Efficiency and Equilibria in Games of Optimal Derivative Design

TL;DR

This paper analyzes optimal derivative design in competitive markets with heterogeneous agents, addressing discontinuities in payoffs, and establishes existence of equilibria and efficient allocations for profit-maximizing and risk-minimizing firms.

Contribution

It extends existing models to a multi-firm setting, introduces tie-breaking rules for discontinuities, and proves existence of Nash equilibria and socially efficient allocations.

Findings

Existence of mixed-strategies Nash equilibria in profit-maximizing firms.

Existence of socially efficient allocations in risk-minimizing firms.

Special tie-breaking rule for entropic risk measure sharing the market proportionally.

Abstract

In this paper the problem of optimal derivative design, profit maximization and risk minimization under adverse selection when multiple agencies compete for the business of a continuum of heterogenous agents is studied. The presence of ties in the agents' best-response correspondences yields discontinuous payoff functions for the agencies. These discontinuities are dealt with via efficient tie--breaking rules. In a first step, the model presented by Carlier, Ekeland & Touzi (2007) of optimal derivative design by profit-maximizing agencies is extended to a multiple--firm setting, and results of Page & Monteiro (2003, 2007, 2008) are used to prove the existence of (mixed-strategies) Nash equilibria. On a second step we consider the more complex case of risk minimizing firms. Here the concept of socially efficient allocations is introduced, and existence of the latter is proved. It is also shown that in the particular case of the entropic risk measure, there exists an efficient "fix--mix" tie-breaking rule, in which case firms share the whole market over given proportions.