Liability Situations with Successive Tortfeasors

Abstract

Given a tort that involves several tortfeasors, an allocation scheme attributes to each of them that part of the damage that reflects their responsibility. We consider successive torts—i.e., torts that involve a causality chain—and show that simple and intuitive principles, which are well-known in the law of tort, uniquely define an allocation scheme. We show that this scheme incentivizes agents to exhibit a certain level of care, creating an efficient prevention of accidents. We further describe the unique rule according to which a liability situation has to be adjusted after a partial settlement such that incentives to settle early are created.

Mathematical Appendix

1.1 Proofs of Theorems 1 and 2

A liability situation is a triple $\left( N,\textbf{S},\Delta \right)$ where N is the (finite) set of tortfeasors, $\textbf{S}=\left( S_k\right) _{k=1}^m$ is a vector of coalitions $S_k\subseteq N$ with $S_k\cap S_l=\emptyset$ for all $k\ne l$, and $\bigcup _{k=1}^mS_k=N$, and $\Delta =\left( \Delta _k\right) _{k=1}^m\in \mathbb R^m$ is a vector of non-negative damages. An allocation scheme is a map f that maps any liability situation $\left( N,\textbf{S},\Delta \right)$ on a vector $\left( f_i\left( N,\textbf{S},\Delta \right) \right) _{i\in N}\in \mathbb R^N_{\ge 0}$ of compensation payments. For a liability situation $\left( N,\textbf{S},\Delta \right)$ denote by $S^*_k$ the members of $S_k$ who were negligent, let $N^*=\bigcup _{k=1}^mS_k^*$, and let

$$\begin{aligned} \Delta ^*_k={\left\{ \begin{array}{ll} \Delta _k &{} \textrm{if}\, \bigcup _{l\le k}S^*_l\ne \emptyset ,\\ 0 &{} \textrm{otherwise} \end{array}\right. } \end{aligned}$$

be the damages that are caused (directly or indirectly) by at least one negligent agent. The mathematical formulation of the axioms 1–5 is as follows.

• Axiom 1. For any liability situation $\left( N,\textbf{S},\Delta \right)$ and any $i\in N$ it holds that $f_{i}(N,\textbf{S},\Delta ) > 0$ if and only if there is k such that $i\in S_k^*$ and $\sum _{l\ge k}\Delta _l>0$.

• Axiom 2. For any liability situation $\left( N,\textbf{S},\Delta \right)$ it holds that $\sum _{i\in N}f_i\left( N,\textbf{S},\Delta \right) = \sum _{k=1}^m\Delta _k^*$.

• Axiom 3. For any liability situation $\left( N,\textbf{S},\Delta \right)$ and any two agents $i,j\in N$ with $i,j\in S_k^*$ for some $k=1,\ldots ,m$ it holds that $f_i\left( N,\textbf{S},\Delta \right) =f_j\left( N,\textbf{S},\Delta \right)$.

• Axiom 4. If $\left( N,\textbf{S},\Delta \right)$ and $\left( N,\textbf{S},\Delta '\right)$ are such that there is $k\le m$ with $\Delta _{h}'=\Delta _{h}$ for all $h\ge k$, then $f_{i}\left( N,\textbf{S},\Delta \right) =f_{i}\left( N,\textbf{S},\Delta '\right)$ for all $i\in \bigcup _{l=k}^mS_l$.

• Axiom 5. For all liability situations $\left( N,\textbf{S},\Delta \right)$ with $\Delta _k=0$ for some k it holds that

  $$\begin{aligned} f\left( N,\textbf{S},\Delta \right) =f\left( N,\textbf{S}_{-k},\Delta _{-k}\right) , \end{aligned}$$

  where

  $$\begin{aligned} S_{-k}&= \left( S_1,\ldots ,S_{k-1},S_k\cup S_{k+1},\ldots ,S_m\right) \\ \Delta _{-k}&=\left( \Delta _1,\ldots ,\Delta _{k-1},\Delta _{k+1},\ldots ,\Delta _m\right) . \end{aligned}$$

Any liability situation $\left( N,\textbf{S},\Delta \right)$ can be naturally associated with a characteristic function form game $\left( N,v^{N,\textbf{S},\Delta }\right)$ by setting

$$\begin{aligned} v^{N,\textbf{S},\Delta }\left( T\right) = \sum _{k:\bigcup _{l=1}^kS_l^*\subseteq T}\Delta _k^*. \end{aligned}$$

for all $T\subseteq N$ (Dehez and Ferey, 2013). In particular, players who are not negligent are null players in this game.

Proof of Theorem 1.

The proposed allocation scheme f is given by

$$\begin{aligned} f_{i}\left( N,\textbf{S},\Delta \right) = {\left\{ \begin{array}{ll} \frac{\Delta ^*_{1}}{\left| N^*\right| } &{} \mathrm {if\,} i\in N^*,\\ 0 &{} \textrm{otherwise}. \end{array}\right. } \end{aligned}$$

Clearly, f satisfies Axioms 1–3. Suppose there is another allocation scheme g that satisfies all three axioms as well. By Axiom 1 $g_i\left( N,\textbf{S},\Delta \right) =0=f_i\left( N,\textbf{S},\Delta \right)$ for all $i\notin N^*$. If there is at least one $i\in N^*$ then, by Axiom 2, $\sum _{i\in N^*}g_i\left( N,\textbf{S},\Delta \right) = \Delta _1^*=\Delta _1$. Hence, by Axiom 3, $g_i\left( N,\textbf{S},\Delta \right) =\frac{\Delta _1}{\left| N^*\right| }=f_i\left( N,\textbf{S},\Delta \right)$. Hence, f and g coincide. Q.E.D.

Note that the Axioms 1–3 are independent. The allocation scheme that splits the damage equally among all players satisfies Axioms 1 and 3, but not 2. The allocation scheme that assigns 50% of the damage according to our allocation scheme and leaves the remaining damage undistributed satisfies Axioms 1 and 3, but not 2. The allocation scheme that assigns the weighed Shapley value to each player satisfies Axioms 1 and 3, but not 2.

Proof of Theorem 2.

We have to prove that the allocation scheme f defined by

$$\begin{aligned} f_{i}\left( N,\textbf{S},\Delta \right) = \sum _{k:i\in \bigcup _{l=1}^kS_k^*}\frac{\Delta ^*_{k}}{\left| \bigcup _{l=1}^kS_k^*\right| }. \end{aligned}$$

for all $i\in N$ is the only allocation scheme that satisfies axioms 1–5. It can easily be seen that

$$\begin{aligned} f\left( N,\textbf{S},\Delta \right) = Sh\left( N,v^{N,\textbf{S},\Delta }\left( S\right) \right) , \end{aligned}$$

(1)

where Sh is the Shapley value (Shapley, 1953). Hence, Axiom 1–3 follow from the null player property, efficiency, and symmetry of the Shapley value. Axiom 5 holds as the two liability situations $\left( N,\textbf{S},\Delta \right)$ and $\left( N,\textbf{S}_{-k},\Delta _{-k}\right)$ are associated with the same characteristic function form game. Axiom 4 is satisfied because of the strong monotonicity of the Shapley value (Young, 1985) as $v^{N,\textbf{S},\Delta }(S)-v^{N,\textbf{S},\Delta }\left( S\setminus \{i\}\right) = v^{N,\textbf{S},\Delta '}(S)-v^{N,\textbf{S},\Delta '}\left( S\setminus \{i\}\right)$ for all such i.

For the uniqueness of f suppose that there is another allocation scheme g that satisfies the axioms as well. For any liability situation $\left( N,\textbf{S},\Delta \right)$ let $I\left( \Delta \right) =\left| \left\{ k:\Delta _k>0\right\} \right|$. If $I\left( \Delta \right) =0$, then there are no positive damages, and Axiom 1 implies that $g_i\left( N,\textbf{S},\Delta \right) =0=f_i\left( N,\textbf{S},\Delta \right)$ for all $i\in N$. Let $\left( N,\textbf{S},\Delta \right)$ be such that $I\left( \Delta \right) \ge 1$, and that the claim is true for all liability situations $\left( N,\textbf{S}',\Delta '\right)$ with $I\left( \Delta '\right) < I\left( \Delta \right)$. By Axiom 5 we can assume without loss of generality that $\Delta _1>0$. Define $\left( N,\textbf{S},\Delta '\right)$ by $\Delta '_1=0$ and $\Delta '_k=\Delta _k$ for all $k\ge 2$. Then

$$\begin{aligned} g_i\left( N,\textbf{S},\Delta \right) =g_i\left( N,\textbf{S},\Delta '\right) =f_i\left( N,\textbf{S},\Delta '\right) =f_i\left( N,\textbf{S},\Delta \right) \end{aligned}$$

for all $i\in \bigcup _{l=2}^m S_l$ by Axiom 4 and the induction hypothesis. Hence, $g_i\left( N,\textbf{S},\Delta \right) =f_i\left( N,\textbf{S},\Delta \right)$ for all $i\in S_1$ by Axioms 2, and 3. Q.E.D.

For the independence of the axioms note that the allocation scheme $f_i^1\left( N,\textbf{S},\Delta \right) = \sum _{k:i\in \bigcup _{l=1}^k S_l}\frac{\Delta ^*_k}{\left| \bigcup _{l=1}^k S_l\right| }$ satisfies all axioms but Axiom 1. Further, $f^2_i\left( N,\textbf{S},\Delta \right) = \frac{1}{2} f_i\left( N,\textbf{S},\Delta \right)$ satisfies all axioms but Axiom 2. If the Shapley value in Eq. (1) is replaced by a weighted Shapley value (Shapley, 1953b; Kalai and Samet, 1987) one obtains an allocation scheme that satisfies all axioms but Axiom 3. The allocation scheme

$$\begin{aligned} f_i^4\left( N,\textbf{S},\Delta \right) = {\left\{ \begin{array}{ll} \frac{1}{\left| \bigcup _{k=1}^m S_k^*\right| }\sum _{k=1}^m\Delta _k^*, &{} \mathrm {if\,} i\in S_k^* \,\,\textrm{for}\,\,\textrm{some}\,\, k=1,\ldots ,m\\ 0, &{} \textrm{otherwise} \end{array}\right. } \end{aligned}$$

satisfies all axioms but Axiom 4.

The allocation scheme

$$\begin{aligned} f_{i}^5\left( N,\textbf{S},\Delta \right) = {\left\{ \begin{array}{ll} \sum _{k:i\in S_k^*}\frac{\Delta _{k}^*}{\left| S_{k}^*\right| }, &{} \mathrm {if\,} i\in S_k^* \,\,\textrm{for}\,\,\textrm{some}\,\, k=1,\ldots ,m\\ 0, &{} \textrm{otherwise} \end{array}\right. } \end{aligned}$$

satisfies all axioms but Axiom 5.

1.2 Proof of Theorem 3

Let $\left( N,\textbf{S},\Delta \right)$ be a (fixed) liability situation. Let $x_i\in \mathbb R$ be agent i’s level of care and denote by $C_i\left( x_i\right)$ the associated private cost of agent i, where $C_i$ is an increasing function. By $p_k\left( \left( x_i\right) _{i\in S_k}\right)$ denote the probability that group $S_k$ causes damage $\Delta _k$ provided that each $i\in S_k$ chooses $x_i$ as his level of care; let $p_k$ be decreasing in all coordinates. Then the expected social costs of the liability situation are given by

$$\begin{aligned} SC\left( x\right) = \sum _{i\in N}C_i\left( x_i\right) + \sum _{k=1}^m p_k\left( \left( x_i\right) _{i\in S_k}\right) \Delta _k. \end{aligned}$$

(2)

Assume that SC has a unique minimum,⁴ and denote the minimizer of SC by $x^*\in \mathbb R^N$. We interpret $x_i^*$ as the standard of care that applies to agent i; in particular, different standards of care may apply to different agents in the liability situation. The adapted axioms then read as follows.

• Axiom 1$^*$. For any liability situation $\left( N,\textbf{S},\Delta \right)$ and any $i\in N$ it holds that $f_{i}(N,\textbf{S},\Delta ) > 0$ if and only if $x_i<x_i^*$ and $\sum _{k:i\in \bigcup _{l=1}^kS_k}\Delta _k>0$.

• Axiom 2$^*$. For any liability situation $\left( N,\textbf{S},\Delta \right)$ it holds that $\sum _{i\in N}f_i\left( N,\textbf{S},\Delta \right) = \sum _{k=1}^m\Delta _k^*$, where

  $$\begin{aligned} \Delta ^*_k={\left\{ \begin{array}{ll} \Delta _k &{} \mathrm {if\, there\, is\,} i\in \bigcup _{l\le k}S_l \mathrm {\,with\,} x_i<x_i^*,\\ 0 &{} \mathrm {otherwise.} \end{array}\right. } \end{aligned}$$

Proof of Theorem 3.

We first show that $x^*$ is a Nash equilibrium. It is obvious that no agent has an incentive to choose $x_i>x_i^*$. Suppose that all agents $j\in N\setminus \{i\}$ chose $x_j=x_j^*$ and assume that $x_i<x_i^*$. Then agent i’s expected payment is

$$\begin{aligned} C_i\left( x_i\right) + \sum _{k:i\in \bigcup _{l=1}^kS_l}p_k\left( \left( x_j\right) _{j\in S_k}\right) \Delta _k. \end{aligned}$$

as i has to recover the full damage alone. As $x^*$ is the unique minimizer of the social cost function in Eq. (2) it holds that

$$\begin{aligned} SC\left( x^*\right)= & {} \sum _{j\in N}C_j\left( x_j^*\right) + \sum _{k=1}^m p_k\left( \left( x^*_j\right) _{j\in S_k}\right) \Delta _k\\< & {} \sum _{j\ne i}C_j\left( x_j^*\right) + C_i\left( x_i\right) + \sum _{k:i\notin \bigcup _{l=1}^kS_l}p_k\left( \left( x^*_j\right) _{j\in S_k}\right) \Delta _k \\{} & {} \qquad + \sum _{k:i\in \bigcup _{l=1}^kS_l}p_k\left( x_i,\left( x^*_j\right) _{j\in S_k\setminus \{i\}}\right) \Delta _k \end{aligned}$$

and therefore

$$\begin{aligned} C_i\left( x_i\right) + \sum _{k:i\in \bigcup _{l=1}^kS_l}p_k\left( x_i,\left( x_j^*\right) _{j\in S_k\setminus \{i\}}\right) \Delta _k> & {} C_i\left( x_i^*\right) + \sum _{k:i\in \bigcup _{l=1}^k S_l} p_k\left( \left( x^*_j\right) _{j\in S_k}\right) \Delta _k\\\ge & {} C_i\left( x_i^*\right) . \end{aligned}$$

Hence, $x_i$ imposes higher expected costs on i than $x_i^*$, so $x^*$ is a Nash Equilibrium.

We now show that $x^*$ is, in fact, the only Nash Equilibrium. For this purpose let x be vector of care levels, and assume that x is a Nash Equilibrium. Let A be the set of agents who choose $x_j=x_j^*$, and let $B=N\setminus A$ the set of agents who choose $x_i<x_i^*$. (Recall that no agent would choose $x_j>x_j^*$ in a Nash Equilibrium.) Then, by Axioms 1$^*$ and 2$^*$, the total expected costs that the agents in B have to bear are

$$\begin{aligned} \sum _{i\in B}C_i\left( x_i\right) + \sum _{k:\bigcup _{l=1}^kS_l\cap B\ne \emptyset }p_k\left( \left( x_j^*\right) _{j\in A\cap S_k},\left( x_i\right) _{i\in B\cap S_k}\right) \Delta _k. \end{aligned}$$

(Recall that $\Delta _k=\Delta _k^*$ for all k with $\bigcup _{l=1}^kS_l\cap B\ne \emptyset$.) By the definition of $x^*$ we have $SC\left( x^*\right) <SC\left( x\right)$ and therefore,

$$\begin{aligned}{} & {} \sum _{i\in B}C_i\left( x_i\right) + \sum _{k:\bigcup _{l=1}^kS_l\cap B\ne \emptyset }p_k\left( \left( x_j^*\right) _{j\in A\cap S_k},\left( x_i\right) _{i\in B\cap S_k}\right) \Delta _k \\{} & {} \qquad > \sum _{i\in B}C_i\left( x_i^*\right) + \sum _{k:\bigcup _{l=1}^kS_l\cap B\ne \emptyset }p_k\left( \left( x_i^*\right) _{i\in S_k}\right) \Delta _k\\{} & {} \qquad \ge \sum _{i\in B}C_i\left( x_i^*\right) . \end{aligned}$$

Since the aggregated expected costs of the agents in B are strictly greater if they choose $\left( x_i^*\right) _{i\in B}$ than if they choose $\left( x_j^*\right) _{j\in B}$, there must be at least one agent $i\in B$ whose private expected costs are strictly greater from choosing $x_i$ than from choosing $x_i^*$. Hence, x cannot be a Nash Equilibrium. Q.E.D.

1.3 Proof of Theorem 4

A settlement amendment scheme is a vector $\left( r_k\right) _{k=1}^m$ with $r_k\ge 0$ for all $k=1,\ldots ,m$ and $\sum _{k=1}^mr_k=1$. If agent i settles and pays an amount $\theta$, the remaining agents face the new liability situation $\left( N\setminus \{i\},\textbf{S}',\Delta '\right)$ with

$$\begin{aligned} S'_k= & {} S_k\setminus \{i\}\\ \Delta '_k= & {} \Delta _k-r_k\theta . \end{aligned}$$

The settlement amendment scheme r promotes settl ements if

$$\begin{aligned} f_j\left( N\setminus \{i\},\textbf{S}',\Delta '\right) > f_j\left( N,\textbf{S},\Delta \right) \end{aligned}$$

for all $j\ne i$ if and only if $\theta <f_i\left( N,\textbf{S},\Delta \right)$.

Proof of Theorem 4.

The proposed settlement amendment scheme is given by

$$\begin{aligned} r_k ={\left\{ \begin{array}{ll} \frac{\Delta _k^*}{\left| \bigcup _{l=1}^kS_l^*\right| }\frac{1}{f_i\left( N,\textbf{S},\Delta \right) } &{} \textrm{if}\, i\in \bigcup _{l=1}^kS_l'\\ 0 &{} \mathrm {otherwise.} \end{array}\right. } \end{aligned}$$

It is easy to verify that r has the desired property; in fact, this follows from the consistency of the Shapley value (Hart and Mas-Colell, 1989).

Let r promote settlements and let i be an agent with $f_i\left( N,\textbf{S},\Delta \right) >0$ who settles at $\theta$. Then $f_j\left( N\setminus \{i\},\textbf{S}',\Delta '\right) \le f_j\left( N,\textbf{S},\Delta \right)$ for all $j\ne i$ if and only if $\theta \ge f_i\left( N,\textbf{S},\Delta \right)$. In case that $\theta = f_i\left( N,\textbf{S},\Delta \right)$ one further obtains

$$\begin{aligned} \sum _{j\ne i}f_j\left( N,\textbf{S}',\Delta '\right) = \sum _{k=1}^m\Delta _k^*-\theta = \sum _{k=1}^m\Delta _k^*-f_i\left( N,\textbf{S},\Delta \right) = \sum _{j\ne i}f_j\left( N,\textbf{S},\Delta \right) . \end{aligned}$$

Hence, in this case it must hold that $f_j\left( N\setminus \{i\},\textbf{S}',\Delta '\right) = f_j\left( N,\textbf{S},\Delta \right)$ for all $j\ne i$. Using the definitions of f and $\left( N\setminus \{i\},\textbf{S}',\Delta '\right)$ this leads to

$$\begin{aligned} \sum _{k:j\in \bigcup _{l=1}^kS_l}\frac{\Delta _k^*}{\left| \bigcup _{l=1}^kS_l\right| }= & {} f_j\left( N,\textbf{S},\Delta \right) = f_j\left( N\setminus \{i\},\textbf{S}',\Delta '\right) \\= & {} \sum _{k:j\in \bigcup _{l=1}^kS_l} \frac{\Delta _k^*-r_kf_i\left( N,\textbf{S},\Delta \right) }{\left| \bigcup _{l=1}^kS_l\setminus \{i\}\right| } \end{aligned}$$

or equivalently

$$\begin{aligned} \sum _{k:j\in \bigcup _{l=1}^kS_l}\frac{r_kf_i\left( N,\textbf{S},\Delta \right) }{\left| \bigcup _{l=1}^kS_l\setminus \{i\}\right| } = \sum _{k:j\in \bigcup _{l=1}^kS_l}\left( \frac{\Delta _k^*}{\left| \bigcup _{l=1}^kS_l\setminus \{i\}\right| }-\frac{\Delta _k^*}{\left| \bigcup _{l=1}^kS_l\right| }\right) \end{aligned}$$

(3)

for all $j\ne i$. If $S_1\ne \{i\}$ these are m linear equations that are linearly independent (as the system is triangular and all diagonal entries are strictly positive), so the solution is unique. If $S_1=\{i\}$ the requirement that $\sum _{k=1}^mr_k=1$ is another constraint that is linearly independent of the $m-1$ independent (non-trivial) equations in (3). Hence, in both cases the solution is unique. As the proposed settlement amendment scheme r solves this linear equation system, it is the only solution. Q.E.D.