2.1 Baseline model and manipulation-proofness

Let $\mathbb{P}$ be a probability measure and let the risk $X$ be a random variable that has some mass at $0$ and is continuous on $(0, M]$ , where it has full support. The probability mass is the probability that no accident occurs. Let the function $c\,:\,[0, M] \rightarrow \mathbb{R}_+$ be the administrative cost of respecting the insurance contract and let $Y\,:\,[0, M]\rightarrow [0, M]$ be the indemnity schedule.Footnote ² The amount $H\geq 0$ denotes the price (premium) of the contract while $W_0\gt 0$ is the initial wealth of the insured and $\rho \geq 0$ is the loading factor. As usual, the function $u\,:\,\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is a twice-differentiable and strictly concave Bernoulli utility function satisfying Inada conditions.

The game proceeds as follows:

1. Stage 1 the insured buys the insurance contract $Y$ at price $H$ ;

2. Stage 2 the loss $x\in [0,M]$ occurs (Nature moves);

3. Stage 3 the insured observes the loss and decides to take hidden action $z\in [0, M - x]$ to augment the damages andFootnote ³

4. Stage 4 the contract is implemented without renegotiation.

The insured’s terminal wealth is therefore

\begin{equation*}W\,:\!=\,W_0 - H - x - z -g(z)+Y(x+z). \end{equation*}

The solution concept is a weak Perfect Bayesian equilibrium (Mas-Colell et al., Reference Mas-Colell, Whinston and Green1995) where we assume that the insured takes the insurer’s favored action whenever indifferent.

By backward induction, the optimal contract solves the program:

(Problem I) \begin{align} \sup _{H \geq 0, Y \in B_+(\mathcal{B}([0,M]))} & \int u(W_0 - H - x - z^*(x) -g(z^*(x))+Y(x+z^*(x)))\,\textrm{d} \mathbb{P}\\[-4pt]\nonumber \end{align}

(LL) \begin{align} \textrm{s.t.}\, & \,0 \leq Y \\[-4pt]\nonumber \end{align}

(B) \begin{align} &\, \forall x,\, Y(x + z^*(x)) \leq x +z^*(x) \\[-4pt]\nonumber \end{align}

(PC) \begin{align} & (1+\rho )\int Y(x+z^*(x))+c(Y(x+z^*(x))) \,\textrm{d} \mathbb{P} \leq H \\[-4pt] \nonumber\end{align}

(IC) \begin{align} & \forall x\, z^*(x) \in \arg \max _z \left\{ Y(x+z)- z -g(z) \right \} \\[10pt] \nonumber\end{align}

where (LL) is the insured’s limited liability constraint, (B) is the boundedness constraint stating that the insurer will never pay more than the observed loss, (PC) is the insurer’s participation constraint, (IC) is the insured’s incentive compatibility constraint and the function

\begin{align*} g(z)= \begin{cases} +\infty &\text{ if } z\lt 0\\[3pt] \beta z &\text{ if } z\geq 0 \end{cases} \end{align*}

represents an extra cost of inflicting damage (Bob buying a sledgehammer).

Let us start with a handy assumption which is common in the literature:

Assumption 1. We assume throughout that feasible contracts are non-decreasing and upper semi-continuous functions: for every $x_0\in [0, M]$ it is

\begin{equation*}\limsup _{x\rightarrow x_0} Y(x)=Y(x_0).\end{equation*}

Of course, this implies that the optimal contract $Y$ is non-decreasing and upper semi-continuous. Assumption1 is standard and is not discussed further. The next statement is standard and will not be proved:

When the loading factor $\rho$ is zero, then Lemma1 states that the contract must be sold at an actuarially fair price.

Let us now inspect the incentive compatibility constraint (IC). This constraint is an optimization program which is ill-behaved. This is because we do not know at this level of generality if the optimal contract $Y$ is continuous. Moreover, as discussed in Section 1.1, we cannot assume away the possibility that $Y$ is discontinuous. We seem to be in trouble now because we cannot differentiate the objective function in (IC) and thus cannot use standard first-order conditions to characterize the (set of) optimal manipulations. Fortunately, the optimization problem of constraint (IC) is more straightforward to characterize than it looks at first sight.

Let $V_Y\,:\,\mathbb{R} \rightarrow \mathbb{R}$ be the value function of the manipulation stage of the game so that

\begin{align*} V_Y(x)= Y\!\left(x+z^*(x)\right)- z^*(x) -g\!\left(z^*(x)\right) \end{align*}

for

\begin{align*} z^*(x) \in \sigma _Y(x)=\arg \max _z \left \{ Y(x+z)- z -g(z) \right \}, \end{align*}

where $\sigma _Y\,:\,[0, M]\rightrightarrows [0, M]$ denotes the optimal choice correspondence of the manipulation stage of the game induced by contract $Y$ .Footnote ⁴

Proof. We first show that $V_Y$ is continuous. Suppose, by contradiction, that there exists a $x_0\in (0, M]$ such that $\lim _{x\uparrow x_0}V_Y(x)\lt V_Y(x_0)$ . We can find an $\varepsilon \gt 0$ small such that for $x=x_0-\varepsilon$ we have $V_Y(x_0)-V_Y(x)\gt (1+\beta )\varepsilon$ . Letting $z^*(x_0)\in \sigma (x_0)$ and setting $z= z^*(x_0)+\varepsilon$ we obtain

\begin{align*} V_Y(x)&\geq Y(x +z)-z-g(z)\\ &=Y(x_0-\varepsilon +z^*(x_0)+\varepsilon )-z^*(x_0)-\varepsilon -g(z^*(x_0)+\varepsilon )\\ &=Y(x_0+z^*(x_0))-z^*(x_0)-\varepsilon - \beta (z^*(x_0)+\varepsilon )\\ &=V_Y(x_0)-(1+\beta )\varepsilon \\ &\gt V_Y(x), \end{align*}

an absurd. Lipschitzianity follows from the same argument assuming the contradiction hypothesis that there are $x,x^{\prime}\in [0, M]$ such that $\vert V_Y(x^{\prime})- V_Y(x)\vert \gt (1+\beta )\vert x^{\prime}-x\vert$ .

Fig. 2 depicts the value function $V_Y$ of the manipulation stage of the game for a given discontinuous contract $Y(x)=(x - \delta ){\unicode{x1D7D9}}_{\{x \geq b\}}$ , where $b \in (0, M)$ , $\delta \in (0, b)$ and $\beta \gt 0$ .Footnote ⁵ In the figure, manipulations always happen in the segment $(a,b)$ , where $a=(\delta +\beta b)/(1+\beta )$ . Clearly, we have

\begin{align*} V_Y(x)= \begin{cases} b-\delta -(1+\beta )x &\text{ if } x\in (a, b)\\[3pt] Y(x) &\text{otherwise}, \end{cases} \end{align*}

and the maximum Lipschitz constant of $V_Y$ is pinned down by $\beta$ .

Figure 2

The value function $V_Y$ of the manipulation stage of the game for $Y(x)=(x - \delta ){\unicode{x1D7D9}}_{\{x \geq b\}}$ for $b\in (0,M)$ , $\delta \in (0,b)$ and $\beta \gt 0$ .

Figure 3

Straight deductibles and constant retention contracts.

Since $V_Y \geq Y$ and $V_Y(x)\gt x$ implies that there is an $x^{\prime}$ such that $ Y(x^{\prime})\gt x^{\prime}$ , it is easy to prove that $V_Y(x)\geq x$ for some $x\in [0,M]$ if and only if $Y(x^{\prime})\geq x^{\prime}$ for some $x^{\prime}\in [0,M]$ . Thus, $V_Y$ satisfies all boundedness constraints whenever $Y$ does. The next Lemma states this observation formally; its proof is omitted since it is immediate.

We are now ready to prove our main result. We say that a contract $Y$ is manipulation-proof if for every $x\in [0,M]$ it is $0 \in \sigma _Y(x)$ .

Proof. Suppose, by contraposition, that $Y$ is optimal but that there exists a set $\chi \subset [0,M]$ such that simultaneously $\mathbb{P}[X\in \chi ]\gt 0$ and for every $x\in \chi$ it is $0\notin \sigma _Y(x)$ . Let $V_Y(x)$ be the value function of the manipulation problem defined by $Y$ , and consider now the alternative contract $(\overline{H}, \overline{Y})$ where $\overline{Y}=V_Y$ . That is, the new indemnity schedule gives realization-by-realization of the same final reimbursement as the original one after manipulations. Since $Y$ is optimal by assumption, it holds that $0\leq Y \leq X$ and $\overline{Y}$ satisfy all the boundedness constraints by Lemma3.

We now verify that the new indemnity schedule $\overline{Y}$ is manipulation-proof. By Assumption1 and Lemma2, $\overline{Y}$ is non-decreasing and Lipschitz with constant $1+\beta$ . By Lipschitzianity, for any two $x,x^{\prime}\in [0, M]$ such that $x\lt x^{\prime}$ we have

\begin{equation*}\overline {Y}(x^{\prime})-\overline {Y}(x) \leq (1+\beta )(x^{\prime}-x),\end{equation*}

or equivalently

\begin{equation*}\overline {Y}(x) \geq \overline {Y}(x^{\prime}) -(1+\beta )(x^{\prime}-x).\end{equation*}

The last inequality implies that for every realization $x\in [0,M]$ we have

\begin{equation*}0 \in \sigma _{\overline {Y}}(x)=\arg \max _z\left \{\overline {Y}(x+z) -z-g(z) \right \}.\end{equation*}

That is, $\overline{Y}$ does not induce manipulations and gives realization-by-realization of the same coverage as $Y$ .

The new contract strictly dominates the original one because

\begin{align*} \int Y\left (x+ z^*(x)\right )+c\left (Y\left (x+z^*(x)\right )\right ) \,\textrm{d} \mathbb{P} \gt \int \overline{Y}(x) + c\left (\overline{Y}(x)\right ) \,\textrm{d} {\mathbb{P}} \end{align*}

and the price $\overline{H}$ of $\overline{Y}$ is strictly smaller than the price $H$ of $Y$ .

We can understand Theorem1 as stating that since the insurer fully prices arson-type risks, contracts that induce arson-type actions will never be offered in equilibrium. Indeed, all extra damage due to arson-type actions must be priced at a higher premium. Of course, the insured will prefer the cheapest contract as he receives realization-by-realization of the same final amount under both contracts. Or, from Bob’s perspective, he was offered two contracts offering the same protection. An expensive one that allows him to take a sledgehammer to his bike and a cheaper one that does not. Bob, being a rational person, chooses the cheaper option.

Characterization.

Corollary1 implies that the family of contracts

\begin{equation*} \{Y\in B_+(\mathcal {B}([0,M]))\,:\,Y(x)=V_Y(x)\}\end{equation*}

consists of functions that are a.e. differentiable.Footnote ⁶ We can rewrite (Problem I) as

(Problem S) \begin{align} \max _{H \geq 0, Y \in B_+(\mathcal{B}([0,M]))} & \int u (W_0 - H - x+Y(x))\,\textrm{d} {\mathbb{P}} \\[-4pt] \nonumber \end{align}

(S1) \begin{align} \textrm{s.t.}\, & \,0 \leq Y \leq X \\[-4pt] \nonumber \end{align}

(S2) \begin{align} & \textrm{slope}(Y)\leq 1+ \beta \\[-4pt] \nonumber \end{align}

(S3) \begin{align} & (1+\rho )\int Y(x)+c(Y(x)) \,\textrm{d} \mathbb{P} = H \\[10pt] \nonumber\end{align}

under the implicit assumption that $Y\in C^0[0,M]$ is a.e. differentiable.Footnote ⁷ Notice how this problem, as rewritten, is almost identical to the problem studied in Spaeter & Roger (Reference Spaeter and Roger1997), except for the extra constraint $\textrm{slope}(Y)\leq 1+\beta$ .

We close this section with an immediate general representation theorem and a few observations that are handy when deriving closed-form solutions. As a consequence of Theorem1, any optimal contract $Y$ can be written as

\begin{align*} Y(x)=\max \{0, \alpha (x)x-d\}, \end{align*}

where $d\geq 0$ is a deductible and $\alpha (x)$ is a non-negative, continuous, and a.e. differentiable function satisfying for every $x\in [0, M]$

\begin{align*} 0 \leq \frac{\partial \alpha (x)x}{\partial x}\leq 1+\beta . \end{align*}

We say that the contract $Y$ is a full insurance contract when $d=0$ and $\alpha (x)=1$ everywhere; is a straight deductible when $d\gt 0$ and $\alpha (x)=1$ everywhere; entails coinsurance when $\alpha (x)=\alpha \in (0,1)$ ; and has upper limit $\gamma \gt 0$ if for every $x$ , $Y(x)\leq \gamma$ , with strict equality for some $x$ . Any contract mixing deductibles, coinsurance, and upper limits are manipulation-proof and can be written as

\begin{align*} Y(x)=\min \left\{\gamma, \max \{0, \alpha x - d\} \right\}. \end{align*}

We obtain the no-sabotage condition when $\beta =0$ . Formally, if $\beta =0$ then for any contract $Y$ the following holds:

1. 1. $\textrm{slope(Y)} \leq 1$ ;

2. 2. the retention function $R(x)=x-Y(x)$ is weakly monotone increasing;

3. 3. the random variables $X,Y$ and $R(X)= X-Y(X)$ are comonotonic.Footnote ⁸

2.2 Some closed-form solutions

We now provide closed-form solutions to our contracting problem.

Full insurance, straight deductibles, and the irrelevance of arson

Lemma4 need not be proved as it is the classic Arrow-Borch-Raviv Theorem (see Dionne et al., Reference Dionne2000). The lemma tells us that (Problem S) becomes interesting only when $c\gt 0$ somewhere. This is because when $c=0$ , the first-best contract never incentivizes Bob to take a sledgehammer to his bike. That is, arson-type risks impact the insurance only when there is a meaningful reason not to provide either full insurance or a straight deductible.

Fixed costs and nuisance claims: arson-type risks reduce welfare

We now consider the case when the administrative costs to deliver the contract are fixed costs. The setting with fixed cost was first introduced in Gollier (Reference Gollier1987) and is well-understood (Carlier & Dana, Reference Dana2003, Reference Carlier and Dana2005; Picard, Reference Picard2000, Reference Picard2013). We consider this case because it is the easiest way to show that arson-type risks hurt the insured. That is, to show that Bob would prefer to be unable to destroy his bike.

We claim that when the insured can freely augment the damage ( $\beta =0$ ), then the optimal contract is a straight deductible.

Proposition 1. Under Assumption 2 , the solution to (Problem S) is a straight deductible: there exists a $d\gt 0$ such that

\begin{equation*}Y(x)= \max \{0, x-d\}.\end{equation*}

We can now rewrite problem (Problem S) as

(Problem FC and A) \begin{align} \max _{d\geq 0} & \int u\! \left(W_0 - H - x +\max \{0, x-d\}\right)\,\textrm{d}\mathbb{P} \\ \textrm{s.t.}\quad & c_0 \mathbb{P}[X\geq d] + \mathbb{E}[X-d\vert X\geq d] = (1+\rho )^{-1}H \nonumber . \end{align}

We want to show that the possibility of arson-type risks hurts the insured, i.e., that Bob would like to commit to leaving his bike intact because he would be offered a better contract. Formally, we will show that absent the incentive compatibility constraint, a contract with an upward jump would improve upon a straight deductible. That is, we want to show that the straight deductible contract in Figure 3(a) is dominated by the discontinuous contract in Figure 3(b).

Once again, Proposition 6 of Carlier & Dana (Reference Carlier and Dana2005) implies that any optimal contract is $1$ -Lipschitz on $\{Y\gt 0\}$ . We can thus consider a contract of the form

\begin{align*} Y_{\delta, b}(x)=(x-\delta ){\unicode{x1D7D9}}_{\{x\geq b\}}=\begin{cases} 0 &\text{ if } x\lt b \\[3pt] x-\delta &\text{ if } x\geq b\end{cases} \end{align*}

for parameter $b\geq 0$ being a threshold, $\delta \geq 0$ being a loss retention parameter and $b \geq \delta$ . The number $b-\delta$ is the magnitude of the jump of $Y_{\delta, b}$ at $b$ and is interpreted as the minimum amount of money that the insured receives if they file a claim. We will refer to contracts with the above form as contracts with constant retention. Clearly, $Y_{\delta, b}$ is not manipulation-proof when $b\gt \delta$ and $Y_{\delta, b}$ is a simple deductible when $b=\delta$ . More importantly, notice that the insured files a reclamation if and only if $Y_{\delta, b}(x)\gt 0$ , so we can partition the interval $[0, M]$ in the two regions $\{Y\gt 0\}$ and $\{Y=0\}=\{Y\gt 0\}^C$ .

Suppose now that the cost $c$ satisfies Assumption2, that the insured cannot use arson-type actions so that there are no incentive compatibility constraints and that we are restricting our search to contracts with constant retention. A few calculations show that we can write the optimization problem as

(Problem FC and No-A) \begin{align} \max _{\delta \geq 0,\, b\geq 0} & \int u(W_0 - H - x +Y_{\delta, b}(x))\,\textrm{d}\mathbb{P} \\[2pt] \nonumber\end{align}

(Positive jump) \begin{align} \textrm{s.t.}\quad & c_0 \mathbb{P}[X\geq b] + \mathbb{E}[X-\delta \vert X\geq b] = (1+\rho )^{-1}H \nonumber \\ & b-\delta \geq 0 . \\[10pt] \nonumber \end{align}

Clearly, the optimization problem with arson-type risks (Problem FC and A) is (Problem FC and No-A) when the Positive jump constraint is binding so $b=\delta$ . It is thus clear that arson-type risks can hurt the insured. We are left to show that arson-type risks always hurt the insured when the fixed cost is strictly positive. This entails showing that the constraint (Positive jump) of (Problem FC and No-A) is an equality if and only if $c_0\gt 0$ . This is a well-known result, being essentially Theorem 2 of Gollier (Reference Gollier1987).

Proof. If $c_0=0$ then (Problem FC and No-A) is the standard Arrow-Borch-Raviv problem for which the solution is a simple deductible, i.e., $Y(x)=\max \{0, x-d\}$ , so $\delta =b=d$ . Conversely, set $\delta =b=d^*\geq 0$ such that $d^*$ solves the reduced problem

\begin{align*} \max _{d\geq 0} & \int u(W_0 - H -x +\max \{0, x-d\})d\mathbb{P} \\ \textrm{s.t.}\quad & c_0 \mathbb{P}[X\geq d] + \mathbb{E}[X-d\vert X\geq d] = (1+\rho )H . \end{align*}

If $c_0=0$ , then we are done. Suppose $c_0\gt 0$ . Then there exists an $\varepsilon \gt 0$ such that the alternative contract

\begin{equation*}Y^{\prime}(x)=(x-d^*+\varepsilon ){\unicode{x1D7D9}}_{\{x\geq d^*\}}\end{equation*}

strictly improves upon the contract $Y(x)=\max \{0, x-d^*\}$ , as desired.

The interpretation is straightforward. Absent arson-type risks, the optimal insurance contract provides the maximum possible protection while eliminating nuisance claims, the small claims which are more costly to administer than the coverage they offer. This is achieved by refusing to reimburse small claims while providing generous protection conditional on the loss being large. This creates a spread in coverage, the discontinuity at the threshold. But Bob wants to exploit this spread by augmenting the damage to his bike. In equilibrium, Bob is never offered such a contract with high protection, and Bob would be strictly better off if he could commit not to buy a sledgehammer.

Continuous costs and the sub-optimality of disappearing deductibles

We now consider the case when $c$ is a continuous function.

Recall that our problem is now almost identical to the one of Spaeter & Roger (Reference Spaeter and Roger1997).

Lemma 6. Under Assumption3, if $Y$ solves the reduced problem

(Reduced problem) \begin{align} \max _{H \geq 0, Y \in B_+(\mathcal{B}([0,M]))} & \int u(W_0 - H - x +Y(x))\,\textrm{d}\mathbb{P} \\ \textrm{s.t.}\, & \,0 \leq Y \leq X \nonumber \\ & (1+\rho )\int Y(x)+c(Y(x)) \,\textrm{d} \mathbb{P} = H \nonumber \end{align}

of Spaeter & Roger (Reference Spaeter and Roger1997) and $Y$ satisfies the slope constraint $\textrm{slope}(Y)\leq (1+\beta )$ (S2) then $Y$ solves (Problem S).

This lemma informs us that the only problematic case that must be handled is when the contract $Y$ found in Spaeter & Roger (Reference Spaeter and Roger1997) does not satisfy constraint (S2) somewhere. Intuitively, it seems natural to attempt flattening $Y$ sufficiently to satisfy $\textrm{slope}(Y)\leq 1+\beta$ , thus solving Problem S. This approach is sometimes fruitful but does not work in some instances, as we explain later. We consider the easiest case when the best contract absent arson-type risks is a completely disappearing deductible. Formally, we say that contract $Y$ is a completely disappearing deductible if there exists a realization $x^{\prime}\in [0, M]$ such that for every $x\geq x^{\prime}$ , it is $Y(x)=x$ .

Proof. The fact that $Y_R$ is a completely disappearing deductible informs us that our optimal contract $Y$ provides as much coverage as possible while not providing full insurance. Since $\beta =0$ , it holds that any optimal contract $Y$ must have a slope lesser or equal to $1$ . We claim that any optimal contract $Y$ is a simple deductible. This follows from the usual argument that if $Y^{\prime}$ is a generalized deductible with a slope lesser or equal to $1$ , then we can find a deductible contract $Y$ such that $Y\geq _{\textrm{cx}}Y^{\prime}$ and $Y$ (weakly) improves upon $Y^{\prime}$ , where $\geq _{\textrm{cx}}$ denotes the convex order (Shaked & Shanthikumar, Reference Shaked and Shanthikumar2007).

Hence, finding $Y$ is equivalent to solving

\begin{align*} \max _{d\geq 0} & \int u(W_0 - H - x +\max \{0, x-d\})\,\textrm{d} {\mathbb{P}} \\ \textrm{s.t.}\quad &(1+\rho )\mathbb{E}[\max \{0, X-d\} +c(\max \{0, X-d\})] = H, \end{align*}

as desired.

As before, Bob’s ability to take a sledgehammer to its bicycles is priced in the insurance contract, and Bob will never be offered a completely disappearing deductible in equilibrium. Again, though Bob would like to purchase such a contract, he cannot. This is because no Bob can commit to being honest. The intuition that the contract-solving problem (Problem S) is simply a “flattening” of the solution to (Reduced problem) is misleading. By additivity of the Lebesgue integral, the insurer’s participation constraint states that the insurer should recoup the cost on average and not realization-by-realization. This, fortunately, gives us some leeway in solving (Problem S). However, this also means that there are some cases where we have to roll up our sleeves and directly attack the problem.