Long-run analysis of the stochastic replicator dynamics in the presence of random jumps

Abstract

A further generalization of the stochastic replicator dynamic derived by Fudenberg and Harris [12] is considered. In particular, a Poissonian integral is introduced to the fitness to simulate the affects of anomalous events. For the two strategy population, an estimation of the long run behavior of the dynamic is derived. For the population with many strategies, conditions for stability to pure strict Nash equilibria, extinction of dominated pure strategies, and recurrence in a neighborhood of an internal evolutionary stable strategy are derived. This extends the results given by Imhof [17].Asymptotic stochastic stability; evolutionarily stable strategy; invariant measure; Lyapunov function; Nash equilibrium; recurrence; jump stochastic differential equation 92D15; 60H10; 60J40; 92D25 IntroductionConsider a two-player symmetric game, where a ij is the payoff to a player using strategy S i against an opponent employing strategy S j , and define A = (a ij ), as the payoff matrix. We define ∆ n = y ∈ R n : y i > 0 for all i and y i = 1 as the n-dimensional simplex and ∆ n as it's closure. Within a population we assume that every individual is programmed to play a pure strategy S i . Let r i (t) be the size of the subpopulation that plays strategy S i at time t, which we denote as the i th subpopulation. Furthermore, define r(t) := (r 1 (t), . . . , r n (t)) T , R(t) := i r i (t) and s(t) := (s 1 (t), . . . , s n (t)) T where s i (t) := r i (t)/R(t) the i th subpopulation frequency of the population . When an agent in the i th subpopulation is randomly matched with another player from the entire population, As(t) i is the average payoff for this individual, and we take this to be the fitness of the player. We assume growth is proportional to fitness:ṙ i (t) = r i (t) As(t) i , and henceṡThis is the replicator dynamic.Foster and Young [30] appear to be the first to use a stochastic differential equation to describe replicator dynamic, which they do by injecting a Brownian term directly into the replicator equation. Considering a biological perspective, Fudenberg and Harris [12] derived a continuous time stochastic replicator dynamic by first assumingfor σ i ∈ R + and the W i (t) are pairwise independent standard Wiener processes. Itô's lemma then yields ds i (t) = j =i s i (t)s j (t)As(t) i − As(t) j dt + σ 2 j s j (t) − σ 2 i s i (t) dt + σ i dW i (t) − σ j dW j (t) .(1)

On the robustness of learning in games with stochastically perturbed payoff observations

“…2 Khasminskii and Potsepun (2006) also consider a Stratonovich-based model while Vlasic (2012) examines the case of random jumps incurred by catastrophic, earthquake-like events 3 Whether the equilibria of the original game are themselves asymptotically stable, depends on the intensity of the noise on different strategies and also on the exact way that the noise enters the process (Mertikopoulos and Viossat, 2016).…”