p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

Abstract

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players are repeatedly and randomly matched to play a symmetric n × n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independently of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p < 1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p < 1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.

Monotone Methods for Equilibrium Selection under Perfect Foresight Dynamics

“…Oyama (2002) shows by direct computation that (2, 2) is absorbing and globally accessible for a small degree of friction. In fact, (2, 2) is a strict MPmaximizer with a strict monotone potential function that is supermodular (Figure 2(b)), while the original game is not supermodular (for any ordering of actions).…”

“…Such payoff functions describe random matching models within a single population, considered in Matsui and Matsuyama (1995), Hofbauer and Sorger (1999), and Oyama (2002), as well as models with nonlinear payoffs, considered in Matsuyama (1991Matsuyama ( , 1992 and Kaneda (1995).…”

“…Matsui and Matsuyama (1995) demonstrate that in 2×2 coordination games, a strict Nash equilibrium is absorbing and globally accessible for any small degree of friction if and only if it is the risk-dominant equilibrium. Beyond 2 × 2 games, Oyama (2002) appeals to the notion of p-dominance to identify (in a single population setting) a class of games where one can explicitly characterize the set of the perfect foresight paths relevant for stability considerations, showing that a p-dominant equilibrium with p < 1/2 is selected. Hofbauer and Sorger (2002) and Kojima (2003) obtain related results based on other risk-dominance concepts in a multiple population setting.…”

Rationalizable foresight dynamics

“…4 See Sorger (1999, 2002), Oyama (2002), and Oyama et al (2003) for more recent developments. a fraction λh of agents are replaced by the same size of entrants.…”

Iterated potential and robustness of equilibria

“…Kajii and Morris (1997) show that if the complete information game has a p-dominant equilibrium with low p, then it is robust to incomplete information, 3 while Ui (2001) shows that in potential games, the potential maximizer is robust to incomplete information. For perfect foresight dynamics, Sorger (1999, 2002) show that a potential maximizer is stable for any small degree of friction, while the p-dominance condition is studied by Oyama (2002) (in a single population setting). 4 Furthermore, Morris and Ui (2005) introduce a generalization of potential and establishes the robustness of generalized potential maximizer to incomplete information.…”