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428.  Selfconfirming Equilibrium and Model Uncertainty

by  Pierpaolo Battigalli, Simone Cerreia-Vioglio, Fabio Maccheroni and Massimo Marinacci

We propose to bring together two conceptually complementary ideas: (1) selfconfirming equilibrium (SCE): at rest points of learning dynamics in a game played recurrently, agents best respond to confirmed 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 Klibanoff, Marinacci and Mukerji (2005), we provide a definition 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 payoffs, but alternative strategies yield payoffs with unknown probabilities; keeping beliefs fixed, 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 different notions of equilibrium are nested in a simple way, from finer to coarser: Nash, Bayesian SCE, Smooth SCE and Waldean SCE. We also prove some equivalence results under special assumptions about the information structure.

Keywords: Selfconfirming equilibrium, conjectural equilibrium, uncertainty, smooth ambiguity.

JEL classification: C72, D80.

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Last updated February 15, 2012