The projection dynamic and the geometry of population games

Abstract

The projection dynamic is an evolutionary dynamic for population games. It is derived from a model of individual choice in which agents abandon their current strategies at rates inversely proportional to the strategies' current levels of use. The dynamic admits a simple geometric definition, its rest points coincide with the Nash equilibria of the underlying game, and it converges globally to Nash equilibrium in potential games and in stable games. JEL classification: C72, C73.

The projection dynamic and the replicator dynamic

“…To keep the presentation self-contained, we briefly review some definitions and results from Lahkar and Sandholm (2008).…”

“…Still, it is possible to construct a matched pair of revision protocols that generate dynamics (R) and (P) throughout the simplex-see Lahkar and Sandholm (2008) for details.…”

“…Our companion paper, Lahkar and Sandholm (2008), established one link between the dynamics' foundations. In general, one provides foundations for an evolutionary dynamic by showing that it is the mean dynamic corresponding to a particular revision protocol-that is, a particular rule used by agents to decide when and how to choose new strategies.…”

“…Microfoundations for this dynamic are provided in Lahkar and Sandholm (2008): there the dynamic is derived from a model of "revision driven by insecurity", in which each agent considers switching strategies at a rate inversely proportional to his current strategy's popularity. Although it is discontinuous at the boundary of the state space, the projection dynamic admits unique forward solution trajectories; its rest points are the Nash equilibria of the underlying game, and it converges to equilibrium from all initial conditions in potential games and in stable games.…”

“…In Lahkar and Sandholm (2008), a principal step in changing the replicator protocol into a projection protocol is to eliminate the former's use of imitation, replacing this with "revision driven by insecurity" and direct (nonimitative) selection of alternative strategies. But if we look only at behavior at interior population states-that is, at states where all strategies are in use-then this one step becomes sufficient for converting replicator protocols into projection protocols.…”