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Contingent fees and endogenous timing in litigation contests


This study examined the contingent fee arrangements and adversarial systems applied in the United States. In the American context, a plaintiff (or a defendant) pays a contingent fee (an hourly fee) to their lawyer. In this adversarial system, lawyers can either be the first or the second mover. Solving the American practice with endogenous timing of litigation efforts, we obtained the following results: (i) if the defendant’s total hourly fee rate is not sufficiently high, the plaintiff’s lawyer is the first mover, with the plaintiff then being the underdog and (ii) if the rate is sufficiently high, the lawyer is the second mover, with the plaintiff then being the favorite. We demonstrated that these results are not ideal for the plaintiff. The equilibrium order of effort increases the equilibrium contingent fee and total legal effort in the trial, making reaching a settlement difficult. However, this improves the justice achieved through litigation if the total hourly fee rate is not significantly high. We suggest that the American practice of contingent fees with endogenous timing of effort is not economically efficient but is suitable for achieving justice.

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  1. Until about two decades ago, every legal system outside the US prohibited contingent fess, with it being suggested that the US should do so as well (Painter, 2000). Recently, the American practice of contingent fees has been either officially or informally introduced in various European and Asian countries. In this context, plaintiffs hire lawyers on a contingent fee basis for many reasons (Baik & Kim, 2007; Bebchuck & Guzman, 1996; Dana & Spier, 1993; Miller, 1987; Park & Lee, 2020). In most civil cases, plaintiffs are individuals, with the defendants often being corporations such as insurance companies. Herein, individuals with less experience in litigation have a greater incentive to prevent their lawyers’ moral hazards and are more likely to face liquidity constraints compared to the larger corporations. Moreover, unlike defendants’ cases, it is easy for plaintiffs to standardize the formula of contingent fees at a certain percentage of the indemnity (Dana & Spier, 1993).

  2. In an adversarial system, the process of lawyers presenting evidence that they have collected, as well as refuting the evidence raised by the other side, is repeated until a court ruling is finalized (Parisi, 2002). Compared to the study by Bumann and Friehe (2013), this study assumes that a given sequential move occurs once.

  3. Lawyers’ legal efforts refer to the time spent by them (such as in collecting and presenting evidence for their clients). However, when a moral hazard occurs, legal efforts by lawyers are smaller than the time spent.

  4. We defined a player with a 50% or greater chance of winning the game as the favorite and their adversary as the underdog (Dixit, 1987).

  5. Under British rule, the defeated litigants must reimburse the costs of the winning litigation parties.

  6. In the asymmetric reimbursement system in environmental conflicts, only polluting firms (defendants) who have lost must reimburse the costs of citizen groups (plaintiffs) and/or their lawyers.

  7. Furthermore, Baik and Lee (2013) found that when players evaluate the prize equally, their delegates are indifferent to their choice of timing.

  8. Our assumptions regarding the success function are adopted in some of the litigation contest literature on delegation contracts (Baik, 2008; Baik & Kim, 2007; Baumann & Friehe, 2012; Park & Lee, 2019, 2020; Wärneryd, 2000). It may be more realistic for each litigant to have a different degree of fault and not know the truth about their relative degree of fault. However, these aspects are not included in this study as it rather focuses on the endogenous timing of the efforts of Delegates 1 and 2.

  9. Miller (1987) explains the settlement range as follows: “The difference between defendant’s loss from trial and plaintiff’s gain from trial may be termed the “litigation differential. The range of offers that it would be rational for defendant to make and for plaintiff to accept may be called the ‘settlement range’” (p. 191).

  10. The second-order condition for this maximization problem is satisfied because Delegate 1’s expected payoff is strictly concave in x1.

  11. Player 2’s expected loss is strictly convex in x2, which satisfies the second-order condition for minimizing Eq. (3).

  12. In the litigation contest without delegation, the probability that Player i wins the case is denoted by wi(y1, y2). The probability of winning is then that wi(y1, y2) = yi/(y1 + y2) for y1 + y2 > 0 and wi(y1, y2) = 1/2 for x1 + x2 = 0. The expected payoff for Player 1 and the expected loss for Player 2, while considering the probability of winning, are G1 = w1vy1 and L2 = w1v + y2, respectively. In the equilibrium of the game with no delegation, the players expend y1Â = y2Â = v/4, and the expected payoff for Player 1 is G1Â = v/4, whereas the expected loss of Player 2 is L1Â = 3v/4. The condition for Player 1 to hire Delegate 1 is that G1Ŝ > G1Â or h ≥ 1.78 and that for Player 2 to hire Delegate 2 is that L2Ŝ < L2Â or h < 3. Thus, the profitable delegation condition under which both players hire delegates are G1Ŝ > G1Â and L2Ŝ < L2Â: 1.78 ≤ h < 3.

  13. As mentioned in Sect. 2, we found that in Baik and Lee’s (2013) study, the equilibrium effort order is not appropriate in terms of measuring justice. Please refer to Sect. 6 for the difference between Baumann and Friehe’s (2013) study and this research.

  14. Baik and Kim (2021) show that when litigants consider the legal ability of lawyers by hiring them in simultaneous games, the plaintiff can be the favorite.

  15. x1*/∂h = v/16 > 0 for 1.78 ≤ h < 2, and ∂x1*/∂h = {3(2)1/2 – 4h1/2}v/(8h5/2) < 0 for 2 ≤ h < 3.

  16. x2*/∂h = – v/16 < 0 for 1.78 ≤ h < 2, and ∂x2*/∂h = – 3(2)1/2v/(8h5/2) < 0 for 2 ≤ h < 3.

  17. π1*/∂h = v/16 > 0 for 1.78 ≤ h < 2, ∂π1*/∂h = {(2 h)1/2 – 1}{3(2)1/2 – 2h1/2}v/(8h5/2) > 0 for 2 ≤ h < 3; ∂G1*/∂h = v/8 > 0 for 1.78 ≤ h < 2, and ∂G1*/∂h = {(2 h)1/2 – 1}v/(2h2) > 0 for 2 ≤ h < 3; ∂L2*/∂h = (4 – h)v/8 > 0 for 1.78 ≤ h < 2, and ∂L2*/∂h = v/(25/2h3/2) > 0 for 2 ≤ h < 3.

  18. The comparative statics with respect to h are as follows: ∂TE Ŝ/∂h = {(1 + h)1/2 – 1}{h + (2 + h)(1 – (1 + h)1/2)}v/{2h2(1 + h)3/2} > 0 for 1.78 ≤ h < 3.

  19. The comparative statics with respect to h are as follows: ∂TE*/∂h = 0 for 1.78 ≤ h < 2 and ∂TE*/∂h = – v/(2h2) < 0 for 2 ≤ h < 3.

  20. We consider the equilibrium total legal expenditure of litigants as a criterion for measuring economic efficiency. In this study, the equilibrium total legal expenditure of the simultaneous move game is TŜ = p1ŜcŜv + hx2Ŝ = {(1 + h)1/2 – 1}{1 + 2 h – (1 + h)1/2}v/{h(1 + h)}, and that of the endogenous timing game is T* = p1*c*v + hx2* = h(6 – h)v/16 for 1.78 ≤ h < 2 and TE* = {3 h – (2 h)1/2}v/(2 h)3/2 for 2 ≤ h < 3. Comparing the equilibrium total effort levels of the two games, we determine that TŜ < T* for 1.78 ≤ h < 3.

  21. Alternatively, if we assume that the contingent fee in the settlement stage is different from that in the trial stage, the settlement range induced in this study may not change. However, as pointed out by a referee, if the contingent fee for the plaintiff’s lawyer is determined before the settlement stage, and the fee is adopted to both settlement and trial stages, the settlement range may be different.

  22. Baumann and Friehe (2012, 2013) do not explicitly derive this; however, by comparing the settlement ranges of the two games using their models, we find that our result is contrary to theirs.

  23. In the American practice, a plaintiff’s lawyer is hired on a contingent fee basis and a defendant’s lawyer is hired on an hourly fee basis, whereas, in European practice, two litigants hire lawyers under hourly fees (Baik and Kim, 2007, 2021; Baumann and Friehe, 2013).

  24. Consider a litigation contest in which litigants hire lawyers on an hourly fee basis. In the contest, the expected payoff for Player 1 is amended as follows: G1 = p1vthx1 for t > 0, where th is Player 1’s total hourly fee rate. We can then solve for the subgame perfect equilibrium of the game if 1/2 < t < 2. Furthermore, we find that the outcomes of the contest are similar to those determined by Baumann and Friehe (2013). The results are available on request.


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Comparison of expected payoffs for Delegate 1 and expected losses for Player 2 in the second stage of the endogenous timing game.

We sequentially compare Delegate 1’s expected payoffs with Player 2’s expected losses in the second stage of the endogenous timing game. Before we consider the comparison, we note the followers’ participation constraints in the 1 M and 2 M subgames. In the 1 M subgame, Delegate 2’s participation constraint is not violated if ch < 2 (see Eq. 9). In the 2 M subgame, the constraint that Delegate 1 participates under ch > 1/2 (see Eq. 12). Thus, the constraint that the two delegates participate under the trial is 1/2 < ch < 2.

First, we compare the expected payoffs for Delegate 1. Using Lemmas 1, 3 and 4, we obtain the following:

$$\begin{gathered} \pi_{{1}}^{S} < \pi_{{1}}^{{{1}M}} {\text{for}}\quad ch \ne {1}, \hfill \\ \pi_{{1}}^{S} = \pi_{{1}}^{{{1}M}} {\text{for}}\quad ch = { 1}, \hfill \\ \end{gathered}$$

wherein π1S − π11M = – c2h(1 − ch)2v/{4(1 + ch)2}. Furthermore, we find the following:

$$\begin{gathered} \pi_{{1}}^{S} > \pi_{{1}}^{{{2}M}} {\text{for 1}}/{2 } < ch < { 1}, \hfill \\ \pi_{{1}}^{S} = \pi_{{1}}^{{{2}M}} \;{\text{for}}\quad ch = { 1}, \hfill \\ \pi_{{1}}^{S} < \pi_{{1}}^{{{2}M}} {\text{for 1 }} < ch < { 2}, \hfill \\ \end{gathered}$$

wherein π1S − π12M = (1 − ch)(4c2h2 + ch − 1)v/{4ch2(1 + ch)2}.

Next, we compare the expected losses of Player 2. Using Lemmas 1, 3 and 4, we find that:

$$\begin{gathered} L_{{2}}^{S} > L_{{2}}^{{{1}M}} {\text{for 1}}/{2 } < ch < { 1}, \hfill \\ L_{{2}}^{S} = L_{{2}}^{{{1}M}}\;{\text{for}}ch = { 1}, \hfill \\ L_{{2}}^{S} < L_{{2}}^{{{1}M}} {\text{for 1 }} < ch < { 2}, \hfill \\ \end{gathered}$$

wherein L2S − L21M = – ch(1 − ch)(c2h2 − ch − 4)v/{4(1 + ch)2}. Furthermore, we find the following:

$$\begin{gathered} L_{{2}}^{S} > L_{{2}}^{{{2}M}} {\text{for}}\quad ch \ne {1}, \hfill \\ L_{{2}}^{S} = L_{{2}}^{{{2}M}} \;{\text{for}}\quad ch = { 1}, \hfill \\ \end{gathered}$$

wherein L2S − L22M = (1 − ch)2v/{4ch(1 + ch)2}.

Finally, we solve for the subgame perfect equilibria in the second stage of the full game. Delegate 1 and Player 2 have two possible strategies: first or second. First represents choosing the first period, and second, choosing the second period. Table

Table 1 Expected payoffs and expected losses depending on the timing of the effort

1 presents Delegate 1’s expected payoffs and Player 2’s expected losses based on Lemmas 1, 3 and 4. Combining (A1)–(A4) and Table 1, we obtain the subgame perfect equilibria in the second stage:

$$\begin{gathered} \left( {First,Second} \right) {\text{for 1}}/{2 } < ch < { 1}, \hfill \\ \left( {First,First} \right) \, \left( {First,Second} \right) \, \left( {Second,First} \right) \, \left( {Second,Second} \right) {\text{for}}\quad ch = { 1}, \hfill \\ \left( {Second,First} \right) {\text{for 1 }} < ch \le {2} \hfill \\ \end{gathered}$$

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Park, SH. Contingent fees and endogenous timing in litigation contests. Eur J Law Econ 54, 453–473 (2022).

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  • Contingent fee
  • Economic efficiency
  • Endogenous timing
  • Justice
  • Litigation contest

JEL Classification

  • D72
  • K41