CARESS Working Paper 01-16 MULTIPLICITY, INSTABILITY AND SUNSPOTS IN GAMES

This paper considers a class of two players games in the unit square for which a similar and high enough responsiveness of each player’s strategy to the other player’s strategy around a Nash equilibrium in pure strategies implies (i ) the existence of at least two other Nash equilibria in pure strategies; (ii ) the non local uniqueness of the strategies of this Nash equilibrium in the sets of rationalizable strategies; and (iii ) the existence of nontrivial correlated equilibria arbitrarily close to this Nash equilibrium. Although a similar result can be shown to follow from Milgrom and Roberts’ (1990) results for supermodular games, the games considered here are not necessarily supermodular, which makes clear that supermodularity is not necessary to obtain it. The simultaneous emergence of phenomena of multiplicity, instability and vulnerability to sunspots studied in this paper parallels similar patterns observed in other frameworks (e.g. overlapping generations economies and finite economies with asymmetric information), and thus hints at the existence of an underlying relation between different avatars of the indeterminacy of the outcome of economies and games that goes beyond the boundaries of any specific framework and may be common to every decision-making problem faced of simultaneous, independent and interrelated optimizers.