Commutativity, comonotonicity, and Choquet integration of self-adjoint operators

In this work, we propose a definition of comonotonicity for elements of [Formula: see text], i.e. bounded self-adjoint operators defined over a complex Hilbert space [Formula: see text]. We show that this notion of comonotonicity coincides with a form of commutativity. Intuitively, comonotonicity is to commutativity as monotonicity is to bounded variation. We also define a notion of Choquet expectation for elements of [Formula: see text] that generalizes quantum expectations. We characterize Choquet expectations as the real-valued functionals over [Formula: see text] which are comonotonic additive, [Formula: see text]-monotone, and normalized

Divergent platforms

Models of electoral competition between two opportunistic, office-motivated parties typically predict that both parties become indistinguishable in equilibrium. I show that this strong connection between the office motivation of parties and their equilibrium choice of identical platforms depends on two—possibly false—assumptions: (1) Issue spaces are uni-dimensional and (2) Parties are unitary actors whose preferences can be represented by expected utilities. I provide an example of a two-party model in which parties offer substantially different equilibrium platforms even though no exogenous differences between parties are assumed. In this example, some voters’ preferences over the 2-dimensional issue space exhibit non-convexities and parties evaluate their actions with respect to a set of beliefs on the electorate

Making the Anscombe-Aumann approach to ambiguity suitable for descriptive applications

The Anscombe-Aumann (AA) model, originally introduced to give a normative basis to expected utility, is nowadays mostly used for another purpose: to analyze deviations from expected utility due to ambiguity (unknown probabilities). The AA model makes two ancillary assumptions that do not refer to ambiguity: expected utility for risk and backward induction. These assumptions, even if normatively appropriate, fail descriptively. This paper relaxes these ancillary assumptions to avoid the descriptive violations, while maintaining AA\xe2\x80\x99s convenient mixture operation. Thus, it becomes possible to test and apply all AA-based ambiguity theories descriptively while avoiding confounds due to violated ancillary assumptions. The resulting tests use only simple stimuli, avoiding noise due to complexity. We demonstrate the latter in a simple experiment where we find that three assumptions about ambiguity, commonly made in AA theories, are violated: reference independence, universal ambiguity aversion, and weak certainty independence. The second, theoretical, part of the paper accommodates the violations found for the first ambiguity theory in the AA model\xe2\x80\x94Schmeidler\xe2\x80\x99s CEU theory\xe2\x80\x94by introducing and axiomatizing a reference dependent generalization. That is, we extend the AA ambiguity model to prospect theory