Diodato Ferraioli

Metastability of logit dynamics for coordination games

Logit dynamics (Blume in Games Econ Behav 5:387–424, 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 with Concurrent Updates for Local Interaction Games

Game Theory is the main tool used to model the behavior of agents that are guided by their own objective in contexts where their gains depend also on the choices made by neighboring agents. Game theoretic approaches have been often proposed for modeling phenomena in a complex social network, such as the formation of the social network itself. We are interested in the dynamics that govern such phenomena. In this paper, we study a specific class of randomized update rules called the logit choice function which can be coupled with different selection rules so to give different dynamics. We study how the logit choice function behave in an extreme case of concurrency.

Mixing Time and Stationary Expected Social Welfare of Logit Dynamics

We study logit dynamics (Blume in Games Econ. Behav. 5:387–424, 1993) for strategic games. This dynamics works as follows: at every stage of the game a player is selected uniformly at random and she plays according to a noisy best-response where the noise level is tuned by a parameter β. Such a dynamics defines a family of ergodic Markov chains, indexed by β, over the set of strategy profiles. We believe that the stationary distribution of these Markov chains gives a meaningful description of the long-term behavior for systems whose agents are not completely rational.Our aim is twofold: On the one hand, we are interested in evaluating the performance of the game at equilibrium, i.e. the expected social welfare when the strategy profiles are random according to the stationary distribution. On the other hand, we want to estimate how long it takes, for a system starting at an arbitrary profile and running the logit dynamics, to get close to its stationary distribution; i.e., the mixing time of the chain.In this paper we study the stationary expected social welfare for the 3-player CK game (Christodoulou and Koutsoupias in Proc. of the 37th Annual ACM Symposium on Theory of Computing (STOC’05), pp. 67–73, ACM, New York, 2005), for 2-player coordination games, and for two simple n-player games. For all these games, we also give almost tight upper and lower bounds on the mixing time of logit dynamics. Our results show two different behaviors: in some games the mixing time depends exponentially on β, while for other games it can be upper bounded by a function independent of β.

Sequential Posted-Price Mechanisms with Correlated Valuations

We study the revenue performance of sequential posted-price mechanisms and some natural extensions for a setting where the valuations of the buyers are drawn from a correlated distribution. Sequential posted-price mechanisms are conceptually simple mechanisms that work by proposing a “take-it-or-leave-it” offer to each buyer. We apply sequential posted-price mechanisms to single-parameter multiunit settings in which each buyer demands only one item and the mechanism can assign the service to at most k of the buyers. For standard sequential posted-price mechanisms, we prove that with the valuation distribution having finite support, no sequential posted-price mechanism can extract a constant fraction of the optimal expected revenue, even with unlimited supply. We extend this result to the case of a continuous valuation distribution when various standard assumptions hold simultaneously (i.e., everywhere-supported, continuous, symmetric, and normalized (conditional) distributions that satisfy regularity, the MHR condition, and affiliation). In fact, it turns out that the best fraction of the optimal revenue that is extractable by a sequential posted-price mechanism is proportional to the ratio of the highest and lowest possible valuation. We prove that a simple generalization of these mechanisms achieves a better revenue performance; namely, if the sequential posted-price mechanism has for each buyer the option of either proposing an offer or asking the buyer for its valuation, then a Ω (1/max { 1,d}) fraction of the optimal revenue can be extracted, where d denotes the degree of dependence of the valuations, ranging from complete independence (d=0) to arbitrary dependence (d = n-1).

Logit dynamics: a model for bounded rationality

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.

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

Social Pressure in Opinion Games

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Imperfect Best-Response Mechanisms

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