William H. Sandholm

Potential Games with Continuous Player Sets

Abstract We study potential games with continuous player sets, a class of games characterized by an externality symmetry condition. Examples of these games include random matching games with common payoffs and congestion games. We offer a simple description of equilibria which are locally stable under a broad class of evolutionary dynamics, and prove that behavior converges to Nash equilibrium from all initial conditions. We consider a subclass of potential games in which evolution leads to efficient play. Finally, we show that the games studied here are the limits of convergent sequences of the finite player potential games studied by Monderer and Shapley [22]. Journal of Economic Literature Classification Numbers: C72, C73, D62, R41.

Evolution in games with randomly disturbed payoffs

We consider a simple model of stochastic evolution in population games. In our model, each agent occasionally receives opportunities to update his choice of strategy. When such an opportunity arises, the agent selects a strategy that is currently optimal, but only after his payoffs have been randomly perturbed. We prove that the resulting evolutionary process converges to approximate Nash equilibrium in both the medium run and the long run in three general classes of population games: stable games, potential games, and supermodular games. We conclude by contrasting the evolutionary process studied here with stochastic fictitious play.

Excess payoff dynamics and other well-behaved evolutionary dynamics

Abstract We consider a model of evolution in games in which agents occasionally receive opportunities to switch strategies, choosing between them using a probabilistic rule. Both the rate at which revision opportunities arrive and the probabilities with which each strategy is chosen are functions of current normalized payoffs. We call the aggregate dynamics induced by this model excess payoff dynamics . We show that every excess payoff dynamic is well-behaved : regardless of the underlying game, each excess payoff dynamic admits unique solution trajectories that vary continuously with the initial state, identifies rest points with Nash equilibria, and respects a basic payoff monotonicity property. We show how excess payoff dynamics can be used to construct well-behaved modifications of imitative dynamics, and relate them to two other well-behaved dynamics based on projections.

Pairwise Comparison Dynamics and Evolutionary Foundations for Nash Equilibrium

We introduce a class of evolutionary game dynamics — pairwise comparison dynamics — under which revising agents choose a candidate strategy at random, switching to it with positive probability if and only if its payoff is higher than the agent’s current strategy. We prove that all such dynamics satisfy Nash stationarity : the set of rest points of these dynamics is always identical to the set of Nash equilibria of the underlying game. We also show how one can modify the replicator dynamic and other imitative dynamics to ensure Nash stationarity without increasing the informational demands placed on the agents. These results provide an interpretation of Nash equilibrium that relies on large numbers arguments and weak requirements on payoff observations rather than on strong equilibrium knowledge assumptions.

Pigouvian pricing and stochastic evolutionary implementation

Abstract We study the implementation of efficient behavior in settings with externalities. A planner would like to ensure that a group of agents make socially optimal choices, but he only has limited information about the agents’ preferences, and can only distinguish individual agents through the actions they choose. We describe the agents’ behavior using a stochastic evolutionary model, assuming that their choice probabilities are given by the logit choice rule. We prove that there is a simple price scheme with the following property: regardless of the realization of preferences, a group of agents subjected to the price scheme will spend the vast majority of time in the long run behaving efficiently. The price scheme defines a game that may possess multiple equilibria, but we are able to obtain a unique and efficient selection from this set because of the stochastic nature of the agents’ choice rule. We conclude by comparing the performance of our price scheme with that of VCG mechanisms.

Evolution and equilibrium under inexact information

Abstract We study a general model of stochastic evolution in games, assuming in which players have inexact information about the games payoffs or the population state. We show that when the population is large, its behavior over finite time spans follows an almost deterministic trajectory. While this result provides a useful description of disequilibrium behavior adjustment, it tells us little about equilibrium play. To address this issue, we establish that the equilibrium behavior of a large population can be approximated by a diffusion. We then propose a new notion of stability called local probabilistic stability (LPS), which requires that a population which begins play in equilibrium settle into a fixed stochastic pattern around the equilibrium. We use the diffusion approximation to prove a simple characterization of LPS. While LPS accords closely with standard deterministic notions of stability at interior equilibria, it is significantly less demanding at boundary equilibria.

Survival of dominated strategies under evolutionary dynamics

We prove that any deterministic evolutionary dynamic satisfying four mild requirements fails to eliminate strictly dominated strategies in some games. We also show that existing elimination results for evolutionary dynamics are not robust to small changes in the specifications of the dynamics. Numerical analysis reveals that dominated strategies can persist at nontrivial frequencies even when the level of domination is not small.

Large population potential games

We offer a parsimonious definition of large population potential games, provide some alternate characterizations, and demonstrate the advantages of the new definition over the existing definition, but also show the equivalence of the two definitions.

Robust permanence and impermanence for stochastic replicator dynamics

Garay and Hofbauer (SIAM J. Math. Anal. 34 (2003)) proposed sufficient conditions for robust permanence and impermanence of the deterministic replicator dynamics. We reconsider these conditions in the context of the stochastic replicator dynamics, which is obtained from its deterministic analogue by introducing Brownian perturbations of payoffs. When the deterministic replicator dynamics is permanent and the noise level small, the stochastic dynamics admits a unique ergodic distribution whose mass is concentrated near the maximal interior attractor of the unperturbed system; thus, permanence is robust against small unbounded stochastic perturbations. When the deterministic dynamics is impermanent and the noise level small or large, the stochastic dynamics converges to the boundary of the state space at an exponential rate.

Local stability under evolutionary game dynamics

We prove that any regular evolutionarily stable strategy (ESS) is asymptotically stable under any impartial pairwise comparison dynamic, including the Smith dynamic; under any separable excess payoff dynamic, including the BNN dynamic; and under the best response dynamic. Combined with existing results for imitative dynamics, our analysis validates the use of regular ESS as a blanket sufficient condition for local stability under evolutionary game dynamics.