Imitation dynamics with payoff shocks

Abstract

We investigate the impact of payoff shocks on the evolution of large populations of myopic players that employ simple strategy revision protocols such as the “imitation of success”. In the noiseless case, this process is governed by the standard (deterministic) replicator dynamics; in the presence of noise however, the induced stochastic dynamics are different from previous versions of the stochastic replicator dynamics (such as the aggregate-shocks model of Fudenberg and Harris in J Econ Theory 57(2):420–441, 1992). In this context, we show that strict equilibria are always stochastically asymptotically stable, irrespective of the magnitude of the shocks; on the other hand, in the high-noise regime, non-equilibrium states may also become stochastically asymptotically stable and dominated strategies may survive in perpetuity (they become extinct if the noise is low). Such behavior is eliminated if players are less myopic and revise their strategies based on their cumulative payoffs. In this case, we obtain a second order stochastic dynamical system where non-equilibrium states are no longer attracting and dominated strategies become extinct (a.s.), no matter the noise level.

Appendix: Auxiliary results from stochastic analysis

In this appendix, we provide an asymptotic growth bound for Wiener processes relying on the law of the iterated logarithm. This result appears in a similar context in Bravo and Mertikopoulos (2014); the proof below is given only for completeness and ease of reference.

Lemma 6.1

Let $W(t) = (W_{1}(t),\cdots ,W_{n}(t))$, $t\ge 0$, be an n-dimensional Wiener processes and let Z(t) be a bounded, continuous process in $\mathbb {R}^{n}$. Then:

$$\begin{aligned} f(t) + \int _{0}^{t} Z(s) \cdot dW(s) \sim f(t) \quad \text {as}\quad t\rightarrow \infty ( a.s.) , \end{aligned}$$

(6.1)

for any function $f:[0,\infty )\rightarrow \mathbb {R}$ such that $\lim _{t\rightarrow \infty } \left( t\log \log t\right) ^{-1/2} f(t) = +\infty$.

Proof

Let $\xi (t) = \int _{0}^{t} Z(s) \cdot dW(s) = \sum _{i=1}^{n} \int _{0}^{t} Z_{i}(s) \,dW_{i}(s)$. Then, the quadratic variation $\rho = [\xi ,\xi ]$ of $\xi$ satisfies:

$$\begin{aligned} d[\xi ,\xi ] = d\xi \cdot d\xi = \sum _{i=1}^{n} Z_{i} Z_{j} \delta _{ij} \,dt \le M \,dt, \end{aligned}$$

(6.2)

where $M = \sup _{t\ge 0} \left\| Z(t) \right\| ^{2} < +\infty$ (recall that Z(t) is bounded by assumption). On the other hand, by the time-change theorem for martingales (Øksendal 2007, Corollary 8.5.4), there exists a Wiener process $\widetilde{W}(t)$ such that $\xi (t) = \widetilde{W}(\rho (t))$, and hence:

$$\begin{aligned} \frac{f(t) + \xi (t)}{f(t)} = 1 + \frac{\widetilde{W}(\rho (t))}{f(t)}. \end{aligned}$$

(6.3)

Obviously, if $\lim _{t\rightarrow \infty } \rho (t) \equiv \rho (\infty ) < +\infty$, $\widetilde{W}(\rho (\infty ))$ is normally distributed so $\widetilde{W}(\rho (t))/f(t) \rightarrow 0$ and there is nothing to show. Otherwise, if $\lim _{t\rightarrow \infty } \rho (t) = +\infty$, the quadratic variation bound (6.2) and the law of the iterated logarithm yield:

$$\begin{aligned} \frac{\big \vert \widetilde{W}(\rho (t))\big \vert }{f(t)} \le \frac{\big \vert \widetilde{W}(\rho (t)) \big \vert }{\sqrt{2\rho (t) \log \log \rho (t)}} \times \frac{\sqrt{2Mt \log \log Mt}}{f(t)} \rightarrow 0 \quad \text {as}\quad t\rightarrow \infty , \end{aligned}$$

(6.4)

and our claim follows. $\square$