## 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.

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

Following Farmer and Pecorino (1999), we show first that for

Similarly, the defendant is not willing to deviate if

**Case 1**.

Thus, when

**Case 2**.

**Case 2.1**:

If

Finally, if

**Case 2.2**:

If

If

The last case is the most problematic. If

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

### B.1 Proof of impossibility of Y ≤ d < 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

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

Hence,

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

Inserting eq. [36] yields:

Then by our initial assumption we have:

which implies

As we already know that

or

which contradicts the initial assumption. Hence, there is no equilibrium with

### B.2 Equilibrium with X , Y > d

We first check whether investment levels

Similarly, the defendant does not deviate if and only is

Knowing that his payoff will be

The plaintiff, on the other hand, is willing to file if and only if the defendant will not defend her case in court or

Equations [39] and [40] represent respectively the defendant’s and the plaintiff’s participation constraints. When they are satisfied, investment levels ^{[28]}

To see that for

which after some algebra reduces to

### B.3 Equilibrium with X , Y < d

We prove first that, for

The rest of the argument is in the text.

### B.4 Equilibrium with X = Y = d

To prove the conditions under which *d* reduces the plaintiff’s payoff if

To see that the plaintiff will not increase *X* above *d* when

for

with equality only for *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 *X* increases. Leaving aside the restriction that the definition in eq. [42] only is valid for *X* reaches *d*. Since the first derivative is negative at *X* implies that the second derivative is negative for all larger values of *X* too. As a consequence, *d* when *d* when

We now turn to the claim that neither of the parties will reduce expenditures below *d* if

into

Inserting this into the second derivative

implies

Hence, for

Since for equal investments the derivative

is positive for

If

By a symmetric argument for the defendant, we find that he also invests more than the plaintiff if

Suppose that the defendant invests

which for

which is obviously positive if

By a corresponding argument, we also get

with strict inequality for

### 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] *d* is some *X* is given by

We call this solution

Since *d*,

To make sure that *d* is the defendant’s best reply to

By continuity, the inequality prevails if *d* slightly increases above

Hence we have

which strictly increases in

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

The symmetric argument works for

To prove the payoffs of eqs [30] and [31], note that eq. [51] implies

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

Finally, we show that for

which is but a rewritten form of eq. [51] and obviously declines in

and thus

where the last inequality follows from the fact that for

## 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

©2015 by De Gruyter