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Rent-Seeking and Litigation: The Hidden Virtues of Limited Fee Shifting

Emanuela Carbonara, Francesco Parisi and Georg von Wangenheim
From the journal Review of Law & Economics

Abstract

In the past couple of decades, scholars have predominantly employed rent-seeking models to analyze litigation problems. In this paper, we build on the existing literature to show how alternative fee-shifting arrangements (e.g., the American rule and English rule with limited fee-shifting) affect parties’ litigation expenditures and their decisions to litigate. Contrary to the prevailing wisdom, we discover that, when fee shifting is limited, the English rule presents some interrelated advantages over the American rule, including the reduction of litigation rates and the possible reduction of expected litigation expenditures. Our results unveil a hidden virtue of limited fee shifting, showing that an increase in such limit may lead to a desirable sorting of socially valuable litigation.

JEL Classification: C72; D72; K41

Appendix A: Subgame-Perfect Nash Equilibrium Under the American Rule

Following Farmer and Pecorino (1999), we show first that for r<1, if the parties reach the litigation stage, investment levels X and Y in [7] are a Nash equilibrium since neither party is willing to deviate. This occurs when exerting the equilibrium effort is at least as good as not exerting any effort (either X=0 or Y=0). The plaintiff is not willing to deviate if ΠP(X,Y)0, i.e., from [8], μV(1+μ)1r1+μ0, which reduces to μr1.

Similarly, the defendant is not willing to deviate if μV(1+μ)2(1+μ+r)<V, i.e., if and only if 1+μ(1r)>0, that is if and only if μ<1r1.

Case 1. r1. It is possible to see that 1+μ(1r)>0 always and that r1<0 if r1. Hence neither parties are willing to deviate and, in case of litigation, payoffs will be ΠP(X,Y) and ΠD(X,Y). If the plaintiff files, the defendant will defend. Knowing that the defendant always litigates, the plaintiff chooses to file.

Thus, when r1, the litigation game has a unique subgame perfect equilibrium in which litigation always occur.

Case 2. r>1. In this case, both r1 and 1r1 are positive, and neither party is willing to deviate if and only if r1<μ<1r1. We now need to distinguish between 1<r2 and r2.

Case 2.1: 1<r2. In this case, r1<1r1. Hence, there exists a range of values for μ such that r1<μ<1r1 and the Nash Equilibrium X,Y exists. In stage 2, the defendant defends if the plaintiff files and, in stage 1, the plaintiff prefers to file.

If μ<r1<1r1, the plaintiff is willing to deviate in the litigation game, hence the Nash equilibrium does not exist. Following Farmer and Pecorino (1999), we assume that the player willing to play the Nash equilibrium solution has a first-mover advantage in a Stackelberg litigation game. If litigation occurs, the player with the first mover advantage sets her legal expenditure at a preemptive level, which is the lowest expenditure that induces the other party to invest 0. In this case, given that the plaintiff is willing to deviate, in the litigation game the defendant has a first-mover advantage. Since PP(0,Y)=0, the plaintiff’s payoff is zero, hence, in stage 1, she chooses not to litigate.

Finally, if μ>1r1, the defendant is willing to deviate in the litigation game. Here the plaintiff would have a first-mover advantage and the defendant would get V. Then the defendant would not defend and in stage 1 the plaintiff would file.

Case 2.2: r>2. In this case, r1>1r1. Again we can have three cases.

If μ>r1>1r1, the defendant is willing to deviate from the litigation Nash equilibrium whereas the plaintiff does not. The plaintiff therefore has a first-mover advantage. As in the previous case the defendant chooses not to defend and, in stage 1, the plaintiff files.

If μ<1r1, the defendant has a first-mover advantage in the litigation game. At stage 1, the plaintiff does not file.

The last case is the most problematic. If 1r1<μ<r1, both the plaintiff and the defendant are willing to deviate from the Nash equilibrium strategies in the litigation game. Therefore it is difficult to determine which player might have the first-mover advantage in a Stackelberg game. In this case, one of the players might be able to commit to a preemptive level of expenditure (so that we are in one of the scenarios described above).

Appendix B: Subgame-Perfect Nash Equilibria Under the English Rule with Limited Fee – Shifting

B.1 Proof of impossibility of Yd<X

We show the impossibility by contradiction. For this, we first assume that both inequalities hold true. Based on the consequential definitions of the parties’ payoffs, we then show that these investments cannot be a Nash equilibrium.

Suppose Yd<X. Then

ΠP=V+dXYr(V+d+Y)Yr+μXrandΠD=μXr(d+V+Y)Yr+μXr

With these payoffs, we get the following first order condition for the defendant:

0=μXr(d+V+Y)rYr1Yr+μXr2μXrYr+μXr=μXrYr+μXr(d+V+Y)rYr1Yr+μXr1

which simplifies to

(35)1=rYr1(d+V+Y)Yr+μXr

It is useful to write this as:

(36)μXr=Y1+r(dr+rVY+rY),

The defendant’s second order condition is:

0μXr(d+V+Y)r(r1)Yr2Yr+μXr2rY2(r1)Yr+μXr3+2μXrrYr1Yr+μXr2=μrXrYr2Yr+μXr3[r(d+V+Y)Yr+μXr(d+VY)Yr+μXr]

If we replace μXr as suggested by eq. [36] inside the brackets, this turns into:

0μrXrYr2Yr+μXr3[r(d+V+Y)Yr+Y1+r(dr+rVY+rY)(d+VY)Yr+Y1+r(dr+rVY+rY)]=μrXrY2r3Yr+μXr3(r1)r(d+V+Y)2

Hence, Yd<X can only be an equilibrium if r<1.

We now turn to the plaintiff’s foc. It is given by:

(37)0=1+rYr(d+V+Y)μX1+rYr+Xrμ2

Inserting eq. [36] yields:

0=1+YμX1+rYr+μXr=1+YXYr(d+V+Y)

Then by our initial assumption we have:

d<X=Y1Yr(d+V+Y)

which implies

(38)0<rd2drV+rVYY2+rY2

As we already know that r<1, inequality [38] also implies

0<rd2drV+rVY

or

Y>dd+VV>d

which contradicts the initial assumption. Hence, there is no equilibrium with Yd<X. By symmetry, we also cannot have an equilibrium with Xd<Y. q.e.d.

B.2 Equilibrium with X,Y>d

We first check whether investment levels Xˆ,Yˆ>d defined by the first-order conditions of maximizing ΠP(X,Y) and ΠD(X,Y) as defined by eqs [13] and [14] constitute a Nash equilibrium of the litigation game. We know from standard rent-seeking theory that it is not sufficient to consider the first-order conditions but that investing nothing may be a better alternative. However, the plaintiff is not willing to deviate if exerting effort X˜ when the defendant invests Y˜ yields a higher payoff than exerting no effort (X=0). This happens when ΠP(Xˆ,Yˆ)>d, that is, when 1+μr>0. The latter inequality is always satisfied for r1. For r>1, it is required that the case presents μ>r1.

Similarly, the defendant does not deviate if and only is ΠD(Yˆ,Xˆ)>Vd, which implies 1+μ(1r)>0, which is always satisfied for r<1 and requires μ<1r1 if r>1.

Knowing that his payoff will be ΠD(Yˆ,Xˆ) in the litigation stage, the defendant is willing to defend in court if and only if ΠD(Xˆ,Yˆ)>V, i.e., if and only if d<d˜D which we can invert to

(39)μ<μˆD(d)(1r)V2dr+(2dr+(r1)V)2+4d(d+V)2d

The plaintiff, on the other hand, is willing to file if and only if the defendant will not defend her case in court or ΠP(Xˆ,Yˆ)>0, that is d<dˆP which we can invert to

(40)μ>μ^P(d)2dr+(r1)V+(2dr+(r1)V)2+4d(d+V)2(d+V)=1/μ^D(d)

Equations [39] and [40] represent respectively the defendant’s and the plaintiff’s participation constraints. When they are satisfied, investment levels Xˆ,Yˆ>d constitute a subgame-perfect Nash equilibrium of the litigation game.[28]

To see that for r1/2 we have dˆD>dˆ we note that the denominator of dˆ is always positive and the same is true for the denominator of dˆD if and only if μ>1+r2r<1. Hence, if the latter condition is violated, dˆD=>dˆ. If the condition holds true, dˆD>dˆ is equivalent to

(41)rμ1/μ+2r=1+1/μr1+μ1+1/μ2r

which after some algebra reduces to r<12+12μ which is satisfied by assumption. Since dˆP is the same as dˆD with μ replaced by 1/μ and dˆ does not change when we replace μ by 1/μ, exactly the same argument also proves that dˆP>dˆ if r1/2. Obviously, if r>1/2, the argument fails if μ or, respectively 1/μ is large enough. And if r>1, the argument for dˆD is reversed for all μ1 and the argument for dˆP is reversed for all μ1, whence min[dˆD,dˆP]<dˆ for all μ.

B.3 Equilibrium with X,Y<d

We prove first that, for r<1, investment levels X˜ and Y˜ obtained in [24] are always Nash equilibria of the litigation game. Following the same procedure adopted for the American rule in Appendix A, the plaintiff does not deviate from the equilibrium X˜,Y˜ if and only if ΠP(X˜,Y˜)>Y˜, which follows from the fact that the plaintiff has to pay the defendant’s legal fees even if her own expenditures are X=0. This implies that the plaintiff does not deviate if and only if μ11r(1r)>0, which is always true if r<1. Similarly, the defendant does not deviate if and only is ΠD(Y˜,X˜)>VX˜, which implies μ11r<1+μ11r, which is always satisfied for r<1. However, as shown in the text, the equilibrium fails to exist if r1.

The rest of the argument is in the text.

B.4 Equilibrium with X=Y=d

To prove the conditions under which X=Y=d is an equilibrium, we first show that increasing the expenditures beyond d reduces the plaintiff’s payoff if ddˆ. We then turn to the condition ddˉ. The argument for the defendant is symmetric and need not be made explicit here.

To see that the plaintiff will not increase X above d when ddˆ, we first consider her payoff

(42)ΠP(X,d)Xd=μXrYr+μXr(V+2d)Xd

for Xd and its first derivative of at X=Y=d:

ΠP(X,Y)XX=Y=d=rμVd((1+μ)22μr)d(1+μ)20duetoddˆ

with equality only for d=dˆ. The plaintiff will thus not increase his expenditures marginally above d.

To show that the plaintiff will not invest any much higher amount, we consider the second derivative of her payoff:

2ΠP(X,Y)X2=rμ(2d+V)Xr2)Yr(Yr+μXr)3(1r)Yr+(1+r)μXr

Obviously, this is strictly negative if r1. If r>1, the second derivative is positive for X=0 and eventually becomes negative as X increases. Leaving aside the restriction that the definition in eq. [42] only is valid for Xd, one can easily check that the right-hand side of [42] becomes d when X=0. Hence this right-hand side must have been increasing in some range before X reaches d. Since the first derivative is negative at X=d, the derivative must have been decreasing, i.e. the second derivative must have been negative somewhere in the range 0<X<d. However, a negative second derivative for any value of X implies that the second derivative is negative for all larger values of X too. As a consequence, (2ΠP(X,d))/(X2)<0 for all Xd which also implies that (ΠP(X,d))/(X)<0 for all Xd. Hence ΠP(X,d)<ΠP(d,d) for all X>d whence the plaintiff will never increase his investments above d when ddˆ. We can make a corresponding argument for the defendant to show that he will never increase her investments above d when ddˆ.

We now turn to the claim that neither of the parties will reduce expenditures below d if ddˉ. For this proof it is helpful to transform the plaintiff’s first-order condition for a payoff maximum with X,Y<d

(43)ΠP(X,Y)X=μrYrXr1(V+X+Y)Yr(Yr+μXr)(Yr+μXr)2=0

into

(44)Yr=μXr1(r(V+X+Y)X).

Inserting this into the second derivative

(45)2ΠP(X,Y)X2=rμXr2Yr(Yr+μXr)3r(V+X+Y)(Yr+μXr)+(V+XY)(Yr+μXr)

implies

(46)2ΠP(X,Y)X2=rμXr2Yr(Yr+μXr)3(1+r)rμXr1(V+X+Y)2

Hence, for r<1, there is no payoff minimum for the plaintiff in 0X<d if Yd.

Since for equal investments the derivative

(47)ΠP(X,Y)XX=Y=rμVY(1++μ2rμ)Y(1+μ)2

is positive for Yddˉ and strictly so, if Y<d, the derivative is also strictly positive for all X<Y and thus the plaintiff will always invest more than the defendant if Y<d and not less than the defendant if Y=d if r<1.

If r1, any extremum in 0X<d cannot be a maximum. Due to Yddˉ we get after some algebra ΠP(0,Y)=Y<ΠP(Y,Y)=μ1+μ(V+2Y)2Y. Hence the plaintiff will again invest not less than the defendant and due to inequality [47] she will invest more unless Y=d.

By a symmetric argument for the defendant, we find that he also invests more than the plaintiff if X<d and the same amount if X=d.

Suppose that the defendant invests Yd. Then for Xd we have

(48)ΠP(X,Y)X=μrYrXr1(V+X+Y)Yr(Yr+μXr)(Yr+μXr)2

which for X=Y simplifies to

(49)ΠP(X,Y)XX=Y=rVY(1+1/μ2r)Y(1+μ)(1+1/μ)

which is obviously positive if 1+1/μ2r0 and positive due to Yddˉ if 1+1/μ2r>0 by the following argument:

μ max [ 1 , 2 r 1 ] implies that YrV1+μ2rrV1+1/μ2r which implies that the derivative is positive and strictly so, if Y<d.

2 r 1 > μ > 1 2 r 1 implies that 1+1/μ2r<0 whence the derivative is positive.

μ min 1 , 1 2 r 1 implies that YrV1+1/μ2r which implies that the derivative is positive and strictly so, if Y<d.

By a corresponding argument, we also get

(50)ΠD(Y,X)YY=X=rVY(1+μ2r)Y(1+μ)(1+1/μ)0

with strict inequality for X<d.

B.5 Equilibrium with X<Y=d for μ<1 or Y<X=d for μ>1

To prove the various claims of Section 4.4, we first recall that according to eq. [48] (ΠP(X,d))/(X)|X=d is positive if ddˆ. This sufficient condition was also necessary, if μ<1, i.e. if the plaintiff is the weaker party (cf. the last alternative in the discussion of eq. [48]). Hence if d>dˆ and the plaintiff is the weaker party, we have (ΠP(X,d))/(X)|X=d<0 and thus the best reply of the plaintiff to the defendant’s choice of d is some X<d (X>d is excluded by Appendix B.1). The first-order condition for the optimal X is given by

(51)ΠP(X,d)X=drX(dr+μXr)2μrXr(V+X+d)X(dr+μXr)=0

We call this solution X˜(d).

Since (ΠP(X,d))/(X)|X=d<0 and limX0(ΠP(X,d))/(X)=+ and the derivative does not display any discontinuity between zero and d, X˜(d) exists. Since the second derivative at X=X˜(d) can easily be shown to be negative for r<1 the solution X˜(d) is unique.

To make sure that (X˜(d),d) is a Nash equilibrium, we have to prove that d is the defendant’s best reply to (X˜(d). We know from the previous appendix, that μ<1 implies

(52)ΠD(Y,d)YY=d>0=ΠP(X,d)XX=difd=dˆ=rV1+1/μ2r

By continuity, the inequality prevails if d slightly increases above dˆ. We thus have

(53)ΠD(Y,X)Y=μXr(rYr1(V+X+Y)YrμXr)(Yr+μXr)2>0

Hence we have Xr<rYr1(V+X+Y)Yr/μ. Inserting this into the second derivative

2ΠD(Y,X)Y2=Yr1μXr(Yr+μXr)2r(VX+Y)Yr2(V+X+Y)(YrμXr)Y(Yr+μXr),

which strictly increases in Xr, yields

(54)2ΠD(Y,X)Y2<Yr2μXr(Yr+μXr)2(1+r)r(V+X+Y)<0

Hence, reducing Y slightly results in a larger and thus still positive first derivative. Reducing Y further step by step always results in ever larger first derivatives and thus negative second derivatives by exactly the argument of eqs [53] through [54]. Hence, the defendant always gains by increasing his investment until he invests d, which completes the proof of d being the best reply to X˜(d).

The symmetric argument works for μ>1 and thus the plaintiff being the stronger party.

To prove the payoffs of eqs [30] and [31], note that eq. [51] implies dr+μX˜(d)r=μrX˜(d)r1(V+X˜(d)+d). Inserting this into the plaintiff’s payoff for μ<1, yields:

(55)ΠP(X˜(d),d)=μX˜(d)rdr+μX˜(d)r(V+X˜(d)+d)X˜(d)d=1rrX˜(d)d

The other expressions in eqs [30] and [31] can be derived accordingly.

Finally, we show that for r1/2, μ<1 implies ΠD(d,X˜(d))>V. Suppose the reverse were true, i.e. 1rX˜(d)dVd. We could then insert this into

(56)0=μVd+1μ(1r)X˜(d)dX˜(d)d1r

which is but a rewritten form of eq. [51] and obviously declines in X˜(d)d. Hence, this would imply

(57)0μVd+1μ(1r)rVdrVd1r

and thus

(58)μ>1rdd+rVrVdr>1rdd+rV

where the last inequality follows from the fact that for r1/2 the equilibrium with (X˜(d),d) may only occur for d<rV. However we know from d>dˉ that μ<1/(rVd+2r1). Some simple algebra shows that this is compatible with inequality [58] only if d>rV. Hence ΠD(d,X˜(d))>V must be true. The proofs for the other claims in the same paragraph follow the same structure and are omitted here.

Acknowledgment

We thank Daniel Pi and Samuel Brylski for their most valuable research and editorial assistance. We are grateful to Giuseppe Dari-Mattiacci, Giuseppe Di Vita, Ted Eisenberg, Luigi Alberto Franzoni, Eric Langlais, Giovanni Battista Ramello, Filippo Roda, Avraham Tabbach, two anonymous referees and to participants to the 8th SIDE – ISLE Conference (Italian Society of Law and Economics) in Rome, to the Workshop In Law and Economics in Hamburg, February 2013, to the 2nd International Workshop on the Economics Analysis of Litigation, Catania, June 2014 and to seminars at the University of Paris Ouest, Nanterre, Dept. of Economics, at the University of Bologna, Dept. of Economics and at the Bucerius Law School, Hamburg, for helpful discussions.

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Published Online: 2015-06-30
Published in Print: 2015-07-01

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