Asymmetric Reimbursement and Contingent Fees in Environmental Conflicts: Observable vs. Unobservable Contracts
Abstract
:1. Introduction
1.1. Background Information
1.2. Related Literature
2. Results
2.1. Framework of Analysis
=1/2 for x1 + x2 = 0,
=p1b − x1.
=p2(1 + ab) − ab − x2.
2.2. The Unobservable-Contract Game
- (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 expendx1* = wδ/[(1 − a)(1 + wδ)2(1 + 2wδ)] and x2* = w/[(1 − a)(1 + wδ)2(1 + 2wδ)].
- (c)
- The probability of winning for player 1 is p1* = wδ/(1 + wδ).
- (d)
- The expected payoff for delegate 1 isπ1* = (wv)2/[(1 − a)(1 + wδ)2(1 + 2wδ)].
- (e)
- The expected payoffs for players areG1* = 2(wδ)2/[(1 + wδ)(1 + 2wδ)] andG2* = [(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 expendx1* = (2 + a)w/[2(2 + a + w)2] and x2* = (2 + a)2w/[2(2 + a + w)2].
- (c)
- The probability of winning for player 1 is p1* = w/(2 + a + w).
- (d)
- The expected payoff for delegate 1 isπ1* = w2/[2(2 + a + w)2].
- (e)
- The expected payoffs for players areG1* = (1 + a)w/[2(2 + a + w)] andG2* = [8 + [12 − (2 + w)2]a + 2(1 − w)a2]/[2(2 + a + w)2].
2.3. The Observable-Contract Game
- (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 expendx1** = θ2w/[(1 − aθ)(1 + wθ)2] and x2** = θw/[(1 − aθ)(1 + wθ)2].
- (c)
- The probability of winning for player 1 is p1** = wθ/(1 + wθ).
- (d)
- The expected payoff for delegate 1 isπ1** = w2θ3/[(1 − aθ)(1 + wθ)2].
- (e)
- The expected payoffs for players areG1** = (1 − θ)wθ/[(1 − aθ)(1 + wθ)] andG2** = [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 expendx1** = (2 + a)w/[2(2 + a + w)2] and x2** = (2 + a)2w/[2(2 + a + w)2].
- (c)
- The probability of winning for player 1 is p1** = w/(2 + a + w).
- (d)
- The expected payoff for delegate 1 isπ1** = w2/[2(2 + a + w)2].
- (e)
- The expected payoffs for players areG1** = (1 + a)w/[2(2 + a + w)] andG2** = [8 + [12 − (2 + w)2]a + 2(1 − w)a2]/[2(2 + a + w)2].
3. Discussion
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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 ∂2G2/∂x22 = (∂2p2/∂x22) (1 + ab) < 0. |
8 | Note that x1 + x2 = bw(ab + 1)[1 + (a + 1)b]/[1 + (a + w)2b]. Further, ∂(x1 + x2)/∂b = [1 + 2a(a + 1)(a + w)2b3 + (2a3 + 4(w + 1)a2 + (2w2 + 2w + 3)a + w2)b2 + 2(2a + 1)b]w/[1 + (a + w)2b]2 > 0 for 0 < θ < 1/(2 + a), which implies that the total efforts are increasing in b. Considering ∂(x1 + x2)/∂b and b* > b** for 0 < θ < 1/(2 + a), this study obtains that if for 0 < θ < 1/(2 + a), then x1* + x2* > x1** + x2**. |
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, ∂(x1* + x2*)/∂a > 0, and ∂(x1** + x2**)/∂a > 0 for 0 < θ < 1/(2 + a). This study uses Maple, a computer program, for comparative static analysis. |
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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
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 StylePark, 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