Abstract
Fundamental results in the theory of extensive form games have singled out the reduced normal form as the key representation of a game in terms of strategic equivalence. In a precise sense, the reduced normal form contains all strategically relevant information. This note shows that a difficulty with the concept has been overlooked so far: given a reduced normal form alone, it may be impossible to reconstruct the game’s extensive form representation.
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Notes
This is especially important for research programs adopting the position that implicit or self-referential equilibrium concepts should be formalized in a framework which clearly specifies the details, order, and dependence of the possible actions and interactions of all involved agents. This is because the latter necessarily results in an extensive form game, which may in principle be susceptible to inessential transformations. An example of such a research program is given by Glycopantis et al. (2001, 2003, 2005, 2009). This also extends to the Nash program (Nash 1953) about giving non-cooperative foundations to cooperative games, because those often lead to extensive form games. A case in point are bargaining games where the extensive form explicitly specifies the bargaining protocol (e.g., Rubinstein 1982; Britz et al. 2010; Glycopantis 2020).
Many of the stronger set-valued refinement concepts always contain a proper equilibrium, e.g., M-stable sets (Mertens 1989, 1991) or equilibrium components with non-zero index (Ritzberger 1994). As far as these sets are contained in connected components of Nash equilibria, they, therefore, guarantee that the probability distribution on plays associated with the solution set generically satisfies backwards induction, since for generic extensive form games the probability distributions on plays are constant across every connected component of Nash equilibria (see Kreps and Wilson 1982).
This concept is slightly weaker than its preference-dependent analogue, precisely because for the latter coincidental payoff ties can render two strategies equivalent.
Such a preference-free extensive form representation amounts to insisting on robustness of the players’ preference ordering over plays against (sufficiently small) payoff perturbations.
For instance, a generic cell-based payoff perturbation of the reduced normal form game can only originate from a trivial extensive form game were all players play simultaneously, while a perturbation of the preferences defined on plays leaves the link between the original extensive form game and its reduced normal form unaffected.
This at least applies if all plays end after finitely many moves. An infinitely repeated game cast in extensive form provides an example of a game tree without terminal nodes. Plays, on the other hand, are still well defined objects. See Alós-Ferrer and Ritzberger (2016).
We assume here non-trivial decisions, where a player active at a node has at least two choices. Some formalisms allow for trivial decision where one player has one and only one choice. Allowing for those does not change the argument.
We are grateful to a referee for inspiring this clarification.
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We thank two anonymous reviewers for helpful comments.
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Alós-Ferrer, C., Ritzberger, K. Reduced normal forms are not extensive forms. Econ Theory Bull 8, 281–288 (2020). https://doi.org/10.1007/s40505-020-00183-8
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DOI: https://doi.org/10.1007/s40505-020-00183-8