Ambiguous events and maxmin expected utility

We study the properties associated to various definitions of ambiguity [L.G. Epstein, J. Zhang, Subjective probabilities on subjectively unambiguous events, Econometrica 69 (2001) 265–306; P. Ghirardato et al., Differentiating ambiguity and ambiguity attitude, J. Econ. Theory 118 (2004) 133–173; K. Nehring, Capacities and probabilistic beliefs: a precarious coexistence, Math. Soc. Sci. 38 (1999) 197–213; J. Zhang, Subjective, ambiguity, expected utility and Choquet expected utility, Econ. Theory 20 (2002) 159–181] in the context of Maximin Expected Utility (MEU). We show that each definition of unambiguous events produces certain restrictions on the set of priors, and completely characterize each definition in terms of the properties it imposes on the MEU functional. We apply our results to two open problems. First, in the context of MEU, we show the existence of a fundamental incompatibility between the axiom of “Small unambiguous event continuity” (Epstein and Zhang, 2001) and the notions of unambiguous event due to Zhang (2002) and Epstein and Zhang (2001). Second, we show that, in the context of MEU, the classes of unambiguous events according to either Zhang (2002) or Epstein and Zhang (2001) are always λλ-systems. Finally, we reconsider the various definitions in light of our findings, and identify some new objects (Z-filters and EZ-filters) corresponding to properties which, while neglected in the current literature, seem relevant to us.