The projection dynamic and the replicator dynamic

Abstract

We investigate a variety of connections between the projection dynamic and the replicator dynamic. At interior population states, the standard microfoundations for the replicator dynamic can be converted into foundations for the projection dynamic by replacing imitation of opponents with "revision driven by insecurity" and direct choice of alternative strategies. Both dynamics satisfy a condition called inflowoutflow symmetry, which causes them to select against strictly dominated strategies at interior states; still, because it is discontinuous at the boundary of the state space, the projection dynamic allows strictly dominated strategies to survive in perpetuity. The two dynamics exhibit qualitatively similar behavior in strictly stable and null stable games. Finally, the projection and replicator dynamics both can be viewed as gradient systems in potential games, the latter after an appropriate transformation of the state space. JEL classification: C72, C73.

Stable games and their dynamics

“…Moreover, the stability results are weaker for these two dynamics: asymptotic stability is guaranteed only for strictly stable games, whereas for null stable games only Lyapunov stability holds; indeed, closed orbits are known to occur in the latter case. See Lahkar and Sandholm (2008) and Sandholm et al (2008) for further discussion.…”

“…5 See Smith (1984), Hofbauer et al (1979), Zeeman (1980, Akin (1990), Aubin (1991), Cressman et al (2001), Nagurney and Zhang (1997), Sandholm et al (2008), and Section 7.2 below. 6 The original papers on consistency are Hannan (1957) and Blackwell (1956).…”

The projection dynamic and the geometry of population games

“…This link and others are explored in the companion paper, Sandholm et al (2008). which tells us that µ k+1 and v k+1 deviate from µ k in the same direction.…”

“…There are in fact many such connections, and they are explored in the companion paper, Sandholm et al (2008).…”

“…6 In contrast, best response and perturbed best response dynamics have agents reconsider their strategies at a fixed rate-see Section 4.2. this point of view, it is surprising that the dynamic with the closest links to the projection dynamic is actually the replicator dynamic (Taylor and Jonker (1978)), which is founded not on direct evaluation of payoffs, but on imitation of opponents. The many connections between these two dynamics are explored in a companion paper, Sandholm et al (2008).…”