Weak Isomorphisms of Extensive Games

Abstract

As Harsanyi & Selten’s (1988) isomorphisms of strategic games, isomorphisms of extensive games can be viewed as a means to identify structurally similar extensive games and to identify corresponding structural elements of these games—players, information sets, actions, and nodes. And it is this emphasis of structural features that distinguishes isomorphisms from considerations of strategic equivalence as Kohlberg & Mertens’ (1986) invariance requirement or Thompson’s (1952) and Elmes & Reny’s (1994) transformations. Concerning strategic games, isomorphisms are bijective mappings between the players’ pure-strategy sets that preserve the player structure and the payoff structure. In extensive games, basically, these mappings can be based either on the action partitions or on the node sets. Actually, both approaches have been adopted in the literature. In a sense, the traditional tree representation of extensive games (see e.g. Selten 1975) corresponds to isomorphisms that are based on bijections of the node sets, whereas the sequence representation (see e.g. Osborne & Rubinstein 1994) corresponds to isomorphisms based on bijections of the action partitions.

While there is an established notion of isomorphisms of strategic games (see Subsection 2.3.4), there is no such notion for extensive games. Mainly, the reason for this gap seems to be the more complex nature of extensive games in connection with the resulting ambiguities in interpreting them. In this chapter, weak isomorphisms of extensive games are introduced and advocated.