Abstract
The comprehensive study of normal form games in Chaps. 2 – 5 has yielded a deeper insight into the relationships between various refinements of the Nash concept. The analysis has also shown that, for ( generic ) normal form games, there is actually little need to refine the Nash concept since, for almost all such games, all Nash equilibria possess all properties one might hope for. 1
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Notes
However, see note 9 to Chap. 1.
It will be clear that ‘generic’ games of perfect information have a unique subgame perfect equilibrium. However, in ‘applications’ ties arise naturally and these give rise to interesting problems, see e. g. O’Neill [1986], Leininger [1986, 1988] and Ponssard [1990c].
Theorem 6.2.4 has been generalized to (a certain class) of perfect information finite-action, infinite-length games in Fudenberg and Levine [1983], and to perfect information games in which the action spaces are continua in Harris [1985], Hellwig and Leininger [1987] and Hellwig, Leininger, Reny and Robson [1991]. As far as this author knows, existence of SPE in (nicely behaved) continuous games of imperfect information has not yet been established, nor are counterexamples known.
See Note 3 to Chap. 1 for a collection of papers in which it is argued that the requirement of subgame perfection is too restrictive.
See also Kreps and Ramey [1987]. In that paper it is also shown that sequential equilibria entail convex structurally consistent assessments, i.e. there exist beliefs that are obtained from a convex combination of b’ s as in (6.3.3) that sustain the equilibrium. Furthermore, Kreps and Ramey give an example illustrating that structural consistency may conflict with sequential rationality: The concept of sequential best reply at u against (b, μ) is based on the assumption that there will be no deviations from b after u, while if Pb(u) = 0, condition (6.3.3) forces the player to assume that there has been a deviation before u; in some games one cannot have a deviation before u without simultaneously having a deviation after u. To put it differently: If at u one believes that b ′ has been played rather than b, why doesn’t one optimize then against b ′ rather than against b ?
Weaker concepts of consistency, leading to correspondingly weaker equilibrium notions are proposed in Fudenberg and Tirole [1989] and Weibull [1990].
McLennan [1989] has characterized the sequential equilibrium concept as the set of fixed points of a certain correspondence from a space of ‘consistent conditional system’ into itself. His framework allows the existence of a sequential equilibrium to be proved by a direct fixed point argument. Kohlberg and Reny [1991] have given a characterization of the Kreps/Wilson consistency concept without using trembles, i. e. their characterization is in terms of the basic data of the extensive form game (i. e. information sets and choices).
Recall from note 5 that structural consistency may conflict with sequential rationality.
Well-behaved imperfect information games in which the action spaces are continua need not have sequential equilibria, see Van Damme [1987, p. 29] for an example.
The relationship between perfectness, sequentiality and (lexicographic) domination is further investigated in Okada [1989].
See Mailath, Samuelson and Swinkels [1990] for a discussion of how also other extensive form equilibrium notions can already be detected in the (reduced) normal form.
Of course one may question whether the perturbations that are considered in Fudenberg et al. [1988] are actually slight perturbations. The results of Fudenberg et al. crucially depend on the fact that a small ex ante uncertainty can become large if an ‘unexpected’ action is observed.
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© 1991 Springer-Verlag Berlin Heidelberg
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van Damme, E. (1991). 6 Extensive Form Games. In: Stability and Perfection of Nash Equilibria. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-58242-4_6
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