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A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements

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Author Info
Jonathan Weinstein
Muhamet Yildiz

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Abstract

Rationalizability is a central solution concept of game theory. Economic models often have many rationalizable outcomes, motivating economists to use refinements of rationalizability, including equilibrium refinements. In this paper we try to achieve a general understanding of when this multiplicity occurs and how one should deal with it. Assuming that the set of possible payoff functions and belief structures is sufficiently rich, we establish a revealing structure of the correspondence of beliefs to sets of rationalizable outcomes. We show that, for any rationalizable action a of any type, we can perturb the beliefs of the type in such a way that a is uniquely rationalizable for the new type. This unique outcome will be robust to further small changes. When multiplicity occurs, then we are in a "knife-edge" case, where the unique rationalizable outcome changes, sandwiched between open sets of types where each of the rationalizable actions is uniquely rationalizable. As an immediate application of this result, we characterize, for any refinement of rationalizability, the predictions that are robust to small misspecifications of interim beliefs. These are only those predictions that are true for all rationalizable strategies, that is, the predictions that could have been made without the refinement. Copyright The Econometric Society 2007.

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File URL: http://hdl.handle.net/10.1111/j.1468-0262.2006.00751.x
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Publisher Info
Article provided by Econometric Society in its journal Econometrica.

Volume (Year): 75 (2007)
Issue (Month): 2 (03)
Pages: 365-400
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Handle: RePEc:ecm:emetrp:v:75:y:2007:i:2:p:365-400

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  1. Andrea Galeotti & Sanjeev Goyal & Matthew O. Jackson & Fernando Vega-Redondo & Leeat Yariv, 2008. "Network Games," Economics Working Papers ECO2008/07, European University Institute. [Downloadable!]
  2. Antonio Penta, 2008. "On the Structure of Rationalizability for Arbitrary Spaces of Uncertainty," PIER Working Paper Archive 09-021, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania, revised 05 Jun 2008. [Downloadable!]
  3. Stephen Morris & Hyun Song Shin, 2007. "Common Belief Foundations of Global Games," Levine's Bibliography 122247000000001638, UCLA Department of Economics. [Downloadable!]
  4. Daisuke Oyama & Olivier Tercieux, 2007. "Robust Equilibria under Non-Common Priors," Levine's Bibliography 843644000000000210, UCLA Department of Economics. [Downloadable!]
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  5. Pierpaolo Battigalli & Alfredo Di Tillio & Edoardo Grillo & Antonio Penta, 2008. "Interactive Epistemology and Solution Concepts for Games with Asymmetric Information," Working Papers 340, IGIER (Innocenzo Gasparini Institute for Economic Research), Bocconi University. [Downloadable!]
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