A general theory of subjective mixtures

Abstract

We provide, and characterize behaviorally, a framework for constructing subjective mixtures which requires neither the Certainty Independence nor the Monotonicity axiom, replacing them with much weaker "local" properties. As we also show by means of examples, this framework provides a purely subjective foundation to most of the recent preference models which employ the Anscombe-Aumann setting. It also allows a subjective formulation of a preference for ambiguity hedging, and as a consequence allows the distinction of the notions of ambiguity aversion and preference for randomization.1 When we need to distinguish between the two papers, we will use GMMSa to refer to and GMMSb to refer to Ghirardato et al. (2003).2 There are, of course, many papers that characterize non-expected utility preferences in purely subjective environments, without explicitly constructing a mixture space structure. For instance, Casadesus-Masanell, Klibanoff, and Ozdenoren (2000) and Alon and Schmeidler (2014) for the Maxmin EU model and Nakamura (1990) for the Choquet EU model. 3 Monotonicity requires that if f (s) g (s) for every state s, then f g . Certainty Independence requires that for all acts f and g , all outcomes x ∈ X , and all λ 4 In Ghirardato and Pennesi ( 2019) we show that in those situations in which the space of outcomes has a productspace structure, subjective mixtures can be derived by an alternative method which makes no assumptions on the existing uncertainty.

Bayesian social aggregation with almost‐objective uncertainty

“…The original reference is Bewley (2002). 13 See Remark 1 of Ghirardato and Pennesi (2020) Theorem 2 establishes a relationship between the belief set underlying the GH representation of the social preferences and the belief sets of the individuals' GH representations. But Theorem 2 applies to any collection of Bewley preferences.…”

“…In fact, a convex X is not necessary to obtain these results. Recently, working in the Savage framework, and generalizing the work of Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), Ghirardato and Pennesi (2020) have shown that if has even one "locally biseparable event," then one can define a "subjective mixture" operation on X for . The aforementioned representation results can then be extended to any monotone, locally biseparable preference using this subjective mixture operation, yielding combined GH/Bewley representations for and .…”

“…Under certain conditions, there is a unique weak* compact, convex set and a utility function that yield both a generalized Hurwicz representation (3) for ≽, and a Bewley representation for ⊵, meaning that 12 In fact, a convex is not necessary to obtain these results. Recently, working in the Savage framework, and generalizing the work of Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), Ghirardato and Pennesi (2020) have shown that if ≽ has even one “locally biseparable event,” then one can define a “subjective mixture” operation on for ⪰. The aforementioned representation results can then be extended to any monotone, locally biseparable preference using this subjective mixture operation, yielding combined GH/Bewley representations for ≽ and ⊵ 13…”

“…1313 See Remark 1 of Ghirardato and Pennesi (2020) for details. However, different agents generally have different subjective mixture operations.…”