# Asymmetric Reimbursement and Contingent Fees in Environmental Conflicts: Observable vs. Unobservable Contracts

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Background Information

#### 1.2. Related Literature

## 2. Results

#### 2.1. Framework of Analysis

_{1}denote delegate 1’s effort level and x

_{2}player 2’s effort level, the probability of player 1 winning v is p

_{1}. According to winner-takes-all, the probability that player 2 wins v is p

_{2}= 1 − p

_{1}. Now, the Tullock-form litigation success function for player 1 is represented by

_{1}(x

_{1}, x

_{2}) = wx

_{1}/(wx

_{1}+ x

_{2}) for x

_{1}+ x

_{2}> 0

=1/2 for x

_{1}+ x

_{2}= 0,

_{1}is:

_{1}= p

_{1}(b − x

_{1}) + (1 − p

_{1})( − x

_{1})

=p

_{1}b − x

_{1}.

_{1}≥ 0. Next, we consider the expected payoffs of each player. Player 1’s expected payoff G

_{1}is given by:

_{1}= p

_{1}[1 − (1 − a)b]

_{2}is given by:

_{2}= p

_{2}(1 − x

_{2}) + (1 − p

_{2})( − ab − x

_{2})

=p

_{2}(1 + ab) − ab − x

_{2}.

#### 2.2. The Unobservable-Contract Game

_{1}with knowledge of b for him and without observing x

_{2}, and player 2 chooses x

_{2}without observing b and x

_{1}.

_{1}and player 2’s decision on x

_{2}, respectively. Then, we consider player 1’s decision on b.

_{1}to maximize π

_{1}given in (1), taking b and x

_{2}as given. Solving for the maximization problem yields the first-order condition: bwx

_{2}/(wx

_{1}+ x

_{2})

^{2}− 1 = 0. Since the litigation success function is concave, the second-order condition is satisfied. From the first-order condition, delegate 1’s best response function x

_{1}(b, x

_{2}) is obtained:

_{1}(b, x

_{2}) = [−x

_{2}+ (bwx

_{2})

^{1/2}]/w.

_{2}to maximize G

_{2}given in (3), taking x

_{1}as given. Solving for the maximization problem yields the first-order condition: (ab + 1)wx

_{1}/(wx

_{1}+ x

_{2})

^{2}− 1 = 0.7 Then, player 2’s best response function x

_{2}(x

_{1}) is

_{2}(x

_{1}) = − wx

_{1}+ [wx

_{1}(ab + 1)]

^{1/2}.

_{2}as given, player 1 selects b to maximize G

_{1}(x

_{1}, x

_{2}) given in (2), using backward induction about x

_{1}(c, x

_{2}) for b. That is, player 1 seeks to maximize

_{1}(b, x

_{2}) = wx

_{1}(b, x

_{2})[1 − (1 − a)b]/[wx

_{1}(b, x

_{2}) + x

_{2}]

_{2}as given. Solving for the first-order condition, we obtain

_{2}− 2(1 − a)b(bwx

_{2})

^{1/2}= 0.

^{*}) and the equilibrium effort levels, (x

_{1}

^{*}, x

_{2}

^{*}), for the game. Without loss of generality, let x

_{1}

^{*}= δx

_{2}

^{*}, where δ is a positive value to be solved for below. Then, we derive (8) and (9) by using (4) and (7):

_{2}

^{*}= b

^{*}w/(1 + wδ)

^{2}

^{*}]x

_{2}

^{*}− 2(1 − a)b

^{*}(b

^{*}wx

_{2}

^{*})

^{1/2}= 0.

_{1}

^{*}= δx

_{2}

^{*}and (8) and (9), we obtain b

^{*}, x

_{1}

^{*}, and x

_{2}

^{*}:

_{1}

^{*}= wδ/[(1 − a)(1 + wδ)

^{2}(1 + 2wδ)],

_{2}

^{*}= w/[(1 − a)(1 + wδ)

^{2}(1 + 2wδ)],

^{*}= 1/[(1 − a)(1 + 2wδ)].

_{1}

^{*}= δx

_{2}

^{*}:

^{1/2}]/[2w[(1 + 8w(1 − a))

^{1/2}− 1 + 2a]],

**Lemma**

**1.**

- (i)
- if 0 < a < [−(3 + 2w) + [(1 + 2w)
^{2}+ 16w)]^{1/2}]/2,- (a)
- The delegation contract is b
^{*}= 1/[(1 − a)(1 + 2wδ)], - (b)
- Delegate 1 and player 2 expendx
_{1}^{*}= wδ/[(1 − a)(1 + wδ)^{2}(1 + 2wδ)] and x_{2}^{*}= w/[(1 − a)(1 + wδ)^{2}(1 + 2wδ)]. - (c)
- The probability of winning for player 1 is p
_{1}^{*}= wδ/(1 + wδ). - (d)
- The expected payoff for delegate 1 isπ
_{1}^{*}= (wv)^{2}/[(1 − a)(1 + wδ)^{2}(1 + 2wδ)]. - (e)
- The expected payoffs for players areG
_{1}^{*}= 2(wδ)^{2}/[(1 + wδ)(1 + 2wδ)] andG_{2}^{*}= [(1 + wδ)[1 − a + (2 − 3a)wδ] − w]/[(1 − a)(1 + wδ)^{2}(1 + 2wδ)].

- (ii)
- if [−(3 + 2w) + [(1 + 2w)
^{2}+ 16w)]^{1/2}]/2 ≤ a < 1,- (a)
- The delegation contract is b
^{*}= 1/2, - (b)
- Delegate 1 and player 2 expendx
_{1}^{*}= (2 + a)w/[2(2 + a + w)^{2}] and x_{2}^{*}= (2 + a)^{2}w/[2(2 + a + w)^{2}]. - (c)
- The probability of winning for player 1 is p
_{1}^{*}= w/(2 + a + w). - (d)
- The expected payoff for delegate 1 isπ
_{1}^{*}= w^{2}/[2(2 + a + w)^{2}]. - (e)
- The expected payoffs for players areG
_{1}^{*}= (1 + a)w/[2(2 + a + w)] andG_{2}^{*}= [8 + [12 − (2 + w)^{2}]a + 2(1 − w)a^{2}]/[2(2 + a + w)^{2}].

^{2}+ 16w)]

^{1/2}]/2, then b

^{*}is less than one half; otherwise, b

^{*}= 1/2. Specifically, (ii) of Lemma 1 says if w ≤ 1, then b

^{*}= 1/2. This implies that if player 1’s objective merit is less than player 2’s, then player 1 pays half of the reward to delegate 1.

#### 2.3. The Observable-Contract Game

_{1}with knowledge of b for him and without observing x

_{2}and player 2 chooses x

_{2}with knowledge of b and without observing x

_{1}. Working backwards, this study solves for a subgame-perfect equilibrium of the two-stage game.

_{1}. The maximization problem derives delegate 1′s best-response function:

_{1}(b) = b

^{2}w(1 + ab)/[1 + b(a + w)

^{2}].

_{2}. Then, we find the best-response function for player 2:

_{2}(b) = bw(1 + ab)

^{2}/[1 + b(a + w)

^{2}].

_{1}(b) = bw[1 − (1 − a)b]/[1 + b(a + w)].

^{**}= [− (1 − a) + [(1 − a)(1 + w)]

^{1/2}]/[(1 − a)(a + w)].

^{**}) and the equilibrium effort levels, (x

_{1}

^{**}, x

_{2}

^{**}), for the observable-contract game. Let x

_{1}

^{**}= θx

_{2}

^{**}, where θ is a positive parameter. Then, using (14), (15), and (17), we derive

^{**}= θ/(1 − aθ),

_{1}

^{**}= θ

^{2}w/[(1 − aθ)(1 + wθ)

^{2}],

_{2}

^{**}= θw/[(1 − aθ)(1 + wθ)

^{2}].

_{1}

^{*}= θx

_{2}

^{*}:

^{1/2}]/[(1 + aw) + a[(1 + a)(1 + 2a + w)]

^{1/2}],

**Lemma**

**2.**

- (i)
- if a < [−(3 + w) + [(1 + w)(9 + w)]
^{1/2}]/2,- (a)
- The delegation contract is b
^{**}= θ/(1 − aθ), - (b)
- Delegate 1 and player 2 expendx
_{1}^{**}= θ^{2}w/[(1 − aθ)(1 + wθ)^{2}] and x_{2}^{**}= θw/[(1 − aθ)(1 + wθ)^{2}]. - (c)
- The probability of winning for player 1 is p
_{1}^{**}= wθ/(1 + wθ). - (d)
- The expected payoff for delegate 1 isπ
_{1}^{**}= w^{2}θ^{3}/[(1 − aθ)(1 + wθ)^{2}]. - (e)
- The expected payoffs for players areG
_{1}^{**}= (1 − θ)wθ/[(1 − aθ)(1 + wθ)] andG_{2}^{**}= [1 − aθ(1 + wθ)^{2}]/[(1 − aθ)(1 + wθ)^{2}].

- (ii)
- if [−(3 + w) + [(1 + w)(9 + w)]
^{1/2}]/2 ≤ a < 1,- (a)
- The delegation contract is b
^{**}= 1/2, - (b)
- Delegate 1 and player 2 expendx
_{1}^{**}= (2 + a)w/[2(2 + a + w)^{2}] and x_{2}^{**}= (2 + a)^{2}w/[2(2 + a + w)^{2}]. - (c)
- The probability of winning for player 1 is p
_{1}^{**}= w/(2 + a + w). - (d)
- The expected payoff for delegate 1 isπ
_{1}^{**}= w^{2}/[2(2 + a + w)^{2}]. - (e)
- The expected payoffs for players areG
_{1}^{**}= (1 + a)w/[2(2 + a + w)] andG_{2}^{**}= [8 + [12 − (2 + w)^{2}]a + 2(1 − w)a^{2}]/[2(2 + a + w)^{2}].

## 3. Discussion

^{*}with b

^{**}in the two environmental conflicts. Further, considering the bounded reimbursement rate (b ≤ 1/2), we obtain: if 0 < θ < 1/(2 + a), then b

^{*}> b

^{**}; and if 1/(2 + a) ≤ θ < 1, then b

^{*}= b

^{**}= 1/2. It is obvious that if b

^{*}> b

^{**}, then x

_{1}

^{*}+ x

_{2}

^{*}> x

_{1}

^{**}+ x

_{2}

^{**}; and if b

^{*}= b

^{**}, then x

_{1}

^{*}+ x

_{2}

^{*}= x

_{1}

^{**}+ x

_{2}

^{**}.8

**Proposition**

**1.**

_{1}= x

_{2}. Then, the winning probability of the plaintiff is w/(1 + w). The equilibrium winning probability for the plaintiff being close to w/(1 + w) makes the fairness of the contest higher [10,11,12]. For example, when w = 1.2, if the game is fair, the probability that player 1 wins should be 54.5%. Using Lemmas 1 and 2, we obtain that w/(1 + w) − p

_{1}

^{*}= w(1 − δ)/[(1 + w)(1 + wδ)] > 0 and w/(1 + w) − p

_{1}

^{**}= w(1 − θ)/[(1 + w)(1 + wθ)] > 0. With a smaller gap, a fairer outcome is reached in this environmental conflict. That is, if p

_{1}

^{*}> p

_{1}

^{**}or δ > θ, then the unobservability contract increases the fairness of the outcome, and vice versa. Using (13) and (21), we obtain that if 0 < θ < 1/(2 + a), then δ > θ; and if 1/(2 + a) ≤ θ < 1, then δ = θ.

**Proposition**

**2.**

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Notes

1 | In the United States, only the remuneration paid by the citizens to the attorney is subject to asymmetric reimbursement, excluding office costs such as postage and fact-finding costs of various documents. Ref. [1] describes it as follows. “Reimbursement could include repaying the cost for discovery, investigation, court costs, and support staffs, but generally is only awarded for attorney’s fees”. |

2 | This paper is narrowly focused on a specific application of contract theory, environmental contests, instead of being broader in nature by creating a unified theory of contracts. |

3 | There are various reasons why plaintiffs hire lawyers under contingent fee [9,13,14,15,16]. In most civil cases, defendants are corporations, such as insurance companies, while plaintiffs are individuals. Accordingly, plaintiffs with less litigation experience have a greater incentive to prevent moral hazard of their attorneys than defendants and are more prone to be confronted with liquidity constraints than defendants. This is also why it is relatively easy for the plaintiffs to bring into line the contingent fee formula at an assured percentage of the compensation for damage, unlike the defendants [14]. On the other hand, it is common for the defendants to have in-house legal resources to invest in civil disputes, unlike the plaintiffs. |

4 | In the United States, the maximum feasible percentage the client can give the attorney is often dictated by legal restrictions on the contingent fee (which prohibit fees exceeding 50 percent). |

5 | As [4] mentioned, player 2 has more information about the case than player 1. Thus, Ref. [4] assumes that w is less than 1. This study assumes that the asymmetry of information is resolved by player 1 hiring delegate 1. Accordingly, this study considers w as player 2’s degree of fault. Thus, w can be greater than 1. |

6 | See [8] for a detailed description of how the delegation contests can be solved with unobservable contracts. |

7 | The second-order condition is satisfied with ∂ ^{2}G_{2}/∂x_{2}^{2} = (∂^{2}p_{2}/∂x_{2}^{2}) (1 + ab) < 0. |

8 | Note that x _{1} + x_{2} = bw(ab + 1)[1 + (a + 1)b]/[1 + (a + w)^{2}b]. Further, ∂(x_{1} + x_{2})/∂b = [1 + 2a(a + 1)(a + w)^{2}b^{3} + (2a^{3} + 4(w + 1)a^{2} + (2w^{2} + 2w + 3)a + w^{2})b^{2} + 2(2a + 1)b]w/[1 + (a + w)^{2}b]^{2} > 0 for 0 < θ < 1/(2 + a), which implies that the total efforts are increasing in b. Considering ∂(x_{1} + x_{2})/∂b and b^{*} > b^{**} for 0 < θ < 1/(2 + a), this study obtains that if for 0 < θ < 1/(2 + a), then x_{1}^{*} + x_{2}^{*} > x_{1}^{**} + x_{2}^{**}. |

9 | Furthermore, the equilibrium contingent fees in the two games increase in a, which makes the equilibrium total effort levels increase: ∂b ^{*}/∂a > 0, ∂b^{**}/∂a > 0, ∂(x_{1}^{*} + x_{2}^{*})/∂a > 0, and ∂(x_{1}^{**} + x_{2}^{**})/∂a > 0 for 0 < θ < 1/(2 + a). This study uses Maple, a computer program, for comparative static analysis. |

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**MDPI and ACS Style**

Park, S.-H.; Settle, C.E.
Asymmetric Reimbursement and Contingent Fees in Environmental Conflicts: Observable vs. Unobservable Contracts. *Games* **2023**, *14*, 55.
https://doi.org/10.3390/g14040055

**AMA Style**

Park S-H, Settle CE.
Asymmetric Reimbursement and Contingent Fees in Environmental Conflicts: Observable vs. Unobservable Contracts. *Games*. 2023; 14(4):55.
https://doi.org/10.3390/g14040055

**Chicago/Turabian Style**

Park, Sung-Hoon, and Chad E. Settle.
2023. "Asymmetric Reimbursement and Contingent Fees in Environmental Conflicts: Observable vs. Unobservable Contracts" *Games* 14, no. 4: 55.
https://doi.org/10.3390/g14040055