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)

Deterministic Monotone Algorithms for Scheduling on Related Machines

We consider the problem of designing monotone deterministic algorithms for scheduling tasks on related machines in order to minimize the makespan. Several recent papers showed that monotonicity is a fundamental property to design truthful mechanisms for this scheduling problem. We give both theoretical and experimental results. First of all we consider the case of two machines when speeds of the machines are restricted to be powers of a given constant c>0. We prove that algorithm Largest Processing Time (LPT) is monotone for any c≥2 while it is not monotone for c≤1.78; algorithm List Scheduling (LS), instead, is monotone only for c>2. In the case of m>2 machines we restrict our attention to the class of “greedy-like” monotone algorithms defined in [Vincenzo Auletta, Roberto De Prisco, Paolo Penna, Giuseppe Persiano, Deterministic truthful approximation mechanisms for scheduling related machines, in: Proceedings of 21st Annual Symposium on Theoretical Aspects of Computer Science. STACS ’04, in: Lecture Notes in Computer Science, vol. 2996, Springer, 2004, pp. 608–619]. It has been shown that greedy-like monotone algorithms can be used to design a family of 2+ε-approximate truthful mechanisms. In particular, in [Vincenzo Auletta, Roberto De Prisco, Paolo Penna, Giuseppe Persiano, Deterministic truthful approximation mechanisms for scheduling related machines, in: Proceedings of 21st Annual Symposium on Theoretical Aspects of Computer Science. STACS ’04, in: Lecture Notes in Computer Science, vol. 2996, Springer, 2004, pp. 608–619], the greedy-like algorithm Uniform is proposed and it is proved that it is monotone when machine speeds are powers of a given integer constant c>0. In this paper we propose a new algorithm, called Uniform_RR, that is still monotone when speeds are powers of a given integer constant c>0 and we prove that its approximation factor is not worse than that of Uniform. We also experimentally compare the performance of Uniform, Uniform_RR, LPT, and several other monotone and greedy-like heuristics

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

Solution of Ulam's Problem on Binary Search with Four Lies

In this paper we determine the minimal number of yes-no queries needed to find an unknown integer between 1 and 1000000 if at most four of the answers may be erroneous

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

The power of verification for one-parameter agents

We initiate the study of mechanisms with verification for one-parameter agents. We give an algorithmic characterization of such mechanisms and show that they are provably better than mechanisms without verification, i.e., those previously considered in the literature. These results are obtained for a number of optimization problems motivated by the Internet and recently studied in the algorithmic mechanism design literature. The characterization can be regarded as an alternative approach to existing techniques to design truthful mechanisms. The construction of such mechanisms reduces to the construction of an algorithm satisfying certain “monotonicity” conditions which, for the case of verification, are much less stringent. In other words, verification makes the construction easier and the algorithm more efficient (both computationally and in terms of approximability)

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 β describes the behavior of a complex system whose individual components act selfishly 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 parameter β. 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 bounded by an exponential in β and in the maximum potential difference. Instead, for games with dominant strategies the mixing time cannot grow arbitrarily with β. Finally, we refine our analysis for a subclass of potential games called graphical coordination games, often used for modeling the diffusion of new technologies. 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 β. Moreover, we consider two specific and popular network topologies, the clique and the ring. For the clique, we prove an almost matching lower bound on the mixing time of the logit dynamics that is exponential in β and in the maximum potential difference, while for the ring we prove that the time of convergence of the logit dynamics to its stationary distribution is significantly shorter