GIACOMO BONANNO and KLAUS NEHRING ON STALNAKER’S NOTION OF STRONG RATIONALIZABILITY AND NASH EQUILIBRIUM IN PERFECT INFORMATION GAMES ABSTRACT. Counterexamples to two results by Stalnaker (Theory and Deci- sion, 1994) are given and a corrected version of one of the two results is proved. Stalnaker’s proposed results are: (1) if at the true state of an epistemic model of a perfect information game there is common belief in the rationality of every player and common belief that no player has false beliefs (he calls this joint condition ‘strong rationalizability’), then the true (or actual) strategy profile is path equiva- lent to a Nash equilibrium; (2) in a normal-form game a strategy profile is strongly rationalizable if and only if it belongs to C ∞ , the set of profiles that survive the iterative deletion of inferior profiles. KEY WORDS: Nash equilibrium, Perfect information games, Stalnaker’s notion Stalnaker (1994, pp. 49–74, Section 6) introduced the notion of Strong Rationalizability and stated a number of results. We provide counterexamples to the two main results (Theorem 3, p. 63, and Theorem 4, p. 64) and give a proof of a corrected version of one of them. First we recall some basic notation and definitions. Let Ŵ =〈N, 〈C i ,u i 〉 i ∈N 〉 be a normal-form game, where N = {1,...,n} is the set of players and, for every i ∈ N , C i is i ’s set of strategies and u i : C → (where C = C 1 ×···× C n ) is i ′ s payoff function. An epistemic model for Ŵ, which we denote by M Ŵ , is a tuple 〈W, a, 〈R i ,P i ,S i 〉 i ∈N 〉 where W is a finite set of states (or possible worlds), a is a designated member of W , representing the true state (or actual world), and, for every player i ∈ N , • R i is a serial, transitive and euclidean relation (the interpreta- tion of xR i y is that at state x player i considers state y possi- ble). We denote the set {y ∈ W : xR i y } by R i (x). • P i : W → (W ) (where (W ) denotes the set of probability distributions on W ) is a function that associates with every w ∈ Theory and Decision 45: 291–295, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.