Symmetry axioms and perceived ambiguity

Peter Klibanoff,Sujoy Mukerji,Kyoungwon Seo

doi:10.1007/s11579-017-0191-2

Abstract

Since at least de Finetti (Annales de l’Institut Henri Poincare 7:1–68, 1937), preference symmetry assumptions have played an important role in models of decision making under uncertainty. In the current paper, we explore (1) the relationship between the symmetry assumption of Klibanoff et al. (KMS) (Econometrica 82:1945–1978, 2014) and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures, shown by KMS (2014) to reflect only perceived ambiguity, and the set of measures (which we will refer to as the Bewley set) developed by Ghirardato et al. (J Econ Theory 118:133–173, 2004), Nehring (Ambiguity in the context of probabilistic beliefs, working paper, 2001, Bernoulli without Bayes: a theory of utility-sophisticated preference, working paper, 2007) and Ghirardato and Siniscalchi (A more robust definition of multiple priors, working paper, 2007, Econometrica 80:2827–2847, 2012). This Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity. Regarding symmetry assumptions, we show that, under relatively mild conditions, a variety of preference symmetry conditions from the literature [including that in KMS (2014)] are equivalent. In KMS (2014), we showed that, under symmetry, the Bewley set and the set of relevant measures are not always the same. Here, we establish a preference condition, No Half Measures, that is necessary and sufficient for the two to be the same under symmetry. This condition is rather stringent. Only when it is satisfied may the Bewley set be interpreted as reflecting only perceived ambiguity and not also taste aspects such as ambiguity aversion.

Highlights

Recent literature on ambiguity or model uncertainty, makes heavy use of models of decision making under uncertainty
In the current paper we explore (1) the relationship between the symmetry assumption of KMS [21] and alternative symmetry assumptions in the literature, and (2) assuming symmetry, the relationship between the set of relevant measures developed by KMS [21] and the set of measures developed by Ghirardato et al [14], Nehring [24,25] and Ghirardato and Siniscalchi [15,16]
This latter relationship is of particular interest because this Bewley set is the main alternative offered in the literature as possibly representing perceived ambiguity

Summary

Introduction

Recent literature on ambiguity (meaning subjective uncertainty about probabilities over states) or model uncertainty (see e.g., surveys by Gilboa and Marinacci [17] and Marinacci [22]), makes heavy use of models of decision making under uncertainty. 4, shows that the possibilities of establishing foundations for practices concerning the way contraints on model uncertainty are incorporated in ambiguity models via theories invoking the Bewley set are limited. This is true in the case of functional forms which allow for non-extreme attitudes to ambiguity that may be varied parametrically in the functional representation (such as in the α-MEU model, the extended MEU with contraction model, and the smooth ambiguity model). In the α-MEU model, the Bewley set can be identified with the set of measures appearing in the model only if α is 0 or 1

Setting and preferences

Relating Event Symmetry to the literature

When do relevant measures and the Bewley set agree?

MEU model

Smooth ambiguity model