The Existence of Subgame-Perfect Equilibrium in Continuous Games with Almost Perfect Information: A Comment

Abstract

HARRIS (1985) HAS SHOWN that subgame-perfect equilibria exist in deterministic continuous games with perfect information.' A recent influential paper by Harris, Reny, and Robson (1995) shows that public randomization ensures the existence of subgameperfect equilibria in continuous games with almost perfect information. The authors exhibit an example of a game with almost perfect information in which no subgameperfect equilibrium exists without public randomization. In addition, their Proposition 36 argues that public randomization is not required in games with perfect information. Contrary to this proposition, we give here an example of a continuous game with perfect information in which no subgame-perfect equilibrium exists if public randomization is not allowed. The result of Harris (1985) implies that our example must be a game in which Nature is an active player. Intrapersonal games for consumers with changing preferences are usually games of this type, as in, for example, stochastic versions of Peleg and Yaari (1973) and Goldman (1980). The example has five stages. In stage 1, player 1 chooses al E [0, 1]. In stage 2, player 2 chooses a2 E [0, 1]. In stage 3, Nature chooses x by randomizing uniformly over the interval [-2 + a, + a2, 2 - a, - a2]. After this, players 3 and 4 move sequentially. The subgame following a history (a,, a2, x) and the associated payoffs for all four players are shown in Figure 1. In the following, let a and ,B denote the probabilities with which players 3 and 4 choose U and u, respectively. Consider first the subgame defined by a, = a2 = 1. Nature's move is degenerate in this subgame: x = 0. The set of subgame-perfect equilibrium paths in the resulting subgame is characterized by three segments of mixing probabilities: a = 0 and /3 E [0, .5]; a E [0, 1] and ,B = .5; and a = 1 and f3 E [.5, 1]. The set of equilibrium expected payoffs for players 1 and 2 implied by these three segments is given by