Third-party funding in a sequential litigation process

Abstract

Third party litigation financing (TPF)—for-profit, nonrecourse funding of litigation by a nonparty—is a new and rapidly developing industry. As a novel phenomenon that involves various normative concerns, TPF has sparked much debate and controversy among scholars and policy-makers, speculating about its potential effects on issues such as the volume of litigation and the quality of claims filed. We develop a game-theoretic model that compares a litigation process with TPF and a “traditional” scheme in which litigation is self-funded. Under the TPF scheme, we decompose the litigation decisions into two parts: the plaintiff is in charge of the legal decisions, while the TPF has the freedom to decide in each stage of the litigation process whether to continue the financial support in the litigation process. Such a setting is characterized by a high level of uncertainty and a degree of asymmetric information between the plaintiff and the TPF. We argue that the divergent interests of the parties to the financing agreement can be aligned by constructing a viable contract that results in the same equilibrium outcome as litigation with no TPF. The contract that achieves these desired results has a few interesting properties. First, it provides a pre-specified remedy to the plaintiff if the TPF funder terminates the financing prior to the conclusion of the litigation process. Second, the contract also specifies the compensation to the TPF funder, which is due upon completion of the litigation process, and it is conditioned upon the awarded verdict.

1 Introduction

“A poor man may have the right upon his side, but be without means to enforce such rights in the courts, and possibly against some powerful adversary. Surely, it cannot be said that in such case it is the intent of the law to prohibit a friend from assisting him with the necessary money to enforce his rights, dependent for his reimbursement solely upon the contingency of securing a portion of the property which may be obtained by the litigation; this being the only security or chance for repayment which the party could give or have¹”

Over a century ago, the Colorado Court of Appeals noted that in many instances, victims are precluded from access to justice and are deprived of compensation for the wrongful damages they have suffered simply because they lack the financial resources necessary to file and litigate their claims. In the past hundred years, the financial barriers have not vanished; indeed, in the United States—as well as in other jurisdictions² —the issue of access to justice is still acute, being restricted by financial, and other, disadvantages. In fact, litigation is not as much a quest for the Holy Grail of truth as it is a financial race (see, for instance, Puri 1998).

As a result of the limited access to justice due to the shortage of funds, various forms of litigation financing have developed over the years. These schemes include such mechanisms as legal aid (Dnes and Rickman 1998; Garoupa and Stephen 2004; Baik and Kim 2007; Lambert and Chappe 2014), public litigation subsidy funds, contingent fee arrangements (Puri 1998; Lambert and Chappe 2014; Kirstein and Rickman 2004) legal cost insurance (Van Velthoven and Van Wijck 2001; Heyes et al. 2004; Friehe 2010; Qiao 2013), crowdfunding (Perry 2018), and other.

One such type of funding—third-party litigation financing (TPF), also known under the monikers ‘claim funding,’ ‘alternative litigation financing,’ ‘litigation lending,’ and ‘third-party funding’— is a new, relatively recent, venue for financing litigation proceedings. This industry has experienced unprecedented growth and widespread recognition in the United States and abroad. As a mechanism that is “playing an increasingly visible role” (Faure and De Mot 2011: 744 ), this litigation financing scheme has instantly ignited the interest of policy-makers and scholars alike; yet it “remains poorly understood” (Shepherd and Stone 2015: 919).

The TPF scheme involves third parties providing non-recourse loans to claim-holders to pursue their claims in exchange for a share of proceeds in the case of success. This mechanism—which some scholars consider to be “one of the most innovative trends in civil litigation financing today” (De Morpurgo 2011: 410)—involves numerous economic incentives, legal considerations and public policy issues (Shamir 2016).

As a novel phenomenon, and one that involves various normative concerns, TPF has sparked much debate and controversy among scholars. With a few notable exceptions of economic analyses (Lysaught and Hazelgrove 2011; Abramowicz and Alper 2013; Demougin and Maultzsch 2014; Abramowicz and Alper 2013; Deffains and Desrieux 2015 Daughety and Reinganum 2013, 2014) and initial empirical research (Abrams and Chen 2013), the vast majority of scholars have engaged in normative policy debates concerning the desirability of the TPF mechanism, speculating about its potential effects on such issues as the volume of litigation (Rubin 2011; Shepherd and Stone 2015), the attorney-client relationship (Kidd 2011; DeStefano 2013; Fischer 2014), the quality of claims filed (Richey 2013), the unconscientiousness of the lending practices (McLaughlin 2006; Barksdale 2007; Hashway 2012) and others.

Currently, the scholars who recognize the shortcomings of the TPF mechanism are divided between those who propose complex solutions—such as the adoption of a one-way fee-shifting rule in favor of defendants in cases funded by TPF (Lysaught and Hazelgrove 2011), allowing the admittance in courts of TPF contracts (Avraham and Wickelgren 2014), mandating the disclosure of the identities of the TPF funders and the details of the arrangements (Sahani 2016; Steinitz 2019), and those who propose prohibiting the TPF mechanism altogether (McLaughlin 2006). However, it is worth noting that both proponents and opponents of this novel funding mechanism agree that it is “the most important civil justice development of this era” (Steinitz 2019: 1073).

The goal of this paper is to contribute to the debate regarding the effects of the TPF. We develop a game-theoretic model that compares a litigation process with TPF with a “traditional” scheme in which litigation is funded by means other than TPF. We first analyze the litigation process when the plaintiff is endowed with the financial means to execute this litigation process and we develop an understanding of when the plaintiff will choose to incur the litigation costs and execute the litigation process. Self-funding by a plaintiff is a more conventional scheme, since the TPF constitutes an alternative route of litigation-funding which developed only recently.

We then compare the outcome of the self-funded litigation with a process in which the TPF provides the financial means. Under the TPF scheme, we decompose the litigation decisions into two parts: the plaintiff (aided by his attorney) is in charge of the legal decisions, while the TPF has the freedom to decide in each stage of the litigation process whether to continue the financial support in the litigation process. Such a setting is characterized by a high level of uncertainty and, in addition, a degree of asymmetric information between the plaintiff and the TPF (e.g., Steinitz 2013). Specifically, in our model, the litigation process is characterized by two sources of uncertainty. First, there is uncertainty regarding the future litigation costs, and, furthermore, there is also uncertainty regarding the monetary outcome of the awarded verdict. In addition to the inherent uncertainty in such a process, we model asymmetric information by assuming that the plaintiff—assisted by his attorney—has better knowledge about the merit of the case (Steinitz 2013:729; Shamir 2016). Such asymmetry of information—which is further enhanced by the fact that the TPF is not privy to the communications protected by the attorney-client privilege (Wendel 2014) —raises the questions whether the plaintiff will have an incentive to mislead the TPF regarding the value and merit of the claim.

Contrary to the conventional modeling approach in which a TPF funder makes a one-time decision whether to provide the funding or not, we follow a more realistic approach and model litigation process as a sequence of stages (Cornell 1990, Grundfest and Huang 2006). Based on an updated valuation, as more information is revealed, this scheme allows the funder to reassess her involvement in a claim; the funder can decide whether to provide the funding for each stage or to withdraw.

Surprisingly, in spite of the complexity of this setting, we argue that the divergent interests of the parties to the financing agreement—i.e., the funder and the plaintiff—can be aligned by constructing a viable contract that results in the same equilibrium outcome as litigation with no TPF. Thus, instead of throwing the baby out with the bath water by designing complex multifaceted regulatory measures that would severely restrain or eliminate the TPF mechanism altogether, this paper suggests that even with the presence of a third-party acting in her own best interest and in the presence of asymmetric information between her and the plaintiff, an equilibrium emerges that would ensure the achievement of an efficient outcome, i.e., an outcome that mimics the results of litigation without the third-party funder. The contract that achieves these desired results has a few interesting properties. First, it provides a pre-specified remedy to the plaintiff if the TPF funder terminates the financing prior to the conclusion of the litigation process. Second, the contract also specifies the compensation to the TPF funder, which is owed upon successful completion of the litigation process, and it is conditioned upon the actual verdict. Our suggested contract stipulates that the TPF funder should be the first claimant to receive her share from the verdict, and the plaintiff is the residual claimant. This ensures credible information sharing regarding the expected outcome of the verdict.

While most of our analysis assumes the plaintiff is risk-neutral, we also evaluate the effect of the TPF mechanism when the plaintiff is risk-averse. We show that in this case, the TPF funder—who is risk-neutral because she can spread the risk across a portfolio of legal claims—will finance meritorious claims, that the plaintiff, due to his risk-aversion, will refrain from litigating. Thus, under the risk-aversion scenario, the TPF mechanism also has a welfare-enhancing effect compared with the self-funded setting as it allows meritorious claims to be litigated. On the other hand, we also claim that under the TPF mechanisms it will be less likely to achieve a settlement compared with a self-funded litigation process with a risk-averse plaintiff.

The remainder of the paper is organized as follows. Section 2 is devoted to a review of the relevant literature. In particular, we concentrate on two bodies of prior research. The first stream of literature applied economic models to capture the various facets of the TPF process. The second body of research relevant to this work proposes treating litigation as an investment project. In Sect. 3, we provide a description of the basic model, i.e., a model with self-funded litigation, and in Sect. 4 we analyze the self-funded litigation process. Section 5 is devoted to the discussion of the TPF scheme. In Sect. 6, we study information asymmetry, and Sect. 7 is devoted to the discussion of the settlement offer. We then discuss two extensions in Sect. 8 and provide some concluding remarks, including future research directions, in Sect. 9.

2 Literature review

This work is closely related to two main streams of research. The first is the research that applied economic models to capture the various facets of the TPF process as well as the incentives of the parties to the TPF contracts. The second part of the Literature Review considers studies that have suggested treating litigation as an investment project – an approach that we also adopt.

2.1 Modeling the TPF process

With respect to the TPF contracts, there are more unknowns than knowns. The industry has often been described as “opaque” , lacking “transparency” (Steinitz 2013:1900 ) and “shrouded in secrecy” (Steinitz 2012: 461), since much of its functioning is governed by confidentiality agreements. This situation led some scholars to suggest different schemes of disclosure rules (Steinitz 2019).

With a few exceptions (Abrams and Chen 2013, Avraham and Sebok 2018), the vast majority of research focused on the normative aspects of TPF (e.g., Barksdale 2007; Lyon 2010; Estevao 2013; Wendel 2018) More recently, scholars have commenced modeling the TPF process. This line of research is comprised of two types of studies. Some research studies focused on the macro level—examining the various effects of the TPF mechanism on the judicial system as a whole, considering the implications for such matters as the volume of lawsuits filed, their nature—i.e., meritorious or frivolous—etc. Other line of research concentrated on the micro level, analyzing the incentives of the individual players in the TPF scheme.

From the perspective of the judicial system as a whole, it has been suggested that TPF may increase the number of lawsuits filed—facilitating both meritorious and frivolous litigation and burdening an already overburdened system. For instance, comparing three alternative schemes of litigation financing—i.e., self-financing by plaintiffs, contingent fees and TPF—Deffains and Desrieux (2015) demonstrate that TPF does not always increase the access to justice, it may decrease the equilibrium settlement amount and cause an increase in the number of frivolous claims.

In light of the potential increase in the volume of litigation, and—specifically, the volume of meritless litigation—scholars have suggested various schemes to mitigate this negative outcome. Lysaught and Hazelgrove (2011) proposed adopting a one-way fee-shifting rule in favor of defendants in cases financed by TPF, arguing that such a rule would reduce the value of speculative litigation, while increasing the value of the lawsuits with a high probability of prevailing. Abramowicz and Alper (2013) proposed using the signals of litigation quality gleaned from the TPF assessment, to screen lawsuits for their quality, allowing only meritorious suits to proceed. Abramowicz 2013 suggested improving the incentive of the TPF companies, inducing them to finance only cases with relatively high probability of success, via the implementation of fee limitation rules.

Other studies have concentrated on the level of individual lawsuits, modeling the effects of TPF on the incentives of the various players—e.g., the plaintiff, defendant and the TPF funder. Avraham and Wickelgren 2014 suggested making the TPF contract admissible in court as a measure to induce the funder to lower the interest rate, benefitting the plaintiffs, while the judge may draw an inference from it about the case value. Based on a model that compares TPF with incentive fees for lawyers, Demougin and Maultzsch (2014) suggest a combination of contingency fees and TPF in order to overcome agency problems and financial constraints of would-be plaintiffs with meritorious suits. Using a signaling model to analyze the effect of TPF on settlement, Daughety and Reinganum (2014) demonstrate that an optimal loan—one that maximizes the joint expected payoff to the funder and the plaintiff—induces full settlement, suggesting that this mechanism may well have the effect of eliminating the inefficiencies associated with information-asymmetry.

These studies constitute the initial steps in modeling the TPF mechanism and provide insights into the significant role that such modeling can play in estimating the potential effects of this scheme on the parties to litigation process funded by TPF, as well as on the judicial system. Our work contributes to this stream of literature by providing a framework that models TPF and suggests that the litigation outcome can be identical to scenarios in which no outside funding is required. We further differ from this past research by modeling the litigation process as a sequential process where the TPF funder is able to update her decisions based on new available information. We next elaborate on the this view of the legal process.

2.2 TPF as an investment project

Traditionally, legal disputes were analyzed as monetizable assets and the funding decisions of litigation process—i.e., the decisions whether lawsuits are worth the investment of funds—were modeled as one-time decisions, made prior to commencing the process. In this scheme, the plaintiff was treated as a one-time decision-maker, who must decide at the onset of litigation whether to finance his lawsuit or to walk away. However, in the past two decades, scholars began to apply finance theory to litigation (Cornell 1990; Bar-Gill 2005; Grundfest and Huang 2005).

This is a more realistic representation of the legal process, because its various stages—such as pretrial motion practice, posttrial motions and adjustments (Hyman et al. 2007,) and appellate rulings—supply litigants with information that allows them to update their expectations and abandon the process, proceed to the trial or settle (Eisenberg and Lanvers 2009). Studies emphasize the fact that parties in litigation can settle at any stage of the litigation and have found that strategic litigants maximize their prospects in litigation not only by choosing if and when to settle and how much to settle for, but also by choosing when and how much to expend on litigation (Cotropia et al. 2017).³. Furthermore, staging is common in contingency arrangements, which typically consist of incremental expenditures by the law-firm, an increasing stake in the proceeds , and a right to exit (Steinitz 2013:1899.) In TPF staged financing allows the funder to reevaluate her investment as more information becomes available, thus addressing information asymmetry, uncertainty and agency costs (Steinitz 2013).

Recognizing the sequential nature of decision-making in litigation, this new school of thought posits that rather than understanding litigation as an asset, a more precise view is that of an option (Cornell 1990: 174.) “[L]awsuits and investment projects have much in common” (Grundfest and Huang 2005: 1267.) This approach examines litigation through the prism of finance theory, viewing it as an option, rather than as an asset. The option-pricing model to legal valuation that views litigation as an option is appropriate, because the plaintiff has the option to settle the case, abandon the case or to continue to trial. In this work, we also emphasize the sequential nature of the litigation process; a process, that at each of its stages, decisions regarding continuing the process, abandoning the litigation process or reaching a settlement are made. Our model also highlights the fact that the value of a litigation process can be viewed as a compound put option.

3 The basic model-self-funded litigation process

Before analyzing the contract between the plaintiff and the TPF funder, we construct a model of the litigation process in which the plaintiff does not face financial constraints that preclude him from financing his lawsuit. This model serves as a benchmark because in this case, the plaintiff chooses the optimal litigation strategy – i.e., the forum to litigate the dispute, the identity of potential defendants and witnesses, the legal doctrine applicable to the case, and other actions, since different alternatives have implications for the potential success, or failure, of litigation. In contrast with the scenario in which the financial or other circumstances force the plaintiff to enter into the contract with the TPF funder in order to finance his lawsuit, no conflict of interest arises in this setting. Therefore, a comparison between the outcome of the benchmark model—i.e., a model without TPF—and a model with TPF can highlight the results of an incentive misalignment between the TPF funder and the plaintiff.

We study a model in which a plaintiff has a claim against a certain defendant. The plaintiff is represented by an attorney able and willing to diligently act on his behalf during the dispute resolution process. Since the focus of this paper is the relationship between the plaintiff and a TPF funder, we assume that there is no inherent conflict of interest between the plaintiff and his attorney. For the purposes of the plaintiff-funder relationship (discussed later), the plaintiff and his counsel act cooperatively. We further assume that all of the parties involved are rational, risk-neutral actors who wish to maximize their payoffs with respect to the dispute (we later relax this assumption and examine the implications of risk-averse plaintiff).

The model that follows highlights the sequential nature of decision-making in a litigation process. Initially, based on the information about the claim available to the plaintiff, a number of litigation strategies can be implemented. For instance, the plaintiff must choose what forum should litigate the dispute, because there are significant differences in procedure, and even outcome, offered by the various dispute resolution bodies. The plaintiff should also decide who are the parties with an actionable claim against them who can be named defendants in the suit, and select what legal doctrine applicable to the facts and circumstances of a particular situation should be implemented, because different legal arguments would have to be made. We denote these available litigation strategies as the group $S=\{s_{0}$, $s_{1},...,s_{n}\}$. Strategy $s_{0}$ represents the “null strategy,” which corresponds to the option of refraining from litigation. For obvious reasons, we are interested in the setting in which the plaintiff does not select this option, deeming litigation to be more profitable. In the latter case, the plaintiff chooses among the non-null strategies that are associated with his evaluation of the litigation process. Each litigation strategy, s, will result in a verdict with a random monetary value of $v_{s}$, if all litigation stages are completed. The monetary verdict value of $v_{s}$ is drawn from a distribution with a mean of ${\bar{v}}_{s}$ and a standard deviation of $\sigma _{s}$. We further denote the cumulative density function (CDF) of $v_{s}$ by G(), and the density function by g(). We assume that once a litigation strategy has been chosen, the plaintiff cannot change it during the course of the trial.

After the initial filing of the lawsuit, the plaintiff decides how to proceed—i.e., at different stages throughout the litigation process, he has to decide whether to file motions, to devote extensive resources to discovery, etc. The litigation process comprises N sequential phases denoted by 1 to N. At the end of each phase, the litigation process is evaluated, and more information is collected about the subsequent stage. Conditional on the results of this evaluation, the plaintiff may choose to withdraw from the litigation or to continue to the next phase. We assume that litigation abandoned in the process does not provide any value to the plaintiff. Each phase t in the litigation process requires a monetary investment of $C_{t}$, but this cost can only be observed upon completing phase $t-1$ and at the beginning of phase t. Furthermore, the cost associated with each phase t is a function of the litigation strategy that the plaintiff has chosen at the beginning of the litigation process. Therefore, before completing phase $t-1$, the cost $C_{t}$ is a random variable, drawn from a distribution with a CDF denoted by $F_{t}^{s}(\cdot ),$ where $s\in S$ is the litigation strategy chosen by the plaintiff at time 0. This modeling framework allows us to capture the fact that information about the litigation is being revealed as the litigation process advances.

Since the null-strategy (denoted by $s_{0})$ has no-value, given that this litigation strategy was not chosen, the plaintiff must make the following choices: (1) decide which litigation strategy to choose during time $t=0;$ (2) at the beginning of each time-period $t>0$, decide whether to withdraw from the litigation process or to continue to the next stage by incurring the realized period t litigation cost of $c_{t}$ (we denote the random variable by $C_{t}$ and its realization by $c_{t})$. At this point in the model, we ignore the ability to reach a settlement, and we later extend our model to incorporate this option as well (Fig. 1).

Fig. 1

Sequence of events

4 Analysis of the self-funded litigation process

4.1 Two litigation stages

To gain some insight into the results of the self-funded litigation process, we start by analyzing the simple litigation process that comprises two steps. Using backward induction, we assume that the plaintiff has chosen the litigation strategy s with the expected reward value of ${\bar{v}}_{s}$. It is important to emphasize that during the second stage of the litigation process, the expenditures incurred in the first stage are already sunk costs. Thus, the plaintiff will continue to litigate if the realized cost $c_{2}$ is lower than the expected reward of ${\bar{v}}_{s};$ the cost that the plaintiff has incurred in order to proceed to the second stage of the litigation process does not affect his decision. Therefore, at the second stage, the plaintiff decides to continue if, and only if:

$$\begin{aligned} c_{2}\le {\bar{v}}_{s}. \end{aligned}$$

We now move to the first stage of the litigation process. When the plaintiff observes the first period realized cost of $c_{1}$, he decides to continue with the litigation process if, and only if, the expected payoff is at least as large as the realized first period cost. Note that at this time, the cost of litigation during the second stage is unknown to the plaintiff. The plaintiff anticipates that during the second litigation stage, he will continue only if the realized value of the random variable $C_{2}$ is lower than ${\bar{v}}_{s}$. Therefore, the payoff of carrying on with litigation is given by:

$$\begin{aligned} \begin{array}{ll} -c_{1}+{\bar{v}}_{s}-c_{2} &{}\text { if }c_{2}\le {\bar{v}}_{s}; \\ -c_{1} &{}\text { if }c_{2}>{\bar{v}}_{s}. \end{array} \end{aligned}$$

If the plaintiff incurs the cost of $c_{1}$, and the realized cost $c_{2}$ is lower than ${\bar{v}}_{s}$, he will complete the litigation process, earning the expected sum of $-c_{1}+{\bar{v}}_{s}-c_{2}$. If the plaintiff incurs the cost of $c_{1}$, but the realized second period cost $c_{2}$ is higher than ${\bar{v}}_{s}$, the plaintiff will not proceed with the litigation process, and his profit will be $-c_{1}$. Thus, the plaintiff will decide to continue during the first litigation stage if:

$$\begin{aligned} E[({\bar{v}}_{s}-C_{2})^{+}]\ge c_{1}, \end{aligned}$$

(1)

where the expectation is taken with respect to the second period litigation cost $C_{2}$, and the function $(x)^{+}=\max (x,0)$. We use the notation $\delta _{t}\in \{0,1\}$ to denote the plaintiff’s decision during stage t, where $\delta _{t}=1$ represents the decision to continue the litigation process during stage t, and $\delta _{t}=0$ represents the decision to abandon litigation during this stage. Therefore, under the self-funded litigation process, the optimal sequence of plaintiff’s decisions given the litigation strategy s is given by the following decision rule:

$$\begin{aligned} \begin{array}{ll} \delta _{1}({\bar{v}}_{s},c_{1})=1 &{}\text { if }c_{1}\le E[({\bar{v}}_{s}-C_{2})^{+}]; \\ \delta _{2}({\bar{v}}_{s},c_{2})=1 &{}\text { if }c_{2}\le {\bar{v}}_{s}\text { and }\delta _{1}=1. \end{array} \end{aligned}$$

During the first stage of the litigation process, the plaintiff incurs the cost of $c_{1}$ only if this is lower than the value of $E[({\bar{v}} _{s}-C_{2})^{+}]$. This term captures the expected reward the plaintiff will receive if he proceeds to the second litigation stage. During the second litigation stage, the plaintiff incurs the cost of $c_{2}$ only if it is lower than the expected monetary value of the verdict of ${\bar{v}}_{s}$ that awaits the plaintiff when the litigation process is completed.

4.2 General number of litigation stages

The argument about the optimal decision rule of the plaintiff can be generalized to N litigation-stages using the following Proposition.

Proposition 1

Suppose that the plaintiff has chosen the litigation strategy $s\in S$ and its expected verdict value is ${\bar{v}}_{s}$. Then,

• (a) (a) The plaintiff will continue to the litigation stage t if and only if stage $t-1$ has been completed and

  $$\begin{aligned} c_{t}\le R_{t}(s,{\bar{v}}_{s}), \end{aligned}$$

  where $R_{t}(s,{\bar{v}}_{s})=E\left[ \left( R_{t+1}(s,{\bar{v}} _{s})-C_{t+1}\right) ^{+}\right]$ and $R_{N}(s,{\bar{v}}_{s})=\bar{v }_{s}$.

• (b) (b) The ex-ante probability to complete the litigation process is $\mathop {\displaystyle \prod }\limits _{t=1}^{N}\Pr (C_{t}\le R_{t}(s,{\bar{v}}_{s}))$.

The term $R_{t}(s,{\bar{v}}_{s})$ represents the expected value of continuing the litigation, given that the litigation process has reached the stage $t-1.~$A plaintiff chooses to continue to litigate if this value exceeds the current litigation costs. A careful examination reveals that the value of $R_{N-1}(s,{\bar{v}}_{s})$ can be viewed as a European put option with the exercise price of ${\bar{v}}_{s}$, and the underlying stock behaves according to the random variable $C_{N}$. According to this view, the litigation process with two remaining periods is equivalent to a European put option. A plaintiff’s decision to incur the cost of $c_{N-1}$ is equivalent to a purchase of that put option. Similarly, a general litigation process with N periods can be viewed as a compound put option; i.e., an option placed on an option. This analysis strengthens the view of a litigation process as a financial instrument, and can explain the phenomenon of investors entering this area and choosing to finance legal claims. While we draw attention to the analogy between the decision to pursue the litigation process and purchasing a (compound) Put option, we leave further examination of this analogy for future research.

Part (b) of the proposition outlines the ex-ante probability to complete the entire legal process. The plaintiff will complete the process if the cost during each period t is lower than the residual value of the process during this period, denoted by $R_{t}(s,{\bar{v}}_{s})$. We further note that $R_{t}(s,{\bar{v}}_{s})\le R_{t+1}(s,{\bar{v}}_{s})$ for any $t\in \{1,N-1\}$, meaning that the residual value of the process increases as the plaintiff reaches closer toward the end of the process.

Following the analysis above, the ex-ante profit of a plaintiff who chooses a litigation strategy s, is denoted by $\pi (s,{\bar{v}}_{s})$ and is given by the following expression:

$$\begin{aligned} \pi (s,{\bar{v}}_{s})= & {} \mathop {\displaystyle \prod }\limits _{t=1}^{N}\Pr (C_{t}\le R_{t}(s, {\bar{v}}_{s}))\left( {\bar{v}}_{s}-\sum _{t=1}^{N}E[C_{t}|C_{t}\le R_{t}(s,{\bar{v}}_{s})]\right) \nonumber \\&-\sum _{t=2}^{N}\left[ \mathop {\textstyle \prod }\limits _{i=1}^{t-1}\Pr (C_{i}\le R_{i}(s, {\bar{v}}_{s}))\Pr (C_{t}>R_{t}(s,{\bar{v}}_{s})) \sum _{i=1}^{t-1}E[C_{i}|C_{i}\le R_{i}(s,{\bar{v}}_{s})]\right] . \end{aligned}$$

(2)

Equation (2) is central to our model and characterizes the plaintiff’s ex-ante profit. It is comprised of two main elements. The first element (first line in the equation) denotes the plaintiff’s profit when the litigation process reaches the final stage and it concluded successfully. This outcome is achieved when during each litigation stage, the litigation costs are lower than the residual value of the claim during this stage (captured by $R_{t}(s,{\bar{v}}_{s}))$. In this case, the plaintiff incurs the costs of all N litigation stages, given by $\sum _{t=1}^{N}E[C_{t}|C_{t}\le R_{t}(s,{\bar{v}}_{s})]$. Note that we take into account the fact that the plaintiff’s decision to continue to the next stage of the litigation process implies that the cost during stage t was lower than $R_{t}(s,{\bar{v}}_{s})$, and, thus, the conditional expected litigation cost of stage t is given by $E[C_{t}|C_{t}\le R_{t}(s, {\bar{v}}_{s})]$. At the end of stage N, the plaintiff receives the reward $v_{s}$, and the expected value of the verdict is given by $\bar{ v}_{s}$.

The second term in Equation (2) captures the plaintiff’s losses when he withdraws from the litigation process without completing it. A litigation process is terminated during stage $t<N$ if all of the previous stages were completed and the cost of litigation during stage t exceeds the residual value of the claim (meaning that $C_{t}>R_{t}(s,{\bar{v}} _{s}))$. The litigation is terminated during stage t with the probability of $\mathop {\textstyle \prod }\limits _{i=1}^{t-1}\Pr (C_{i}\le R_{i}(s,{\bar{v}}_{s}))\Pr (C_{t}>R_{t}(s,{\bar{v}}_{s}))$. In this case, the plaintiff incurs the litigation costs of all of the previous stages given by $\sum _{i=1}^{t-1}E[C_{i}|C_{i}\le R_{i}(s,{\bar{v}}_{s})]$. Litigation terminated during the first litigation stage carries no costs to the plaintiff.

During time $t=0$, the plaintiff must decide on the optimal litigation strategy out of the set of S possible options. This problem can be formulated as

$$\begin{aligned} \Pi ^{*}=\max _{s\in S}\pi (s,{\bar{v}}_{s}). \end{aligned}$$

We denote the optimal litigation strategy by $s^{*}$, such that $s^{*}\in \arg \max \pi (s,{\bar{v}}_{s})$,⁴ and the expected reward of this strategy by ${\bar{v}}_{s}^{*}$. To avoid trivial cases, we assume the existence of a strategy that provides the plaintiff with a positive outcome, such that the litigation process will be executed.

5 Third-party legal funding

We now assume that the plaintiff lacks the resources to finance his litigation process. Assuming that the financing is provided by an independent third-party, we evaluate the consequences of such an arrangement.

Explicitly, we assume that the plaintiff and his attorney are in charge of the legal process, and therefore, they choose the litigation strategy $s\in S$. During every period, the litigation cost $C_{t}$ is being realized with the realized cost of $c_{t}$, and the TPF funder must decide whether to finance this litigation stage or to abandon the litigation process. If the TPF funder elects to discontinue financing the litigation, she cannot resume financing at a later stage; in such an event, the value of the outcome is zero (no verdict is awarded) minus all costs incurred during the process. It is important to emphasize the de-composition of the decision rights among the multiple parties in our setting: whereas the litigation strategy is determined by the plaintiff, the decision whether to fund each stage in the process is made by the TPF funder.

The sequence of events of the litigation process with TPF is as follows:

1. The plaintiff chooses a litigation strategy s and estimates the expected verdict reward to be ${\bar{v}}_{s}$.

2. The plaintiff shares the chosen litigation strategy s with the TPF funder. We assume that the litigation strategy s is revealed truthfully, since otherwise, during the litigation process, the TPF funder can infer that the plaintiff chose a different litigation strategy than the one reported at the beginning of the game. We further assume that the distribution of the litigation costs is common knowledge also to the TPF funder. Therefore, based on the litigation strategy, the TPF funder can estimate the litigation costs, since the distribution $F_{t}^{s}(\cdot )$ is assumed to be common knowledge and it is a function of the chosen litigation strategy.

3. The plaintiff shares with the TPF funder the value of the expected verdict ${\bar{v}}_{s}$. In order to better understand the properties of the suggested contract, we first assume that information regarding the expected reward ${\bar{v}}_{s}$ is shared truthfully. We then relax this assumption, and show that the suggested contract induces the voluntary truthful revelation of the value of ${\bar{v}}_{s}$.

4. The TPF funder decides whether to accept a contract $\mathbb {k}$ offered to her by the plaintiff.

5. Upon acceptance of this contract, the TPF funder makes separate decisions at each litigation stage whether she wishes to incur the realized cost $c_{t}$ of this litigation stage, or to terminate the financing. We denote the decision rule of the TPF funder during the litigation stage t and given the contract $\mathbb {k}$, when the chosen litigation strategy is s and the expected verdict is ${\bar{v}}_{s}$ by $\delta _{t}^{TPF}(s,{\bar{v}} _{s},c_{t}|\mathbb {k})$, such that

$$\begin{aligned} \begin{array}{ll} \delta _{t}^{TPF}(s,{\bar{v}}_{s},c_{t}|\mathbb {k})=1 &{}\text { if the litigation stage }t\text { is funded; }\\ \delta _{t}^{TPF}(s,{\bar{v}}_{s},c_{t}|\mathbb {k})=0 &{}\text { if the litigation stage }t\text { is not funded.} \end{array} \end{aligned}$$

Before observing the litigation cost $c_{t}$, the decision rule of the TPF, denoted by $\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})$, can take the values of either 0 or 1.

We further use the notation $\pi ^{p}(s,{\bar{v}}_{s},\mathbb {k})$ to describe the plaintiff’s ex-ante profit when the chosen litigation strategy is s, the expected value of this litigation strategy is ${\bar{v}}_{s}$ and the contract is $\mathbb {k}$. Similarly, we use the notation of $\pi ^{TPF}(s,{\bar{v}}_{s},\mathbb {k})$ to denote the TPF funder’s payoff .

We next define some desired properties of a contract $\mathbb {k}$ between the plaintiff and the TPF funder:

Definition 1

(a) A contract $\mathbb {k}$ is efficient if:

(i) It induces the plaintiff to choose the litigation strategy $s^{*}$ (i.e., the chosen litigation strategy in the case without the the TPF).

(ii) The decision rule of the TPF funder is identical to the decision rule of the plaintiff (i.e., $\delta _{t}^{TPF}(s,{\bar{v}}_{s},c_{t}|\mathbb {k})=$ $\delta _{t}({\bar{v}}_{s},c_{t})$ for every t and possible $c_{t})$.

(b) The contract $\mathbb {k}$ satisfies the participation constraints of the TPF funder and the plaintiff if it promises each one an ex-ante positive profit (i.e., $\pi ^{TPF}(s,{\bar{v}}_{s},\mathbb {k})\ge 0$ and $\pi ^{p}(s, {\bar{v}}_{s},\mathbb {k})\ge 0)$.

Definition 1 characterizes a few desired properties of a contract. Part (a) defines contract efficiency. An efficient contract induces the plaintiff and the TPF funder to make the same decisions as the benchmark model of self-funded litigation, i.e., a model that analyzes litigation without the TPF. This requirement implies that in the TPF contract, the plaintiff chooses the same litigation strategy $s^{*}$ as in the benchmark case, and the TPF funder’s decisions, given the contract $\mathbb {k}$ and the information she has received, are identical to the decisions the plaintiff would have made in the self-funded litigation. Part (b) of Definition 1 states that both parties–the plaintiff and the TPF funder–accept the contract, because it provides them with an ex-ante positive payoff.

The range of possible contracts (not necessarily meeting the criteria in Definition 1) between the TPF funder and the plaintiff is vast. We focus on one specific family of contracts, called Termination-Compensation (referred to hereafter as TC. The Termination-Compensation contract has the following properties:

1. The TC contract stipulates that the TPF funder must compensate the plaintiff in the amount of $D_{t}(s,{\bar{v}}_{s})$ if the former decides to cease financing the litigation process during the litigation stage t. While under this contract, the TPF funder has the discretion to discontinue financing the litigation process at any stage, under this contract she becomes liable for some of the damage imposed on the plaintiff due to the termination of financing the litigation process (and consequently, termination of the legal process). The term $D_{t}(s,{\bar{v}}_{s})$ outlines the amount of compensation the TPF funder would undertake in the event of termination during time t.

2. Upon successful resolution of the litigation, the TPF funder is entitled to a payment of $P(s,v_{s})$, that can depend (as we see later) on the outcome of the verdict - $v_{s}$. The TC specifies the amount the TPF funder will receive from the recoveries of the claim. Since the outcome of the verdict is a random variable, the contract states that the first claimant is the TPF funder, and the plaintiff is the residual claimant. Therefore, the payoff of the plaintiff, given successful completion of the litigation process, is $max(v_{s}-$ $P(s,v_{s}),0)$.

Under this family of contracts, the ex-ante payoff of the parties is given by:

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k})= & {} \mathop {\displaystyle \prod }\limits _{t=1}^{N}\Pr (\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=1)E\left[ \left( v_{s}-P(s, {\bar{v}}_{s})\right) ^{+}\right] \nonumber \\&+\sum _{t=1}^{N}\left[ \mathop {\textstyle \prod }\limits _{i=1}^{t-1}\Pr ((\delta _{t}^{TPF}(s, {\bar{v}}_{s},C_{t}|\mathbb {k})=1))\Pr ((\delta _{t}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=0))D_{t}(s,{\bar{v}}_{s})\right] ; \end{aligned}$$

(3)

$$\begin{aligned} \pi ^{TPF}(s,{\bar{v}}_{s},\mathbb {k})= & {} \mathop {\displaystyle \prod }\limits _{t=1}^{N}\Pr (\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=1))\left( E\left[ P(s,\bar{v }_{s})\right] -\sum _{i=1}^{N}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1)]\right) \nonumber \\&-\sum _{t=1}^{N}\left[ \begin{array}{c} \mathop {\textstyle \prod }\limits _{i=1}^{t-1}\Pr (\delta _{t}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1))\Pr ((\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=0)) \\ \left( \sum _{i=1}^{t-1}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1)]+D_{t}(s,{\bar{v}}_{s})\right) . \end{array} \right] \end{aligned}$$

(4)

Equation (3) provides a characterization of the plaintiff’s payoff under the TC contract. The term $\Pr (\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=1)$ describes the probability that the TPF funder would execute stage t; thus, the term $\mathop {\displaystyle \prod }\limits _{t=1}^{N}\Pr (\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=1)$ represents the probability of the successful completion of the entire litigation process given the contract $\mathbb {k}$ and the information the TPF funder has received about the claim. In this event, the plaintiff receives the payoff of $max\left( \left( v_{s}-P(s,{\bar{v}}_{s})\right) ,0\right)$, which is the outcome of the verdict minus the payment to the TPF funder (if this residual reward is positive). This sum is a random variable that depends on the actual outcome of the trial. If the realized value of the claim is lower than the promised payment to the TPF funder, the plaintiff will not receive anything from the verdict. In the second part of the plaintiff’s payoff function, the term $\mathop {\textstyle \prod }\limits _{i=1}^{t-1}\Pr ((\delta _{t}^{TPF}(s,{\bar{v}}_{s},C_{t}|\mathbb {k})=1))\Pr ((\delta _{t}^{TPF}(s, {\bar{v}}_{s},C_{t}|\mathbb {k})=0))$ represents the probability of terminating the litigation process during stage t. In this event, the plaintiff receives a compensation in the amount of $D_{t}(s,{\bar{v}} _{s})$ from the TPF funder. Similarly, Equation (4) represents the TPF funder’s payoff. The first term is the payoff upon the successful completion of the litigation process, whereas the second term is the payoff in the event of the termination of the litigation process. Based on the payoff functions characterized above, we define the function $\pi _{t}^{TPF}(s,{\bar{v}}_{s},\mathbb {k})$ as the payoff function of the TPF funder, starting from the litigation stage t.

A contract $\mathbb {k}$, that belongs to the family of TC contracts, is efficient (see Definition 1), and satisfies the participation constraints if the following conditions are satisfied:

$$\begin{aligned} \arg \max _{s\in S}\pi ^{p}(s,{\bar{v}}_{s},\mathbb {k})= & {} s^{*}; \end{aligned}$$

(5)

$$\begin{aligned} -D_{t}(s,{\bar{v}}_{s})< & {} -c_{t}+\pi _{t+1}^{TPF}(s,{\bar{v}} _{s},\mathbb {k})\text { iff }\delta _{t}({\bar{v}}_{s},c_{t})=1\text { for every }c_{t}\text { and }t; \end{aligned}$$

(6)

$$\begin{aligned} \pi ^{TPF}(s,{\bar{v}}_{s},\mathbb {k})\ge & {} 0 \end{aligned}$$

(7)

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k})\ge & {} 0 \end{aligned}$$

(8)

Equation (5) states that the plaintiff maximizes his profit by choosing the same litigation strategy as the benchmark. Equation (6) ensures that the incentives of the TPF funder are aligned with the decisions that the plaintiff would have reached in the same situation under the self-funded litigation process. Finally, Equations (7) and (8) ensure that both the TPF funder and the plaintiff agree to enter into the contract.

We are now ready to state the main result of this Section.

Proposition 2

There exists a TC contract that satisfies conditions (5)-(8). This contract has the following structure:

$$\begin{aligned} D_{t}(s,{\bar{v}}_{s})= & {} D(s,{\bar{v}}_{s})=\pi (s,{\bar{v}}_{s}) \text { for every period }t; \\ P(s,{\bar{v}}_{s})= & {} \left\{ \begin{array}{ll} v_{s} &{}\text { if }v_{s}\le x; \\ x &{} o/w. \end{array} \right. \end{aligned}$$

where x is chosen such that

$$\begin{aligned} E\left[ P(s,{\bar{v}}_{s})\right] ={\bar{v}}_{s}-D(s,{\bar{v}}_{s}). \end{aligned}$$

Proposition 2 shows that in spite of the fact that under TPF the decision rights are distributed between the plaintiff and the TPF funder, it is possible to draft a contract that results in the optimal sequence of litigation decisions. This contract induces the plaintiff to choose the optimal litigation strategy. Based on the chosen litigation strategy, the TPF funder’s decision to fund each litigation stage–or to terminate the funding–mimics the decisions that would have been made by the plaintiff, if the latter had had the financial resources at his disposal. Proposition 2 demonstrates the existence of a contract that achieves these desired properties. This Proposition also reveals that the structure of such a contract is relatively simple. According to the terms of this contract, the plaintiff chooses an optimal litigation strategy. Thereafter, if the TPF funder chooses to terminate her litigation financing, the plaintiff is entitled to receive compensation in a fixed pre-specified amount. In the contract characterized in Proposition 2, the compensation for terminating the litigation process is fixed and it is independent of the litigation stage in which the process is terminated. If the TPF funder provides the funding for all the litigation stages, she is entitled to receive the full value of the verdict, up to a prespecified amount. Any amount that is awarded above the pre-specified value of x is awarded to the plaintiff. The payment to the TPF funder is determined such that the plaintiff will receive the same amount (in expectation) regardless of whether the litigation process was terminated or concluding successfully.

It is interesting to observe the way in which the litigation cost risk is allocated between the plaintiff and the TPF funder under the proposed contract. If the litigation process is terminated, the plaintiff receives the same payoff, that equals the ex-ante value of the litigation process without TPF. Therefore, from the plaintiff’s perspective, not only does the TPF provides him with the financial resources to fund the claim, but it also eliminates the risk of litigation cost – it guarantees the plaintiff a fixed reward if the litigation cost is higher than anticipated. According to the contract characterized above, it is the TPF funder who bears the risk of the litigation cost. This result suggests that a plaintiff may use the TPF mechanism not only to finance the claim, but also to reduce the volatility of the litigation cost.

In contrast with the risk of litigation cost, the plaintiff still incurs the verdict risk. While the expected verdict is ${\bar{v}}_{s}$, the actual outcome of the verdict $v_{s}$ is stochastic, and the court may award the plaintiff a lower amount than the expected verdict reward. Under the proposed contract, the first claimant is the TPF funder. The TPF funder receives the total verdict award up to a sum that ensures the recovery of the expected costs of the TPF funder. The plaintiff receives the portion of the verdict award that exceeds the pre-specified payment to the TPF funder, such that the payment to the plaintiff is zero if the awarded verdict is lower than the maximum promised payment to the TPF Funder. The fact that the TPF funder is the first claimant to receive the payment from the verdict, and the plaintiff is the residual claimant, will prove to be crucial in the next Section, as we discuss the incentives of the plaintiff to report the expected value of the verdict in a truthful manner.

In order to better understand the intuition behind the result of Proposition 2, note that according to the contract characterized in this proposition, the plaintiff receives the ex-ante value of the claim. As a result, the plaintiff chooses the litigation strategy s that maximizes the overall value of the litigation process; therefore, this is the same strategy that the plaintiff chooses absent financial constraints. Such an outcome (that the plaintiff extracts all surplus of the legal claim) can be supported when the plaintiff has the bargaining power to make a take it or leave it offer to the TPF funder or when the market for funding legal claims is competitive.

One of the main properties of the contract characterized in Proposition 2 is the use of damage compensation that the TPF funder must pay the plaintiff when the former decides to terminate funding the litigation process. This compensation is used to align the incentives of the TPF funder and the plaintiff, inducing the former to make the same financing decisions as the latter would have made absent financial constraints. In a self-funded case, the plaintiff would incur the litigation cost of $c_{t}$ during the litigation stage t only if the following inequality holds $0\le -c_{t}+R_{t}(s,v_{s})$ (see Proposition 1). The plaintiff evaluates the expected recoveries of the claim against the current litigation cost. However, when the TPF funder faces the same dilemma, the expected payment of the claim is lower than $R_{t}(s,v_{s})$, since both the plaintiff and TPF funder share the rewards gained from the lawsuit. Since the TPF funder receives only a portion of the recoveries, in order to align her incentives to make the same funding decisions as the benchmark, the contract stipulates that when terminating the contract, the TPF funder should pay some compensation fee to the plaintiff.

6 Information asymmetry regarding $v_{s}$

So far, we have assumed that when the plaintiff chooses a litigation strategy s, the expected value of this claim (${\bar{v}}_{s})$ is known to the TPF funder. This assumption is valid when the TPF funder has the resources to validate the value of ${\bar{v}}_{s}$ based on the information provided by the plaintiff. This is true in many cases, since the TPF hires expert attorneys that confirm the estimate provided by the plaintiff. However, an alternative assumption is that the plaintiff has better information regarding the value of the claim since he has intimate knowledge regarding the details of the underlying matter; this is information that cannot be validated by the TPF funder. Therefore, if it is deemed to be in his best interest, the plaintiff can attempt to manipulate the value of the claim and report an expected value of ${\widetilde{v}}_{s}$ that may differ from ${\bar{v}}_{s}$. In this Section ,we explore the implications of information asymmetry on the contract signed between the plaintiff and the TPF funder. We further assume that the plaintiff can manipulate the value of the expected verdict, but cannot manipulate the actual litigation strategy that was chosen (s). We adopt this assumption since upon declaring a specific litigation strategy, the TPF funder can verify that the plaintiff follows the aforementioned litigation strategy. All the actions that are defined by the legal strategy (such as what forum should litigate the dispute, who are the parties with an actionable claim against them who can be named defendants in the suit, what legal doctrine is applicable to the case) can be easily verified by the TPF funder by observing the way the case is managed. Therefore, we assume that the plaintiff will not manipulate the chosen strategy but can only manipulate the value of the legal process since this information cannot be easily verified by the TPF funder. This is in contrast with the outcome of the litigation process that cannot be observed prior to the completion of the litigation process.

6.1 Analysis of the incentives for information manipulation

Let $\pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\widetilde{v}}_{s})$ be the ex-ante profit of the plaintiff when he chooses the litigation strategy of s; he observes that its expected value is ${\bar{v}}_{s}$, but he reports this value to be ${\widetilde{v}}_{s}$. We use the following definition to explore the incentives of the plaintiff to truthfully reveal the expected value of the claim.

Definition 2

A contract $\mathbb {k}$ induces truthful information-sharing if

$$\begin{aligned} {\bar{v}}_{s}\in \arg \max _{{\widetilde{v}}_{s}}\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\widetilde{v}}_{s}), \end{aligned}$$

i.e., if the plaintiff finds it in his best interest to report the true value of the claim.

The following Proposition shows another important property of the TC contract.

Proposition 3

Assume that $\pi (s,{\bar{v}}_{s})\ge 0$, then the contract TC characterized in Proposition 2 induces truthful information revelation of the claim’s expected value of ${\bar{v}}_{s}$.

Proposition 3 suggests that the contract TC, that is characterized in Proposition 2, induces the plaintiff to truthfully reveal the expected value of the claim. Even when the setting is characterized by information asymmetry, the information shared by the plaintiff and the TPF funder is not distorted and is shared truthfully.

This result is due to the fact that inflating the value of the claim has a number of mixed effects that the plaintiff must take into consideration. On the one hand, reporting a higher value of the expected verdict benefits the plaintiff when the litigation process is terminated, since the compensation the TPF funder commits to pay the plaintiff in the case of termination increases with the reported expected value of the claim (note that the termination compensation $D(s,{\widetilde{v}}_{s})$ is an increasing function of the reported value of the claim ${\widetilde{v}}_{s})$. However, on the other hand, inflating the value of a claim has two adverse effects that can hurt the plaintiff. First, the amount owed to the TPF funder upon successful completion of the litigation process, would also increase. Proposition 2 suggests that the payment to the TPF funder upon successful completion of the litigation process when the plaintiff announces the value of $\widetilde{ v}_{s}$ is determined such that $E\left[ P(s,{\widetilde{v}}_{s})\right] = {\widetilde{v}}_{s}-D(s,{\widetilde{v}}_{s})$. This value is increasing in ${\widetilde{v}}_{s}$, and, thus, the plaintiff would have to provide a higher fee to the TPF funder conditioned upon successful completion of the case. Since the plaintiff is the residual claimant, a higher payoff to the TPF funder implies a lower residual payment to the plaintiff (the expected payment the plaintiff receives upon successful completion of the litigation process is ${\bar{v}}_{s}-{\widetilde{v}}_{s}+D(s,{\widetilde{v}}_{s})$, and this value is decreasing in ${\widetilde{v}}_{s}$). Furthermore, anticipating a higher payment, there is a higher probability that the TPF funder would choose to see the litigation through by providing the financing for all litigation stages. Proposition 3 demonstrates that these countervailing incentives results in the plaintiff choosing to voluntarily reveal the true value of the claim over inflating the value of the expected verdict.

In a similar manner, if the plaintiff chooses to deflate the value of the claim he needs to consider the following countervailing effects. First, deflating the value of the claim would encourage the TPF funder to terminate the process when observing a lower cost than the plaintiff would choose under the self-funded scenario. In this case, there is a higher probability that the plaintiff will receive the termination compensation payment of $D(s, {\widetilde{v}}_{s})$, but since this compensation is increasing with ${\widetilde{v}}_{s}$, deflating the value of ${\widetilde{v}}_{s}$ results in a lower compensation to the plaintiff. On the other hand, if the litigation process is completed, the plaintiff will receive a higher expected payment (since the expected payment to the TPF funder will be lower), but the probability of observing such an event is low due to the deflated value of the verdict’s expected reward of ${\widetilde{v}}_{s}$. Therefore, the proposition shows that deflating the value of claim is not an equilibrium either, and the plaintiff will share the claim’s value truthfully.

Another way to understand the intuition behind this result, is that the TC contract elicits the TPF funder to make the same decisions as in the self-funded litigation scenario, and the plaintiff is able to extract the entire expected surplus from the process. Therefore, the plaintiff would want the TPF funder to follow the optimal sequence of decisions; i.e., mimic the decisions he would have made in the self-funded litigation process.

Two important aspects regarding the suggested contract should be emphasized. First, the plaintiff receives a fixed sum whenever the TPF funder decides to terminate the litigation process. In addition when the litigation process is completed successfully, the expected payoff to the plaintiff is also equal to the same termination payment. However, although in expectation the plaintiff receives the same amount regardless of the outcome of the litigation process, this contract is not equivalent to simply selling the litigation claim to the TPF funder up-front. When selling up-front the litigation process, the plaintiff is not liable for the outcome of the verdict. In this case, truthful information sharing cannot be achieved as an equilibrium, since the plaintiff will have an incentive to inflate the value of the claim in order to receive a higher up-front payment for selling the rights to his claim. In the suggested TC contract, the plaintiff is accountable for the shared information because he receives the residual payment from the verdict. Therefore, as discussed above, inflating the claim’s expected verdict results in a higher probability for completing the litigation process, and, consequently, it results in the plaintiff receiving a smaller share of the verdict’s outcome.

The second important aspect is related to the condition given in Proposition 3. The condition asserts that this proposition holds only when the net value of the claim is positive. If this is not the case, truthful revelation of the expected verdict cannot be supported in equilibrium. The intuition behind this condition is as follows: suppose that the claim’s expected value is negative, then under the self-funded setting, this claim will not be litigated, and the monetary outcome to the plaintiff is zero. However, if the plaintiff can convince the TPF funder that the expected verdict is positive, the TPF funder will pursue this litigation process. In this scenario there is a strictly positive probability that the plaintiff will receive a positive payment (for example, if the litigation process is terminated or if the actual verdict is higher than expected). As a result, the plaintiff will have an incentive to inflate the value of the claim in order to convince the TPF funder to finance the litigation process. Therefore, although Proposition 3 presents an important result about aligning the incentives to share information truthfully, this result is valid only when the net payoff of the claim is positive. However it should be noted that although the plaintiff may have better information about the merits of his claim, the TPF funder has resources to evaluate the claim with some accuracy.⁵ If the information available to the TPF funder is sufficient to determine that the net payoff of the claim is positive, then, according to our result, the TPF funder can rely on the information provided by the plaintiff to assess the expected value of the verdict.

6.2 A numerical example

To better understand the result presented in Proposition 3, we analyze the following simple numerical example. Assume that the litigation process is comprised of only two stages. It is common knowledge that the litigation cost during each stage is drawn from a uniform distribution over [0, 10];  The plaintiff knows that the verdict’s monetary outcome has the expected value of $E[v_{s}]={\bar{v}}_{s}=12$. We further assume that although the TPF funder does not know ${\bar{v}}_{s}$, she believes that the net payoff of the claim is positive, such that the condition of Proposition 3 holds.

We first analyze this outcome under the benchmark of self-funded litigation process. During the second litigation stage, since the cost cannot exceed 10 and the expected verdict reward is 12, the plaintiff will choose to execute this stage for any second litigation stage cost realization. In anticipation of the decision to continue during the second litigation stage, during the first litigation stage, the plaintiff evaluates the continuation value of $R_{2}(s,{\bar{v}}_{s})=12-5=7$. Therefore, during the first litigation stage, the plaintiff will choose to execute this stage if the realized first period cost is $c_{1}\le 7$, an outcome that occurs with the probability of 0.7. Therefore, the ex-ante value of this litigation process is given by:

$$\begin{aligned} \pi (s,{\bar{v}}_{s})=0.3*0+0.7*\left[ -E\left[ C_{1}|C_{1}\le 7 \right] -E\left[ C_{2}\right] +{\bar{v}}_{s}\right] =0.7*\left[ -3.5-5+12\right] =2.45. \end{aligned}$$

Now assume the TPF funder funds the litigation process, and the information regarding ${\bar{v}}_{s}$ is shared truthfully. According to the contract terms outlined above, the terms of the contract are defined such that

$$\begin{aligned} D(s,{\bar{v}}_{s})= & {} 2.45 \\ E[P(s,{\bar{v}}_{s})]= & {} 12-2.45=9.55, \end{aligned}$$

and as shown in Proposition 2. this contract results in the same decisions as the self-funded case; the expected payoff to the plaintiff in this case is 2.45, and the expected payoff to the TPF funder is zero, since $\pi ^{TPF}(s,{\bar{v}}_{s},\mathbb {k},{\bar{v}}_{s})=0.3*(-2.45)+0.7*[-3.5-5+9.55]=0$.

6.2.1 Incentives for inflating verdict estimate

Now assume that the plaintiff decides to inflate the value of the expected verdict and reports that ${\widetilde{v}}_{s}=14$ instead of ${\bar{v}} _{s}=12$. Under this report, the optimal decisions are to always execute the second litigation stage (given that the process has reached this stage), and since $R_{2}(s,{\widetilde{v}}_{s})=14-5=9$, to execute the first litigation stage if $c_{1}\le 9$, an outcome that happens with the probability of 0.9 . In this case, the ex-ante estimate of the claim’s value is

$$\begin{aligned} \pi (s,{\widetilde{v}}_{s})=0.1*0+0.9*\left[ -E\left[ C_{1}|C_{1}\le 9 \right] -E\left[ C_{2}\right] +{\widetilde{v}}_{s}\right] =0.9*\left[ -4.5-5+14\right] =4.05. \end{aligned}$$

Note that in this case, $\pi (s,{\widetilde{v}}_{s})>\pi (s,{\bar{v}}_{s})$. According to the contract TC, the terms of the contract are defined such that

$$\begin{aligned} D(s,{\widetilde{v}}_{s})= & {} 4.05; \\ E[P(s,{\widetilde{v}}_{s})]= & {} 14-4.05=9.95, \end{aligned}$$

and in this case the expected payoff to the plaintiff is given by

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\widetilde{v}}_{s})=0.1*4.05+0.9*\left( 12-9.95\right) =2.25. \end{aligned}$$

(9)

The first term in Equation (9) denotes the payoff to the plaintiff when the litigation process is terminated. In this case, the plaintiff receives the sum of 4.05 but it occurs only with probability of 0.1, and the second term is the payoff to the plaintiff when the litigation process is completed. Note that under this manipulation, the payoff to the plaintiff is lower than his payoff under truthful information sharing when the litigation process is completed. Under this manipulation, the estimated net value of the verdict is higher (4.05 compared with the true value of 2.45), and the plaintiff will be better-off if the litigation process is terminated during stage 1. However, due to this high value, there is a lower probability of terminating the litigation process. Furthermore, when the litigation process is completed successfully the plaintiff will be entitled to a lower residual payment ($12--9.95$ compared with $12--9.55$), because the TPF funder is the first to receive her payment, and the payment is higher than in the truth-telling case. Therefore, in this example, the plaintiff does not have an incentive to inflate the value of the claim. We next examine whether the plaintiff has an incentive to deflate the value of the expected verdict.

6.2.2 Incentives for deflating verdict estimate

Now assume that the plaintiff reports that the verdict’s expected outcome is ${\widetilde{v}}_{s}=10$ instead of ${\bar{v}}_{s}=12$. In this case, the second litigation stage will always be executed (as previously), and $R_{2}(s,{\widetilde{v}}_{s})=10-5=5$; therefore, the first litigation stage will be executed only if $c_{1}\le 5$, which happens with the probability of 0.5. The ex-ante net value of the claim is estimated to be

$$\begin{aligned} \pi (s,{\widetilde{v}}_{s})=0.5*0+0.5*\left[ -E\left[ C_{1}|C_{1}\le 5 \right] -E\left[ C_{2}\right] +{\widetilde{v}}_{s}\right] =0.5*\left[ -2.5-5+10\right] =1.25. \end{aligned}$$

Therefore, the contract terms in this case, will be defined in the following manner:

$$\begin{aligned} D(s,{\widetilde{v}}_{s})= & {} 1.25; \\ E[P(s,{\widetilde{v}}_{s})]= & {} 10-1.25=8.75. \end{aligned}$$

Under this contract, and knowing that the true expected value of the verdict is 12, the net payoff to the plaintiff is given by

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\widetilde{v}}_{s})=0.5*1.25+0.5*\left( 12-8.75\right) =2.25. \end{aligned}$$

(10)

In Equation (10) the first term is the plaintiff’s payoff when the litigation process is terminated, and the second term is his payoff when the litigation process is completed. Under this manipulation, the plaintiff will receive a higher payoff if the litigation process is completed (an expected payment of $12-8.75=3.25)$, but this happens only with the probability of 0.5, compared with completion probability of 0.7 under the truth-telling scenario. With the complement probability, the litigation process is terminated during the first litigation process, and then the compensation to the plaintiff is low due to the information deflation. As a result, in this case, the plaintiff is also better-off announcing the true value of the expected verdict.

7 Settlement offers

So far, we have overlooked the important issue of settlement offers. We now discuss the effect of the contract TC on the decision whether to accept a settlement offered during the litigation process. We assume that the defendant may offer a settlement at every litigation stage t. A settlement during time t is offered after the litigation cost of this stage, i.e., $c_{t}$ has been realized, but before this litigation cost was incurred. We denote a settlement offer made during the litigation stage t by $o_{t}$.

Under our benchmark case, in a self-funded litigation process, a settlement offer of $o_{t}$ during time t will be accepted by the plaintiff if

$$\begin{aligned} o_{t}\ge R_{t}(s,{\bar{v}}_{s})-c_{t}. \end{aligned}$$

(11)

Recall that the RHS of equation (11) denotes the expected value to the plaintiff of incurring the litigation cost of $c_{t}$ and continuing to the next litigation stage. If the defendant is willing to offer a settlement in the amount of $o_{t}$ that exceeds the value of $R_{t}(s,{\bar{v}}_{s})-c_{t}$, a risk-neutral plaintiff will accept this settlement offer. The decision whether to accept the settlement offer of $o_{t}$ depends on a few factors. First, it depends on the realized cost of litigation during stage t (i.e., $c_{t})$ since by accepting the settlement offer, the plaintiff (or TPF funder) avoids this cost. Second, it depends on the value of $R_{t}(s,{\bar{v}}_{s})$ which takes into consideration the distribution of legal costs during future litigation stages, and the expected verdict value. Note that this decision is not influenced by the value of past costs, since these are viewed as sunk costs.

We now turn to evaluate the effect of TPF funding on the incentives of the plaintiff and TPF funder to accept an offered settlement. We define a settlement outcome as being efficient in the following way.

Definition 3

A settlement outcome is efficient if the following conditions are met: (a) a settlement offer is accepted if, and only if, it is accepted in the benchmark case (i.e., according to Equation 11); (b) the settlement offer is accepted by both the plaintiff and the TPF funder.

According to our definition, a settlement is efficient if it results in accepting only offers that would have been accepted in the benchmark setting (self-funded litigation). In addition, a settlement satisfies the incentive rationality requirement if both the plaintiff and the TPF funder are better-off accepting the settlement offer over continuing the litigation process. The next Proposition shows that the contract TC can result in an efficient incentive-rational settlement process.

Proposition 4

Assume that when a settlement offer $o_{t}$ during litigation stage t,is accepted, the plaintiff receives the value of $D(s,{\bar{v}}_{s})$ and the TPF funder receives the payoff of $o_{t}-$ $D(s,{\bar{v}}_{s})$. Then, a settlement process is efficient and satisfies the incentive rationality constraint.

Proposition 4 also sheds light on the value of the TC contract demonstrating that this contract can result in an efficient settlement. If the value of a settlement offer of $o_{t}$ is allocated between the plaintiff and the TPF funder so that the former receives the payoff of $D(s,{\bar{v}}_{s})$ and the latter the residual payoff of $o_{t}-$ $D(s,{\bar{v}}_{s})$, then the contract will result in accepting the same settlement offers as in the self-funded litigation process. According to the contract TC, the plaintiff receives the sum of $D(s, {\bar{v}}_{s})$ if the litigation process is terminated mid-term, and he receives the expected sum of ${\bar{v}}_{s}-E\left[ P(s,{\bar{v}}_{s}) \right]$ if the litigation process is completed successfully. Proposition 2 shows that in our contract, $D(s,{\bar{v}}_{s})={\bar{v}}_{s}-E\left[ P(s,{\bar{v}}_{s})\right]$; thus, the plaintiff receives the same expected payoff regardless of whether the litigation process was terminated or completed. As a result, the plaintiff accepts any settlement offer, in which he receives at least the sum of $D(s,{\bar{v}}_{s})$. Proposition 4 suggests that this, indeed, will be the plaintiff’s share in any settlement offer.

When the TPF funder considers whether to accept a settlement offer of $o_{t}$ , she would agree to accept the settlement offer if

$$\begin{aligned} -c_{t}+\pi _{t+1}^{TPF}(s,{\bar{v}}_{s})\le \max \{\left( o_{t}-D(s, {\bar{v}}_{s})\right) ,-D(s,{\bar{v}}_{s})\}=o_{t}-D(s,{\bar{v}} _{s}). \end{aligned}$$

(12)

The LHS describes the payoff the TPF funder receives from continuing litigation and the RHS denotes the value the TPF funder receives if the settlement offer is accepted. Note that in addition to continuing the litigation process, the TPF funder can either terminate the litigation process and pay $D(s,{\bar{v}}_{s})$, or if a settlement offer is accepted, she receives the sum of $o_{t}-D(s,{\bar{v}}_{s})$. Based on the results of Proposition 2, since the TPF funder chooses to continue the litigation process in the same cases the plaintiff would have chosen in a self-funded litigation process, $-c_{t}+\pi _{t+1}^{TPF}(s,{\bar{v}} _{s})+D(s,{\bar{v}}_{s})=-c_{t}+ R_{t}(s,{\bar{v}}_{s})$. Based on this property, the condition of the TPF funder to accept the settlement offer can be expressed as $-c_{t}+ R_{t}(s,{\bar{v}}_{s})\le o_{t}$, which is identical to the condition of accepting the settlement offer in the self-funded litigation (Equation 11).

Thus, the Proposition suggests that the TC contract characterized in Proposition 2 mimics the optimal sequence of litigation decisions, including the decision to settle. Proposition 4 asserts that in order to reach a settlement, the TPF funder and the plaintiff should mutually agree to accept the settlement offer. This raises the issue regarding the TPF funder’s control over the litigation process. The American version of the TPF resembles the features of passive investment: formally, the TPF funder is prohibited from interfering with litigation (Coffee 2010: 340-341; Steinitz 2010: 1279–1280; Sebok 2013: 2941-42), and must take a back seat with respect to the case management and decision-making (DeStefano 2012: 2796). The plaintiff, assisted by his legal counsel, retains the control over the management of the legal proceedings, including the power to accept or reject any settlement offers, thereby deciding on whether to risk continued litigation and costs—usually to be borne primarily by the funder—or to settle.

Nevertheless, while in the United States TPF funders “agree not to make any decisions about the lawsuits ...and agree not to interfere in the relationship between the plaintiffs/borrowers and their lawyers” (Martin 2008: 109), the researchers have long been skeptical about the level of the funders’ detachment and passivity. Given the funders’ financial investment and direct economic interest in the outcome of litigation, their desire to exercise some degree of control is natural. Even if the funder surrenders all control, she will de facto preserve some influence over the management of the legal proceedings due to her being a savvy player and because she is providing funds to pursue the litigation (DeStefano 2012: 2827; DeStefano 2014: 322). This situation further reinforces our conclusion, that in order to achieve a settlement the TPF funder and the plaintiff should mutually agree to accept the settlement offer, is aligned with reality.

8 Extensions

In this section we discuss two extensions to our model. The first one studies the implications of our setting when the plaintiff is risk averse (and the TPF funder is still risk neutral), and the second extension demonstrates that the ability to renegotiate will not change the outcome outlined in the main model.

8.1 Risk aversion

So far, we have assumed that the plaintiff is risk neutral. An alternative assumption is to assume that the plaintiff is risk averse. When the plaintiff is risk averse we assume that he has a concave increasing utility function denoted by u() with respect to the payoff. In this case, the plaintiff maximizes his expected utility, and not the expected payoff, as was assumed in the paper so far. We continue to assume that the TPF funder is risk neutral, because it is plausible that she invests in multiple claims, which allows her to diversify the investment portfolio.

Full analysis of the case of risk aversion is beyond the scope of this paper; however, in this extension we discuss two important aspects that are related to risk aversion. We assume that information about the expected value of the verdict, ${\bar{v}}_{s}$, is common knowledge (as in Proposition 2), and that the plaintiff and the TPF funder contract using the TC contract.⁶ The following result reveals two important aspects of risk aversion

Corollary 1

Assume that a risk averse plaintiff contracts with a risk neutral TPF funder. Then:

(a) There are claims, that the plaintiff will not want to pursue under the self-funded setting, but the TPF funder will finance.

(b) There are settlement offers that will be accepted under the self-funded scenario, but will not be accepted by the TPF funder.

The corollary states two important differences between a case that is self-financed by a risk averse plaintiff, and a case that is funded by a risk neutral TPF funder. A TPF funder is willing to finance any case that has a positive net-payoff. However, due to the risk aversion level of the plaintiff, there are claims that the plaintiff will prefer not to pursue, even though they have a positive net payoff. Therefore, as opposed to the conclusion of Propositions 2 that suggests that the same claims will be pursued under the self-funded setting and the TPF funding, when the plaintiff is risk averse, the TPF mechanism can increase the volume of litigated claims. On the one hand, this has a negative effect, since the increase in the number of litigated claims would create additional burden on an already over-congested system. On the other hand, this result suggests that a risk averse plaintiff will not pursue meritorious claims because of litigation cost uncertainty. Therefore, introducing the TPF option can be viewed as welfare enhancing, since it provides the plaintiff with the means to achieve justice in a meritorious case. Thus, the fact that the outcome of the TPF mechanism is not identical with the case of financially non-constrained risk-averse plaintiff does not imply inefficiency; to the contrary, the TPF mechanism results in meritorious claims being pursued, while absent this mechanism a risk-averse plaintiff may refrain from pursuing such claims.

The second part of the corollary discusses the effect of the TPF funding mechanism on the possibility of reaching a settlement. A settlement offer guarantees a certain reward, and due to the risk aversion, a plaintiff may prefer to accept a settlement offer (which represents a certain monetary reward absent any risk) even if the expected payoff from continuing the litigation is higher. As opposed to the self-funded litigation process, under the TPF mechanism, the TPF funder will accept a settlement offer only if the settlement offer is (weakly) higher than the expected reward from continuing the litigation process. As stated above, analyzing the dynamics between a risk-averse plaintiff and the TPF funder when they receive a settlement offer is beyond the scope of this paper. However, if both–the plaintiff and the TPF funder–should agree to accept the settlement offer, introducing the TPF option will result in fewer settlement offers being accepted. This finding echoes the concerns that some scholars have raised – introducing the TPF option will result in less settlements, consequently creating additional congestion on the judicial system.

The results introduced in this subsection also suggest that a risk-averse plaintiff may prefer to use the TPF mechanism even when he is not financially-constrained. Under the TPF mechanism outlined in this paper, the plaintiff receives the ex-ante value of the litigation process without incurring any risk related to the litigation costs.

8.2 Renegotiation

In the main model, we have assumed that once the terms of the contract were determined, the plaintiff and the TPF funder are not able to modify these terms. In this extension we evaluate the implications of this assumption. Specifically, we allow the plaintiff and the TPF funder to renegotiate the terms of the contracts during any period in the model. We assume that to modify the terms of the contract, both the plaintiff and the TPF funder must agree to it. This implies that the renegotiated contract must make both parties better-off compared with the original contract. The following result illustrates that no renegotiation is possible in our setting.

Proposition 5

Renegotiation will not alter the terms of the contract.

The intuition behind the proposition is as follows. Assume that the TPF funder finds the litigation cost to be too expensive and, thus, wishes to terminate the litigation process. According to the signed contract, upon terminating the legal process, the TPF funder must provide the plaintiff with the compensation of $D(s,{\bar{v}}_{s})$. The TPF funder can try and convince the plaintiff not to terminate the legal process if she is given a higher reward if the litigation is continued. However, the plaintiff will agree to award the TPF funder a higher reward for completing the legal process if by continuing the process his payoff will be higher than when terminating it. However, note that according to the contract, by terminating the process the plaintiff receives the sum of $D(s,{\bar{v}}_{s}),\$and this is also the expected payoff that the plaintiff receives if the legal process is completed. Consequently, awarding a higher payoff for the TPF funder means that the payoff of the plaintiff upon completing the legal process will be lower than $D(s,{\bar{v}}_{s});$ thus, any offer to re-negotiate the terms of the contract will be rejected by the plaintiff. In a similar manner, it is possible to show that any offer coming from the plaintiff and makes him better-off compared with the original contract will be rejected by the TPF funder since it must reduce her expected payoff.

9 Concluding remarks

The subject of third-party litigation funding has stimulated substantial research and ignited a heated debate among scholars and public policy-makers. It appears that the outcome of this debate is far from settled, with the commentators suggesting a wide range of measures from placing a blanket ban on the TPF financing mechanism (Rubin 2011), to restricting the TPF to certain cases (McLaughlin 2006), or to placing various limitations on the use of the TPF instrument (Estevao 2013; Martin 2008). Still other observers concentrate on the ‘rules of the game’ advocating the adoption of various procedural measures (Lysaught and Hazelgrove 2012), structural channels (Estevao 2013), or regulation (Couture 2014).

Most of the relevant research has examined the implications of the TPF mechanism at the normative level—i.e., its effects on the judiciary-system, such as the number and the merits of the lawsuits filed (Rubin 2011; Richey 2013), its potential manipulative use to influence the legal precedent (Rubin 2011; Lysaught and Hazelgrove 2012; Deffains and Desrieux 2014), etc. In contrast with this perspective, we examine the effect of TPF at the micro level and analyze the decision-making of the parties to a TPF agreement—i.e., the plaintiff and the TPF funder—during the management of one lawsuit.

More specifically, we decompose the decisions into two components: choices determined by the plaintiff and decisions made by the TPF funder. In our model the plaintiff dictates the litigation strategy of the lawsuit, whereas the TPF funder decides whether to continue funding the claim at each litigation stage. In spite of this decomposition of the decision rights between two independent self-interested parties, we are able to characterize a contract that induces the parties to make the same decisions as if those decisions had been made solely by the plaintiff, without the need for external funding.

This contract provides a pre-specified remedy to the plaintiff if the TPF funder terminates the financing prior to the conclusion of the litigation. The contract also specifies the compensation to the TPF funder upon successful completion of the litigation process. This special contract aligns the TPF funder’s decisions to continue financing the litigation process with the decisions the plaintiff would have made in the absence of financial constraints, i.e., had the plaintiff been able to finance his litigation without the TPF.

We further demonstrate that even if the plaintiff can manipulate the value of the claim—e.g., exaggerate its value in order to entice the TPF funder to provide the financing—this contract induces the plaintiff to voluntarily reveal the true value of his claim. The plaintiff would choose to be truthful because manipulating the value of the claim results in opposing effects that cancel each other out. More specifically, while overstating the value of the lawsuit ensures a higher compensation damage, it also results in a higher payment that the plaintiff would have to pay to the TPF funder upon successful completion of the litigation process.

In addition to the contribution of a viable TPF contract that prompts the parties to mimic litigation without TPF, our model provides insight into the sequential aspect of a litigation process. We model a litigation process as a sequence of stages; at each stage, the TPF funder can decide whether to continue the process—i.e., to keep financing the lawsuit—or to terminate the funding. This is more realistic and characteristic of the traditional approach, which treats the TPF funder as a one-time decision-maker, who at the onset of litigation must make the decision whether to fund the lawsuit. In our model, the TPF funder can reevaluate her decision to fund each litigation stage separately, based on new information that comes to light in the course of the litigation process; this approach provides the TPF funder with broader freedom as she enters the litigation funding venture.

We also provide a unique view of the litigation process as a financial derivative. We demonstrate that the decision whether or not to fund the final litigation stage is equivalent to purchasing a put option. In the same vein, the decision to fund an earlier litigation stages is equivalent to purchasing the option to purchase a put option. This view of a litigation process as a financial instrument opens the door to additional analysis regarding the value of a sequential litigation process and has been viewed as creating “exciting possibilities” (Steinitz 2013: 1908).

A few possible ways to extend this research exist. First, in this work we assume that the plaintiff has superior information compared with the TPF funder. It is possible to envision situations in which actually the opposite holds – the TPF funder, due to its legal expertise, has the better ability to evaluate the merits of the legal case. In this scenario, the TPF funder will need to decide whether to share in a truthful manner its evaluation of the case with the plaintiff. Second, in this work, we have assumed that attorney’s incentives are perfectly aligned with the plaintiff’s. However, previous empirical research has shown that this is not always the case. Therefore, another possible venue for research is to model the way the attorney’s incentives are influenced by the relationship between the plaintiff and the TPF funder. For example, Can the contract, suggested in this paper, alter the incentives of the attorney to exert effort during the trial? Can this contract alter the incentives of the attorney to share truthfully its assessment regarding the case with the plaintiff? Finally, this paper considers a situation in which the expected value of the verdict is not observed by the TPF funder. An alternative is that it is the effort level, exerted by the plaintiff and the attorney during the legal process, that is not observed by the TPF funder. In this case, the question is whether it is possible to characterize a contract that will provide the plaintiff and the attorney with the incentives to exert the optimal effort level in spite being funded by the TPF funder.

Appendix-proofs

Proof of Proposition 1. The proof uses backward induction. We start with period N. In this case, based on the definition of $R_{N}={\bar{v}} _{s}$, the plaintiff chooses to execute the final stage only if $c_{N}\le {\bar{v}}_{s}$, which is indeed the optimal decision.

We now assume that the decision rule is optimal for the case of $t+1$, and examine the optimality of the decision rule for the case of period t. Note that based on the optimality of period $t+1$ the profit of plaintiff starting at period $t+1$ is

$$\begin{aligned} \begin{array}{ll} R_{t+1}(s,{\bar{v}}_{s})-c_{t+1} &{} \,\hbox {if} R_{t+1}(s,{\bar{v}}_{s})\ge c_{t+1}; \\ 0 &{} R_{t+1}(s,{\bar{v}}_{s})<c_{t+1}. \end{array} \end{aligned}$$

and the ex-ante payoff of the plaintiff starting at period $t+1$ is therefore given by

$$\begin{aligned} E\left[ \left( R_{t+1}(s,{\bar{v}}_{s})-c_{t+1}\right) ^{+}\right] , \end{aligned}$$

which by definition equals $R_{t}(s,{\bar{v}}_{s})$. Therefore, for a plaintiff observing the litigation cost of $c_{t}$ it is optimal to continue the process if $c_{t}\le$ $R_{t}(s,{\bar{v}}_{s})$. Thus, the Proposition is correct for period t, and by induction it is correct for any period.

Proof of Proposition 2. (a) We first show that under the TC contract, the plaintiff chooses the optimal litigation strategy $s^{*}$. Note that if the litigation process is terminated during any period $t<N$, the plaintiff receives the compensation of $D(s,{\bar{v}}_{s})=\pi (s, {\bar{v}}_{s})$. If the litigation process is completed successfully, the plaintiff receives the reward of $v_{s}$ minus the payment to the TPF funder. The expected payment to the TPF funder is $E\left[ P(s,v_{s})\right] ={\bar{v}}_{s}-D(s,{\bar{v}}_{s})={\bar{v}}_{s}-$ $\pi (s,\bar{ v}_{s})$. Therefore, the expected net payoff to the plaintiff when the litigation process is completed is

$$\begin{aligned} {\bar{v}}_{s}-E\left[ P(s,v_{s})\right] ={\bar{v}}_{s}-\left( {\bar{v}}_{s}-\pi (s,{\bar{v}}_{s})\right) =\pi (s,{\bar{v}}_{s}). \end{aligned}$$

Therefore, according to the TC contract, the plaintiff receives the same expected reward, regardless of the litigation outcome, and this payoff is $\pi (s,{\bar{v}}_{s})$. When the plaintiff needs to determine the optimal litigation strategy, he maximizes the following objective function of $\max _{s}\pi (s,{\bar{v}}_{s})$, and by definition $s^{*}\in \arg \max \pi (s,{\bar{v}}_{s})$.

(b) We next show that TPF funder receives the same decisions as the plaintiff in the benchmark. The proof is conducted by backward induction. For period N, the TPF funder continues the litigation process if

$$\begin{aligned} -D(s,{\bar{v}}_{s})\le E\left[ P(s,v_{s})\right] -c_{N}. \end{aligned}$$

The LHS denotes the compensation that the TPF funder pays the plaintiff if the litigation process is terminated. The RHS is the payoff to the TPF funder if she continues the litigation process. In the latter case, she incurs the cost of $c_{N}$ and receives the payment of $P(s,v_{s})$. Note that $E\left[ P(s,v_{s})\right] ={\bar{v}}_{s}-$ $D(s,{\bar{v}}_{s})$. Therefore, the TPF funder continues if

$$\begin{aligned} -D(s,{\bar{v}}_{s})\le & {} {\bar{v}}_{s}-D(s,{\bar{v}}_{s})-c_{N} \\ 0\le & {} {\bar{v}}_{s}-c_{N}, \end{aligned}$$

and this is the same decision rule for the plaintiff in the benchmark case.

We next assume that the Proposition holds for period $t+1$ and examine period t. In period t, a plaintiff in the benchmark case would fund this stage if the following condition is satisfied

$$\begin{aligned} c_{t}\le & {} \underset{*}{\underbrace{\mathop {\displaystyle \prod }\limits _{j=t+1}^{N}\Pr (C_{j}\le R_{j}(s,{\bar{v}}_{s}))}}\left( {\bar{v}} _{s}-\sum _{j=t+1}^{N}E[C_{j}|C_{j}\le R_{j}(s,{\bar{v}}_{s})]\right) \\&-\sum _{j=t+1}^{N}\left[ \underset{**}{\underbrace{ \mathop {\textstyle \prod }\limits _{i=t+1}^{j}\Pr (C_{i}\le R_{i}(s,{\bar{v}}_{s}))\Pr (C_{j}>R_{j}(s,{\bar{v}}_{s}))}}\sum _{i=t+1}^{j-1}E[C_{i}|C_{i}\le R_{i}(s,{\bar{v}}_{s})]\right] . \end{aligned}$$

Under the TPF case, the TPF funder would find it beneficial to continue to the next stage if

$$\begin{aligned} -D(s,{\bar{v}}_{s})\le & {} \underset{***}{\underbrace{ \mathop {\displaystyle \prod }\limits _{j=t+1}^{N}\Pr (\delta _{j}^{TPF}(s,{\bar{v}} _{s},C_{j}|\mathbb {k})=1))}}\left( E\left[ P(s,v_{s})\right] - \sum _{i=t+1}^{N}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}}_{s},C_{i}|\mathbb {k})=1)]\right) \nonumber \\&-\sum _{j=t+1}^{N}\left[ \begin{array}{c} \underset{****}{\underbrace{\mathop {\textstyle \prod }\limits _{i=t+1}^{j}\Pr (\delta _{i}^{TPF}(s,{\bar{v}}_{s},C_{i}|\mathbb {k})=1))\Pr ((\delta _{j}^{TPF}(s,{\bar{v}}_{s},C_{j}|\mathbb {k})=0))}} \\ \left( \sum _{i=t+1}^{j-1}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1)]+D_{t}(s,{\bar{v}}_{s})\right) \end{array} \right] \nonumber \\&-c_{t}. \end{aligned}$$

(13)

Based on the induction assumption that the plaintiff and the TPF funder receive the same funding decisions for any period $t+1,...,N$, the probability of successfully completing the litigation process or terminating it at a certain point are identical. Also using the fact that $E\left[ P(s,v_{s})\right] ={\bar{v}}_{s}-D(s,{\bar{v}}_{s})$, we can express Equation (13) in the following way

$$\begin{aligned} -D(s,{\bar{v}}_{s})\le & {} \mathop {\displaystyle \prod }\limits _{j=t+1}^{N}\Pr (\delta _{j}^{TPF}(s,{\bar{v}}_{s},C_{j}|\mathbb {k})=1))\left( {\bar{v}}_{s}-D(s, {\bar{v}}_{s})-\sum _{i=t+1}^{N}E[C_{i}|\delta _{i}^{TPF}(s,v_{s},C_{i}|\mathbb {k})=1)]\right) \\&-\sum _{j=t+1}^{N}\left[ \begin{array}{c} \mathop {\textstyle \prod }\limits _{i=t+1}^{j}\Pr (\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{i}|\mathbb {k})=1))\Pr ((\delta _{j}^{TPF}(s,{\bar{v}}_{s},C_{j}|\mathbb {k})=0)) \\ \left( \sum _{i=t+1}^{j-1}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1)]+D_{t}(s,{\bar{v}}_{s})\right) \end{array} \right] \\&-c_{t} \\= & {} \mathop {\displaystyle \prod }\limits _{j=t+1}^{N}\Pr (\delta _{j}^{TPF}(s,{\bar{v}} _{s},C_{j}|\mathbb {k})=1))\left( {\bar{v}}_{s}-\sum _{i=t+1}^{N}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}}_{s},C_{i}|\mathbb {k})=1)]\right) \\&-\sum _{j=t+1}^{N}\left[ \begin{array}{c} \mathop {\textstyle \prod }\limits _{i=t+1}^{j}\Pr (\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{i}|\mathbb {k})=1))\Pr ((\delta _{j}^{TPF}(s,{\bar{v}}_{s},C_{j}|\mathbb {k})=0)) \\ \left( \sum _{i=t+1}^{j-1}E[C_{i}|\delta _{i}^{TPF}(s,{\bar{v}} _{s},C_{t}|\mathbb {k})=1)]\right) \end{array} \right] \\&-c_{t}-D(s,{\bar{v}}_{s}) \end{aligned}$$

Note that the value of $-D(s,{\bar{v}}_{s})$ appears on both sides of the inequality, and, thus, cancels. Furthermore, after these manipulations, the decision rule of the plaintiff indeed coincides with the decision rule of the TPF funder. This concludes the proof that the contract satisfies Equation (6). Finally, note that $\pi (s^{*},v_{s^{*}})>0$ .

Proof of Proposition 3. We denote by $\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\widetilde{v}}_{s})$ the payoff of the plaintiff when he reports the claim value of ${\widetilde{v}}_{s}$ and by $\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\bar{v}}_{s})$ when he truthfully reports the claim value of ${\bar{v}}_{s}$. Truthful information sharing is the strategic choice of the plaintiff when

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\bar{v}}_{s})-\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\widetilde{v}}_{s})\ge 0\text { for every possible }{\bar{v}} _{s}\text { and }{\widetilde{v}}_{s}. \end{aligned}$$

Recall that $\pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\bar{v}}_{s})=D(s, {\bar{v}}_{s})$. Denote by $z({\widetilde{v}}_{s})$ the probability to successfully completing all the litigation phases given the announced value of ${\widetilde{v}}_{s}$. Therefore,

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\widetilde{v}}_{s})= & {} z({\widetilde{v}} _{s})E\left[ \left( v_{s}-P(s,{\widetilde{v}}_{s}\right) )^{+}\right] +\left( 1-z({\widetilde{v}}_{s})\right) D(s,{\widetilde{v}}_{s}) \\= & {} z({\widetilde{v}}_{s})\left( {\bar{v}}_{s}-{\widetilde{v}}_{s}+D(s, {\widetilde{v}}_{s})\right) +\left( 1-z({\widetilde{v}}_{s})\right) D(s, {\widetilde{v}}_{s}). \end{aligned}$$

Therefore,

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\bar{v}}_{s})-\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\widetilde{v}}_{s})= & {} z({\widetilde{v}}_{s})\left( D(s,{\bar{v}} _{s})-{\bar{v}}_{s}+{\widetilde{v}}_{s}-D(s,{\widetilde{v}}_{s})\right) \nonumber \\&+\left( 1-z({\widetilde{v}}_{s})\right) \left( D(s,{\bar{v}}_{s})-D(s, {\widetilde{v}}_{s})\right) \nonumber \\= & {} z({\widetilde{v}}_{s})\left( {\widetilde{v}}_{s}-{\bar{v}}_{s}\right) +D(s, {\bar{v}}_{s})-D(s,{\widetilde{v}}_{s}), \end{aligned}$$

(14)

and we need to show that this value is non-negative. Denote the ex-ante expected litigation cost given the claim value of ${\widetilde{v}}_{s}$ by $E[C|{\widetilde{v}}_{s}]$. Therefore,

$$\begin{aligned} \pi (s,{\widetilde{v}}_{s})= & {} D(s,{\widetilde{v}}_{s})=z({\widetilde{v}}_{s}) {\widetilde{v}}_{s}-E[C|{\widetilde{v}}_{s}]; \\ \pi (s,{\bar{v}}_{s})= & {} D(s,{\bar{v}}_{s})=z({\bar{v}}_{s}) {\bar{v}}_{s}-E[C|{\bar{v}}_{s}], \end{aligned}$$

and note that $\pi (s,{\bar{v}}_{s})=D(s,{\bar{v}}_{s})=z({\bar{v}} _{s}){\bar{v}}_{s}-E[C|{\bar{v}}_{s}]\ge z({\widetilde{v}} _{s})v_{s}-E[C|{\widetilde{v}}_{s}]=\pi (s,{\widetilde{v}}_{s})$. Therefore,

$$\begin{aligned} D(s,{\widetilde{v}}_{s})-D(s,{\bar{v}}_{s})\le z({\widetilde{v}}_{s})\left( {\widetilde{v}}_{s}-{\bar{v}}_{s}\right) , \end{aligned}$$

which implies that

$$\begin{aligned} \pi ^{p}(s,{\bar{v}}_{s},\mathbb {k},{\bar{v}}_{s})-\pi ^{p}(s,{\bar{v}} _{s},\mathbb {k},{\widetilde{v}}_{s})= & {} z({\widetilde{v}}_{s})\left( {\widetilde{v}} _{s}-{\bar{v}}_{s}\right) -\left( D(s,{\widetilde{v}}_{s})-D(s,{\bar{v}} _{s})\right) \\\ge & {} z({\widetilde{v}}_{s})\left( {\widetilde{v}}_{s}-{\bar{v}}_{s}\right) -z({\widetilde{v}}_{s})\left( {\widetilde{v}}_{s}-{\bar{v}}_{s}\right) =0. \end{aligned}$$

Further note that choosing another strategy ${\widetilde{s}}$ instead of the strategy s cannot improve the situation for the plaintiff since according to the suggested contract the TPF just recovers its costs, and the plaintiff receives the entire surplus from the legal claim - $\pi (s,{\bar{v}}_{s})$ . Therefore, when choosing the strategy the plaintiff solves for $max_{s}\pi (s,{\bar{v}}_{s})$, and by definition $s^{*}\in \arg \max \pi (s, {\bar{v}}_{s})$

Proof of Proposition 4. Note that for any stage $\hbox {{ /}{t}}$ in which the litigation process is terminated the plaintiff receives the sum of $D(s, {\bar{v}}_{s})$. When the litigation process is completed successfully, the plaintiff receives the sum of max($v_{s}-P(s,v_{s}),0)$ which according to Proposition 2, its expected value equals $D(s,{\bar{v}} _{s}).$ Therefore, without a settlement, the plaintiff receives the sum of $D(s,{\bar{v}}_{s})$ regardless of the outcome of the litigation process. Thus, the plaintiff would agree to any settlement that gives him at least this value.

Under the benchmark case, a plaintiff would agree to a settlement offer of $o_{t}$ if $-c_{t}+$ $R_{t}(s,{\bar{v}}_{s})\le o_{t}$. Based on Proposition 2, we have that $\underset{*}{\underbrace{-c_{t}+R_{t}(s, {\bar{v}}_{s})=-c_{t}+\pi _{t+1}^{TPF}(s,{\bar{v}}_{s})+D(s,\bar{v }_{s})}}$. Under the TC contract, if a settlement is offered, the TPF funder receives the sum of $o_{t}-D(s,{\bar{v}}_{s})$; thus she would agree to this settlement offer if

$$\begin{aligned} o_{t}-D(s,{\bar{v}}_{s})\ge -c_{t}+\pi _{t+1}^{TPF}(s,{\bar{v}}_{s}), \end{aligned}$$

where the RHS represents the TPF funder’s payoff from continuing the case and the LHS the payoff from accepting the settlement offer. Using $*$, this equation can be expressed as

$$\begin{aligned} o_{t}-D(s,{\bar{v}}_{s})\ge & {} -c_{t}+R_{t}(s,{\bar{v}}_{s})-D(s, {\bar{v}}_{s}) \\ o_{t}\ge & {} -c_{t}+R_{t}(s,{\bar{v}}_{s}). \end{aligned}$$

Note that this is the condition that ensures that the plaintiff would agree to accept the settlement offer of $o_{t}$ under the benchmark case of self-funded litigation process.

Proof of Corollary 1. The results are based on Jensen’s inequality. By Jensen’s Inequality, for a strictly concave utility function, $E\left[ u(x)\right] <u\left[ E(x)\right]$. A risk neutral TPF funder will fund any claim such that $E(x)\ge 0$, where x represents the stochastic payoff of the claim. Assume that $E(x)=0$, then $E\left[ u(x)\right] <0$, and, thus, a risk-averse plaintiff will not fund this claim under the self-funded litigation setting.

In a similar manner, there exists $\underline{o}_{t}$, such that $u( \underline{o}_{t})= E\left[ u(x)\right]$, and, thus, the risk-averse plaintiff will choose to accept any settlement offer with $o_{t}>\underline{o }_{t}$. However, due to the risk-aversion, we have that $\underline{o} _{t}<E(x)$.

Proof of Proposition 5. Assume that the TPF funder wishes to negotiate the terms of the contract. The plaintiff would agree only if he receives the amount of x which is higher than what he expects to receive based on the original contract. According to the original contract, the plaintiff receives the amount of $D(s,{\bar{v}}_{s})$ if the legal process is terminated, and receives the same expected amount if the legal process is completed successfully (see Proposition 2). Therefore, the plaintiff will refuse to any renegotiation offer that results in a lower amount of $D(s,{\bar{v}}_{s})$. However, the TPF funder cannot do any better if she does not offer the plaintiff a lower amount than $D(s, {\bar{v}}_{s})$ when the case is completed, and therefore the contract outlined in Proposition 2 is renegotiation-proof. Similar arguments can be made regarding an offer to re-negotiate that comes from the plaintiff.