Mathias Staudigl

Stochastic stability in asymmetric binary choice coordination games

A recent literature in evolutionary game theory is devoted to the question of robust equilibrium selection under noisy best-response dynamics. In this paper we present a complete picture of equilibrium selection for asymmetric binary choice coordination games in the small noise limit. We achieve this by transforming the stochastic stability analysis into an optimal control problem, which can be solved analytically. This approach allows us to obtain precise and clean equilibrium selection results for all canonical noisy best-response dynamics which have been proposed so far in the literature, among which we find the best-response with mutations dynamics, the logit dynamics and the probit dynamics.

Evolution of Social Networks

Modeling the evolution of networks is central to our understanding of large communication systems, and more general, modern economic and social systems. The research on social and economic networks is truly interdisciplinary and the number of proposed models is huge. In this survey we discuss a small selection of modeling approaches, covering classical random graph models, and game-theoretic models to analyze the evolution of social networks. Based on these two basic modeling paradigms, we introduce co-evolutionary models of networks and play as a potential synthesis.

Constrained interactions and social coordination

We consider a co-evolutionary model of social coordination and network formation where agents may decide on an action in a 2x2 - coordination game and on whom to establish costly links to. We fi nd that a payo ff dominant convention is selected for a wider parameter range when agents may only support a limited number of links as compared to a scenario where agents are not constrained in their linking choice. The main reason behind this result is that under constrained interactions agents face a trade-off between the links they have and those they would rather have.

Co-evolutionary dynamics and Bayesian interaction games

Recently there has been a growing interest in evolutionary models of play with endogenous interaction structure. We call such processes co-evolutionary dynamics of networks and play. We study a co-evolutionary process of networks and play in settings where players have diverse preferences. In the class of potential games we provide a closed-form solution for the unique invariant distribution of this process. Based on this result we derive various asymptotic statistics generated by the co-evolutionary process. We give a complete characterization of the random graph model, and stochastically stable states in the small noise limit. Thereby we can select among action profiles and networks which appear jointly with non-vanishing frequency in the limit of small noise in the population. We further study stochastic stability in the limit of large player populations.

Sample Path Large Deviations for Stochastic Evolutionary Game Dynamics

We study a model of stochastic evolutionary game dynamics in which the probabilities that agents choose suboptimal actions are dependent on payoff consequences. We prove a sample path large deviation principle, characterizing the rate of decay of the probability that the sample path of the evolutionary process lies in a prespecified set as the population size approaches infinity. We use these results to describe excursion rates and stationary distribution asymptotics in settings where the mean dynamic admits a globally attracting state, and we compute these rates explicitly for the case of logit choice in potential games.

Large Deviations and Stochastic Stability in the Small Noise Double Limit, I: Theory

We consider a model of stochastic evolution under general noisy best response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. Our analysis focuses on behavior in the small noise double limit: we first take the noise level in agents’ decisions to zero, and then take the population size to infinity. We show that in this double limit, escape from and transitions between equilibria can be described in terms of solutions to continuous optimal control problems. These are used in turn to characterize the asymptotics of the the stationary distribution, and so to determine the stochastically stable states. The control problems are tractable in certain interesting cases, allowing analytical descriptions of the escape dynamics and long run behavior of the stochastic evolutionary process.

Large Deviations and Stochastic Stability in the Small Noise Double Limit, II: The Logit Model

We describe the large deviations properties, stationary distribution asymptotics, and stochastically stable states of stochastic evolutionary processes based on the logit choice rule, focusing on behavior in the small noise double limit. These aspects of the stochastic evolutionary process can be characterized in terms of solutions to certain minimum cost path problems. We solve these problems explicitly using tools from optimal control theory. The analysis focuses on three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, but our approach can be applied to other classes of games and other choice rules.

Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities

We examine a class of stochastic mirror descent dynamics in the context of monotone variational inequalities (including Nash equilibrium and saddle-point problems). The dynamics under study are formulated as a stochastic differential equation, driven by a (single-valued) monotone operator and perturbed by a Brownian motion. The system’s controllable parameters are two variable weight sequences, that, respectively, pre- and post-multiply the driver of the process. By carefully tuning these parameters, we obtain global convergence in the ergodic sense, and we estimate the average rate of convergence of the process. We also establish a large deviations principle, showing that individual trajectories exhibit exponential concentration around this average.

A Limit Theorem for Markov Decision Processes

In this paper we prove a deterministic approximation theorem for a sequence of Markov decision processes with finitely many actions and general state spaces as they appear frequently in economics, game theory and operations research. Using viscosity solution methods no a-priori differentiabililty assumptions are imposed on the value function. Applications for this result can be found in large deviation theory, and some simple economic problems.