Comparing All-or-Nothing and Proportionate Damages: A Rent-Seeking Approach

Jef De Mot; Thomas J. Miceli

doi:10.1515/rle-2014-0058

Published by De Gruyter February 21, 2015

Comparing All-or-Nothing and Proportionate Damages: A Rent-Seeking Approach

Jef De Mot and Thomas J. Miceli

From the journal Review of Law & Economics

https://doi.org/10.1515/rle-2014-0058

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Abstract

This paper compares the all-or-nothing and proportionate damage rules for allocating damages in tort cases under evidentiary uncertainty. The focus is on how the two rules affect litigation expenditures by plaintiffs and defendants. The results of simulation experiments show that the expected judgment at trial is higher under the all-or-nothing rule for cases where the defendant did not take adequate care, but the judgment is higher under the proportionate rule when the defendant took more than adequate care. As for litigation expenditures, assuming equal costs of litigation, overall expenditures are higher under the all-or-nothing rule, except for very weak and very strong cases.

Keywords: All-or-nothing rule; proportionate damages; litigation costs; rent-seeking

JEL Classification: K13; K41

Appendix

With respect to the all-or-nothing rule, the first-order conditions for X and Y are:

FXF+(1−F)YJ−XF2(XF+(1−F))2J=CpXF(1−F)(XF+(1−F)Y)2J=Cd

From these conditions, it follows that

Y∗=CpCdX∗

Putting this result back into the first-order conditions gives us the equilibrium expenditures for the plaintiff and defendant, respectively:

CpX∗=CpF(1−F)CdF+(1−F)CpCd2JCdY∗=CpF(1−F)CdF+(1−F)CpCd2J

The expected value of trial for the plaintiff under this rule is

PX∗,Y∗J−CpX=FF+(1−F)CpCdJ−CpF(1−F)CdF+(1−F)CpCd2J

while the expected cost of trial for the defendant is

PX∗,Y∗J+CdY∗=FF+(1−F)CpCdJ+CpF(1−F)CdF+(1−F)CpCd2J

Under the proportional rule, the antiderivative of

P0XFP0XF+(1−P0)Y(1−F)

equals

XF((XF−Y(1−F))P0−Y(1−F)ln((XF−Y(1−F))P0+Y(1−F))(XF−Y(1−F))2

The first-order conditions are:

2XF2−F(1−F)Y+YF(1−F)(ln(Y(1−F)−ln(XF)))−YF(1−F)(XF−Y(1−F))2J−2XF(XF−Y(1−F))+XYF(1−F)(ln(Y(1−F)−ln(XF))(XF−Y(1−F))3FJ=Cp−XF(1−F)+XF(1−F)(ln(Y(1−F)−ln(XF)))+XF(1−F)(XF−Y(1−F))2J+2XF(XF−Y(1−F))+XYF(1−F)(ln(Y(1−F)−ln(XF))(XF−Y(1−F))3(1−F)J=−Cd

from which it follows that

Y∗∗=CpCdX∗∗

Putting this result back in the first-order conditions gives us the Nash equilibrium effort levels of the two parties:

X∗∗=2F−CpCd(1−F)+F+CpCd(1−F)lnCpCd1−FFCp(1−F)CpCd−F3CpCdF(1−F)JY∗∗=2F−CpCd(1−F)+F+CpCd(1−F)lnCpCd1−FFCd(1−F)CpCd−F3CpCdF(1−F)J

The resulting total expenditures of the plaintiff and the defendant, respectively, are

CpX∗∗=2F−CpCd(1−F)+F+CpCd(1−F)lnCpCd1−FF(1−F)CpCd−F3CpCdF(1−F)JCdY∗∗=2(F−CpCd(1−F))+(F+CpCd(1−F))lnCpCd1−FF((1−F)CpCd−F)3CpCdF(1−F)J

The expected judgment equals:

E(J)=F+(1−F)CpCdlnCpCd1−FF−1F−(1−F)CpCd2FJ

The expected value of trial for the plaintiff is

F+(1−F)CpCd(lnCpCd1−FF−1)F−(1−F)CpCd2FJ−2F−CpCd(1−F)+F+CpCd(1−F)lnCpCd1−FF(1−F)CpCd−F3CpCdF(1−F)J

while the expected loss of the defendant equals:

F+(1−F)CpCdlnCpCd1−FF−1F−(1−F)CpCd2FJ+2F−CpCd(1−F)+F+CpCd(1−F)lnCpCd1−FF(1−F)CpCd−F3CpCdF(1−F)J

Acknowledgement

We acknowledge the helpful comments of a reviewer and Francesco Parisi (the editor).

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Published Online: 2015-2-21

Published in Print: 2015-3-1

Comparing All-or-Nothing and Proportionate Damages: A Rent-Seeking Approach

Abstract

Appendix

Acknowledgement

Bibliography

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