SUPPLEMENT TO “BELIEF-FREE EQUILIBRIA IN GAMES WITH INCOMPLETE INFORMATION”

THIS SUPPLEMENT CONTAINS some omitted details on the existence of belieffree equilibria for two families of games that are studied in the literature on reputation. Namely, for the specific cases considered in footnotes 15 and 19, we claim that it is possible to find games arbitrarily close to the respective original reputation games such that V ∗ has nonempty interior. In this supplementary material, we explain this in greater detail. Consider first a one-sided incomplete information game Γ with known-own payoffs, where player 2’s payoff matrix is u2, while player 1’s payoff is u1 in state j = 1 and −u2 in state j = 2. In footnote 15 of the paper, we claim that there exists a game Γ̂ arbitrarily close to Γ for which the set of belief-free equilibria is nonempty. Let us start with a two-player full information game where u1 and u2 are players’ payoff matrixes, and assume that the set of individually rational payoffs of this game has nonempty interior (otherwise the question of reputation is trivial). Consider a complete information zero-sum two-player game Γ 0, where player 2’s payoff matrix is u2 with value v1 = v, v2 = −v. Let (s∗ 1 s∗ 2) denote a saddle point of this game. Let Mi denote the highest feasible payoff for player i and let a denote the action profile attaining this payoff. First, we shall show that there always exists a perturbation of payoffs in Γ 0 that generates a full information game Γ ′ arbitrarily close to Γ 0 and whose set of individually rational payoffs has nonempty interior. To this purpose we perturb payoffs in such a way that (s∗ 1 s ∗ 2) remains an equilibrium of Γ ′, but there exists a feasible payoff Pareto dominating (s∗ 1 s ∗ 2). 1. s∗ i is not completely mixed, for some i = 1 2: (a) Mi > vi: Let s′ i denote some action assigned zero probability by s ∗ i , and increase u−i(s′ i s−i) by ε > 0 for all s−i. Call u ′ the new payoff matrix. Since player i is not using s′ i, s ∗ −i remains a best reply to s ∗ i , and since i’s payoff matrix has not changed, s∗ i also remains a best reply to s ∗ −i. So s ∗ remains an equilibrium. Because player i does not use s′ i, it means that ui(s ′ i s ∗ −i) ≤ vi and so u−i(s′ i s ∗ −i)+ ε > v−i, while also ui(s′ i s∗ −i)+u−i(s i s∗ −i)+ ε > 0 (since the game is zero sum), that is, ui(s ′ i s ∗ −i)+u−i(s i s∗ −i) > 0. As a denotes an action profile such that ui(a) = Mi, there exists a mixture λa + (1 − λ)(s′ i s∗ −i) that strictly improves upon the Nash equilibrium (s∗ 1 s ∗ 2). (b) Mi = vi: This means that player i is getting his maximal payoff from playing s∗ i independently of player −i’s action, so that any strategy profile (s∗ i s−i), s−i ∈ A−i, is a saddle point. So pick one action s−i and consider the game in which ui(si s−i) = ui(si s−i) + ε for all si ∈ Ai and some ε > 0, and