Reduced normal forms are not extensive forms
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Reduced normal forms are not extensive forms Carlos Alós‑Ferrer1 · Klaus Ritzberger2 Received: 15 April 2020 / Accepted: 9 May 2020 © Society for the Advancement of Economic Theory 2020
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. Keywords Reduced normal forms · Extensive form games JEL Classification C72
1 Introduction Most game-theoretic solution concepts are defined in the normal form, that is, as (sets of) strategy profiles, e.g., iteratively undominated strategies, rationalizable strategies (Bernheim 1984; Pearce 1984), curb sets (Basu and Weibull 1991), tenable blocks (Myerson and Weibull 2015), and of course Nash equilibrium (Nash 1950, 1951). A number of refinements of Nash equilibrium, on the other hand, rely on the extensive form representation, e.g., subgame perfection (Selten 1965), quasiperfect equilibrium (van Damme 1984), or sequential equilibrium (Kreps and Wilson 1982). Some refinements are defined in the agent normal form and then translated to the extensive form game (Selten 1975).
* Carlos Alós‑Ferrer carlos.alos‑[email protected] Klaus Ritzberger [email protected] 1
Department of Economics, Zurich Center for Neuroeconomics (ZNE), University of Zurich, Blümlisalpstrasse 10, 8006 Zurich, Switzerland
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Department of Economics, Royal Holloway, University of London, H220 Horton Building, Egham, Surrey TW20 0EX, UK
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C. Alós‑Ferrer, K. Ritzberger
All known refinement concepts that are defined in the extensive form suffer from a fragility, though. An extensive form game can often be written down in different ways, reflecting the inessential transformations introduced by Thompson (1952) and Elmes and Reny (1994) (see also Battigalli et al. 2020). And yet, these transformations do affect refinement criteria that are based on the extensive form. That is, a subgame perfect or sequential equilibrium may cease to be so after an inessential (Thompson) transformation; or a mere Nash equilibrium may become subgame perfect or sequential after such a transformation. Since these transformations of the extensive form only generate different “framings” of the same problem, this fragility appears undesirable, at least from a rationalistic viewpoint.1 In the normal form, two pure strategies of a player are strategically equivalent if they give the same payoffs to all players for all strategy profiles among the opponents. The (pure-strategy) reduced normal form is obtained by collapsing all strategically equivalent strategies to single representatives. Two different extensive form representations can be obtained from each other through inessential transforma
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