Higher Order Game Dynamics

TL;DR

This paper extends continuous-time game dynamics to higher orders, analyzing how strategies evolve when acceleration and higher derivatives influence payoffs, leading to new insights on strategy extinction and convergence rates.

Contribution

It introduces a broad class of higher order game dynamics, generalizing first order models like replicator dynamics, and demonstrates their implications for strategy elimination and equilibrium convergence.

Findings

Strictly dominated strategies become extinct as fast as in first order dynamics.

Weakly dominated strategies also become extinct for higher order (n>1) dynamics.

Higher order dynamics accelerate convergence to strict equilibria.

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>1. 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 of evolutionary game theory, and we show that con- vergence to strict equilibria in n-th order dynamics is n orders as fast as in first order.