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.

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.