# 1573 Robust Multiplicity with a Grain of Naiveté Aviad

In an important paper, Weinstein and Yildiz (2007) show that if players have an infinite depth of reasoning and this is commonly believed, types generically have a unique rationalizable action in games that satisfy a richness condition. We show that this result does not extend to environments where players may have a finite depth of reasoning, or think it is possible that the other player has a finite depth of reasoning, or think that the other player may think that is possible, and so on, even if this so-called “grain of naiveté” is arbitrarily small. More precisely, we show that even if there is almost common belief in the event that players have an infinite depth of reasoning, there are types with multiple rationalizable actions, and the same is true for “nearby” types. Our results demonstrate that both uniqueness and multiplicity are robust phenomena when we relax the assumption that it is common belief that players have an infinite depth, if only slightly. ∗This paper supersedes Heifetz and Kets (2011). We thank Sandeep Baliga, Eddie Dekel, Alessandro Pavan, David Pearce, Marciano Siniscalchi, and Satoru Takahashi for valuable input, and we are grateful to seminar audiences at Hebrew University, Northwestern University, Tel Aviv University, NYU, Penn State, and conference participants at the SING8 conference in Budapest for helpful comments. We thank Luciano Pomatto for excellent research assistance. †Department of Management and Economics, Open University of Israel. E-mail: aviadhe@openu.ac.il. Phone: +972-9-778-1878. ‡Kellogg School of Management, Northwestern University. E-mail: w-kets@kellogg.northwestern.edu. Phone: +1-505-204 8012.

[1]  K. Parthasarathy Introduction to Probability and Measure , 1979 .

[2]  Philip H. Dybvig,et al.  Bank Runs, Deposit Insurance, and Liquidity , 1983, Journal of Political Economy.

[3]  M. Obstfeld Rational and Self-Fulfilling Balance-of-Payments Crises , 1984 .

[4]  S. Zamir,et al.  Formulation of Bayesian analysis for games with incomplete information , 1985 .

[5]  D. Monderer,et al.  Approximating common knowledge with common beliefs , 1989 .

[6]  A. Rubinstein The Electronic Mail Game: Strategic Behavior Under "Almost Common Knowledge" , 1989 .

[7]  H. Carlsson,et al.  Global Games and Equilibrium Selection , 1993 .

[8]  Eddie Dekel,et al.  Hierarchies of Beliefs and Common Knowledge , 1993 .

[9]  A. Heifetz The bayesian formulation of incomplete information — The non-compact case , 1993 .

[10]  M. Obstfeld Models of Currency Crises with Self-Fulfilling Features , 1995 .

[11]  D. Stahl,et al.  On Players' Models of Other Players: Theory and Experimental Evidence , 1995 .

[12]  A. Heifetz,et al.  Topology-Free Typology of Beliefs , 1998 .

[13]  Self-fulfilling debt crises , 1998 .

[14]  Colin Camerer,et al.  Iterated Dominance and Iterated Best-Response in Experimental P-Beauty Contests , 1998 .

[15]  Miguel A. Costa-Gomes,et al.  Cognition and Behavior in Normal-Form Games: An Experimental Study , 1998 .

[16]  Itay Goldstein,et al.  Demand Deposit Contracts and the Probability of Bank Runs , 2002 .

[17]  Frank Heinemann,et al.  American Economic Association Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks : Comment , 2015 .

[18]  S. Morris,et al.  Global Games: Theory and Applications , 2001 .

[19]  S. Morris,et al.  Does One Soros Make a Difference? A Theory of Currency Crises with Large and Small Traders , 2001 .

[20]  S. Baliga,et al.  Arms Races and Negotiations , 2002 .

[21]  D. Bergemann,et al.  Robust Mechanism Design , 2003 .

[22]  Pierpaolo Battigalli,et al.  Rationalization and Incomplete Information , 2003 .

[23]  S. Morris,et al.  Liquidity Black Holes , 2003 .

[24]  Peter Ockenfels,et al.  Measuring Strategic Uncertainty in Coordination Games , 2004, SSRN Electronic Journal.

[25]  Colin Camerer,et al.  A Cognitive Hierarchy Model of Games , 2004 .

[26]  R. Nagel,et al.  THE THEORY OF GLOBAL GAMES ON TEST: EXPERIMENTAL ANALYSIS OF COORDINATION GAMES WITH PUBLIC AND PRIVATE INFORMATION , 2004 .

[27]  Stephen Morris,et al.  Topologies on Types , 2005 .

[28]  A. Pavan,et al.  Signaling in a Global Game: Coordination and Policy Traps , 2002, Journal of Political Economy.

[29]  D. Fudenberg,et al.  Interim Correlated Rationalizability , 2007 .

[30]  A. Cabrales,et al.  Equilibrium selection through incomplete information in coordination games: an experimental study , 2007 .

[31]  Muhamet Yildiz,et al.  A Structure Theorem for Rationalizability with Application to Robust Predictions of Refinements , 2007 .

[32]  R. Aumann,et al.  Unraveling in Guessing Games : An Experimental Study , 2007 .

[33]  V. Crawford,et al.  Level-k Auctions: Can a Non-Equilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private-Value Auctions? , 2007 .

[34]  Pierpaolo Battigalli,et al.  Interactive Epistemology and Solution Concepts for Games with Asymmetric Information , 2008 .

[35]  Yi-Chun Chen,et al.  Uniform Topologies on Types , 2009 .

[36]  Pierpaolo Battigalli,et al.  Interactive Epistemology and Solution Concepts for Games with Asymmetric Information , 2008 .

[37]  W. Kets Bounded Reasoning and Higher-Order Uncertainty , 2012 .

[38]  A. Heifetz,et al.  All Types Naive and Canny , 2012 .

[39]  W. Kets Finite Depth of Reasoning and Equilibrium Play in Games with Incomplete Information , 2013 .

[40]  Miguel A. Costa-Gomes,et al.  Structural Models of Nonequilibrium Strategic Thinking: Theory, Evidence, and Applications , 2013 .