The Nash equilibrium condition (2.12) requires that each player’s strategy be optimal from the ex ante point of view. Ex ante optimality implies that the strategy is also ex post optimal at each information set that is reached with positive probability in equilibrium, but, as the game of Figure 2 illustrates, such ex post optimality need not hold at the unreached information sets. The example suggests imposing ex post opti- mality as a necessary requirement for self-enforcingness but, of course, this requirement is meaningful only when conditional expected payo¤s are well-de…ned, i.e. when the information set is a singleton. In particular, the suggestion is feasible for games with perfect information, i.e. games in which all information sets are singletons, and in this case one may require as a condition for s¤ to be self-enforcing that it satis…es
uih(s¤)¸ uih(s¤nsi) for all i; all si 2 Si all h 2 Hi: (3.1)
Condition (3.1) states that at no decision point h can a player gain by deviating from s¤
if after h no other player deviates from s¤. Obviously, equilibria satisfying (3.1) can be
found by rolling back the game tree in a dynamic programming fashion, a procedure al- ready employed in Zermelo (1912). It is, however, also worthwhile to remark that already in Von Neumann and Morgenstern (1944) it was argued that this backward induction procedure was not necessarily justi…ed as it incorporates a very strong assumption of “persistent” rationality. Recently, Hart (1999) has shown that the procedure may be justi…ed in an evolutionary setting. Adopting Zermelo’s procedure one sees that, for perfect information games, there exists at least one Nash equilibrium satisfying (3.1) and that, for generic perfect information games, (3.1) selects exactly one equilibrium. Furthermore, in the latter case, the outcome of this equilibrium is the unique outcome that survives iterated elimination of weakly dominated strategies in the normal form
of the game. (Each elimination order leaves at least this outcome and there exists a sequence of eliminations that leaves nothing but this outcome, cf. Moulin (1979).) Selten (1978) was the …rst paper to show that the solution determined by (3.1) may be hard to accept as a guide to practical behavior. (Of course, it was already known for a long time that in some games, such as chess, playing as (3.1) dictates may be infeasible since the solution s¤ cannot be computed.) Selten considered the …nite repetition of
the game from Figure 2, with one player 2 playing the game against a sequence of di¤erent players in each round and with players always being perfectly informed about the outcomes in previous rounds. In the story that Selten associates with this game, player 2 is the owner of a chain store who is threatened by entry in each of …nitely many towns. When entry takes place (r1 is chosen), the chain store owner either acquiesces
(chooses r2) or …ghts entry (chooses l2). The backward induction solution has players
play (r1; r2) in each round, but intuitively, we expect player 2 to behave aggressively
(choose l2) at the beginning of the game with the aim of inducing later entrants to stay
out. The chain store paradox is the paradox that even people who accept the logical validity of the backward induction reasoning somehow remain unconvinced by it and do not act in the manner that it prescribes, but rather act according to the intuitive solution. Hence, there is an inconsistency between plausible human behavior and game- theoretic reasoning. Selten’s conclusion from the paradox is that a theory of perfect rationality may be of limited relevance for actual human behavior and he proposes a theory of limited rationality to resolve the paradox. Other researchers have argued that the paradox may be caused more by the inadequacy of the model than by the solution concept that is applied to it. Our intuition for the chain store game may derive from a richer game in which the deterrence equilibrium indeed is a rational solution. Such richer models have been constructed in Kreps and Wilson (1982b), Milgrom and Roberts (1982) and Aumann (1992). These papers change the game by allowing a tiny probability that player 2 may actually …nd it optimal to …ght entry, which has the consequence that, when the game still lasts for a long time, player 2 will always play as if it is optimal to …ght entry which forces player 1 to stay out.
underlies (3.1), i.e. players are forced to believe that even at information sets h that can be reached only by many deviations from s¤, behavior will be in accordance with
s¤. This assumption that forces a player to believe that an opponent is rational even
after he has seen the opponent make irrational moves has been extensively discussed and criticized in the literature, with many contributions being critical (see, for exam- ple, Basu (1988, 1990), Ben Porath (1993), Binmore (1987), Reny (1992ab, 1993) and Rosenthal (1981)). Binmore argues that human rationality may di¤er in systematic ways from the perfect rationality that game theory assumes, and he urges theorists to build richer models that incorporate explicit human thinking processes and that take these systematic deviations into account. Reny argues that (3.1) assumes that there is common knowledge of rationality throughout the game, but that this assumption is self-contradicting: Once a player has “shown” that he is irrational (for example, by playing a strictly dominated move), rationality can no longer be common knowledge and solution concepts that build on this assumption are no longer appropriate. Au- mann and Brandenburger (1995) however argue that Nash equilibrium does not build on this common knowledge assumption. Reny (1993), on the other hand, concludes from the above that a theory of rational behavior cannot be developed in a context that does not allow for irrational behavior, a conclusion similar to the one also reached in Selten (1975) and Aumann (1987b). Aumann (1995), however, disagrees with the view that the assumption of common knowledge of rationality is impossible to maintain in extensive form games with perfect information. As he writes “The aim of this paper is to present a coherent formulation and proof of the principle that in P I games, common knowledge of rationality implies backward induction” (p. 7) (see also Aumann (1998) for an application to Rosenthal’s centipede game; the references in that paper provide further information, also on other points of view).
We now leave this discussion on backward induction in games with perfect information and move on to discuss more general games. Selten (1965) notes that the argument leading to (3.1) can be extended beyond the class of games with perfect information. If the game g admits a subgame °, then the expected payo¤s of s¤in ° depend only on what
s¤prescribes in °. Denote this restriction of s¤to ° by s¤
all other parts of the game have become strategically irrelevant, hence, Selten argues that, for s¤to be self-enforcing, it is necessary that s¤
° be self-enforcing for every subgame
°. Selten de…ned a subgame perfect equilibrium as an equilibrium s¤ of g that induces
a Nash equilibrium s¤
° in each subgame ° of g and he proposed subgame perfection as
a necessary requirement for self-enforcingness. Since every equilibrium of a subgame of a …nite game can be “extended” to an equilibrium of the overall game, it follows that every …nite extensive form game has at least one subgame perfect equilibrium.
Existence is, however, not as easily established for games in which the strategy spaces are continuous. In that case, not every subgame equilibrium is part of an overall equilibrium: Players moving later in the game may be forced to break ties in a certain way, in order to guarantee that players who moved earlier indeed played optimally. (As a simple example, let player 1 …rst choose x 2 [0; 1] and let then player 2, knowing x, choose y 2 [0; 1]. Payo¤s are give by u1(x; y) = xy and u2(x; y) = (1¡x)y. In the unique subgame perfect
equilibrium both players choose 1 even though player 2 is completely indi¤erent when player 1 chooses x = 1:) Indeed, well-behaved continuous extensive form games need not have a subgame perfect equilibrium, as Harris et al. (1995) have shown. However, these authors also show that, for games with almost perfect information (“stage” games), existence can be restored if players can observe a common random signal before each new stage of the game which allows them to correlate their actions. For the special case where information is perfect, i.e. information sets are singletons, Harris (1985) shows that a subgame perfect equilibrium does exist even when correlation is not possible (see also Hellwig et al. (1990)).
Other chapters of this Handbook contain ample illustrations of the concept of subgame perfect equilibrium, hence, we will not give further examples. It su¢ces to remark here that subgame perfection is not su¢cient for self-enforcingness, as is illustrated by the game from Figure 3.
@ @ @ @ @ ¡¡ ¡¡ ¡ 1 r1 l1 (2; 2) l2 r2 t 3; 1 1; 0 b 0; 1 0; x l2 r2 l1 2; 2 2; 2 r1t 3; 1 1; 0 r1b 0; 1 0; x