and push for a high settlement offer. We find, in contrast, the Benson’s 1979 assertion above, that taking bargaining costs into account improves the plaintiff’s lawyer’s
credibility when rejecting low offers: her rejections are not costless the “hard bargain- ing effect”. However, the effects of more high offers on settlement timing can be offset
by the fact that hard bargaining means rejecting low offers, so that the overall settle- ment timing results are ambiguous. Further, the plaintiff often fails to benefit from this
strategy because his lawyer attempts to offload the bargaining costs faced through 1 the settlement offer accepted the “cost effect”, and 2 the contingent fee required to
cover her opportunity costs. Thus, although the model confirms the potential for hard bargaining under contingent fees, it highlights several forces that mitigate against any
benefits this might have for the plaintiff.
The paper is structured as follows. The next section describes the model of pretrial bargaining and solves for the Perfect Bayesian Equilibrium strategies of the plaintiff’s
lawyer and the defendant, both when the former does and does not face case costs. The third section then introduces price competition amongst plaintiffs’ lawyers to make the
contingent fee percentage endogenous and allow us to compare the outcomes when the lawyer does and does not bear costs. Section IV presents the results on settlement timing
and the plaintiff’s payoffs, and considers the effects of case characteristics. Section V concludes the paper.
II. The Model and Equilibrium Strategies
The Model The model follows Spier 1992 in adapting Fudenberg and Tirole 1983 to the civil
litigation context. Unlike Spier, however, we are interested in looking at contingent fees and the way bargaining strategies differ with the plaintiff’s lawyer’s costs of litigation.
Hence, we also draw on Miller 1987.
A plaintiff has filed a suit against a defendant for damages, x, caused to him in an accident. Being ill-informed about legal procedure, the plaintiff has retained a lawyer to
assess the strength of his case and negotiate on his behalf.
7
To keep things simple, we assume that the defendant has not hired a lawyer.
8
It is accepted that the defendant is liable for the accident, but only the plaintiff’s side knows the extent of the damage he
has suffered. His opponent knows that this damage is either high or low x and x, respectively, where x . x . 0, and has priors such that Prx 5 x 5 p and Prx 5 x 5
1 2 p.
9
All parties plaintiff, defendant, lawyer are risk neutral and have common discount factor d [ 0, 1.
We simplify the pretrial negotiations by assuming that bargaining takes place over two rounds, before trial see Figure 1; this is sufficient to illustrate the dynamics of interest
without complicating the analysis.
10
In the first period, the defendant makes a settle-
7
The plaintiff’s role is purely passive: he accepts all his lawyer’s advice. Given the relative ignorance of legal matters apparently displayed by many litigants [see Harris et al. 1984; Genn 1987], this assumption does not seem
unreasonable. It is made by Miller 1987 and Gravelle and Waterson 1993.
8
Alternatively, there is no conflict between defendant and lawyer. Either assumption allows us to focus on the role of costs and contingent fees without needing to take account of a double agency problem.
9
The fact that both the xs are strictly positive implies that only quantum is at issue rather than liability. See Genn 1987, p. 75 and Kritzer 1990, p. 143 for confirmation that this is a common situation. Setting x 5 0 would place
liability at issue instead.
10
It is easy to generalize the bargaining in many ways: for example, to any finite number of pretrial periods, to continuous prior distributions and to unequal discount factors across the parties and their lawyers [see Cramton
297 N. R
ICKMAN
ment offer based on his prior beliefs. If this is accepted, the case ends. If it is rejected, Period 2 commences and the defendant having updated his beliefs in response to the
rejection makes a second offer. If trial is reached, in Period 3, the defendant pays x if the plaintiff is “high-damage” and x if the plaintiff is “low-damage”, i.e., the court makes
the correct decision.
11
The defendant’s strategies are a pair of settlement offers S
1
in Period 1 and S
2
in Period 2. If S
1
is rejected, the defendant uses Bayes’ rule to update his beliefs to the posterior {mS
1
, 1 2 mS
1
}, where mS
1
[ Prx uS
1
rejected. The plaintiff’s lawyer’s strategies consist of a pair of acceptance probabilities as responses to S
1
and S
2
. A lawyer representing a low-damage plaintiff plays a
1
S
1
and a
2
S
2
, where a
t
S
t
5 1 implies acceptance and a
t
S
t
5 0 implies rejection of an offer, t 5 1, 2. Similarly, a high- damage lawyer’s strategies are a
1
S
1
and a
2
S
2
[ [0, 1].
1984; Spier 1992. Our results are robust to such amendments. Both Spier 1992 and Daughety and Reinganum 1994 interpret the finite horizon model as a postfiling one, where a trial date has already been set.
11
An exogenous probability of court error could be added without altering the main results. Rubinfeld and Sappington 1987 also show that a lawyer’s willingness to sink costs can act as a signal to the court and, thereby,
endogenise its probability of error. We ignore this possibility to focus on the lawyer’s bargaining with the opponent.
F
IG
. 1. Two-period litigation model.
298 Contingent fees and litigation settlement
The plaintiff pays a fraction a [ 0, 1 of his settlement amount or trial award to his lawyer. Thus, the plaintiff’s lawyer’s payoff is contingent upon her performance for
her client.
12
We also assume that there are per period costs associated with bargaining, c Ä 0 for the plaintiff’s side and h Ä 0 for the defendant. For example, these may be
the ongoing disbursements required to prepare the case and include such items as expert witness fees, filing, and travel costs [see Kritzer et al. 1984]. In common with
Benson 1979 and Miller 1987 inter alia, the plaintiff’s lawyer pays c. In the Lawyer’s Costs section below, we shall compare the results when c 5 0 and when c 5 C . 0. First,
we compute the equilibrium for any value of c.
Finally, we assume the American rule for allocating legal costs is in use; thus, each side bears its own costs. Of course, with the court outcome known in advance, assuming a
degree of cost shifting as under the English rule would be of quantitative significance only; no strategic considerations or qualitative properties of the equilibria would be
affected.
13
Perfect Bayesian Equilibrium We compute the Perfect Bayesian Equilibrium PBE. Let S
2
be the highest settlement offer that the defendant would make in Period 2. Because, under the American cost
rule, and with concern for disbursements, the plaintiff’s lawyer expects to receive dax 2 c at trial, this settlement offer must solve aS
2
5 dax 2 c. Arguing similarly for the lowest Period 2 settlement offer that the defendant can realistically make S
2
, we have S
2
5 d
S
x 2 c
a
D
and S
2
5 d
S
x 2 c
a
D
. 1
Similarly, define the highest and lowest Period 1 settlement offers to be S
1
5 d
S
S
2
2 c
a
D
and S
1
5 d
S
S
2
2 c
a
D
. 2
Notice that, excluding d, two factors can lower the settlement offer: being based on x rather than x, and a higher degree of future cost extraction higher c or lower a.
These will form the basis of the “hard bargaining” and “cost” effects we describe below. The defendant’s problem is to design a sequence of offers {S
1
, S
2
} to maximize his payoff given his beliefs updated in Period 2 about the plaintiff’s damage level. Each
type of plaintiff’s lawyer plays payoff-maximizing acceptance strategies, bearing in mind their influence on the defendant’s offers, via Bayes’ rule. The PBE is solved by back-
wards induction. We provide an intuitive account of the proof here; formal details are in Fudenberg and Tirole 1983, 1991 and the Appendix.
In one-shot litigation, the defendant chooses his strategy on the basis of which offer
12
Modeling the contingent fee as a simple share parameter, a, ignores a variety of alternative contingent arrange- ments, some of which may be preferable to the lawyer and client [see Halpern and Turnbull 1983; Hay 1995;
Rubinfeld and Scotchmer, 1993]. However, as noted earlier, our aim is not to investigate the optimal contingent fee arrangement but, instead, to see how the lawyer’s response to having a financial interest in the case can affect
settlement timing and payoffs.
13
See Shavell 1982 for an analysis of the effects of different cost allocation rules on settlement behavior. Donohue 1990 and Smith 1992 consider the interaction between cost rules and contingent fees, in models with no explicit
pretrial bargaining.
299 N. R
ICKMAN
gives him the highest payoff. Offering S
2
guarantees settlement, saves the defendant trial costs h, and allows him to extract future costs in the settlement offer ca.
However, it means paying unnecessarily high damages if the plaintiff is low damage. Thus, there is a critical value of p that makes the defendant indifferent between S
2
and S
2
. Calling this l
2
, we have l
2
S
2
1 1 2 l
2
dx 1 h [ S
2
, so
l
2
; h 1
c a
x 2 x 1 h 1 c
a .
3
Now consider the beginning of the two-period litigation. We can again calculate a critical expression, l
1
, such that when p , l
1
, the defendant offers S
1
and settles the case: there is too little prospect of facing a low-damage plaintiff to justify a low offer
and the associated costs of its rejection. When p . l
1
, the defendant offers S
1
. The high-damage lawyer rejects this, anticipating a high-damage offer in Period 2 or a
high-damage trial award. The low-damage lawyer faces a predicament. If she accepts S
1
for sure, the defendant will assume that any rejection of this offer is from a high-damage lawyer f m 5 0 and will, therefore, offer S
2
. As the low-damage lawyer would prefer this to S
1
, her original acceptance cannot constitute part of an equilib- rium. If, instead, the low-damage lawyer rejects S
1
with probability one, the defendant has no new information with which to update his beliefs f m 5 p. He, therefore,
offers S
2
, and the low-damage lawyer regrets rejecting S
1
it has no chance of generating S
2
. Hence, the low-damage lawyer’s equilibrium strategy must be to mix in Period 1 such that the defendant is made indifferent between offering S
2
and S
2
in the next period, i.e., the lawyer must induce m 5 l
2
. Calling this mixing probability a, we have from m 5 l
2
and Bayes’ rule
14
a 5 1 2 ~1 2 p h 1
c a
p~ x 2 x .
4 Finally, it can be shown that the critical first period expression, l
1
, again weighs up the costs and benefits to the defendant of a low initial offer: it is given by
l
1
; h 1
c a
a
F
d~ x 2 x 1 h 1 c
a
G
. 5
The outcome is depicted in Figure 2. In summary, we have demonstrated the following:
14
In fact, a
1
[ [0, a] can support this equilibrium. We focus on a
1
5 a because it is what Rasmusen 1989, p. 240 describes as the Pareto optimal focal point: it maximizes the first period settlement probability.
300 Contingent fees and litigation settlement
P
ROPOSITION
1: In the PBE, when p , l
1
, S
1
is offered and accepted. When p . l
1
, S
1
is offered and the probability of instant settlement is pa. The defendant offers S
2
in Period 2, generating a subsequent settlement probability of p1 2 a and leaving a trial probability of
1 2 p. From Figure 2 we can define p
ˆ such that l
1
p ˆ [ p
ˆ . Thus, we could equivalently state that the defendant offers S
1
when p , p ˆ and S
1
when p . p ˆ .
15
Lawyer’s Costs and Credibility We now examine the effects of the lawyer’s costs on the PBE. Specifically, we compare
the results when all lawyers have c 5 0 with those when all lawyers have c 5 C . 0: for simplicity, the two situations are assumed not to coexist. The lawyers’ “type” is common
knowledge.
16
Using obvious notation to distinguish settlement offers, mixing probabil- ities and values of l when c 5 0 and when c 5 C, we have
15
We can easily solve for p ˆ . Define g [ h 1 ca[1 1 dx 2 x 1 h 1 ca]. Then p
ˆ 5 g[dx 2 x
2
1 g]
21
[ 0, 1.
16
As noted in the Introduction, we can interpret the situation when c 5 C as involving a conflict of interest between plaintiff and lawyer. When c 5 0, the lawyer faces exactly the same bargaining incentives as her client would if he were
conducting the litigation. Accordingly, no conflict is present and the lawyer receives and accepts exactly the same settlement offers as the plaintiff would. When c 5 C, however, the lawyer faces costs of bargaining that her client would
not. It is now possible that her negotiating position will differ from that which her non-cost-bearing client would choose.
F
IG
. 2. The perfect Bayesian equilibrium.
301 N. R
ICKMAN
P
ROPOSITION
2: Holding a constant, S
t
. S
t, C
; S
t
. S
t C
; a . a
C
; l
t C
. l
t
; t 5 1, 2. P
ROOF
: This follows directly from setting c 5 0 and c 5 C . 0 in 1–5 and comparing the outcomes.■
To understand this result, begin with S
t
. When c 5 0, there are no marginal costs of continuing to bargain. In one sense, this strengthens the plaintiff’s lawyer’s bargaining
position and, for any given damage level, forces a higher settlement offer from the defendant [see 1 and 2]. In turn, this lowers l
2
[i.e., l
2 C
. l
2
] because the defendant has no costs to extract in his settlement offer, so his relative cost of a
low-damage offer in Period 2 falls. The value of a shows how a lawyer representing a low-damage plaintiff reacts to this. Because her intention is to induce a settlement offer
based on x, bearing no costs means that she must “work hard” to convince the defendant that he faces a high-damage plaintiff in Period 2. She does this by accepting
an offer that a high-damage plaintiff’s lawyer would certainly reject i.e., S
1
with higher probability because this makes the rejections she plays more credible signals that she
indeed represents a high-damage plaintiff: hence, a . a
C
. Finally, the tendency for “zero cost” lawyers to accept low-damage first period offers more frequently increases
the set of parameters for which such offers are forthcoming: hence, l
1 C
. l
1
. Thus, bearing costs has two effects on a contingent fee lawyer’s behavior. First, she
accepts lower offers given the level of damages the cost effect. However, second, when representing a low-damage plaintiff she is able to bargain hard because costly actions convey
more credible signals than costless ones. Because the cost-bearing lawyer risks more by rejecting an offer, the rejection is a more credible signal that she represents a high-
damage plaintiff, even though she does not the hard bargaining effect.
Having established that cost-bearing and contingent fees can produce this effect in a dynamic setting, we now wish to see how this influences the timing and size of settle-
ment. Before moving on we can, as before, define p ˆ
and p ˆ
C
as the critical values of p that determine whether the defendant offers S
1
or S
1
at the start of bargaining. It follows from Proposition 2 that p
ˆ
C
. p ˆ
.
17
III. Price Competition Among Lawyers
