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The Robustness Of Equilibrium Analysis: The Case Of Undominated Nash Equilibrium

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  • Takashi Kunimoto

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Abstract

I consider a strategic game form with a finite set of payoff states and employ undominated Nash equilibrium (UNE) as a solution concept under complete information. I propose notions of the proximity of information according to which the continuity of UNE concept is considered as the robustness criterion. I identify a topology (induced by what I call d?) with respect to which the undominated Bayesian Nash equilibrium (UBNE) correspondence associated with any game form is upper hemi-continuous at any complete information prior. I also identify a slightly coarser topology (induced by what I call d??) with respect to which the UBNE correspondence associated with some game form exhibits a failure of the upper hemi-continuity at any complete information prior. In this sense, the topology induced by d? is the coarsest one. The topology induced by d?? is also used in both Kajii and Morris (1998) and Monderer and Samet (1989, 1996) with some additional restriction. I apply this robustness analysis to the UNE implementation. Appealing to Palfrey and Srivastava’s (1991) canonical game form, I show, as a corollary, that almost any social choice function is robustly UNE implementable relative to d?. I show, on the other hand, that only monotonic social choice functions can be robustly UNE implementable relative to d??. This clarifies when Chung and Ely’s Theorem 1 2003) applies.

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Bibliographic Info

Paper provided by McGill University, Department of Economics in its series Departmental Working Papers with number 2006-26.

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Length: 36 pages
Date of creation: Sep 2006
Date of revision:
Handle: RePEc:mcl:mclwop:2006-26

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  1. Levine, David & Kreps, David & Fudenberg, Drew, 1988. "On the Robustness of Equilibrium Refinements," Scholarly Articles 3350444, Harvard University Department of Economics.
  2. Matthew, Jackson O. & Palfrey, Thomas R. & Srivastava, Sanjay., 1990. "Undominated Nash Implementation in Bounded Mechanism," Working Papers 754, California Institute of Technology, Division of the Humanities and Social Sciences.
  3. T. Börgers, 2010. "Weak Dominance and Approximate Common Knowledge," Levine's Working Paper Archive 378, David K. Levine.
  4. Palfrey, Thomas R & Srivastava, Sanjay, 1991. "Nash Implementation Using Undominated Strategies," Econometrica, Econometric Society, vol. 59(2), pages 479-501, March.
  5. Monderer, Dov & Samet, Dov, 1989. "Approximating common knowledge with common beliefs," Games and Economic Behavior, Elsevier, vol. 1(2), pages 170-190, June.
  6. Eric Maskin, 1998. "Nash Equilibrium and Welfare Optimality," Harvard Institute of Economic Research Working Papers 1829, Harvard - Institute of Economic Research.
  7. Samuelson, L., 1989. "Dominated Strategies And Common Knowledge," Papers 1-89-3, Pennsylvania State - Department of Economics.
  8. Dirk Bergemann & Stephen Morris, 2005. "Robust Implementation: The Role of Large Type Spaces," Levine's Bibliography 784828000000000116, UCLA Department of Economics.
  9. Samuelson, Larry, 1992. "Dominated strategies and common knowledge," Games and Economic Behavior, Elsevier, vol. 4(2), pages 284-313, April.
  10. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
  11. Kajii, Atsushi & Morris, Stephen, 1998. "Payoff Continuity in Incomplete Information Games," Journal of Economic Theory, Elsevier, vol. 82(1), pages 267-276, September.
  12. Dirk Bergemann & Stephen Morris, 2003. "Robust Mechanism Design," Cowles Foundation Discussion Papers 1421, Cowles Foundation for Research in Economics, Yale University.
  13. Rubinstein, Ariel, 1989. "The Electronic Mail Game: Strategic Behavior under "Almost Common Knowledge."," American Economic Review, American Economic Association, vol. 79(3), pages 385-91, June.
  14. Drew Fudenberg & Eddie Dekel, 1987. "Rational Behavior with Payoff Uncertainty," Working papers 471, Massachusetts Institute of Technology (MIT), Department of Economics.
  15. Kim-Sau Chung & Jeffrey C. Ely, 2003. "Implementation with Near-Complete Information," Econometrica, Econometric Society, vol. 71(3), pages 857-871, 05.
  16. Atsushi Kajii & Stephen Morris, . "The Robustness of Equilibria to Incomplete Information," Penn CARESS Working Papers ed504c985fc375cbe719b3f60, Penn Economics Department.
  17. Matthew O. Jackson & Sanjay Srivastava, 1992. "Characterizations of Game Theoretic Solutions which Lead to Impossibility Theorems," Discussion Papers 1004, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  18. Barton L. Lipman, 1997. "Finite Order Implications of Common Priors," Game Theory and Information 9703005, EconWPA.
  19. Brandenburger, Adam & Dekel, Eddie, 1987. "Rationalizability and Correlated Equilibria," Econometrica, Econometric Society, vol. 55(6), pages 1391-1402, November.
  20. Aumann, Robert & Brandenburger, Adam, 1995. "Epistemic Conditions for Nash Equilibrium," Econometrica, Econometric Society, vol. 63(5), pages 1161-80, September.
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