Litigation and settlement under loss aversion

Abstract

In this paper, we investigate how loss aversion affects people’s behavior in private litigation. We find that a loss-averse plaintiff demands a higher settlement for intermediate claims to maintain her threat to proceed to trial following rejection compared to a loss-neutral plaintiff. For larger claims, a loss-averse plaintiff demands a lower offer to increase the settlement probability as loss pains her extra in trial. We also investigate how various policies affect loss-averse litigants’ settlement decisions. Only a reduction in the asymmetry of information about trial odds uniformly leads to higher settlement rates.

Full text

Vol.:(0123456789)
European Journal of Law and Economics (2023) 56:369–402
https://doi.org/10.1007/s10657-023-09777-6
1 3
Litigation andsettlement underloss aversion
CédricArgenton1 · XiaoyuWang2
Accepted: 3 July 2023 / Published online: 15 August 2023
© The Author(s) 2023
Abstract
In this paper, we investigate how loss aversion affects people’s behavior in private
litigation. We find that a loss-averse plaintiff demands a higher settlement for inter-
mediate claims to maintain her threat to proceed to trial following rejection com-
pared to a loss-neutral plaintiff. For larger claims, a loss-averse plaintiff demands
a lower offer to increase the settlement probability as loss pains her extra in trial.
We also investigate how various policies affect loss-averse litigants’ settlement deci-
sions. Only a reduction in the asymmetry of information about trial odds uniformly
leads to higher settlement rates.
Keywords Settlement· Loss aversion· Asymmetric information
JEL Classification D82· K51
1 Introduction
Private litigation is one of the most important aspects of a modern legal system.
In (the 12-month period ending in) 2016, about 20.3 million new civil cases were
filed in State courts in the US, to which one must add 274,555 new civil cases filed
in federal district courts.1 However, most cases do not go to trial: they are dropped,
We are grateful to Jan Boone, Eric van Damme, Bruno Deffains, Rosa Ferrer, Jan Potters, Florian
Schütt, Bert Willems, Eyal Zamir as well as conference participants at the 35th EALE annual
conference (Milan, 2018), seminar audiences at Tilburg University, and anonymous referees for
helpful comments and suggestions. All remaining errors are our own.
Xiaoyu Wang: Most of the research was done during the author’s PhD stage at Tilburg University.
* Cédric Argenton
c.argenton@tilburguniversity.edu
1 CentER & TILEC, Tilburg University, Tilburg, TheNetherlands
2 School ofEconomics, Shandong University, Jinan, China
1 Data for State courts are taken from the State Court Caseload 2016, produced by the Natio nal Cente r
for State Courts (consulted on September 3, 2018). Data for US courts are taken from the Federal Judi-
cial Caseload Statistics, 2016, produced by the Admin istra tive Offic e of the US Courts (consulted on
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resolved by motions, or settled in some way. Data on US State courts show that over
96 percent of civil cases do not go to trial (Ostrom etal., 2001).2
In other jurisdictions, settlement rates, although smaller, are also significant. In
Japan, 255,113 civil cases were settled or resolved by Alternative Dispute Resolu-
tions (ADR) while 404,094 were resolved with a final decision by the court after
a full trial in 2011. In Germany, 1.37 million out of 2.85 resolved civil cases were
settled or resolved by ADR in 2010.3 For civil cases in China, 2.79 million civil
cases were settled or mediated while 4.71 million reached the stage of trial in 2017.4
For an extensive review regarding settlement rates across the world, see Chang and
Klerman (2021). In these and other jurisdictions, given the high costs associated
with the judicial system, public authorities actively try to encourage alternative
modes of dispute resolution, other than trial.5 Fostering settlements is typically part
of such a strategy, but it requires understanding how parties to a dispute behave in
the first place.
In this paper, we study how loss aversion affects litigants’ choices about settle-
ment in private litigation from a theoretical perspective. Given how pervasive loss
aversion is, we are interested in how it affects litigants’ decisions such as filing a law-
suit and choosing a settlement offer, and which policies (e.g. fee-shifting rules or an
in-court settlement regime) could promote settlement. We first show that, contrary
to the existing literature and what first intuition may suggest, loss-averse plaintiffs
are not uniformly likely to settle more often, or for less, than loss-neutral litigants.
Second, regarding policy changes, we find that only a reduction in the asymmetry of
information about trial odds uniformly leads to higher settlement rates.
Specifically, we incorporate loss aversion into the model of Nalebuff (1987),
which builds upon Bebchuk (1984). An uninformed plaintiff makes a settlement
offer to a defendant who is privately informed about his likelihood of losing in court.
The settlement offer has a screening function: a defendant with a weaker case will
accept the offer while a stronger defendant will prefer trial. Following Nalebuff
(1987), if the offer is rejected, the plaintiff can drop the suit and save trial costs. In
this case, the plaintiff faces a credibility constraint: her settlement offer is credible
3 Data for Japan and Germany were collected from an OECD database.
4 The Law Year Book of China 2017, p. 1161.
5 For instance, the US Department of Justice has set up an Office of Dispute Resolution whose mission
is to promote the effective use of alternative modes of dispute resolution throughout the Department but
also in litigation. The Ministry of Justice of the UK stated “The aim (of the civil justice system reform)
is to deliver a system that is affordable and promotes the early resolution of disputes.” Specific measures
include U.S. Federal Rules of Civil Procedure Rule 16, the German § 278 Zivilprozessordnung (ZPO),
UK Civil Procedure Rule Part 36 and Directive 2008/52/EC from the European Union.
Footnote 1 (continued)
September 3, 2018). For State courts, cases classified as having either a civil or a domestic relations
nature were both taken up.
2 Similarly, data on federal courts demonstrate that, for fiscal year 2016, almost 99 percent of civil cases
were resolved without trial. (Federal Judicial Caseload Statistics, 2016, produced by the Administrative
Office of the US Courts. See this link, consulted on September 3, 2018.) It is not necessarily straightfor-
ward to compute settlement rates and one should approach those aggregate figures with caution (Eisen-
berg & Lanvers, 2009).
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only if she can maintain her threat to proceed to trial following rejection. Such a
credibility constraint is realistic and has played a central role in the subsequent lit-
erature about litigation, particularly in studies that aim to explain the prevalence of
negative expected-value lawsuits (see Bebchuk 1996, e.g.).6
Although it is intuitive that loss aversion makes for a weaker plaintiff who set-
tles for less, in our model this only happens when the stake is high enough. When
stakes are intermediate, the need to remain credible induces a loss-averse plaintiff
to ask for more.7 The intuition is as follows. Because trial brings more disutility to
a loss-averse plaintiff than to a loss-neutral plaintiff, the former needs to face more
favorable odds at trial to meet the credibility constraint and be true to her threat
to proceed, following the rejection of her offer. Consequently, she has to further
increase the offer amount so as to face more rejections and, therefore, a weaker pool
of defendants at trial.
We discuss changes to the environment or policy rules (level and allocation of
trial costs, underlying uncertainty and in-court settlement regime). We find that the
effects on settlement rates and litigation costs differ depending on whether the cred-
ibility constraint is binding or not. As a result, many rules or policies foster settle-
ments in some range of stakes (judgments), but discourage them for other values.
Only a reduction in the asymmetry of information (in the form of a shrinkage in the
support of the distribution of defendant types) can uniformly reduce trial costs. We
interpret this result as showing that changes or procedures that encourage early dis-
covery are the only ones that decisively promote settlements. The fact that reducing
information asymmetry fosters settlements is known at least since Bebchuk (1984);
what our model highlights is that a lot of the other policies that have been proposed
in the literature have different comparative statics, depending on the claim size.
Compared to Bebchuk (1984), we identify two new mechanisms brought by loss
aversion. First, fee-shifting may have ambiguous effects when loss aversion plays a
role. In particular, when the level of loss-aversion is high for the plaintiff, the Eng-
lish rule may encourage settlement. Second, a higher plaintiff win rate, which has no
effect on the settlement rate in Bebchuk (1984), leads to either a higher settlement
rate or a lower one in our model, depending on the stake.
There is a large literature on the behavior of private litigants. At first, it took for
granted that litigants behave as rational expected utility maximizers (and, more
often than not, as expected wealth maximizers). The economic theories of litiga-
tion started with Landes (1971) and Gould (1973), who focused on the divergence
in expectations about trial outcomes between the plaintiff and the defendant.8
6 Bebchuk (1996) uses a multi-stage, complete-information model of bargaining over settlement with
the added feature that litigants allocate different portions of their total litigation costs and decide whether
to continue at each stage. This divisibility means that, at a late stage, a plaintiff who does not have to
spend a lot of continuation costs may have a credible threat to proceed to the next stage and, by backward
induction, that may make that threat credible in earlier stages as well.
7 When the stakes are low, given the litigation costs, the plaintiff does not introduce a lawsuit in the first
place.
8 See also Posner (1973); Shavell (1982). This literature is still active: Vasserman and Yildiz (2019)
revisit the negotiation dynamics created by excess optimism in the presence of public information arrival.
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Subsequently, scholars started to use asymmetric information to model litigation and
settlement behavior. In a typical tort litigation setting, the plaintiff may indeed have
private information about the damages she has suffered while the defendant may
have private information about his liability for the accident. Fournier and Zuehlke
(1989), Ramseyer and Nakazato (1989), Farber and White (1991), Waldfogel (1998),
Osborne (1999) and Sieg (2000) all provide empirical evidence for the existence and
the importance of asymmetric information in various litigation environments. P’ng
(1983) and Reinganum and Wilde (1986) use a signaling model where the informed
party moves first by making a settlement offer. Bebchuk (1984) adopted a screening
set-up where the uninformed party makes a settlement offer to the informed one.9
The main prediction of that model, that cases reaching trial should disproportion-
ately be made up of cases favorable to the defendant, is borne out in many con-
texts.10 However, some of the predictions of that model, for instance, that shifting
legal fee to the party that loses in court (so-called English rule) should decrease
the settlement rate, are not consistent with the extant empirical evidence. On the
contrary, our model shows why loss aversion makes it a distinct possibility (see
Sect.4.1 for further discussion).
More recent work has tried to move away from standard expected utility maxi-
mization and to incorporate more behaviorally-relevant decision-theoretic founda-
tions such as self-serving bias (e.g. Babcock and Loewenstein, 1997; Farmer and
Pecorino, 2002), fairness considerations (e.g. Farmer and Pecorino, 2004), salience
(e.g. Friehe and Pham, 2020), etc. Zamir (2014) and Robbennolt (2014) and Zamir
and Teichman (2018) are good points of entry into this literature, especially when it
comes to the impact of loss aversion.
Starting with Kahneman and Tversky’s prospect theory (Kahneman & Tversky,
1979), numerous studies have established that decision makers evaluate options
based on gains and losses in comparison with a reference point. The evaluation is
asymmetric: losses loom larger than same-sized gains. Loss aversion is observed in
many real-world contexts, as well as laboratory or field experiments. It has proven
to be a powerful explaining tool. For instance, combining loss aversion and myopia,
Benartzi and Thaler (1995) provided an explanation to the equity premium puzzle.
Camerer etal. (1997) used loss aversion to make sense of cab drivers’ decisions on
their daily working hours. Genesove and Mayer (2001) found that it explained the
behavior of sellers on the housing market in Boston in the 1990s. Several studies
(Thaler, 1980; Knetsch & Sinden, 1984; Thaler & Johnson, 1990) used loss aversion
to explain the fact that people place higher value on objects which they already have
compared to those they do not have (the endowment effect). Loss aversion also helps
explain the sunk cost fallacy and the escalation of commitment (Arkes & Blumer,
1985).
9 Spier (1992) extended the framework of Bebchuk (1984) by allowing multiple periods of bargaining
to explain the “U-shaped” time pattern of settlement. Schweizer (1989), Spier (1994) and Klerman etal.
(2018) explored litigation games with two-sided asymmetric information.
10 For the case of medical malpractice in the US, see the review by Peters (2009), which shows that
cases with objective evidence of negligent or deficient care are more likely to settle.
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Loss aversion has been shown to have an important impact on legal theories as
well. For example, Zamir and Ritov (2010) used loss aversion to explain the popu-
larity of contingent-fee arrangements with lawyers, which cost their clients two or
three times more than an hourly-rate or fixed-amount arrangement. Wistrich and
Rachlinski (2012) found that loss aversion and the sunk-cost fallacy led experienced
lawyers to prolong litigation, which hurt their clients.
Some authors have already attempted at introducing loss aversion in settlement
models. The leading line of literature uses loss aversion in combination with regret
models11 to explain why people are more likely to settle.12 The intuition runs as fol-
lows: if anticipated regret drives behavior, then a party does not want to take the
chance of going to trial, possibly experiencing a large loss and suffering from the
fact that she could have settled instead. Thus, her willingness to accept a settlement
increases ex ante. In effect, loss aversion magnifies the intensity of regret. We show
that loss aversion does not necessarily lead to more settlements once the decision to
drop the lawsuit following settlement offer rejection is taken into account. Langlais
(2010) also introduces a kind of (non-linear) loss aversion (in the form of disap-
pointment aversion) in the Bebchuk (1984) model and reports ambiguous results
about the likelihood of settlement. We focus on the possibility of dropping the law-
suit after rejection of the settlement offer and offer the testable prediction that the
effect of loss aversion on the likelihood of settlement depends on the stake.13
Our model is particularly adapted to tort or employment cases where a one-time
victim sues a repeated or well-informed defendant. In such cases the plaintiff is usu-
ally a layperson, likely to exhibit loss aversion and, lacking the experience or the
expertise, to be less informed than the defendant about her chances of success.
Section2 introduces our baseline model and the main results. Section3 discusses
comparative statics regarding litigation costs and the distribution of defendant types.
Section 4 discusses the possibility of fostering settlements by using fee-shifting
rules or an in-court settlement system. Section5 offers a discussion of our results.
2 Baseline model
2.1 Setup
In this section, we introduce our litigation model, featuring asymmetric information
and a loss-averse plaintiff. We assume that there are two players, the plaintiff and the
defendant. For convenience only, in what follows we take the plaintiff to be female
and the defendant male. The plaintiff sues the defendant for compensation (or stake)
11 See, for instance, Bell (1982) or Loomes and Sugden (1982).
12 See Zamir (2015, p. 87 ff.) Zamir (2015) and Guthrie (1999). Zamir (2015, p. 89–90) writes: “Regret
and loss-aversion thus explain why so many litigants (plaintiffs and defendants alike) prefer to settle their
cases despite the prevalence of asymmetric information, strategic considerations, and a host of psycho-
logical barriers to settlement.”
13 We further discuss the difference with Langlais (2010) in the discussion section.
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W, which is assumed to be fixed and commonly known to both players at the begin-
ning of the game.14 If they proceed to the trial stage, the plaintiff will pay fixed
litigation costs
Cp⩾0
and the defendant will pay
Cd⩾0
. Those represent the direct
and opportunity costs associated with introducing, supporting and defending a for-
mal lawsuit. If the two parties manage to work out a settlement before trial, they will
save litigation costs
Cp
and
Cd
. If the plaintiff does not drop the suit after settlement
fails, trial will follow. A more detailed description of the timing comes later in this
section. For the time being, we introduce the key assumptions of the model.
Asymmetric information In a civil dispute, the defendant often has more infor-
mation regarding the existence of liability (for instance, whether negligence could
be proven in court). We assume the defendant to have private information about
the strength of his case. To be specific, he knows his probability of losing in court,
which is randomly drawn from a continuous distribution on the support [0,1] at the
beginning of the game and is denoted by q. The plaintiff, on the other hand, only
knows the distribution of q, represented by a probability density function
f(
⋅
)
and
a corresponding cumulative function
F(
⋅
)
. Based on this limited information, she
makes a unique settlement offer to the defendant.15
Loss-averse plaintiff The plaintiff’s preferences are represented by a reference-
dependent utility function with loss aversion. We use Kahneman and Tversky’s
value function of income w with respect to a fixed reference point, o16:
In the gain domain, the utility is the difference between the actual income w and the
reference income o. In the loss domain, the difference is multiplied by the loss aver-
sion coefficient,
𝜇
(
𝜇⩾1
). This coefficient describes the importance of loss aversion
in the plaintiff’s preferences: for
𝜇=1
, the plaintiff is a standard expected utility
maximizer; for
𝜇>1
, losses loom larger in her assessment of uncertain prospects,
and the more so, the higher
𝜇
.17 In what follows, we assume o to be equal zero. That
is, we assume that the reference point is the status quo prior to starting litigation.
It is assumed to be exogenous and constant during the litigation period. Although
(1)
u
(w
|
o)=
{
w−oif w⩾
o
𝜇(w−o)if w<
o
15 One could of course consider the contract-theoretical case where the plaintiff attempts to screen
defendants by offering them to choose their preferred option in a menu of settlement amounts and con-
tinuation probabilities, so that they truthfully reveal their type. It is however hard to think of a situation
where the plaintiff could commit herself to proceed to trial with a given probability. On the contrary, we
believe that the plaintiff can always choose to drop the lawsuit after her offer has been rejected and that is
what we model.
16 Another standard formulation is to assume the utility function contains two parts: the intrinsic utility
(expected wealth) and the reference dependent utility. It leads to a re-scaled utility function similar to the
one we use. All results remain unchanged.
17 We assume
𝜇
to be common knowledge. If
𝜇
were part of the plaintiff’s private information, the set-
tlement offer S would carry a signaling function as well as a screening one. That is an interesting topic
which lies outside the scope of this paper.
14 In practice, certainly in liability cases, the exact extent of harm of harm is not known with certainty
and its assessment is part of the dispute. We abstract from this aspect.
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the specification of an exogenous reference point is not self-evident, we note that
the use of this reference point (as opposed to, say, the situation before the faulty
action taken by the defendant) is supported by some experimental evidence (Zamir
& Ritov, 2012).18
We assume that utility is linear in w. That is, we assume that the plaintiff is risk-
neutral in the gain and loss domains, respectively, and isolate the effects of loss aver-
sion. In practice, individuals are likely to exhibit both risk and loss aversion. For
simplicity, we circumvent the differences in risk attitudes and focus exclusively on
the fact that losses loom larger than gains. In the discussion section, we elaborate on
the changes which risk aversion would bring to our analysis.
Whether the defendant also exhibits loss aversion (which may be an empirical
issue if, for instance, it is a corporation or an insurance company in an individual
tort case) is immaterial to our analysis. Winning at trial, losing or accepting the set-
tlement offer, the defendant would always find himself in the loss domain. Every
payment he makes would be multiplied by coefficient
−𝜇
in his utility function, so
the level of
𝜇
would not matter for his decisions as long as it is non-zero.19
Timing and choices The exact specification of our game is represented in Fig.1.
Compared to Bebchuk (1984), one noticeable feature of our game is the following:
if the settlement offer is rejected, then the plaintiff has the chance to drop the suit. In
that case, she does not have to pay litigation costs
Cp
but receives nothing from the
defendant. It means that in the pre-trial settlement phase of the game, she has to look
at the credibility of her implicit threat to actually proceed with trial in case her set-
tlement offer is rejected, a point first made by Nalebuff (1987).
We solve this game of incomplete information for perfect Bayesian equilibria.
Before going into the actual analysis, we survey the key decisions to be made by the
litigants, according to backward induction.
Dropping the suit In Stage 4, the plaintiff decides whether to drop the suit or not
given that her offer has been rejected. Depending on the amount of the rejected offer
19 We ignore the possibility that defendants might be risk-seeking in the loss domain to focus on the sole
effect of loss aversion on the part of plaintiffs.
18 As people internalize losses and adapt their reference point over time, the time interval between
the event that caused litigation and the settlement decision is likely to make the plaintiff view the pre-
settlement status as the reference point, especially if legal procedures take long. This point of view is
supported by the literature on hedonic adaptation to injuries (e.g. Bronsteen et al. (2008)). However,
Korobkin and Guthrie (1994) experimentally showed that it is possible to manipulate the perception of
plaintiffs to be operating in the gain or loss domain. In any case, if the plaintiff were basing her utility on
the situation before the accident (e.g. Belton etal. (2014) and Korobkin and Guthrie (1997)), under com-
pensatory damages she would experience utility only in the loss domain (for, in the best case, she can
only hope to be fully compensated for the harm suffered), in which case loss aversion would not play any
role and our analysis would be moot. Another potential reference point candidate is an expectation-based
reference point (e.g. Bell (1985); Loomes and Sugden (1986); Kőszegi and Rabin (2006)). However, the
experimental tests of the model’s key predictions are mixed (Marzilli Ericson & Fuster, 2011; Heffetz &
List, 2014; Abeler et al., 2011; Gneezy etal., 2017) and Cerulli-Harms et al. (2019). Particularly, Hef-
fetz (2018) found that reference points are “sunk-in (rather than merely lagged) beliefs”, which means
the expectation needs “some sense of internalization or getting used to” to form a salient reference point.
Moreover, using the expectation-based reference point brings the question of how sophisticated the liti-
gants are since the choice of a settlement offer affects the expected pool of defendants, and thus is sup-
posed to modify the reference point used for assessing outcomes.
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and the acceptance strategies of the various defendants, she updates her belief about
the defendant’s type q. Then, she makes herdecision by comparing the expected util-
ity of trial (formally defined later) and that of dropping the suit, which is assumed to
be zero.
Accepting the offer In Stage 3, the defendant decides whether to accept the offer
or not. The decision will depend on whether a trial is likely to follow or not and,
in case it is, on the expected trial costs compared to those of accepting the settle-
ment offer. For convenience, we assume that the defendant accepts the offer when
indifferent.20
Making an offer In Stage 2, anticipating the defendant’s acceptance/rejection
behavior and her own decisions regarding pursuing the lawsuit in case her offer
is rejected, the plaintiff chooses a settlement amount that maximizes her expected
utility. At this stage, she faces a credibility constraint: an offer that is too generous
might be rejected only by the more serious defendant types, preventing her from
rationally continuing with litigation after rejection and thus demolishing the cred-
ibility of her threat to proceed to trial.
Bringing the lawsuit If the plaintiff always gets negative utility from trial, she will
drop the suit in Stage 4. She thus gets zero utility from bringing the lawsuit and is
indifferent between bringing it or not. For convenience, we assume that she will not
bring the lawsuit in the first place. (It is also likely that, in reality, merely filing a
lawsuit already comes at a cost.)
2.2 Formal solution
A perfect-Bayesian equilibrium (PBE) for our game consists of a strategy pro-
file and a belief system regarding the defendant’s type. A strategy profile
𝛽={L,S∗
,d(S),r(S,q)}
specifies the following actions:
L∈{sue, notsue}
is the
plaintiff’s decision about whether to bring the lawsuit or not in Stage 1;
S∗
is the
offer made by the plaintiff in Stage 2;
d(S)∈[0, 1]
is the plaintiff’s probability of
dropping the suit in Stage 4 after offer S has been rejected; r(S,q) is the probability
that a type-q defendant rejects offer S in Stage 3. The plaintiff’s belief is specified by
𝜎(q,S)
, which describes the probability density function of facing a type q defendant
after offer S is rejected (Stage 4).21
We start by remarking that an offer with
d(S)=1
(i.e. the plaintiff drops the
suit with probability 1 following rejection) is always rejected in equilibrium. If the
20 Throughout, we assume that the settlement offer is binding if accepted. As remarked by an anony-
mous referee, since acceptance of the settlement signals to the plaintiff that the defendant is relatively
weak, she may have an incentive to renege on the settlement offer. We rule out this possibility.
21 In all rigor, the plaintiff should also hold a belief at her two first information sets (Stage 1 and Stage
2). However, at that point, she does not have any other information than the prior distribution of Nature’s
move so that her equilibrium beliefs must correspond to her prior belief. Hence, we do not formally spec-
ify those beliefs as part of our description of the equilibrium.
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plaintiff drops the suit in Stage 4 when all defendants reject her offer, it must be
because the expected utility of the lawsuit against the average defendant is negative.
If so, the same was true in Stage 1 and, by assumption, the plaintiff then does not
bring the lawsuit in the first place. In what follows, we focus on equilibria where a
lawsuit is introduced.22
Next, we show that the defendant’s equilibrium strategy exhibits a cut-off prop-
erty on and off the equilibrium path.
Lemma 1 In a perfect Bayesian equilibrium, for an offer S with
d(S)∈[0, 1)
, if a
type
q
defendant weakly prefers rejecting to accepting, then (i) a type
q
defendant
strictly prefers rejecting

S>S
and (ii) a defendant with
q<q
strictly prefers reject-
ing S to accepting it.
Proof see the appendix.
◻
Notice that sequential rationality requires that Lemma 1 holds for equilibrium
offer
S∗
as well as any other offer S off the equilibrium path. By definition of a PBE,
the belief of the plaintiff following rejection (Stage 4) must therefore be consistent
with the defendant’s equilibrium strategies.
Now we discuss subgames with
d(S)=0
, when the credibility constraint is not
binding. Directly from Lemma1, for an offer with
d(S)=0
, the defendant’s equilib-
rium choice is characterized by a cut-off type p(S)23:
The defendant of type q will reject S for sure if
q<p(S)
; if
q>p(S)
, he will accept
S for sure. Moreover, we have
p�(S)=1∕W>0
: if the plaintiff increases the offer
amount, the probability of rejection will increase as defendants with weaker cases
shift to rejecting.
From the plaintiff’s perspective, the probability of trial is therefore F(p(S)) and
her expected utility is (with subscript p standing for plaintiff):
(2)
p(S)=
S−Cd
W
(3)
U
p(S)=[1−F(p(S))]S+∫
p(S)
0
[
qW −Cp−(𝜇−1)(1−q)Cp
]
f(q)
dq
22 The plaintiff benefits from bringing the lawsuit if W is large enough. The exact threshold
W
is defined
later in Eq. (9).
23 With
d(S)=0
, one can restrict the range of equilibrium offer
S∗
to
[Cd,W+Cd]
without loss of gen-
erality.
S∗=Cd
is the highest offer that is accepted with probability 1 by all defendant types. In equi-
librium any choice
S∗
<
Cd
is strictly dominated by
S∗=Cd
because the latter brings the plaintiff more
and is also accepted for sure.
S∗=W+Cd
is the lowest offer rejected with probability 1 by all defendant
types. Any choice
S∗
>
W+Cd
leads to the same outcome as
S∗=W+Cd
. Thus, if there are equilibria
in which offer S* is rejected for sure and the plaintiff drops the lawsuit, there are also equilibria in which
she makes an even higher offer, but all those equilibria are outcome-equivalent: by our assumptions, the
lawsuit is not introduced in the first place.
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qW −Cp
is the expected income from trial. Loss aversion introduces an asymmetry
between settlement and trial in the plaintiff’s choice: settlement is a sure gain while
trial might lead to a loss (given the existence of trial costs). Faced with a type q
defendant, the plaintiff loses with probability
(1−q)
and the loss
Cp
is amplified by
loss aversion. The solution to the first-order condition,
Sfoc
, is given by:
Rewriting the above using
p�(S)=1∕W
, we have:
The right-hand side of the first-order condition is decreasing in p(S). For it to
uniquely pin down an interior solution
p(
Sfoc
)
and thus
Sfoc
, we make the following
assumptions on the distribution of q:
Assumptions A: For the p.d.f.
f(
⋅
)
and the corresponding c.d.f.
F(
⋅
)
, we have that:
1. 1
f(0)
>
𝜇C
p
+C
d
W
;
2.
f(q)
1−F(q)
is increasing in q;
3. The concavity of
f(q)
1−F(q)
does not change in [0,1]:





𝜕2

f(q)
(1−F(q))

𝜕q2





has a constant
sign for
q∈[0, 1]
.
The first assumption guarantees that the marginal benefit of asking for more is high
enough at
p(S)=0
, ruling out the corner solution
S=Cd
. Any distribution with a thin
left tail satisfies it. Along with the second assumption, which is the standard monotone
hazard rate property, it guarantees that an interior solution exists. The third assumption
is about the curvature of the hazard rate and guarantees uniqueness. Log-concave dis-
tributions satisfy the second and the third assumptions, and most (truncations of) com-
mon distributions exhibit the third property.
Proposition 1 Under assumptions A, the first-order condition (5) has a unique solu-
tion in p(S) as well as in S.
Proof see the appendix.
◻
In first-order condition (4), the left-hand side is the marginal benefit of further
increasing S. If the plaintiff increases the offer amount, she will extract more from
defendant types in [p(S),1], the ones who settle. If the offer is accepted, the plaintiff’s
(4)
1
−F
(
p
(
S
foc))
=f
(
p
(
S
foc))
p
�(
S
foc)(
Cp+Cd
)
+(𝜇−1)f
(
p
(
Sfoc
))
p�
(
Sfoc
)(
1−p
(
Sfoc
))
C
p
(5)
1
−F
(
p
(
Sfoc
))
f
(
p
(
Sfoc
))
=
(
Cp+Cd
)
W+(𝜇−1)
(
1−p
(
Sfoc
))
C
p
W
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payoff is marginally increased by 1. The right-hand side denotes the marginal cost of
increasing S. For a marginally higher offer, the marginal defendant(with type p(S)), will
shift from accepting to rejecting. The plaintiff bears the full costs of this shift, which
are the litigation costs
Cp+Cd
multiplied by the intensity of the marginal shift. This
trade-off is well-known since Bebchuk (1984). The second term on the right-hand side
is new and results from loss aversion: against the marginal type p(S), the plaintiff’s los-
ing probability is
(1−p(S))
, which costs her
(𝜇−1)Cp
in addition.
The above only applies to offers with
d(S)=0
. For this condition to hold, the trial
stage utility must be non-negative. With the cut-off property of defendant’s rejection
decision (Lemma 1), the plaintiff’s expected trial stage utility is defined as below:
Utrial
p
(S)=𝔼
[
q
|
q<p(S)
]
W−Cp−(𝜇−1)
(
1−𝔼
[
q
|
q<p(S)
])
C
p
Fig. 1 Timing of the litigation game.Not all subgames are represented on the tree. For the plaintiff’s first
two information sets (dotted circles), she has only prior information about the defendant’s type. For the
plaintiff’s third information set (Stage 4), she updates her belief based on the rejection of her offer
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1 3
where
𝔼
denotes the (conditional) mathematical expectation corresponding to the
plaintiff’s equilibrium beliefs about which defendant types accept or reject offer S.
Thus, the plaintiff’s objective in Stage 2 becomes24:
Utrial
p
(S
)
is the expected utility at trial stage. It is non-decreasing in S over
[
C
d
,W+C
d]
: when the plaintiff increases the offer amount, the expected winning
probability
𝔼[
q
|
q<p(S)
]
becomes higher as the marginal type p(S) becomes higher.
Demanding more in the settlement, the plaintiff pushes weaker defendant types
to trial, which means that she faces a more favorable pool of defendants in court.
Therefore, the credibility constraint puts a lower bound on the settlement offer. The
lower bound, denoted by
S,
is the unique solution to the following equation:
Therefore, the equilibrium settlement offer
S∗
is given by:
The defendant rejects offer
S∗
if and only if
q<p(S∗)
. If
S∗
is rejected, the plaintiff’s
belief about the defendant’s type is updated accordingly:
A technicality, which already arose in Nalebuff (1987), concerns very low offers.
The full characterization of our PBE, which contains information about off-equilib-
rium situations and the corresponding belief systems, is available in the appendix.
2.3 Comparison withatraditional plaintiff
We now compare a loss-averse plaintiff’s choices (
S∗
and d(S)) with those of a loss-
neutral plaintiff (
𝜇=1
). We use subscript tp to denote such a ‘traditional plaintiff’.
Her objective is (assuming that bringing the lawsuit is profitable):
max
S
Up(S)s.t.U
trial
p(S)
≥0
(6)
Utrial
p
(S)=𝔼
[
q
|
q<p(S)
]
W−Cp−(𝜇−1)
(
1−𝔼
[
q
|
q<p(S)
])
Cp=
0
(7)
S∗
=max
(
S
foc
,S
)
𝜎
(q,S∗)=
{
f(q)∕F(p(S∗)) if q∈
[
p(S∗),1
]
0 otherwise
max
SUtp(S)=
∫
p(S)
0
(
qW −Cp
)
f(q)dq +(1−F(p(S)))
S
s
.t.Utrial
tp (S)=
∫p(S)
0
(
qW −Cp
)
f(q)
F
(
p
(
S
))
dq ⩾0
24 It means that in calculating the plaintiff’s utility of bringing the lawsuit, we assume that W is high
enough such that trial is profitable for the plaintiff against the average defendant. If
Utrial
p
(S)<
0
for any
S, the plaintiff does not bring the lawsuit and drop it after rejection as we assumed.
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This is a similar constrained maximization problem: the plaintiff chooses a settle-
ment offer that maximizes her expected utility given that her trial stage utility is
non-negative if this offer is rejected. Similar to (5) and (6), we can solve for
Sfoc
tp
by
(5
′
) and
Stp
by (6
′
):
The solution is
Comparing the settlement offers
(
S
∗
and
S∗
tp)
and the probabilities of trial
(
F(p(S∗
))
and
F
p

S∗
tp

, we find that the result depends on the claim W. We have the fol-
lowing proposition:
Proposition 2 Compared with a traditional plaintiff, there exist unique values
W
tp ,
W
and

W
(
Wtp <W<
W
)
such that
1. For small claims
(
Wtp ⩽W<W
)
, a loss-averse plaintiff does not file a lawsuit
while a traditional plaintiff does.
2. For big claims
(
W⩾

W
)
1) a loss-averse plaintiff demands a smaller settlement;
2) the probability of trial is lower; 3) total expected litigation costs are lower.
3. For medium claims
(
W⩽W<

W
)
, 1) a loss-averse plaintiff demands a higher
settlement offer to make her threat to litigate credible; 2) the probability of trial
is higher; 3) total expected litigation costs are higher.
Proof see the appendix.
◻
The three critical values for W are defined as follows:
(5’)
1
−F

p

Sfoc
tp

=f

p

Sfoc
tp

p�

Sfoc
tp

Cp+Cd

(6’)
U
trial
tp
(
Stp
)
=0⇒𝔼
[
q
|
qp
(
Stp
)]
W−Cp=
0
(7’)
S
∗
tp =max
(
Sfoc
tp ,Stp
)
(8)
W
tp =
C
p
𝔼
[q]
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1 3
W
tp is the minimum compensation level which incentivizes the traditional plaintiff
to introduce the lawsuit.
W
is the minimum compensation level which incentivizes
the loss-averse plaintiff to introduce the lawsuit:

W
is implicitly defined as:
It is the compensation level at which the loss-averse plaintiff and the traditional
plaintiff choose the same settlement offer. The existence and uniqueness of

W
is
established in the proof of Proposition2 (see the appendix).
For the plaintiff, there are two scenarios that affect her settlement offer. First,
the offer optimizes her expected utility at the moment that the offer is made.
As the plaintiff becomes (more) loss averse, the utility cost of losing in court
becomes higher. To avoid this increased utility cost, she reduces the offer
amount to increase the probability of acceptance. The second scenario is that
after the settlement offer is rejected, the plaintiff’s expected utility from trial is
negative. It is not credible for her to proceed to trial and it implies that the offer
is rejected by all defendant types. Increasing the amount, her offer is rejected by
defendant types who have higher probability of losing in court, thus increasing
the expected value from trial. For the offer to be credible, the expected value
must be at least zero. A (more) loss-averse plaintiff counts the utility cost of
losing in court more and needs a higher offer (i.e. a weaker pool of defendants
who reject) to maintain a credible threat to proceed to trial. Depending on which
scenario we are in, a (more) loss averse plaintiff can thus make a higher or lower
settlement offer. For small damage claims, the plaintiff does not file the lawsuit.
For high claims, the first scenario applies. For intermediate damage claims, we
are in the second scenario: it is optimal to file a lawsuit and the credibility con-
straint is binding.
One might have thought that loss aversion makes for weaker plaintiffs who
sue less often and, when they do, always settle for less. Proposition2 shows that
the need to remain credible induces loss-averse plaintiffs to ask for more than
loss-neutral plaintiffs when stakes are of medium size. That is because, under
loss aversion it becomes harder to settle intermediate claims than big claims and
total litigation costs go up in that case. An interesting, testable implication is
that the presence of loss aversion will shift the composition of lawsuits that pro-
ceed to trial away from small and large stakes, and towards intermediate stakes,
diminishing the dispersion of judgments.
(9)
W
=(1+(𝜇−1)(1−𝔼[q]))
C
p
𝔼
[q]
S(
W
)
=S
foc
tp (
W
)
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2.4 A numerical example
Figures 2 and 3 give a numerical example of the probabilities of trial
F(p(S∗))
(
F
(
p
(
S∗
tp
)))
, and settlement offers
S∗
(
S∗
tp
) under different claims W when q follows
a truncated normal distribution on [0,1].25
The results from Proposition 2 are clear from the figures.

Wp
(

Wtp
)
features a
kink in the
S∗(W)
(
S∗
tp(W)
)
curve. For intermediate W, the optimal settlement offer
is determined by the credibility constraint that trial stage utility should be non-nega-
tive; for larger W, the optimal offer is determined by the first-order condition. For
W⩽W<

W
, the loss-averse plaintiff demands a higher settlement to make sure that
she will not drop the case if her offer is rejected. For
W⩾

W
, the loss-averse plain-
tiff demands a lower settlement offer to increase the probability of settlement. Both
results come from the fact that the loss-averse plaintiff suffers additional utility loss
when she loses in trial.
For
W<Wtp
, neither a traditional plaintiff nor a loss-averse plaintiff finds it prof-
itable to bring a lawsuit. For
W
tp
⩽W<W
, a traditional plaintiff brings a lawsuit
whereas a loss-averse plaintiff does not. Again, the intuition is that it is harder for a
loss-averse plaintiff to profitably go to trial: she endures additional utility loss if she
loses in trial compared to a traditional plaintiff. Thus, compensation W has to be
higher for the loss-averse plaintiff to bring a lawsuit.
3 Comparative statics
We now go over some of the comparative statics of our model. We first examine
what happens when trial costs change before looking at the role of the underlying
uncertainty about the winner of a trial (distribution of q). We are interested in char-
acterizing the effects on litigation costs. From the point of view of economic wel-
fare, there is no reason for having a narrow concern for litigation costs, as deter-
rence and precedent-setting certainly have social value. However, given their high
administrative costs, judicial systems often try to foster alternative dispute resolu-
tion mechanisms. It is therefore of interest to look at litigation costs.
3.1 Litigation costs
3.1.1 Plaintiff’s litigation costs
Cp
When the credibility constraint is not binding, an increase in
Cp
leads to a lower
probability of trial. Higher
Cp
means larger losses. So, the effect under loss aversion
25 The untruncated distribution has mean 0.5 and standard deviation 0.2. We use this distribution for all
following numerical examples unless specified otherwise.
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is bigger. The plaintiff thus prefers a higher settlement probability to avoid the loss.
The net effect on total litigation costs is ambiguous (there are fewer trials but each
trial now costs more to the plaintiff).
When the credibility constraint is binding, an increase in
Cp
leads to a higher
probability of trial because the plaintiff has to further increase the offer amount to
keep her threat to proceed to trial credible. Thus, contrary to the standard model,
such an increase in the plaintiff’s litigation costs might decrease the probability of
settlement due to the credibility constraint. (This effect was already observed in
Nalebuff (1987) but is magnified by the presence of loss aversion.) If
Cp
becomes
too high for the plaintiff to profit from litigation, the probability of trial drops to
zero, as the lawsuit is simply not introduced. Figure4 gives an illustration.
3.1.2 Defendant’s litigation costs
Cd
The effects of higher
Cd
also depend on W. When W is intermediate and the credibil-
ity constraint is binding,
Cd
does not affect the plaintiff’s choice of
p(S∗)
. It is deter-
mined by Eq. (6),
Utrial
p
(S
∗
)=
0
, and
Cd
plays no part in it. As in Nalebuff (1987),
S∗
increases as
Cd
does so as to keep
p(S∗)
unchanged and Eq. (6) satisfied.26
If W is high enough such that the credibility constraint is not binding, an increase
in
Cd
lowers
p(S∗)
as the marginal benefit of settlement becomes higher for the
plaintiff from first-order condition (5). It translates into a lower probability of trial.
However, when trial takes place, litigation costs are higher because
Cd
is higher. The
effects on
S∗
and total litigation costs are thus ambiguous.
Figure5 gives an illustration.
26 The reason is that the curve of function p(S) is shifted downward. In effect, the plaintiff takes advan-
tage of the defendant’s increased willingness to absorb a high settlement in the presence of high costs.
Fig. 2 Probabilities of
trial for different plaintiffs
(
𝜇
=
2, Cp
=
4, Cd
=
2
)
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3.2 Distribution ofq
In this subsection, we modify the assumption that the support of q is [0,1]. Instead,
we assume the support to be
[q,q]
with
q>0
and
q<1
. We consider two distribu-
tional changes regarding q.
First, we consider a rightward shift to distribution
G(
⋅
)
over support
[q+𝜀,q+𝜀]
with
g(x+𝜀)=f(x)
for
x∈[q,q]
. The new distribution
G(
⋅
)
first-order stochasti-
cally dominates distribution
F(
⋅
)
, which means that the plaintiff unambiguously
faces a pool of weaker defendants.
Under the new distribution
G(
⋅
)
, the plaintiff’s credibility constraint is less
restrictive as the overall winning probability is higher. It translates into a lower
settlement offer and a lower probability of trial. When the credibility constraint
is not binding, the loss-neutral plaintiff asks for a higher settlement offer but
the probability of trial stays the same (first-order condition (5
′
)). However, for
the loss-averse plaintiff, the probability of trial will increase under distribution
G(
⋅
)
. Intuitively, as the plaintiff’s overall probability of losing is lower under
G(
⋅
)
, the effect of loss aversion becomes smaller and, as a result, he chooses to
bargain more aggressively. This effect was not present in Nalebuff (1987) and
is specifically related to the presence of loss aversion.
Proposition 3 When the distribution of defendant’s types switches from F to G:
1.
W
decreases;
2.
p(S)
decreases;
3.
p(
Sfoc
)
increases by more than
𝜀
;
4.

W
decreases.
Proof see the appendix
◻
Thus, a loss-averse plaintiff sues for a wider range of claims, settles intermediate
claims more often, but settles high claims less often. So, again, the effect on litiga-
tion costs depends on the size of the claim. Figure6 gives an illustration.
Second, we consider a mean-preserving truncation of
F(
⋅
)
. Formally, for F with sup-
port
[
q,q
]
⊂[0, 1
]
and for a small
𝜀>0
, define
q�=q+𝜀
and
q′<q
such that
𝔼
[q]=𝔼
[
q
|
q∈
[
q�
,q�
]]
. (Such a
q′
can always be found.) Take

G
to be the truncation
of F on
[
q′
,q′
]
. Then,

G
is a mean-preserving truncation of F and second-order stochas-
tically dominates F. Such a change captures a reduction in the degree of information
asymmetry between the two parties (leaving the average odds unchanged). In practice,
it means “extreme” cases are eliminated from the distribution.
Proposition 4 When the distribution of defendant’s types switches from F to a mean-
preserving truncation

G
:
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1.
W
is not affected;
2.
p(S)
decreases;
3.
p(
Sfoc
)
decreases.
Proof see the appendix.
◻
Thus, there is no difference to the amount of lawsuits filed but fewer proceed to
trial, independently of the judgment size W. This result is similar to the one in Beb-
chuk (1984): the effect on
S∗
is ambiguous but the probability of trial is for sure
lower. A mean-preserving truncation of the distribution of q means that the plaintiff
has more precise information about the defendant’s type. Therefore, a mutually ben-
eficial settlement becomes more likely.
Figure7 gives an illustration.
4 Extensions: fostering settlements
Imagine again that society considers total trial costs to be of concern. What can it
do to reduce the volume and costs of trials? Procedural rules about the allocation of
trial costs or other ways to foster settlements have been extensively discussed in the
literature. In our model, what happens when some of those rules are implemented?
We consider fee-shifting rules and in-court settlements in turn.
4.1 Fee‑shifting rules
In the baseline model, we assumed that the court enforced the so-called American
rule in the allocation of litigation costs: each party pays for their own legal expenses
regardless of the trial outcome. Now, we consider the English rule, which provides
that the loser in court pays for both parties’ litigation costs. It is equivalent to mov-
ing to an environment with
WEN =
W
+
C
p+
C
d
,
CEN
p=
Cp
+
C
d
and
CEN
d
=
0
under
the American rule. In practice, the English rule amounts to raising the stakes for the
plaintiff both on the income and the cost sides. Fee-shifting has been extensively
studied, theoretically, experimentally and empirically.27
For intermediate claims where the credibility constraint is binding, shifting to the
English rule has ambiguous effects. Under the American rule,
W
, the lowest com-
pensation that incentivizes the plaintiff to sue, is given by:
Under the English rule, we have:
W
=
[
1+(𝜇−1)(1−𝔼[q])
]C
p
𝔼
[p]
27 See Kritzer (2001); Spier (2007) and Helmers etal. (2019) for reviews of the literature.
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The relative size of
W
and
WEN
depends on
𝜇
,
Cp
,
Cd
as well as the unconditional
expectation of q. If the credibility constraint is not binding, the likelihood of settle-
ment is unambiguously lower under the English rule if the plaintiff is not loss-averse
(Bebchuk, 1984,Proposition 6). With loss aversion, fee-shifting may have ambigu-
ous effects: if the level of loss-aversion is high for the plaintiff, then the English rule
may encourage settlement. From the first-order condition (5), we have the following
comparison:
For
𝜇=1
, fee-shifting unambiguously decreases the right side of Eq. (5). From the
increasing hazard rate property, p
(
Sfoc
)
increases as a result, leading to a lower set-
tlement rates. For Eq. (5
EN
), the right-hand side might become smaller if
𝜇
and
Cd
are large. Intuitively, if the heavy cost is shifted to the plaintiff and the effect of loss
aversion is large, then the plaintiff might prefer settling with a higher probability.
Figure8 illustrates such a possibility.
Obviously, the net effect on the total number of trials will depend on the distribu-
tion of W. However, a decrease in the number of suits proceeding to trial is possible.
This finding is important, because some of the available experimental or empirical
evidence about the impact of fee-shifting (Anderson and Rowe, 1995; Hughes and
Snyder, 1995; Kritzer, 2001, Helmers etal., 2019) reports an increase in settlement
rates upon the adoption of the English rule, which our model rationalizes, contrary
to Bebchuk’s (1984) or Nalebuff’s (1987).
W
EN =
𝜇(1−𝔼[q])
𝔼
[q](
Cp+Cd
)
(5)
1
−F
(
p
(
Sfoc
))
f
(
p
(
Sfoc
))
=
(
Cp+Cd
)
W+(𝜇−1)
(
1−p
(
Sfoc
))
Cp
W
1
−F

p

Sfoc
EN

f

p

Sfoc
EN

=

Cp+Cd

W+Cp+Cd
+(𝜇−1)1−pSfoc
EN Cp+Cd
W+Cp+Cd5EN

Fig. 3 Settlement offers
from different plaintiffs
(
𝜇
=
2, Cp
=
4, Cd
=
2
)
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4.2 An in‑court settlement regime
From Proposition 2, we can see that the plaintiff’s binding credibility constraint
leads to a higher offer amount and thus higher probabilities of trial for medium-
range claims under loss aversion. The constraint results from the plaintiff’s lack of
commitment power. If the plaintiff could credibly commit to trial in case her offer
is rejected, then she (as well as the defendant) would benefit: she would be able to
make a lower settlement offer that suits herself better. To achieve this, one may think
of moving from the out-of-court settlement regime which we have studied so far to
Fig. 4 The effect of higher
Cp
on trial probabilities and on litigation costs.The shift is from
Cp
=
2
to
Cp
=
4
, and other parameters are the same as before
Fig. 5 The effect of higher
Cd
on trial probabilities and on litigation costs. The shift is from
Cd
=
2
to
Cd
=
4
, and other parameters are the same as before
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an in-court settlement regime. If it costs the judicial system much less to register set-
tlements than to conduct full trials, this could decrease total litigation costs.
Suppose indeed that the legal system does not allow a plaintiff to drop a suit
outside court. Then, even a settlement necessitates to go, and pay, for trial.28 In an
(extreme) in-court settlement regime, the plaintiff pays
Cp
at the time she introduces
the lawsuit: she will use the court’s and her lawyer’s services even if she settles with
the defendant as this has to be agreed by the court. This will remove the credibility
constraint. The loss-averse plaintiff chooses S to maximize the following:
The optimal settlement offer (
S∗
in
) is characterized by the following first-order
condition:
Our assumptions on
F(
⋅
)
guarantees that we have a unique interior solution
S∗
in ∈[Cd,W+Cd)
. It is straightforward to show that
S∗
in
>S
foc
:
Cp
has been paid up-
front so saving
Cp
is no longer an advantage associated to settlement, compared to
trial. The credibility constraint no longer plays a role because giving up trial means
a sure loss of
Cp
anyway. Therefore, for intermediate values of W at which the cred-
ibility constraint was binding in the out-of-court settlement regime, we may have
S∗
in <S∗
for the loss-averse plaintiff.
For the lowest W that incentivizes a loss-averse plaintiff to sue (
Win
), we have
W
in
⩽W
. Intuitively, at
W=W
in the in-court settlement system, the plaintiff could
bring the lawsuit and ask for
S⩾W+Cd
. This brings her the same utility as in the
out-of-court settlement regime. It is possible that she can do better because the cred-
ibility constraint is no longer playing a role.
A special case arises when
Cd⩾Cp
. The plaintiff can bring the lawsuit, pay
Cp
and ask for
S=Cd
as soon as
W⩾0
. The defendant will accept the offer whatever
his type is. We have
W
in
⩽W
in general and
W
in
⩽0
if
Cd⩾Cp
: under the in-court
settlement regime, the plaintiff brings more small-claim lawsuits and sometime even
lawsuits with negative expected values. Cases with negative expected value become
profitable in the in-court settlement regime, provided the defendant’s costs are high
enough. That is consistent with the results in Bebchuk (1996). Figure 9 gives an
illustration for
Cd⩾Cp
cases.
Thus, the effect of requiring the plaintiff to settle in-court (at a cost) would have
an ambiguous effect on the volume of litigation: the net effect would again depend
on the distribution of claims.
(10)
U
in-court
p(S) =[1−F(p(S))](S−Cp)+∫
p(S)
0
[
q(W−Cp)−(𝜇−1)(1−q)Cp
]
f(q)
dq
1
−F
(
p
(
S
∗
in
))
f
(
p
(
S∗
in))
=Cd
W+(𝜇−1)
(
1−p
(
S∗
in
))
Cp
W
(
5in-court
)
28 In many jurisdictions, settlements have to be agreed by courts in some circumstances. In the US, typi-
cally: class actions, domestic relations cases involving the division of debt/assets and child custody/sup-
port, civil rights cases, and especially any civil action where the plaintiff is a minor or disabled person.
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1 3
5 Discussion
Our goal in this paper was to show how loss aversion theoretically affects people’s
behavior in (civil) litigation, in particular with regards to settlements, in the pres-
ence of a realistic credibility constraint about the plaintiff’s threat to proceed to trial.
We have shown how loss aversion might lead to fewer suits for small claims, a lower
settlement probability for medium claims, and a higher settlement probability for
large claims. This translates into the testable prediction that loss aversion should
lead to relatively more medium-size claims reaching trial stage.
Coming back to Langlais’ (2010) results, we can now point out the sources of
differences. Beyond the presence of the credibility constraint, our model differs
in the specification of the utility function and the choice of the party proposing
the settlement. Langlais (2010) introduces a form of disappointment aversion that
distorts the plaintiff’s perception of the winning rate to:
𝛽
is the coefficient of disappointment aversion. The plaintiff is the informed party
and the uninformed defendant makes the settlement offer. The plaintiff who is indif-
ferent between accepting or rejecting settlement offer S is thus characterized by:
The derivative of
q
with respect to S, which is fixed at 1/W in our model, is now
given by a complicated function of
q
and the following first-order condition for an
interior solution for S (the equivalent of our Eq. (5)):
𝜎
(q)=
q
1+
𝛽
−
𝛽
q
<q
.
𝜎(q)W−Cp=S.
Fig. 6 FOSD shift of distribution (
𝜇
=
2, Cp
=
4, Cd
=
2
). The shift is from a truncated normal distribu-
tion with mean 0.5 and support [0.2,0.8] to a truncated normal distribution with mean 0.7 and suppor t
[0.4,1]. The untruncated distribution has standard deviation 0.2
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European Journal of Law and Economics (2023) 56:369–402
The non-monotonicity of the right-hand side is the cause of the fundamental ambi-
guity about the effect on the likelihood of settlement in Langlais (2010).
In our model, due to loss aversion’s effect on the plaintiff’s credibility con-
straint, policies aiming at reducing the number of costly trials and at fostering
settlements may have different effects for claims of different sizes. In our modeled
environment, the only change which unambiguously leads to fewer trials across
the board is the reduction in the degree of information asymmetry about trial
odds. Thus, rules and policies that encourage access to informed legal advice or
discovery at an early stage seem to be the best way to foster settlements.29 Asym-
metric uncertainty is the cause of the inefficiency. So, it is not surprising that a
decrease in asymmetry improves welfare, which is a common result of litigation
models featuring asymmetric information. What our model shows is that many
of the other, often-floated proposals do not uniformly increase the likelihood of
settlements.
Our analysis was based on a number of simplifying assumptions. We now elabo-
rate on a number of them.
In our analysis, we assumed that both litigants were risk-neutral. Risk aversion
and loss aversion are similar in one respect: the decision-maker puts extra weight
on the worst outcomes. If the plaintiff were to display some risk aversion in addition
to loss aversion (in the sense that the first term in her utility function would now
be a concave function), this would reinforce our results since, for any distribution
1−F(q)
f
(
q
)
=
(C
p
+C
d
W+q−𝜎(q)
)
q
𝜎
(
q
)
1−q
1
−
𝜎
(
q
)
.
Fig. 7 Mean preserving truncation of distribution (
𝜇
=
2, Cp
=
4, Cd
=
2
). The shift is from a truncated
normal distribution with mean 0.5 and support [0,1] to a truncated normal distribution with mean 0.5 and
support [0.1,0.9]. The untruncated distribution has standard deviation 0.2
29 Waldfogel (1998) and Huang (2009) provide some empirical evidence to this effect.
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European Journal of Law and Economics (2023) 56:369–402
1 3
of plaintiff win rates, the plaintiff’s expected utility of going to trial would now be
lower, which would again tighten the credibility constraint for medium claims and
lead to more settlements for large claims.
If we include the prospect theory insight that people are risk-averse in the gain
domain and risk-loving in the loss domain, then it will become more difficult for the
litigants to reach a settlement. For one, risk aversion makes it harder for the plain-
tiff to commit to trial and will tighten the credibility constraint. Second, in the loss
domain, risk-loving makes the defendant more willing to accept the gamble of a trial
and less willing to accept a settlement. For medium stakes, the prediction will thus
be that settlement is less likely, just like in our model. For large stakes, however, this
risk preference pattern may increase or decrease the likelihood of settlement, com-
pared to the case of risk neutrality (contrary to our unambiguous prediction of more
settlements under loss aversion).
In our baseline model, we assumed that litigation costs and judgments were fixed
and focused on the effect of loss aversion on the plaintiff’s choice of a settlement
offer. In reality, the size of the judgment may be unknown and governed by a sta-
tistical distribution. If W is random but the lowest possible value is larger than or
equal to
Cp
, results are unchanged in our model since the plaintiff remains in the
gain domain upon winning at trial in all circumstances. (By linearity, her expected
utility remains the same.) If, on the contrary, W can assume values lower than
Cp
,
then the plaintiff may suffer a loss even when she wins a trial. Under loss aversion,
this means that the plaintiff’s expected utility derived from trial will decrease. In
our model, this will generate a lower trial probability for the loss-averse plaintiff for
high stakes and a higher trial probability for intermediate stakes, thus magnifying
our results.
This is assuming that the distribution of the stake, W, is independent of the dis-
tribution of the plaintiff win rate, q. In practice, the two might be (positively) cor-
related. For instance, a strong case, from the point of view of the plaintiff, may
translate into a higher chance of winning the trial or a higher judgment. When intro-
ducing positive correlation and comparing two distributions, one has to be careful
with keeping the expected value of suing the (unconditional) average defendant
unchanged. When doing so, it can be shown that in the presence of positive correla-
tion and, again, assuming that the lowest possible value for W is higher than
Cp
, the
likelihood of settlement go down for medium-size claims and go up for large claims.
Indeed, the conditional expectation of qW for those defendants that reject a given
settlement offer is systematically lower than under statistical independence, which
decreases the expected utility of trial for the plaintiff.
If litigation costs were random (with unchanged means), it is again straightfor-
ward to see that, in our linear model, nothing would change as long as the plaintiff
remained in the gain domain upon winning her trial. (The defendant remains in the
loss domain in any case.) If
Cp
were possibly to assume so high values as to bring
the plaintiff into the loss domain even upon winning her trial, then this would lead
her to decrease her valuation of her prospects at trial. Thus, the credibility constraint
would be tightened for intermediate stakes (leading to a lower settlement probabil-
ity), while the likelihood of settlement would go up for high stakes, which would
again reinforce our results.
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Note that our model considers two stages (offer and trial) and assumes that all
costs are to be borne at the trial stage. This allows us to consider the trial costs
as fixed when comparing a loss-neutral plaintiff to a loss-averse plaintiff. Bebchuk
(1996) argues that litigation can be a multi-stage process with various fractions of
total costs to be spent by the parties at each stage. If that is true, then the intro-
duction of loss aversion on the plaintiff side may change her decisions to sustain
the case at various stages. The introduction of loss aversion in the Bebchuk (1996)
model is therefore an interesting question but one that lies outside the scope of this
paper.30
More generally, litigation costs can be viewed as the outcome of endogenous
decisions made by the parties involved and therefore, the introduction of loss aver-
sion might affect litigation spending. If the plaintiff can choose how much to spend
on a case, some exploratory results show that litigation expenditure is affected by
loss aversion in an ambiguous manner, as two countervailing forces appear: the
incentive to avoid losing (increased spending raises the winning probability and thus
prevents a distressing loss) and the incentive to cut losses (greater spending results
in larger losses if the plaintiff loses in trial). We expect the loss-averse plaintiff to
spend more for cases with higher fixed costs because the incentive to avoid losing
is stronger, by comparison with a loss-neutral plaintiff, but we leave this interesting
issue to further research.
Appendix
Proof of Lemma 1 In the proper subgame after the plaintiff makes offer S, by
assumption the defendant expects the plaintiff to pursue the case with some posi-
tive probability if he rejects the offer (
d(S)∈[0, 1)
). The defendant will compare the
Fig. 8 Probabilities of trial under different rules (
𝜇
=
2, Cp
=
4, Cd
=
2
)
30 We thank an anonymous referee for raising that point.
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394
European Journal of Law and Economics (2023) 56:369–402
1 3
outcome of accepting the offer and that of rejecting it. For a defendant with type q,
the expected utility from rejecting the offer is (subscript d stands for defendant):
The expected utility from accepting the offer is
−S
. As
Utrial
d
(q
)
is decreasing in q,
Utrial
d
(q)⩾−
S
implies that
Utrial
d
(q)>−
S
for
q<q
and
Utrial
d
(q)>−

S
for

S>S
.
◻
Proof of Proposition1 the existence and uniqueness of
Sfoc
Recall first-order condition (5):
Define function
H(
⋅
)
as the following:
H(x) is continuous and twice differentiable by our assumptions. We have:
This means that
H(x)=0
has at least one solution at [0, 1] (by the intermediate
value theorem). For the solution to be unique, a sufficient condition is that the sec-
ond-order derivative of function
H(
⋅
)
has a constant sign over [0,1]. Because the
loss-aversion part
−(
𝜇−1)x
C
p
W
is linear in x, the condition is met if and only if the
second-order derivative of the reversed hazard rate function
1−F(x)
f(x)
has a constant
sign over [0,1], which is our assumption A3. The uniqueness is established in the
following manner.
Under the three assumptions 1)
H(0)<0
; 2)
1−F(x)
f(x)
is decreasing in x on [0,1];
and 3) the second-order derivative of
1−F(x)
f(x)
has a constant sign on [0,1]. The graph
of function H(x) has therefore three possible shapes over [0,1]: 1) monotonically
increasing; 2) increasing in
[0, x]
and decreasing in
(x,1]
where
x∈[0, 1]
and
H�(x)=0
; 3) decreasing in
[0, x]
and increasing in
(x,1]
where
x∈[0, 1]and H�(x)=0
.
In form 1), we can directly apply the intermediate value theorem on [0,1]. With
monotonicity, it is clear that a unique root exists. In form 2) we can apply the inter-
mediate value theorem on
[0, x]
and in form 3) on
[x,1]
. With monotonicity over the
Utrial
d
(q)=−(1−d(S))
(
qW +C
d),
1
−F
(
p
(
S
foc))
f
(
p
(
Sfoc
))
=
(Cp+Cd)
W+(𝜇−1)
(
1−p
(
Sfoc
))
Cp
W
H
(x)≡
(
Cp+Cd
)
W+(𝜇−1)(1−x)Cp
W−1−F(x)
f(x),x∈[0, 1
]
=(𝜇Cp+Cd)
W
−(𝜇−1)xCp
W
−1−F(x)
f(x)
H
(1)=
(
𝜇Cp+Cd
)
W−(𝜇−1)Cp
W=
(Cp+Cd)
W>
0
H
(0)=
(𝜇Cp+Cd)
W
−1
f(0)
<0 by assumption A1
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relevant sub-interval, a unique root is also guaranteed. The corresponding settlement
offer S can be recovered from p(S) in (2).
◻
Full description of the PBE
Note that the plaintiff cannot commit to trial if an offer
S<S
is rejected. One
has to specify what happens following such offers. Pure strategies are excluded: if
the plaintiff dropped the suit after rejection, then all defendants should reject offer
S<S
, and the plaintiff would face the universe of defendants in Stage 4. However,
we have assumed, for the time being, that it was profitable for her to sue the average
defendant, in which case she should not drop the suit. The equilibrium in those sub-
games therefore features mixed strategies: the defendant rejects offer S if
q<p(S)
and accepts it if
q⩾p(S)
. The plaintiff, who is indifferent between giving up the
lawsuit and not, proceeds to trial with probability
S∕
S.31 For the plaintiff, an offer
S<S
is dominated by
S
: offering
S<S
, she received the same trial stage utility
of 0 but the settlement amount is strictly smaller than
S
for an equal probability of
acceptance. So, those subgames are never reached in equilibrium.
For
W⩾W
, our PBE is described by the following assessment
(𝛽,𝜎)
:
𝛽
=
{
sue, S∗=max
(
Sfoc ,S
)
,d(S)=
{
1−S∕SifS<S
0 ifS⩾S,r(S,q)=
{
0q⩾p(max(S,S))
1q<p(max(S,S))
};
𝜎
(q,S)=
{
f(q)∕F(p(S)) if q∈[p(S),1]
0 otherwise if S⩽S⩽W+Cd
;
𝜎
(q,S)=
{
f(q)∕F(p(S)) if q∈[p(S),1]
0 otherwise if 0 ⩽S<S
;
Fig. 9 Probabilities of trial and litigation costs under different settlement systems (
𝜇
=
2, Cp
=
4, Cd
=
2
)
31 For subgames with
S
<
S
, the lower S is, the more likely the plaintiff drops the suit following rejec-
tion. In all such subgames, defendant with type
p(S)
remains the one who is indifferent between accept-
ing and rejecting the offer. The plaintiff is thus indifferent between proceeding to trial and giving up
lawsuit.
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396
European Journal of Law and Economics (2023) 56:369–402
1 3
𝜎(q,S)=f(q)if S
>
W+Cdor S
<
0.
Proof of Proposition2 The loss-averse plaintiff’s trial stage utility is:
We have
dUtrial
p
∕dp(S)>
0
. And since
dp(S)∕dS ⩾0
, we have
dUtrial
p
∕dS ⩾
0
. This
means the maximum of
Utrial
p
is achieved at
p(S)=1
with
S
⩾
W+Cd
for any given
W. We use
U
trial
p
to denote this maximum:
We also have that
dUtrial
p
∕dW >
0
. By definition of
W
,
Utrial
p=𝔼[q]W−Cp−(𝜇−1)
(1−𝔼[q])Cp=0
. So, for
W<W
,
Utrial
p
(S)<
0
for any S. Hence, the credibility con-
straint cannot be met and the plaintiff will drop the suit for sure if a settlement offer
is rejected. All types of defendants reject the offer. Therefore, the loss-averse plain-
tiff will not bring the lawsuit. A similar proof goes for the traditional plaintiff for
W<W
tp . As
W
tp =
C
p
𝔼
[p]
<W , a loss-averse plaintiff does not file a lawsuit while a
traditional plaintiff does for
W∈[W
tp
,W)
. This proves part 1.
To prove parts 2 and 3, we state the following two lemmas.
Lemma 2
S>S
tp ;
S
foc <S
foc
tp
.
Proof The proof of this lemma directly follows the definition of the relevant settle-
ment offers. For
S
and
S
tp , we have:
As
𝜇>1
and
𝔼[q|q<x]
is weakly increasing in
x∈[0, 1]
, we have
S>Stp
. Moreo-
ver,
S(W)
and
S
tp
(W)
are implicitly defined from the above equations. By the implicit
functions theorem, they are both continuous functions.
For
Sfoc
and
Sfoc
tp
, using
p�(S)=1∕W
, we
Similarly,
Sfoc(W)
and
Sfoc
tp
(W
)
are implicitly defined from the above first-order (5)
and (5
′
) and they are thus continuous functions of W. To see that
Sfoc
is smaller, we
Utrial
p
(S)=𝔼[q
|
q<p(S)]W−Cp−(𝜇−1)(1−𝔼[q
|
q<p(S)])C
p
U
trial
p
=𝔼[q]W−Cp−(𝜇−1)(1−𝔼[q])C
p
(6)
𝔼[q|q<p(S)]W−Cp=(𝜇−1)(1−𝔼[q|q<p(S)])Cp
(6’)
𝔼[q|q<p(S
tp
)]W−C
p
=0
(5)
1
−F
(
p
(
S
foc))
=f
(
p
(
S
foc))(
Cp+Cd
)
∕W
+(𝜇−1)f
(
p
(
Sfoc
))(
1−p
(
Sfoc
))
C
p
∕W
(5’)
1
−F

p

Sfoc
tp

=f

p

Sfoc
tp

Cp+Cd

∕W
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European Journal of Law and Economics (2023) 56:369–402
check what happens on the margin if the loss-averse plaintiff chooses
Sfoc
tp
instead.
The first-order derivative becomes:
Therefore,
S
foc <S
foc
tp
.
◻
To continue with the proof, we define two more critical values of W. For the loss-
averse plaintiff,

Wp
is defined as in
S
foc
(

Wp
)
=S
(

Wp
)
; for the traditional plaintiff,

Wtp
is defined in
S
foc
tp
(

Wtp
)
=Stp(

Wtp
)
.

Wp
(

Wtp
) is the lowest compensation at which
the constraint
Utrial
p
(S)⩾
0
(
Utrial
tp
(S)⩾
0
) is slack for the loss-averse (traditional)
plaintiff.
To see that

Wp
is uniquely defined, we use the fact that for
W⩾W
, we have:
At
W=W
, an equilibrium offer cannot be lower than
W+Cd
due to the credibility
constraint: we have p(S)=1
>
p(S
foc)
.32 For
W
→
∞,p(S)
→
0
and
p(Sfoc)
→
1
.
From the intermediate value theorem, we can find

Wp
such that p
(
Sfoc
)
=p(S
)
and
S
foc
(

Wp
)
=S(

Wp
)
. A similar proof goes for the traditional plaintiff. At
W
=

W
tp
, we
have
Sfoc
tp (

Wtp)=Stp (

Wtp
)
.
Lemma 3
1. At
W
=

W
tp
,
p
(S∗)>p
(
S∗
tp
)
and
S∗>S∗
tp
; for
W
>
�
W
tp
,
S
∗
tp
=S
foc
tp
. For
W
∈[Wtp ,

Wtp
]
, the
Utrial
tp
(S)⩾
0
constraint is binding for the traditional plaintiff;
2. At
W
=

W
p
,
p(S∗)<p(S∗
tp)
and
S∗<S∗
tp
; for
W
>
�
W
p
,
S∗=Sfoc
. For
W
∈[W,

W
p]
,
the
U
trial
p
(S)⩾
0
constraint is binding for the loss-averse plaintiff;
3.
�
Wtp
<
�
W
p
.
Proof At
W
=

W
tp
,
S
∗
tp =Stp =S
foc
tp (definition of

Wtp
). By Lemma 2, we have
S
>Stp =S
foc
tp >S
foc
. Therefore,
S∗>S∗
tp
and p(S∗)>p
(
S∗
tp
)
.
At
W
=

W
p
,
S∗=S=Sfoc
(definition of

Wp
). By Lemma 2, we have
S
tp <S=Sfoc <S
foc
tp . Therefore,
S∗<S∗
tp
, and
p(S∗)>p(S∗
tp)
.
U
�
p
(
Sfoc
tp
)
= −(𝜇−1)f
(
p
(
Sfoc
tp
))(
1−p
(
Sfoc
tp
))
Cp∕W<
0
dp(
S
foc)
∕dW >0, dp(S)∕dW <
0
32 For
W=W
, the plaintiff’s expected utility from bringing the lawsuit and making a credible offer is
zero. She is indifferent between bringing it and not. For simplicity, we assume that she brings the lawsuit,
makes a credible offer and pursue the case to trial. We assume the same thing for traditional plaintiff for
W=W
tp.
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398
European Journal of Law and Economics (2023) 56:369–402
1 3
Since
Sfoc
tp >Stp at

Wp
and
Sfoc
tp =Stp at

Wtp
, it means the credibility constraint is
binding at

Wtp
but not binding at

Wp
. We have
�
Wtp
<
�
W
p
.
◻
Directly from Lemma 2 and Lemma 3, for
W
⩽W<
�
W
tp
, the credibility con-
straint is binding for both plaintiffs. We have
S∗=S
,
S∗
tp
=S
tp ,
S>S
tp ⟹
S∗>S∗
tp
;
for
W
>
�
W
p
, neither is binding and we have
S∗=Sfoc
,
S
∗
tp
=S
foc
tp ,
S
foc <S
foc
tp ⟹S∗<S∗
tp
.
Now we show that in interval
[

W
tp
,

W
p]
, there exist

W
such that two plaintiffs
make the same settlement offer. By Lemma3, for
W
∈[

W
tp
,

W
p]
, we have
S
∗
tp
=S
foc
tp ,
S∗=S
and thus
p(S∗)=p(S)
and
p(
S∗
tp
)
=p
(
Sfoc
tp
)
.
At
W
=

W
tp
, p
(
S∗
tp
)
<p(S∗
)
; at

Wp
, p
(
S∗
tp
)
>p(S∗
)
. By the implicit function the-
orem, p
(
S∗
tp
)
and
p(S∗)
, as functions of W, are continuous in
W
∈[

W
tp
,

W
p]
and we
have the following monotonicity results from (5
′
) and (6):
From the intermediate value theorem,
p(S∗)
and
p(S∗
tp)
intersect at a unique point in
(

W
tp
,

W
p)
. We use

W
to denote this intersection. At

W
, we have
S
=S∗=S∗
tp
=S
foc
tp
.
The loss-averse plaintiff’s credibility constraint is binding while the traditional
plaintiff’s is not. In sum, for
W
⩽W<

W , we have:
Total expected litigation costs are higher if the plaintiff is loss-averse because the
probability of trial is higher. For
W>
W
, we have:
Thus, total litigation costs are smaller if the plaintiff is loss-averse.
End of proof Proposition 2.
Proof of Proposition3
1.
W
is defined by:
W
=(1+(𝜇−1)(1−𝔼[q]))
C
p
𝔼[q]
.
𝔼[q]
is higher under G since G
first-order stochastically dominates F. Thus,
W
is lower under G.
2.
p(S)
is the unique solution to
𝔼[q|q
<
p(S)]W−Cp−(
𝜇
−1)(1−𝔼[q|q
<
p(S)])
Cp
=
0
. Under G,
𝔼[q|q<h]
is higher for any
h>q+𝜀
. Since the LHS is increas-
ing in
𝔼[q|q<p(S)]
and
E[q|q<p(S)]
increases in
p(S)
, a lower
p(S)
is needed
for the equality to remain true.
dp(
Sfoc
tp
)
∕dW >
0
dp(S)∕dW
<
0
S
∗>S∗
tp,p(S∗)>p
(
S∗
tp
)
if W⩽W<
W
S
∗<S∗
tp,p(S∗)<p
(
S∗
tp
)
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399
1 3
European Journal of Law and Economics (2023) 56:369–402
3. Under F,
Sfoc
is given by (5). The RHS of Eq. (5) decreases in
p(Sfoc )
and
1−G(q+𝜀)
g(q+𝜀)
=
1−F(q)
f(q).
If the shift from F to G increases
p(Sfoc )
by
𝜀
, we must have
for
𝜇>1.
As the inverse hazard rate of F is assumed to be decreasing, the first
order condition under distribution G can then only be met with equality if
p(Sfoc )
is increased by more than
𝜀
.
4. It follows from 2 and 3 that

W
decreases.
◻
Proof of Proposition4
1.
W
is defined by:
W
=(1+(𝜇−1)(1−𝔼[q]))
C
p
𝔼[q]
.
𝔼[q]
is the same under

G
. Thus,
W
is not affected.
2.
p(S)
is the unique solution to
𝔼[q|q
<
p(S)]W−Cp−(
𝜇
−1)(1−𝔼[q|q
<
p(S)])
Cp
=
0
. Under

G
,
𝔼[q|q<h]
is higher for any
h>a+𝜀
. Since the LHS is increas-
ing in
𝔼[q|q<p(S)]
and
𝔼[q|q<p(S)]
increases in
p(S)
, one needs a lower
p(S)
for the equality to remain true.
3.
p(Sfoc)
under

G
is given by
1
−

G
(
p
(
S
foc))
g
(
p
(
Sfoc
))
=p�(Sfoc)(Cp+Cd)+(𝜇−1)p�(Sfoc )(1−p(Sfoc))C
p
.
Now,
g(q)=f(q)∕[F(b�)−F(a�)]
,

G(q)=0
for
q∈[a,a�]
, and

G(q)=
[F(q)−F(a′)]∕[F(b′)−F(a′)]
for
q∈(a�
,b�]
. Thus, for a given q, the LHS is
lower than in (4) while the RHS is unchanged. As the RHS is decreasing in
p(
S
foc)
, it calls for a decrease in p
(
S
foc)
for the equality to be maintained.
◻
Funding No specific funding was received for this research.
Declarations
Conflict of interest The authors are not aware of any conflict of interest pertaining to this piece of research.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Com-
mons licence, and indicate if changes were made. The images or other third party material in this article
are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the
material. If material is not included in the article’s Creative Commons licence and your intended use is
not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission
directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen
ses/ by/4. 0/.
1
−G
(
p
(
S
foc)
+𝜀
)
f
(
p
(
Sfoc
)
+𝜀
)
>
(Cp+Cd)
W+(𝜇−1)
(
1−
[
p
(
Sfoc
)
+𝜀
])
Cp
W
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400
European Journal of Law and Economics (2023) 56:369–402
1 3
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