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Multistage Situations

Joseph Greenberg
Dov Monderer
Benyamin Shitovitz

Abstract

We introduce and analyze "multistage situations," which generalize "multistage games" (which, in turn, generalize "repeated games"). One reason for this generalization is to avoid the perhaps unrealistic constraint--inherent to noncooperative games--that the set of strategy tuples must be a Cartesian product of the strategy sets of the players. Another reason is that in most economic and social activities (e.g., in sequential bargaining without a rigid protocol) the "rules of the game" are rather amorphous; the procedures are rarely pinned down. Such social environments can, however, be represented as multistage situations and be effectively analyzed through the theory of social situations. The paper contributes to the theory of social situations within the framework of multistage situations (e.g., the existence of a largest conservative stable standard of behavior which yields a definition that extends "subgame perfect equilibrium paths"), and to the theory of multistage games (e.g., the existence of $\varepsilon$-generalized perfect equilibrium, for all $\varepsilon > 0$). It also provides an equivalence theorem between subgame perfection and the largest conservative stable standard of behavior for multistage games. The usefulness of our approach is further illustrated by our notion of "$k$-rationality" whereby (at each substitution) players look only $k$ steps ahead.