Using Mathematica’s Global Optimization Techniques To Understand Legal Rules

Towards A Better Liability Insurance Contract

Seth J. Chandler

University of Houston Law Center

Abstract

The structure of optimal insurance contracts has been the subject of some scrutiny in the economic analysis of law. Existing studies generally suffer, however, from two crippling assumptions. First, they assume that utility functions are concave over their entire domain. While this may be true if only positive levels of wealth are considered, it is almost certainly not true once one admit the possibility of negative levels of wealth—situations in which liabilities exceed assets. Yet it is precisely over this negative and positive domain that liability insurance is intended to operate. Second, most studies have assumed that the insurer’s obligation under an insurance contract is unable to be conditioned upon the statistical distribution of losses for the particular adverse event that materializes. Instead, they have assumed that insurance contracts must exist as they tend now to be written: offering a formula indemnity whose only variable is the size of the ultimate loss visited upon the insured.

This article takes advantage of modern numerical optimization techniques to begin removing these limitations.  Building on the new optimization techniques contained in Mathematica, proprietary add-on packages, and recent work on “particle swarms,” the article studies whether insureds would be likely to prefer a policy that was predetermined to have “per occurrence” limits that varied according to projected characteristics of any lawsuit filed against the insured. The article finds that, in theory, insureds would prefer such a distribution-variant insurance contract, provided the transaction costs in measuring the relevant features of the distribution involved were not particularly high. It also asserts that court decisions and statutes have come up with a crude and low-cost version of this idea already. By modifying current liability insurance contracts, such that the nominal limit effectively does not apply if the plaintiff has made a settlement offer that would resolve the case by the insurer paying no more than those nominal limits, the court in fact creates a crude but often rational surrogate for the sort of distribution-variant contract the insured would in fact prefer. The study of distribution-variant insurance contracts thus suggests how courts might sculpt the duty to settle to better accomodate the interests of the insured for whom the pure system has too-high transaction costs. The Mathematica algorithms developed here also establish several little-known facts about optimal insurance policies and show some of the comparative strengths and weaknesses of various numerical optimization methods. It appears unlikely that analytic methods are capable of coping with a problem of this complexity.