Imitation Dynamics with Payoff Shocks

TL;DR

This paper studies how payoff shocks influence the evolution of strategies in large populations, revealing stability of equilibria under noise and the effects of player myopia on strategy extinction.

Contribution

It introduces a new stochastic model of imitation dynamics with payoff shocks and analyzes stability properties, highlighting differences from previous models and effects of player foresight.

Findings

Strict equilibria are stochastically stable regardless of shocks.

High noise can sustain non-equilibrium states and dominated strategies.

Less myopic players lead to extinction of dominated strategies regardless of noise.

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, 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 whose attracting states coincide with the game's strict equilibria and where dominated strategies become extinct (a.s.), no matter the noise level.