# Pairwise counter-monotonicity

## TL;DR

This paper explores pairwise counter-monotonicity, an extremal negative dependence concept, establishing its properties, relationships with other dependence structures, and its significance in risk sharing among non-risk-averse agents.

## Contribution

It introduces a stochastic representation and invariance property for pairwise counter-monotonicity, linking it to negative association, joint mix dependence, and risk sharing problems.

## Key findings

- Pairwise counter-monotonicity implies negative association.
- It is equivalent to joint mix dependence under certain conditions.
- This dependence structure is crucial in optimal risk sharing for non-risk-averse agents.

## Abstract

We systematically study pairwise counter-monotonicity, an extremal notion of negative dependence. A stochastic representation and an invariance property are established for this dependence structure. We show that pairwise counter-monotonicity implies negative association, and it is equivalent to joint mix dependence if both are possible for the same marginal distributions. We find an intimate connection between pairwise counter-monotonicity and risk sharing problems for quantile agents. This result highlights the importance of this extremal negative dependence structure in optimal allocations for agents who are not risk averse in the classic sense.