Schedule of the Workshop "Evolutionary Dynamics and Market Behavior"

Mario Bravo: Reinforcement learning with restrictions on the action set Abstract (with Mathieu Faure)

Abstract: Consider a 2-player normal-form game repeated over time. We introduce an adaptive learning procedure, where the players only observe their own realized payoff at each stage. We assume that agents do not know their own payoff function, and have no information on the other player. Furthermore, we assume that they have restrictions on their own action set such that, at each stage, their choice is limited to a subset of their action set. We prove that the empirical distributions of play converge to the set of Nash equilibria for zero-sum and potential games, and games where one player has two actions.

Rida Laraki: Higher Order Game Dynamics
(Joint work with Panayotis Mertikopoulos) https://sites.google.com/site/ridalaraki/

Abstract: Continuous-time game dynamics are typically first order systems where payoffs determine the growth rate of the players' strategy shares. In this paper, we investigate what happens beyond first order by viewing payoffs as higher order forces of change, specifying e.g. the acceleration of the players' evolution instead of its velocity (a viewpoint which emerges naturally when it comes to aggregating empirical data of past instances of play). To that end, we derive a wide class of higher order game dynamics, generalizing first order imitative dynamics, and, in particular, the replicator dynamics. We show that strictly dominated strategies become extinct in $n$-th order payoff-monotonic dynamics $n$ orders as fast as in the corresponding first order dynamics; furthermore, in stark contrast to first order, weakly dominated strategies also become extinct for $n\geq2$. All in all, higher order payoff-monotonic dynamics lead to the elimination of weakly dominated strategies, followed by the iterated deletion of strictly dominated strategies, thus providing a dynamic justification of the well-known epistemic rationalizability process of Dekel and Fudenberg (1990). Finally, we also establish a higher order analogue of the folk theorem and we show that convergence to strict equilibria in $n$-th order dynamics is $n$ orders as fast as in first order.

Panayotis Mertikopoulos: Entropy-driven game dynamics, quantal responses and Hessian Riemannian structures

Abstract: A key property of the replicator dynamics is their equivalence to the exponential weight algorithm in continuous time, i.e. the learning process in which players select an action with probability exponentially proportional to the action's aggregate payoff over time. By considering a more general quantal response framework which extends the logit model above, we derive a new class of entropy-driven game dynamics that is intimately related to the notion of a Hessian Riemannian metric on the simplex (viz. a metric obtained by taking the Hessian of an entropy-like function). In this setting, dominated strategies become extinct and the folk theorem of evolutionary game theory continues to hold, but the rate of extinction of dominated strategies (or of convergence to strict equilibria) changes with the entropy used. In the case of random matching in two player games, we show that time averages converge if the dynamics are permanent, and we establish a sufficient condition for permanence. If there is time, we will also discuss the global properties of these dynamics in potential, contractive and conservative games, and we will explore some links with smooth best-reply dynamics.

Marius Ochea: Heterogenous Heuristics in 3x3 Bimatrix Population Games

Abstract: We numerically investigate population-level evolutionary dynamics resulting from individual-level, adaptive play both under homogenous ( "self-play") and heterogenous ( "mixed play") scenarios. In a class of bimatrix 3x3 normal form games (Sparrow and van Strien 2009, GEB), that includes Rock-Paper-Scissors as a special case, rich limit behavior unfolds as certain heuristics ' parameters are modulated.

Gregory Roth: "stochastic persistence of interacting structured populations" (with Sebastian Schreiber).

Abstract: I will illustrate our result through an evolutionary rock-scissors-paper game with some spatial structure in the population.

Ryoji Sawa: Stochastic stability in large population and small mutation limits for general normal form games

Abstract: We consider models of stochastic evolution in normal form games with K ≥ 2 strategies in which agents employ best response with mutations. Forming the dynamic as a Markov chain with state space being the set of best responses, we overcome the technical difficulty which arises with the large population. We study the long run behavior of the resulting Markov process for four limiting cases: the small noise limit, the large population limit, and the both orders of double limits. We characterize conditions with which selection results of the both orders of double limits coincide.