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.
Appendix A: Subgame-Perfect Nash Equilibrium Under the American Rule
Following Farmer and Pecorino (1999), we show first that for , if the parties reach the litigation stage, investment levels and in  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 or ). The plaintiff is not willing to deviate if , i.e., from , , which reduces to .
Similarly, the defendant is not willing to deviate if , i.e., if and only if , that is if and only if .
Case 1. . It is possible to see that always and that if . Hence neither parties are willing to deviate and, in case of litigation, payoffs will be and . If the plaintiff files, the defendant will defend. Knowing that the defendant always litigates, the plaintiff chooses to file.
Thus, when , the litigation game has a unique subgame perfect equilibrium in which litigation always occur.
Case 2. . In this case, both and are positive, and neither party is willing to deviate if and only if . We now need to distinguish between and .
Case 2.1: . In this case, . Hence, there exists a range of values for such that and the Nash Equilibrium exists. In stage 2, the defendant defends if the plaintiff files and, in stage 1, the plaintiff prefers to file.
If , 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 , the plaintiff’s payoff is zero, hence, in stage 1, she chooses not to litigate.
Finally, if , the defendant is willing to deviate in the litigation game. Here the plaintiff would have a first-mover advantage and the defendant would get . Then the defendant would not defend and in stage 1 the plaintiff would file.
Case 2.2: . In this case, . Again we can have three cases.
If , 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 , 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 , 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
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 . Then
With these payoffs, we get the following first order condition for the defendant:
which simplifies to
It is useful to write this as:
The defendant’s second order condition is:
If we replace as suggested by eq.  inside the brackets, this turns into:
Hence, can only be an equilibrium if .
We now turn to the plaintiff’s foc. It is given by:
Inserting eq.  yields:
Then by our initial assumption we have:
As we already know that , inequality  also implies
which contradicts the initial assumption. Hence, there is no equilibrium with . By symmetry, we also cannot have an equilibrium with . q.e.d.
B.2 Equilibrium with
We first check whether investment levels defined by the first-order conditions of maximizing and as defined by eqs  and  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 when the defendant invests yields a higher payoff than exerting no effort (). This happens when , that is, when . The latter inequality is always satisfied for . For , it is required that the case presents .
Similarly, the defendant does not deviate if and only is , which implies , which is always satisfied for and requires if .
Knowing that his payoff will be in the litigation stage, the defendant is willing to defend in court if and only if , i.e., if and only if which we can invert to
The plaintiff, on the other hand, is willing to file if and only if the defendant will not defend her case in court or , that is which we can invert to
Equations  and  represent respectively the defendant’s and the plaintiff’s participation constraints. When they are satisfied, investment levels constitute a subgame-perfect Nash equilibrium of the litigation game.
To see that for we have we note that the denominator of is always positive and the same is true for the denominator of if and only if . Hence, if the latter condition is violated, . If the condition holds true, is equivalent to
which after some algebra reduces to which is satisfied by assumption. Since is the same as with replaced by and does not change when we replace by , exactly the same argument also proves that if . Obviously, if , the argument fails if or, respectively is large enough. And if , the argument for is reversed for all and the argument for is reversed for all , whence for all .
B.3 Equilibrium with
We prove first that, for , investment levels and obtained in  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 if and only if , which follows from the fact that the plaintiff has to pay the defendant’s legal fees even if her own expenditures are . This implies that the plaintiff does not deviate if and only if , which is always true if . Similarly, the defendant does not deviate if and only is , which implies , which is always satisfied for . However, as shown in the text, the equilibrium fails to exist if .
The rest of the argument is in the text.
B.4 Equilibrium with
To prove the conditions under which is an equilibrium, we first show that increasing the expenditures beyond d reduces the plaintiff’s payoff if . We then turn to the condition . 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 , we first consider her payoff
for and its first derivative of at :
with equality only for . 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:
Obviously, this is strictly negative if . If , the second derivative is positive for and eventually becomes negative as X increases. Leaving aside the restriction that the definition in eq.  only is valid for , one can easily check that the right-hand side of  becomes when . Hence this right-hand side must have been increasing in some range before X reaches d. Since the first derivative is negative at , the derivative must have been decreasing, i.e. the second derivative must have been negative somewhere in the range . 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, for all which also implies that for all . Hence for all whence the plaintiff will never increase his investments above d when . We can make a corresponding argument for the defendant to show that he will never increase her investments above d when .
We now turn to the claim that neither of the parties will reduce expenditures below d if . For this proof it is helpful to transform the plaintiff’s first-order condition for a payoff maximum with
Inserting this into the second derivative
Hence, for , there is no payoff minimum for the plaintiff in if .
Since for equal investments the derivative
is positive for and strictly so, if , the derivative is also strictly positive for all and thus the plaintiff will always invest more than the defendant if and not less than the defendant if if .
If , any extremum in cannot be a maximum. Due to we get after some algebra . Hence the plaintiff will again invest not less than the defendant and due to inequality  she will invest more unless .
By a symmetric argument for the defendant, we find that he also invests more than the plaintiff if and the same amount if .
Suppose that the defendant invests . Then for we have
which for simplifies to
which is obviously positive if and positive due to if by the following argument:
implies that which implies that the derivative is positive and strictly so, if .
implies that whence the derivative is positive.
implies that which implies that the derivative is positive and strictly so, if .
By a corresponding argument, we also get
with strict inequality for .
B.5 Equilibrium with for or for
To prove the various claims of Section 4.4, we first recall that according to eq.  is positive if . This sufficient condition was also necessary, if , i.e. if the plaintiff is the weaker party (cf. the last alternative in the discussion of eq. ). Hence if and the plaintiff is the weaker party, we have and thus the best reply of the plaintiff to the defendant’s choice of d is some ( is excluded by Appendix B.1). The first-order condition for the optimal X is given by
We call this solution .
Since and and the derivative does not display any discontinuity between zero and d, exists. Since the second derivative at can easily be shown to be negative for the solution is unique.
To make sure that is a Nash equilibrium, we have to prove that d is the defendant’s best reply to . We know from the previous appendix, that implies
By continuity, the inequality prevails if d slightly increases above . We thus have
Hence we have . Inserting this into the second derivative
which strictly increases in , yields
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  through . Hence, the defendant always gains by increasing his investment until he invests d, which completes the proof of d being the best reply to .
The symmetric argument works for and thus the plaintiff being the stronger party.
To prove the payoffs of eqs  and , note that eq.  implies . Inserting this into the plaintiff’s payoff for , yields:
The other expressions in eqs  and  can be derived accordingly.
Finally, we show that for , implies . Suppose the reverse were true, i.e. . We could then insert this into
which is but a rewritten form of eq.  and obviously declines in . Hence, this would imply
where the last inequality follows from the fact that for the equilibrium with may only occur for . However we know from that . Some simple algebra shows that this is compatible with inequality  only if . Hence must be true. The proofs for the other claims in the same paragraph follow the same structure and are omitted here.
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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