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.

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, 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 Resolutions (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 settlement 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 lawsuit 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 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 literature 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 settles 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 credibility constraint is binding or not. As a result, many rules or policies foster settlements 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 discovery 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 English 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 litigation started with Landes (1971) and Gould (1973), who focused on the divergence in expectations about trial outcomes between the plaintiff and the defendant.⁸ 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 disproportionately be made up of cases favorable to the defendant, is borne out in many contexts.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 maximization and to incorporate more behaviorally-relevant decision-theoretic foundations 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).

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 popularity 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 follows: 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 disappointment aversion) in the Bebchuk (1984) model and reports ambiguous results about the likelihood of settlement. We focus on the possibility of dropping the lawsuit 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 usually 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) W, which is assumed to be fixed and commonly known to both players at the beginning of the game.¹⁴ If they proceed to the trial stage, the plaintiff will pay fixed litigation costs $C_p \geqslant 0$ and the defendant will pay $C_d \geqslant 0$. Those represent the direct and opportunity costs associated with introducing, supporting and defending a formal lawsuit. If the two parties manage to work out a settlement before trial, they will save litigation costs $C_p$ and $C_d$. 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 information 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 $f (\cdot )$ and a corresponding cumulative function $F (\cdot )$. 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:

$$\begin{aligned} u (w|o)= & {} \left\{ \begin{array}{lll} w - o &{} {\text {if}} &{} w \geqslant o\\ \mu (w - o) &{} {\text {if}} &{} w < o \end{array}\right. \end{aligned}$$

(1)

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 aversion coefficient, $\mu$ ($\mu \geqslant 1$). This coefficient describes the importance of loss aversion in the plaintiff’s preferences: for $\mu = 1$, the plaintiff is a standard expected utility maximizer; for $\mu > 1$, losses loom larger in her assessment of uncertain prospects, and the more so, the higher $\mu$.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 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 aversion. 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 settlement offer, the defendant would always find himself in the loss domain. Every payment he makes would be multiplied by coefficient $- \mu$ in his utility function, so the level of $\mu$ would not matter for his decisions as long as it is non-zero.¹⁹

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

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 $C_p$ 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 settlement 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 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 utility 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 settlement 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 credibility 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 profile and a belief system regarding the defendant’s type. A strategy profile $\beta = \{ L, S^{*}, d (S), r (S, q) \}$ specifies the following actions: $L \in \{ {\text {sue}}, {\text {not}} {\text {sue}} \}$ 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; $d (S) \in [0, 1]$ 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 $\sigma (q, S)$, 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 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 property on and off the equilibrium path.

Lemma 1

In a perfect Bayesian equilibrium, for an offer S with $d (S) \in [0, 1)$, if a type ${\tilde{q}}$ defendant weakly prefers rejecting to accepting, then (i) a type ${\tilde{q}}$ defendant strictly prefers rejecting ${\tilde{S}} > S$ and (ii) a defendant with $q < {\tilde{q}}$ strictly prefers rejecting S to accepting it.

Proof

see the appendix. $\square$

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 equilibrium choice is characterized by a cut-off type p(S)²³:

$$\begin{aligned} p (S)= \; & {} \frac{S - C_d}{W} \end{aligned}$$

(2)

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):

$$\begin{aligned} U_p (S)= \; & {} [1 - F (p (S))] S + \int _0^{p (S)} \left[ qW - C_p - (\mu - 1) (1 - q) C_p\right] f (q) dq \end{aligned}$$

(3)

$q W - C_p$ 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 $C_p$ is amplified by loss aversion. The solution to the first-order condition, $S^{foc}$, is given by:

$$\begin{aligned} 1 - F \left( p \left( S^{foc}\right) \right)= \; & {} f \left( p \left( S^{foc}\right) \right) p' \left( S^{foc}\right) \left( C_p + C_d\right) \nonumber \\{} & {} + (\mu - 1) f \left( p \left( S^{foc}\right) \right) p' \left( S^{foc}\right) \left( 1 - p \left( S^{foc}\right) \right) C_p \end{aligned}$$

(4)

Rewriting the above using $p' (S) = 1 / W$, we have:

$$\begin{aligned} \frac{1 - F \left( p \left( S^{foc}\right) \right) }{f \left( p \left( S^{foc}\right) \right) }= & {} \frac{\left( C_p + C_d\right) }{W} + (\mu - 1) \left( 1 - p \left( S^{foc}\right) \right) \frac{C_p}{W} \end{aligned}$$

(5)

The right-hand side of the first-order condition is decreasing in p(S). For it to uniquely pin down an interior solution $p \left( S^{foc}\right)$ and thus $S^{foc}$, we make the following assumptions on the distribution of q:

Assumptions A: For the p.d.f. $f (\cdot )$ and the corresponding c.d.f. $F (\cdot )$, we have that:

• 1. $\dfrac{1}{f (0)} > \dfrac{\mu C_p + C_d}{W}$;

• 2. $\dfrac{f (q)}{1 - F (q)}$ is increasing in q;

• 3. The concavity of $\frac{f (q)}{1 - F (q)}$ does not change in [0,1]: $\left( \dfrac{\partial ^2 \left( \frac{f (q)}{(1 - F (q))} \right) }{\partial q^2}\right)$ has a constant sign for $q \in [0, 1]$.

The first assumption guarantees that the marginal benefit of asking for more is high enough at $p (S) = 0$, ruling out the corner solution $S = C_d$. 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 distributions satisfy the second and the third assumptions, and most (truncations of) common distributions exhibit the third property.

Proposition 1

Under assumptions A, the first-order condition (5) has a unique solution in p(S) as well as in S.

Proof

see the appendix. $\square$

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 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 $C_p + C_d$ 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 losing probability is $(1 - p (S))$, which costs her $(\mu - 1) C_p$ 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:

$$\begin{aligned} U_p^{trial} (S)= \; & {} {\mathbb {E}} \left[ q|q< p (S)\right] W - C_p - (\mu - 1) \left( 1 -{\mathbb {E}}\left[ q|q < p (S)\right] \right) C_p \end{aligned}$$

where ${\mathbb {E}}$ 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:

$$\begin{aligned} \max _S U_p (S) \hspace{0.27em} s.t.U_p^{trial} (S) \ge 0 \end{aligned}$$

$U_p^{trial} (S)$ is the expected utility at trial stage. It is non-decreasing in S over $\left[ C_d, W + C_d\right]$: when the plaintiff increases the offer amount, the expected winning probability ${\mathbb {E}} \left[ q|q < p (S)\right]$ 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 ${\underline{S}},$ is the unique solution to the following equation:

$$\begin{aligned} U_p^{trial} ({\underline{S}}) ={\mathbb {E}} \left[ q|q< p ({\underline{S}})\right] W - C_p - (\mu - 1) \left( 1 -{\mathbb {E}}\left[ q|q < p ({\underline{S}})\right] \right) C_p = 0 \end{aligned}$$

(6)

Therefore, the equilibrium settlement offer $S^{*}$ is given by:

$$\begin{aligned} S^{*} = \max \left( S^{foc}, {\underline{S}}\right) \end{aligned}$$

(7)

The defendant rejects offer $S^{*}$ if and only if $q < p \left( S^{*}\right)$. If $S^{*}$ is rejected, the plaintiff’s belief about the defendant’s type is updated accordingly:

$$\begin{aligned} \sigma \left( q, S^{*}\right)= & {} \left\{ \begin{array}{ll} f (q) / F \left( p \left( S^{*}\right) \right) &{} {\text {if}} \; q \in \left[ p \left( S^{*}\right) , 1\right] \\ 0 &{} {\text {otherwise}} \end{array}\right. \end{aligned}$$

A technicality, which already arose in Nalebuff (1987), concerns very low offers. The full characterization of our PBE, which contains information about off-equilibrium 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 ($\mu = 1$ ). We use subscript tp to denote such a ‘traditional plaintiff’. Her objective is (assuming that bringing the lawsuit is profitable):

$$\begin{aligned} \max _S U_{tp} (S)= & {} \int _0^{p (S)} \left( qW - C_p\right) f (q) dq + \left( 1 - F (p (S))\right) S\\ s.t.U_{tp}^{trial} (S)= & {} \int _0^{p (S)} \left( q W - C_p\right) \frac{f (q)}{F (p (S))} d q \geqslant 0 \end{aligned}$$

This is a similar constrained maximization problem: the plaintiff chooses a settlement 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 $S_{tp}^{foc}$ by (5$'$) and $\underline{S_{tp}}$ by (6$'$):

$$\begin{aligned} 1 - F \left( p \left( S_{tp}^{foc}\right) \right)= f \left( p \left( S_{tp}^{foc}\right) \right) p' \left( S_{tp}^{foc}\right) \left( C_p + C_d\right)\end{aligned}$$

(5')

$$\begin{aligned}U_{tp}^{trial} \left( \underline{S_{tp}}\right) = 0\Rightarrow {\mathbb {E}} \left[ q|q p \left( \underline{S_{tp}}\right) \right] W - C_p = 0\end{aligned}$$

(6')

The solution is

$$\begin{aligned} S_{tp}^{*}&= \max \left( S_{tp}^{foc}, \underline{S_{tp}}\right)\end{aligned}$$

(7')

Comparing the settlement offers $\big (S^{*}$ and $S_{tp}^{*}\big )$ and the probabilities of trial $\bigg (F \left( p \left( S^{*}\right) \right)$ and $F \left( p \left( S_{tp}^{*}\right) \right) \bigg )$, we find that the result depends on the claim W. We have the following proposition:

Proposition 2

Compared with a traditional plaintiff, there exist unique values $\underline{W_{tp}}$, ${\underline{W}}$ and ${\tilde{W}}$ $\left( \underline{W_{tp}}< {\underline{W}} < {\tilde{W}}\right)$ such that

• 1. For small claims $\left( \underline{W_{tp}} \leqslant W < {\underline{W}}\right)$, a loss-averse plaintiff does not file a lawsuit while a traditional plaintiff does.

• 2. For big claims $\left( W \geqslant {\tilde{W}}\right)$ 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 $\left( {\underline{W}} \leqslant W < {\tilde{W}}\right)$, 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. $\square$

The three critical values for W are defined as follows:

$$\begin{aligned} \underline{W_{tp}}= & {} \frac{C_p}{{\mathbb {E}} [q]} \end{aligned}$$

(8)

$\underline{W_{tp}}$ is the minimum compensation level which incentivizes the traditional plaintiff to introduce the lawsuit. ${\underline{W}}$ is the minimum compensation level which incentivizes the loss-averse plaintiff to introduce the lawsuit:

$$\begin{aligned} {\underline{W}} = \left( 1 + (\mu - 1) (1 -{\mathbb {E}}[q])\right) \frac{C_p}{{\mathbb {E}} [q]} \end{aligned}$$

(9)

${\tilde{W}}$ is implicitly defined as:

$$\begin{aligned} {\underline{S}} \left( {\tilde{W}}\right) = S_{tp}^{foc} \left( {\tilde{W}}\right) \end{aligned}$$

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 ${\tilde{W}}$ 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 constraint 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 proceed to trial away from small and large stakes, and towards intermediate stakes, diminishing the dispersion of judgments.

2.4 A numerical example

Figures 2 and 3 give a numerical example of the probabilities of trial $F \left( p \left( S^{*}\right) \right)$ $\left( F \left( p \left( S_{tp}^{*}\right) \right) \right)$, and settlement offers $S^{*}$($S_{tp}^{*}$) under different claims W when q follows a truncated normal distribution on [0, 1].²⁵

Fig. 2

Probabilities of trial for different plaintiffs ($\mu = 2, C_p = 4, C_d = 2$)

Fig. 3

Settlement offers from different plaintiffs ($\mu = 2, C_p = 4, C_d = 2$)

The results from Proposition 2 are clear from the figures. $\widetilde{W_p}$ $\left( \widetilde{W_{tp}}\right)$ features a kink in the $S^{*} (W)$ $\left( S_{tp}^{*} (W)\right)$ curve. For intermediate W, the optimal settlement offer is determined by the credibility constraint that trial stage utility should be non-negative; for larger W, the optimal offer is determined by the first-order condition. For ${\underline{W}} \leqslant W<$ ${\tilde{W}}$, the loss-averse plaintiff demands a higher settlement to make sure that she will not drop the case if her offer is rejected. For $W \geqslant {\tilde{W}}$, the loss-averse plaintiff 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 $W < \underline{W_{tp}}$, neither a traditional plaintiff nor a loss-averse plaintiff finds it profitable to bring a lawsuit. For $\underline{W_{tp}} \leqslant W < {\underline{W}}$, 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 characterizing the effects on litigation costs. From the point of view of economic welfare, there is no reason for having a narrow concern for litigation costs, as deterrence and precedent-setting certainly have social value. However, given their high administrative costs, judicial systems often try to foster alternative dispute resolution mechanisms. It is therefore of interest to look at litigation costs.

3.1 Litigation costs

3.1.1 Plaintiff’s litigation costs $C_p$

When the credibility constraint is not binding, an increase in $C_p$ leads to a lower probability of trial. Higher $C_p$ means larger losses. So, the effect under loss aversion 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 $C_p$ 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 $C_p$ 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.

Fig. 4

The effect of higher $C_p$ on trial probabilities and on litigation costs. The shift is from $C_p = 2$ to $C_p = 4$, and other parameters are the same as before

3.1.2 Defendant’s litigation costs $C_d$

The effects of higher $C_d$ also depend on W. When W is intermediate and the credibility constraint is binding, $C_d$ does not affect the plaintiff’s choice of $p (S^{*})$. It is determined by Eq. (6), $U_p^{trial} \left( S^{*}\right) = 0$, and $C_d$ plays no part in it. As in Nalebuff (1987), $S^{*}$ increases as $C_d$ does so as to keep $p \left( S^{*}\right)$ unchanged and Eq. (6) satisfied.²⁶

If W is high enough such that the credibility constraint is not binding, an increase in $C_d$ lowers $p \left( S^{*}\right)$ 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 $C_d$ is higher. The effects on $S^{*}$ and total litigation costs are thus ambiguous.

Figure 5 gives an illustration.

Fig. 5

The effect of higher $C_d$ on trial probabilities and on litigation costs. The shift is from $C_d = 2$ to $C_d = 4$, and other parameters are the same as before

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 $[{\underline{q}}, {\bar{q}}]$ with ${\underline{q}} > 0$ and ${\bar{q}} < 1$. We consider two distributional changes regarding q.

First, we consider a rightward shift to distribution $G (\cdot )$ over support $[{\underline{q}} + \varepsilon , {\bar{q}} + \varepsilon ]$ with $g (x + \varepsilon ) = f (x)$ for $x \in [{\underline{q}}, {\bar{q}}]$. The new distribution $G (\cdot )$ first-order stochastically dominates distribution $F (\cdot )$, which means that the plaintiff unambiguously faces a pool of weaker defendants.

Under the new distribution $G (\cdot )$, 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 $G (\cdot )$. Intuitively, as the plaintiff’s overall probability of losing is lower under $G (\cdot )$, 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. ${\underline{W}}$ decreases;

• 2. $p ({\underline{S}})$ decreases;

• 3. $p \left( S^{foc}\right)$ increases by more than $\varepsilon$;

• 4. ${\tilde{W}}$ decreases.

Proof

see the appendix $\square$

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 litigation costs depends on the size of the claim. Figure 6 gives an illustration.

Fig. 6

FOSD shift of distribution ($\mu = 2, C_p = 4, C_d = 2$). The shift is from a truncated normal distribution with mean 0.5 and support [0.2,0.8] to a truncated normal distribution with mean 0.7 and support [0.4,1]. The untruncated distribution has standard deviation 0.2

Second, we consider a mean-preserving truncation of $F (\cdot )$. Formally, for F with support $\left[ {\underline{q}}, {\bar{q}}\right] \subset [0, 1]$ and for a small $\varepsilon > 0$, define ${\underline{q}}' = {\underline{q}} + \varepsilon$ and ${\bar{q}}' < {\bar{q}}$ such that ${\mathbb {E}} [q] ={\mathbb {E}} \left[ q | q \in \left[ {\underline{q}}', {\bar{q}}'\right] \right]$. (Such a ${\bar{q}}'$ can always be found.) Take ${\tilde{G}}$ to be the truncation of F on $\left[ {\underline{q}}', {\bar{q}}'\right]$. Then, ${\tilde{G}}$ is a mean-preserving truncation of F and second-order stochastically 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 ${\tilde{G}}$:

• 1. ${\underline{W}}$ is not affected;

• 2. $p ({\underline{S}})$ decreases;

• 3. $p \left( S^{foc}\right)$ decreases.

Proof

see the appendix. $\square$

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 Bebchuk (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 beneficial settlement becomes more likely.

Figure 7 gives an illustration.

Fig. 7

Mean preserving truncation of distribution ($\mu = 2, C_p = 4, C_d = 2$). 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

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 moving to an environment with $W^{EN} = W + C_p + C_d$, $C_p^{EN} = C_p + C_d$ and $C_d^{EN} = 0$ 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, ${\underline{W}}$, the lowest compensation that incentivizes the plaintiff to sue, is given by:

$$\begin{aligned} {\underline{W}}= & {} \left[ 1 + (\mu - 1) (1 -{\mathbb {E}} [q])\right] \dfrac{C_p}{{\mathbb {E}} [p]} \end{aligned}$$

Under the English rule, we have:

$$\begin{aligned} {\underline{W}}^{{\text {EN}}}= & {} \frac{\mu (1 -{\mathbb {E}} [q])}{{\mathbb {E}} [q]} \left( C_p + C_d\right) \end{aligned}$$

The relative size of ${\underline{W}}$ and ${\underline{W}}^{{\text {EN}}}$ depends on $\mu$, $C_p$, $C_d$ as well as the unconditional expectation of q. If the credibility constraint is not binding, the likelihood of settlement 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 ambiguous 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:

$$\begin{aligned} \frac{1 - F \left( p \left( S^{foc}\right) \right) }{f \left( p \left( S^{foc}\right) \right) } = \dfrac{\left( C_p + C_d\right) }{W} + (\mu - 1) \left( 1 - p \left( S^{foc}\right) \right) \frac{C_p}{W} \end{aligned}$$

(5)

$$\begin{aligned} \frac{1 - F \left( p \left( S^{foc}_{\text {EN}}\right) \right) }{f \left( p \left( S^{foc}_{\text {EN}}\right) \right) } = \dfrac{\left( C_p + C_d\right) }{W + C_p + C_d} + (\mu - 1) \left( 1 - p \left( S^{foc}_{\text {EN}}\right) \right) \frac{C_p + C_d}{W + C_p + C_d} \qquad \left( {5^{\textrm{EN}}}\right) \end{aligned}$$

For $\mu = 1$, fee-shifting unambiguously decreases the right side of Eq. (5). From the increasing hazard rate property, $p \left( S^{foc}\right)$ increases as a result, leading to a lower settlement rates. For Eq. (5$^{{\textrm{EN}}}$), the right-hand side might become smaller if $\mu$ and $C_d$ 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.

Fig. 8

Probabilities of trial under different rules ($\mu = 2, C_p = 4, C_d = 2$)

Obviously, the net effect on the total number of trials will depend on the distribution 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).

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 an in-court settlement regime. If it costs the judicial system much less to register settlements 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 $C_p$ 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:

$$\begin{aligned} U_p^{\text {in-court}} (S) \; = & {} [1 - F (p (S))] (S - C_p) + \int _0^{p (S)} \left[ q (W - C_p) - (\mu - 1) (1 - q) C_p\right] f (q) d q \end{aligned}$$

(10)

The optimal settlement offer ($S_{in}^{*}$) is characterized by the following first-order condition:

$$\begin{aligned} \frac{1 - F \left( p \left( S_{in}^{*}\right) \right) }{f \left( p \left( S_{in}^{*}\right) \right) }&= \dfrac{C_d}{W} + (\mu - 1) \left( 1 - p \left( S_{in}^{*}\right) \right) \frac{C_p}{W}\qquad \qquad \qquad \qquad \qquad \qquad \left( {5^{\text {in-court}}}\right) \end{aligned}$$

Our assumptions on $F (\cdot )$ guarantees that we have a unique interior solution $S_{in}^{*} \in [C_d, W + C_d)$. It is straightforward to show that $S_{in}^{*} > S^{foc}$: $C_p$ has been paid up-front so saving $C_p$ 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 $C_p$ anyway. Therefore, for intermediate values of W at which the credibility 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 ($\underline{W_{in}}$), we have $\underline{W_{in}} \leqslant {\underline{W}}$. Intuitively, at $W = {\underline{W}}$ in the in-court settlement system, the plaintiff could bring the lawsuit and ask for $S \geqslant W + C_d$. This brings her the same utility as in the out-of-court settlement regime. It is possible that she can do better because the credibility constraint is no longer playing a role.

A special case arises when $C_d \geqslant C_p$. The plaintiff can bring the lawsuit, pay $C_p$ and ask for $S = C_d$ as soon as $W \geqslant 0$. The defendant will accept the offer whatever his type is. We have $\underline{W_{in}} \leqslant {\underline{W}}$ in general and $\underline{W_{in}} \leqslant 0$ if $C_d \geqslant C_p$: 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 $C_d \geqslant C_p$ cases.

Fig. 9

Probabilities of trial and litigation costs under different settlement systems ($\mu = 2, C_p = 4, C_d = 2$)

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.

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 presence 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:

$$\begin{aligned} \sigma (q) = \frac{q}{1 + \beta - \beta q} < q. \end{aligned}$$

$\beta$ is the coefficient of disappointment aversion. The plaintiff is the informed party and the uninformed defendant makes the settlement offer. The plaintiff who is indifferent between accepting or rejecting settlement offer S is thus characterized by:

$$\begin{aligned} \sigma \left( {\hat{q}}\right) W - C_p = S. \end{aligned}$$

The derivative of ${\hat{q}}$ with respect to S, which is fixed at 1/W in our model, is now given by a complicated function of ${\hat{q}}$ and the following first-order condition for an interior solution for S (the equivalent of our Eq. (5)):

$$\begin{aligned} \frac{1 - F \left( {\hat{q}}\right) }{f \left( {\hat{q}}\right) } = \left( \frac{C_p + C_d}{W} + {\hat{q}} - \sigma \left( {\hat{q}}\right) \right) \frac{{\hat{q}}}{\sigma \left( {\hat{q}}\right) } \frac{1 - {\hat{q}}}{1 - \sigma \left( {\hat{q}}\right) }. \end{aligned}$$

The non-monotonicity of the right-hand side is the cause of the fundamental ambiguity 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 constraint, 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 Asymmetric 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 elaborate 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 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 plaintiff 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, compared 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 statistical distribution. If W is random but the lowest possible value is larger than or equal to $C_p$, 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 $C_p$, 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 distribution of the plaintiff win rate, q. In practice, the two might be (positively) correlated. 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 introducing 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 correlation and, again, assuming that the lowest possible value for W is higher than $C_p$, 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 straightforward 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 $C_p$ 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 probability), while the likelihood of settlement would go up for high stakes, which would again reinforce our results.

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 introduction 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 aversion 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 positive probability if he rejects the offer ($d (S) \in [0, 1)$). The defendant will compare the 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):

$$\begin{aligned} U_d^{trial} (q) = - (1 - d (S)) \left( qW + C_d\right), \end{aligned}$$

The expected utility from accepting the offer is $- S$. As $U_d^{trial} (q)$ is decreasing in q, $U_d^{trial} \left( {\tilde{q}}\right) \geqslant - S$ implies that $U_d^{trial} (q) > - S$ for $q < {\tilde{q}}$ and $U_d^{trial} \left( {\tilde{q}}\right) > - {\tilde{S}}$ for ${\tilde{S}} > S$. $\square$

Proof of Proposition 1

the existence and uniqueness of $S^{{\text {foc}}}$

Recall first-order condition (5):

$$\begin{aligned} \frac{1 - F \left( p \left( S^{foc}\right) \right) }{f \left( p \left( S^{foc}\right) \right) }= & {} \frac{(C_p + C_d)}{W} + (\mu - 1) \left( 1 - p \left( S^{foc}\right) \right) \frac{C_p}{W} \end{aligned}$$

Define function $H (\cdot )$ as the following:

$$\begin{aligned} H (x)\equiv \; & {} \frac{\left( C_p + C_d\right) }{W} + (\mu - 1) (1 - x) \frac{C_p}{W} - \frac{1 - F (x)}{f (x)}, x \in [0, 1]\\ = \; & {} \frac{\left( \mu C_p + C_d\right) }{W} - (\mu - 1) x \frac{C_p}{W} - \frac{1 - F (x)}{f (x)} \end{aligned}$$

H(x) is continuous and twice differentiable by our assumptions. We have:

$$\begin{aligned} H (1) \; = \; & {} \frac{\left( \mu C_p + C_d\right) }{W} - (\mu - 1) \frac{C_p}{W} = \frac{(C_p + C_d)}{W} > 0\\ H (0) \; = \; & {} \frac{\left( \mu C_p + C_d\right) }{W} - \frac{1}{f (0)} < 0 \,\, {\text {by}}\,\, {\text {assumption}} \,\,A 1 \end{aligned}$$

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 second-order derivative of function $H (\cdot )$ has a constant sign over [0, 1]. Because the loss-aversion part $- (\mu - 1) x \frac{C_p}{W}$ is linear in x, the condition is met if and only if the second-order derivative of the reversed hazard rate function $\frac{1 - F (x)}{f (x)}$ 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) $\frac{1 - F (x)}{f (x)}$ is decreasing in x on [0, 1]; and 3) the second-order derivative of $\frac{1 - F (x)}{f (x)}$ 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 $[0, {\hat{x}}]$ and decreasing in $({\hat{x}}, 1]$ where ${\hat{x}} \in [0, 1]$ and $H' ({\hat{x}}) = 0$; 3) decreasing in $[0, {\tilde{x}}]$ and increasing in $({\tilde{x}}, 1]$ where ${\tilde{x}} \in [0, 1] \; {\text {and}} \; H' ({\tilde{x}}) = 0$.

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 intermediate value theorem on $[0, {\hat{x}}]$ and in form 3) on $[{\tilde{x}}, 1]$. With monotonicity over the relevant sub-interval, a unique root is also guaranteed. The corresponding settlement offer S can be recovered from p(S) in (2).$\square$

Full description of the PBE

Note that the plaintiff cannot commit to trial if an offer $S < {\underline{S}}$ 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 $S < {\underline{S}}$, 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 subgames therefore features mixed strategies: the defendant rejects offer S if $q < p ({\underline{S}})$ and accepts it if $q \geqslant p ({\underline{S}})$. The plaintiff, who is indifferent between giving up the lawsuit and not, proceeds to trial with probability $S / {\underline{S}}$.³¹ For the plaintiff, an offer $S < {\underline{S}}$ is dominated by ${\underline{S}}$: offering $S < {\underline{S}}$, she received the same trial stage utility of 0 but the settlement amount is strictly smaller than ${\underline{S}}$ for an equal probability of acceptance. So, those subgames are never reached in equilibrium.

For $W \geqslant {\underline{W}}$, our PBE is described by the following assessment $(\beta , \sigma )$:

$\beta = \left\{ {\text {sue}}, S^{*} = \max \left( S^{{\text {foc}}}, {\underline{S}}\right) , d (S) = \left\{ \begin{array}{ll} 1 - S / {\underline{S}} &{} {\text {if}} S< {\underline{S}}\\ 0 &{} {\text {if}} S \geqslant {\underline{S}} \end{array}\right. , r (S, q) = \left\{ \begin{array}{ll} 0 &{} q \geqslant p (\max (S, {\underline{S}}))\\ 1 &{} q < p (\max (S, {\underline{S}})) \end{array}\right. \right\} ;$

$\sigma (q, S) = \left\{ \begin{array}{ll} f (q) / F (p (S)) &{} {\text {if}} \; q \in [p (S), 1]\\ 0 &{} \; {\text {otherwise}} \end{array}\right. {\text {if}} \; {\underline{S}} \leqslant S \leqslant W + C_d;$

$\sigma (q, S) = \left\{ \begin{array}{ll} f (q) / F (p ({\underline{S}})) & {\text {if}} \; q \in [p ({\underline{S}}), 1]\\ 0 & \; {\text {otherwise}} \end{array}\right. {\text {if}} \; 0 \leqslant S < {\underline{S}};$

$\sigma (q, S) = f (q) ~ ~ {\text {if}} \; S > W + C_d \; \text { or }\; S < 0.$

Proof of Proposition 2

The loss-averse plaintiff’s trial stage utility is:

$$\begin{aligned} U_p^{trial} (S) ={\mathbb {E}} [q|q< p (S)] W - C_p - (\mu - 1) (1 -{\mathbb {E}}[q|q < p (S)]) C_p \end{aligned}$$

We have $dU_p^{trial} / dp (S) > 0$. And since $dp (S) / dS \geqslant 0$, we have $dU _p^{trial} / dS \geqslant 0$. This means the maximum of $U_p^{trial}$ is achieved at $p (S) = 1$ with $S \geqslant W + C_d$ for any given W. We use $\overline{U_p^{trial}}$ to denote this maximum:

$$\begin{aligned} \overline{U_p^{trial}} ={\mathbb {E}} [q] W - C_p - (\mu - 1) (1 -{\mathbb {E}}[q]) C_p \end{aligned}$$

We also have that $dU_p^{trial} / dW > 0$. By definition of ${\underline{W}}$, $\overline{U_p^{trial}} ={\mathbb {E}} [q] {\underline{W}} - C_p - (\mu - 1) (1 -{\mathbb {E}} [q]) C_p = 0$. So, for $W < {\underline{W}}$, $U_p^{trial} (S) < 0$ for any S. Hence, the credibility constraint 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 plaintiff will not bring the lawsuit. A similar proof goes for the traditional plaintiff for $W < \underline{W_{tp}}$. As $\underline{W_{tp}} = \frac{C_p}{{\mathbb {E}} [p]} < {\underline{W}}$, a loss-averse plaintiff does not file a lawsuit while a traditional plaintiff does for $W \in [\underline{W_{tp}}, W)$. This proves part 1.

To prove parts 2 and 3, we state the following two lemmas.

Lemma 2

${\underline{S}} > \underline{S_{tp}}$; $S^{foc} < S_{tp}^{foc}$.

Proof

The proof of this lemma directly follows the definition of the relevant settlement offers. For ${\underline{S}}$ and $\underline{S_{tp}}$, we have:

$$\begin{aligned} {\mathbb {E}} [q|q< p ({\underline{S}})] W - C_p&= (\mu - 1) (1 -{\mathbb {E}}[q|q < p ({\underline{S}})]) C_p \end{aligned}$$

(6)

$$\begin{aligned} {\mathbb {E}} [q|q < p (\underline{S_{tp}})] W - C_p&= 0\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

(6')

As $\mu > 1$ and ${\mathbb {E}} [q|q < x]$ is weakly increasing in $x \in [0, 1]$, we have ${\underline{S}} > \underline{S_{tp}}$. Moreover, ${\underline{S}} (W)$ and $\underline{S_{tp}} (W)$ are implicitly defined from the above equations. By the implicit functions theorem, they are both continuous functions.

For $S^{foc}$ and $S_{tp}^{foc}$, using $p' (S) = 1 / W$, we

$$\begin{aligned} 1 - F \left( p \left( S^{foc}\right) \right)&= f \left( p \left( S^{foc}\right) \right) \left( C_p + C_d\right) / W \\&\quad + (\mu - 1) f \left( p \left( S^{foc}\right) \right) \left( 1 - p \left( S^{foc}\right) \right) C_p /W \end{aligned}$$

(5)

$$\begin{aligned} 1 - F \left( p \left( S_{tp}^{foc}\right) \right)&= f \left( p \left( S_{tp}^{foc}\right) \right) \left( C_p + C_d\right) / W\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

(5')

Similarly, $S^{foc} (W)$ and $S_{tp}^{foc} (W)$ are implicitly defined from the above first-order (5) and (5$'$) and they are thus continuous functions of W. To see that $S^{foc}$ is smaller, we check what happens on the margin if the loss-averse plaintiff chooses $S_{tp}^{foc}$ instead. The first-order derivative becomes:

$$\begin{aligned} U_p' \left( S_{tp}^{foc}\right) = - (\mu - 1) f \left( p \left( S_{tp}^{foc}\right) \right) \left( 1 - p \left( S_{tp}^{foc}\right) \right) C_p / W < 0 \end{aligned}$$

Therefore, $S^{foc} < S_{tp}^{foc}$. $\square$

To continue with the proof, we define two more critical values of W. For the loss-averse plaintiff, $\widetilde{W_p}$ is defined as in $S^{foc} \left( \widetilde{W_p}\right) = {\underline{S}} \left( \widetilde{W_p}\right)$; for the traditional plaintiff, $\widetilde{W_{tp}}$ is defined in $S_{tp}^{foc} \left( \widetilde{W_{tp}}\right) = \underline{S_{tp}} (\widetilde{W_{tp}})$. $\widetilde{W_p}$ ($\widetilde{W_{tp}}$) is the lowest compensation at which the constraint $U_p^{trial} (S) \geqslant 0$ ($U_{tp}^{trial} (S) \geqslant 0$) is slack for the loss-averse (traditional) plaintiff.

To see that $\widetilde{W_p}$ is uniquely defined, we use the fact that for $W \geqslant {\underline{W}}$, we have:

$$\begin{aligned} dp \left( S^{foc}\right) / dW > 0, dp ({\underline{S}}) / dW < 0 \end{aligned}$$

At $W = {\underline{W}}$, an equilibrium offer cannot be lower than $W + C_d$ due to the credibility constraint: we have $p ({\underline{S}}) = 1 > p (S^{foc})$.³² For $W \rightarrow \infty , p ({\underline{S}}) \rightarrow 0$ and $p (S^{foc}) \rightarrow 1$. From the intermediate value theorem, we can find $\widetilde{W_p}$ such that $p \left( S^{foc}\right) = p ({\underline{S}})$ and $S^{foc} \left( \widetilde{W_p}\right) = {\underline{S}} (\widetilde{W_p})$. A similar proof goes for the traditional plaintiff. At $W = \widetilde{W_{tp}}$, we have $S_{tp}^{foc} (\widetilde{W_{tp}}) = \underline{S_{tp}} (\widetilde{W_{tp}})$.

Lemma 3

• 1. At $W = \widetilde{W_{tp}}$, $p \left( S^{*}\right) > p \left( S_{tp}^{*}\right)$ and $S^{*} > S_{tp}^{*}$; for $W > \widetilde{W_{tp}}$, $S_{tp}^{*} = S_{tp}^{foc}$. For $W \in [\underline{W_{tp}}, \widetilde{W_{tp}}]$, the $U_{tp}^{trial} (S) \geqslant 0$ constraint is binding for the traditional plaintiff;

• 2. At $W = \widetilde{W_p}$, $p \left( S^{*}\right) < p (S_{tp}^{*})$ and $S^{*} < S_{tp}^{*}$; for $W > \widetilde{W_p}$, $S^{*} = S^{foc}$. For $W \in [{\underline{W}}, \widetilde{W_p}]$, ${\text {the}} U_p^{trial} (S) \geqslant 0$ constraint is binding for the loss-averse plaintiff;

• 3. $\widetilde{W_{tp}} < \widetilde{W_p}$.

Proof

At $W = \widetilde{W_{tp}}$, $S_{tp}^{*} = \underline{S_{tp}} = S_{tp}^{foc}$ (definition of $\widetilde{W_{tp}}$). By Lemma 2, we have ${\underline{S}}> \underline{S_{tp}} = S_{tp}^{foc} > S^{foc}$. Therefore, $S^{*} > S_{tp}^{*}$ and $p (S^{*}) > p \left( S_{tp}^{*}\right)$.

At $W = {\tilde{W}}_p$, $S^{*} = {\underline{S}} = S^{foc}$ (definition of $\widetilde{W_p}$). By Lemma 2, we have $\underline{S_{tp}}< {\underline{S}} = S^{foc} < S_{tp}^{foc}$. Therefore, $S^{*} < S_{tp}^{*}$, and $p \left( S^{*}\right) > p (S_{tp}^{*})$.

Since $S_{tp}^{foc} > \underline{S_{tp}}$ at $\widetilde{W_p}$ and $S_{tp}^{foc} = \underline{S_{tp}}$ at $\widetilde{W_{tp}}$, it means the credibility constraint is binding at $\widetilde{W_{tp}}$ but not binding at $\widetilde{W_p}$. We have $\widetilde{W_{tp}} < \widetilde{W_p}$. $\square$

Directly from Lemma 2 and Lemma 3, for ${\underline{W}} \leqslant W < \widetilde{W_{tp}}$, the credibility constraint is binding for both plaintiffs. We have $S^{*} = {\underline{S}}$, $S_{tp}^{*} = \underline{S_{tp}}$, ${\underline{S}}> \underline{S_{tp}} \Longrightarrow S^{*} > S_{tp}^{*}$; for $W > \widetilde{W_p}$, neither is binding and we have $S^{*} = S^{foc}$, $S_{tp}^{*} = S_{tp}^{foc}$, $S^{foc}< S_{tp}^{foc} \Longrightarrow S^{*} < S_{tp}^{*}$.

Now we show that in interval $[\widetilde{W_{tp}}, \widetilde{W_p}]$, there exist ${\tilde{W}}$ such that two plaintiffs make the same settlement offer. By Lemma 3, for $W \in [\widetilde{W_{tp}}, \widetilde{W_p}]$, we have $S_{tp}^{*} = S_{tp}^{foc}$, $S^{*} = {\underline{S}}$ and thus $p \left( S^{*}\right) = p ({\underline{S}})$ and $p \left( S_{tp}^{*}\right) = p \left( S_{tp}^{foc}\right)$.

At $W = \widetilde{W_{tp}}$, $p \left( S_{tp}^{*}\right) < p \left( S^{*}\right)$; at $\widetilde{W_p}$, $p \left( S_{tp}^{*}\right) > p \left( S^{*}\right)$. By the implicit function theorem, $p \left( S_{tp}^{*}\right)$ and $p \left( S^{*}\right)$, as functions of W, are continuous in $W \in [\widetilde{W_{tp}}, \widetilde{W_p}]$ and we have the following monotonicity results from (5$'$) and (6):

$$\begin{aligned} dp \left( S_{tp}^{foc}\right) / dW> & {} 0\\ dp ({\underline{S}}) / dW< & {} 0 \end{aligned}$$

From the intermediate value theorem, $p (S^{*})$ and $p (S_{tp}^{*})$ intersect at a unique point in $(\widetilde{W_{tp}}, \widetilde{W_p})$. We use ${\tilde{W}}$ to denote this intersection. At ${\tilde{W}}$, we have ${\underline{S}} = S^{*} = S_{tp}^{*} = S_{tp}^{foc}$. The loss-averse plaintiff’s credibility constraint is binding while the traditional plaintiff’s is not. In sum, for ${\underline{W}} \leqslant W < {\tilde{W}}$, we have:

$$\begin{aligned} S^{*}> S_{tp}^{*}, \; p \left( S^{*}\right) > p \left( S_{tp}^{*}\right)& \; {\text {if}} \quad &{\underline{W}} \leqslant W < {\tilde{W}} \end{aligned}$$

Total expected litigation costs are higher if the plaintiff is loss-averse because the probability of trial is higher. For $W > {\tilde{W}}$, we have:

$$\begin{aligned} S^{*}< S_{tp}^{*}, \; p \left( S^{*}\right) < p \left( S_{tp}^{*}\right) \end{aligned}$$

Thus, total litigation costs are smaller if the plaintiff is loss-averse.

End of proof Proposition 2.

Proof of Proposition 3

• 1. ${\underline{W}}$ is defined by: ${\underline{W}} = (1 + (\mu - 1) (1 -{\mathbb {E}}[q])) \frac{C_p}{{\mathbb {E}} [q]}$. ${\mathbb {E}} [q]$ is higher under G since G first-order stochastically dominates F. Thus, ${\underline{W}}$ is lower under G.

• 2. $p ({\underline{S}})$ is the unique solution to ${\mathbb {E}} [q|q< p ({\underline{S}})] W - C_p - (\mu - 1) (1 -{\mathbb {E}}[q|q < p ({\underline{S}})]) C_p = 0$. Under G, ${\mathbb {E}} [q|q < h]$ is higher for any $h > {\underline{q}} + \varepsilon$. Since the LHS is increasing in ${\mathbb {E}} [q|q < p ({\underline{S}})]$ and $E [q|q < p ({\underline{S}})]$ increases in $p ({\underline{S}})$, a lower $p ({\underline{S}})$ is needed for the equality to remain true.

• 3. Under F, $S^{foc}$ is given by (5). The RHS of Eq. (5) decreases in $p (S^{{\text {foc}}})$ and $\frac{1 - G (q + \varepsilon )}{g (q + \varepsilon )} = \frac{1 - F (q)}{f (q)}.$ If the shift from F to G increases $p (S^{{\text {foc}}})$ by $\varepsilon$, we must have

  $$\begin{aligned} \frac{1 - G \left( p \left( S^{foc}\right) + \varepsilon \right) }{f \left( p \left( S^{foc}\right) + \varepsilon \right) } > \frac{(C_p + C_d)}{W} + (\mu - 1) \left( 1 -\left[ p \left( S^{foc}\right) + \varepsilon \right] \right) \frac{C_p}{W} \end{aligned}$$

  for $\mu > 1.$ 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 $p (S^{{\text {foc}}})$ is increased by more than $\varepsilon$.

• 4. It follows from 2 and 3 that ${\tilde{W}}$ decreases.

$\square$

Proof of Proposition 4

• 1. ${\underline{W}}$ is defined by: ${\underline{W}} = (1 + (\mu - 1) (1 -{\mathbb {E}}[q])) \frac{C_p}{{\mathbb {E}} [q]}$. ${\mathbb {E}} [q]$ is the same under ${\tilde{G}}$. Thus, ${\underline{W}}$ is not affected.

• 2. $p ({\underline{S}})$ is the unique solution to ${\mathbb {E}} [q|q< p ({\underline{S}})] W - C_p - (\mu - 1) (1 -{\mathbb {E}}[q|q < p ({\underline{S}})]) C_p = 0$. Under ${\tilde{G}}$, ${\mathbb {E}} [q|q < h]$ is higher for any $h > a + \varepsilon$. Since the LHS is increasing in ${\mathbb {E}} [q|q < p ({\underline{S}})]$ and ${\mathbb {E}} [q|q < p ({\underline{S}})]$ increases in $p ({\underline{S}})$, one needs a lower $p ({\underline{S}})$ for the equality to remain true.

• 3. $p (S^{foc})$ under ${\tilde{G}}$ is given by $\frac{1 - {\tilde{G}} \left( p \left( S^{foc}\right) \right) }{{\tilde{g}} \left( p \left( S^{foc}\right) \right) } = p' \left( S^{foc}\right) (C_p + C_d) + (\mu - 1) p' \left( S^{foc}\right) \left( 1 - p \left( S^{foc}\right) \right) C_p$. Now, ${\tilde{g}} (q) = f (q) / [F (b') - F (a')]$, ${\tilde{G}} (q) = 0$ for $q \in [a, a']$, and ${\tilde{G}} (q) = [F (q) - F (a')] / [F (b') - F (a')]$ for $q \in (a', b']$. Thus, for a given q, the LHS is lower than in (4) while the RHS is unchanged. As the RHS is decreasing in $p \left( S^{f o c}\right)$, it calls for a decrease in $p \left( S^{foc}\right)$ for the equality to be maintained.

$\square$