increase in a monotone and submodular way with respect to the set of initial adopters of that alternative? (3) To what extent does the competition among alternatives affect the total number of agents adopting one of the alternatives

Two-way Greedy: Algorithms for Imperfect Rationality

The realization that selfish interests need to be accounted for in the design of algorithms has produced many contributions in computer science under the umbrella of algorithmic mechanism design. Novel algorithmic properties and paradigms have been identified and studied. Our work stems from the observation that selfishness is different from rationality; agents will attempt to strategize whenever they perceive it to be convenient according to their imperfect rationality. Recent work has focused on a particular notion of imperfect rationality, namely absence of contingent reasoning skills, and defined obvious strategyproofness (OSP) as a way to deal with the selfishness of these agents. Essentially, this definition states that to care for the incentives of these agents, we need not only pay attention about the relationship between input and output, but also about the way the algorithm is run. However, it is not clear what algorithmic approaches must be used for OSP. In this paper, we show that, for binary allocation problems, OSP is fully captured by a combination of two well-known algorithmic techniques: forward and reverse greedy. We call two-way greedy this algorithmic design paradigm. Our main technical contribution establishes the connection between OSP and two-way greedy. We build upon the recently introduced cycle monotonicity technique for OSP. By means of novel structural properties of cycles and queries of OSP mechanisms, we fully characterize these mechanisms in terms of extremal implementations. These are protocols that ask each agent to consistently separate one extreme of their domain at the current history from the rest. Through the connection with the greedy paradigm, we are able to import a host of approximation bounds to OSP and strengthen the strategic properties of this family of algorithms. Finally, we begin exploring the power of two-way greedy for set systems

Election Manipulation in Social Networks with Single-Peaked Agents

Several elections run in the last years have been characterized by attempts to manipulate the result of the election through the diffusion of fake or malicious news over social networks. This problem has been recognized as a critical issue for the robustness of our democracy. Analyzing and understanding how such manipulations may occur is crucial to the design of effective countermeasures to these practices. Many studies have observed that, in general, to design an optimal manipulation is usually a computationally hard task. Nevertheless, literature on bribery in voting and election manipulation has frequently observed that most hardness results melt down when one focuses on the setting of (nearly) single-peaked agents, i.e., when each voter has a preferred candidate (usually, the one closer to her own belief) and preferences of remaining candidates are inversely proportional to the distance between the candidate position and the voter's belief. Unfortunately, no such analysis has been done for election manipulations run in social networks. In this work, we try to close this gap: specifically, we consider a setting for election manipulation that naturally raises (nearly) single-peaked preferences, and we evaluate the complexity of election manipulation problem in this setting: while most of the hardness and approximation results still hold, we will show that single-peaked preferences allow to design simple, efficient and effective heuristics for election manipulation