The Robustness of Equilibria to Incomplete Information

Abstract

A number of papers have shown that a strict Nash equilibrium action pro¯le of a game may never be played if there is a small amount of incomplete information (see, for example, Carlsson and van Damme (1993a)). We present a general approach to analyzing the robustness of equilibria to a small amount of incomplete information. A Nash equilibrium of a complete information game is said to be robust to incomplete information if every incomplete information game with payo®s almost always given by the complete information game has an equilibrium which generates behavior close to the Nash equilibrium. We show that an open set of games has no robust equilibrium and examine why we get such di®erent results from the re¯nements literature. We show that if a game has a unique correlated equilibrium, it is robust. Finally, a natural many-player many-action generalization of risk dominance is shown to be a su±cient condition for robustness. ¤ We are grateful to Eddie Dekel, Drew Fudenberg and George Mailath for valuable comments.

Iterated Potential and Robustness of Equilibria

“…Following Kajii and Morris (1997), we say that a * is robust if, for small ε > 0, every ε-elaboration of g has a Bayesian Nash equilibrium σ with σ P (a * ) close to 1. Definition 3.3.…”

“…We will need the following lemma, the proof of which mimics that of Lemma B in Kajii and Morris (1997).…”

“…In bold strokes, two types of methods have proved to be powerful in identifying equilibria that are robust in various aspects: namely, the potential method (due to Monderer and Shapley (1996); see also Blume (1993), Sorger (1999, 2002), Ui (2001)) and the risk-dominance method (due to Harsanyi and Selten (1988); see also Kandori, Mailath, and Rob (1993), Young (1993), Matsui and Matsuyama (1995), Morris, Rob, and Shin (1995), and Kajii and Morris (1997)). These criteria, however, are very demanding and such concepts fail to exist in many games.…”

“…The first one is the socalled "robustness to incomplete information" test as originally defined by Kajii and Morris (1997). To motivate this approach, consider an analyst who plans to model some strategic situation by a particular complete information game.…”

“…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).…”