Stochastic Perturbations of Discontinuous Dynamics with Applications to Game Theory

The Nash equilibrium is a key concept in game theory and in economic theory. However, as a static notion it does not help us to understand how the equilibrium is reached or how an equilibrium is selected when a game has multiple equilibria. Thus, dynamical models that are based on best responding, i.e., models where the players are trying to maximize their payoffs in order the reach an equilibrium, need to be considered. In this work we consider stochastic dynamics based on best replying, both evolutionary ones and more general best-response dynamics, and provide asymptotic results for them, such as a characterization of the stable outcomes of evolutionary games. This work consists of four parts. In the first part we establish the stability properties of evolutionary models in games with perfect information. Such games have a unique backward induction equilibrium, which we show to be the unique evolutionary stable outcome. In the second and third parts we consider the Matching Pennies game, whose unique Nash equilibrium is completely mixed. In particular, in the second part we show that the mixed equilibrium in the Matching Pennies game is the unique evolutionary stable outcome in simple evolutionary models, and in the third part we extend this result, showing that the equilibrium is the unique stable outcome in more general best-response dynamics. Results for this simple game were never obtained before. In the last part we develop approximation tools that enable us to analyze general discontinuous stochastic models, and these tools, too, were not available before. Thus, using the results in this work, we are able to analyze stochastic best-response models in games using known results about the convergence of the deterministic bestresponse dynamic (i.e., fictitious play).