Selfconfirming Equilibrium and Model Uncertainty

We propose to bring together two conceptually complementary ideas: (1) selfcon…rming equilibrium (SCE): at rest points of learning dynamics in a game played recurrently, agents best respond to con…rmed beliefs, i.e., beliefs consistent with the evidence they accumulated, and (2) ambiguity aversion: agents, other things being equal, prefer to bet on events with known rather than unknown probabilities and, more generally, distinguish objective from subjective uncertainty, a behavioral trait captured by their ambiguity attitudes. Using as a workhorse the “smooth ambiguity” model of Klibano¤, Marinacci and Mukerji (2005), we provide a de…nition of “Smooth SCE”which generalizes the traditional concept of Fudenberg and Levine (1993a,b), here called Bayesian SCE, and admits Waldean (maxmin) SCE as a limit case. We show that the set of equilibria expands as ambiguity aversion increases. The intuition is simple: by playing the same strategy in a stable state an agent learns the implied objective probabilities of payo¤s, but alternative strategies yield payo¤s with unknown probabilities; keeping beliefs …xed, increased aversion to ambiguity makes such strategies less appealing. In sum, by combining the SCE and ambiguity aversion ideas a kind of “status quo bias”emerges: in the long run, the uncertainty related to tested strategies disappears, but the uncertainty implied by the untested ones does not. We rely on this core intuition to show that di¤erent notions of equilibrium are nested in a simple way, from …ner to coarser: Nash, Bayesian SCE, Smooth SCE and Waldean SCE. We also prove some equivalence results under special assumptions about the information structure. Keywords: Selfcon…rming equilibrium, conjectural equilibrium, model uncertainty, smooth ambiguity. JEL classification: C72, D80. Chi lascia la via vecchia per la via nuova, sa quel che perde ma non sa quel che trova1 Part of this research was done while the …rst author was visiting the Stern School of Business of New York University, which he thanks for its hospitality. We thank Nicodemo De Vito, Ignacio Esponda, Eduardo Feingold, Faruk Gul, Johannes Hörner, Yuchiro Kamada, Wolfgang Pesendorfer and Bruno Strulovici for some useful discussions, as well as seminar audiences at Napoli, NYU, Penn, Princeton and Yale. The authors gratefully acknowledge the …nancial support of the European Research Council (advanced grant BRSCDP-TEA). 1 Italian proverb “Those who leave the old road for a new one, know what they leave but do not what they will …nd.”