Anticomonotonicity for Preference Axioms: The Natural Counterpart to Comonotonicity

TL;DR

This paper explores anticomonotonicity as a natural counterpart to comonotonicity, revealing its role in strengthening classical models and providing new properties and models through AC restrictions.

Contribution

It introduces anticomonotonicity restrictions, showing they reinforce existing axioms and offer new insights and models in preference theory.

Findings

AC restrictions strengthen classical models like expected utility and no-arbitrage.

AC restrictions reveal where classical axioms are most critically tested.

Examples show AC can weaken some axioms and create new properties.

Abstract

Comonotonicity (``same variation'') of random variables minimizes hedging possibilities and has been widely used, e.g., in Gilboa and Schmeidler's ambiguity models. This paper investigates anticomonotonicity (``opposite variation''; abbreviated ``AC''), the natural counterpart to comonotonicity. It minimizes leveraging rather than hedging possibilities. Surprisingly, AC restrictions of several traditional axioms do not give new models. Instead, they strengthen the foundations of existing classical models: (a) linear functionals through Cauchy's equation; (b) Anscombe-Aumann expected utility; (c) as-if-risk-neutral pricing through no-arbitrage; (d) de Finetti's bookmaking foundation of Bayesianism using subjective probabilities; (e) risk aversion in Savage's subjective expected utility. In each case, our generalizations show where the critical tests of classical axioms lie: in the AC cases (maximal hedges). We next present examples where AC restrictions do essentially weaken existing axioms, and do provide new properties and new models.