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Ex post moral hazard and Bayesian learning in insurance.


We study a dynamic insurance market with asymmetric information and ex post moral hazard. In our model, the insurance buyer's risk type is unknown to the insurer; moreover, the buyer has the option of not reporting losses. The insurer sets premia according to the buyer's experience rating, computed via Bayesian estimation based on buyer's history of reported claims. Accordingly, the buyer has strategic incentive to withhold information about losses. We construct an insurance market information equilibrium model and show that a variety of reporting strategies are possible. The results are illustrated with explicit computations in a two-period risk-neutral case study.


Buyers of casualty and property insurance possess varying levels of risk. Risk type is used by the insurer to set premia and is the main parameter in pricing the resulting insurance contract. Unfortunately, the intrinsic riskiness of the buyer is typically unknown from the point of view of the insurer. This leads ex ante to problems of moral hazard and adverse selection. Namely, higher-risk buyers will attempt to enter into contracts designed for low-risk buyers, and once they obtain insurance, all buyers have little incentive to act prudently. However, many insurance contracts (notably automobile insurance) have a recurring nature. Thus, the issue of information asymmetry is partially mitigated by implementing an experience-rating or bonus-malus system (see Lemaire, 1995, for a comprehensive review of such insurance schemes), through which the insurer gives incentives to the buyer to act in the "best" behavior. Through such incentives, a self-serving buyer may reveal his (1) risk type or exercise an optimal amount of preventive efforts.

A second level of ex post information asymmetry arises in connection with reporting losses. After an insurable event occurs, the buyer has the option of not reporting the loss in the hopes of signaling that he is of a lower risk type and, hence, deserves a lower future premium. If the gain from lowering his perceived riskiness (and the corresponding future premia) outweighs the cost for settling the loss out-of-pocket, the buyer will not report the loss. This might happen, for instance, if the risk loading on the insurance is high enough and leads to ex post moral hazard. The presence of this nonreporting option has serious implications since it dramatically alters the information received by the insurer. Instead of acting to reveal his risk type, the buyer strategically manipulates reports. Although the insurer is not necessarily hurt by underreporting and may even encourage it to reduce processing costs, she needs a learning mechanism to infer the risk type of the buyer based on submitted claims. This is necessary in order to correctly determine the premium and the corresponding experience-rating update. In a competitive insurance market, failure to properly infer the buyer's risk profile would immediately result in losses. Thus, a rational insurer recognizes that nonreporting occurs and acts accordingly.

The converse of nonreporting is insurance fraud whereby the buyer may manufacture false claims. The information asymmetry of insurance fraud is usually resolved by claim verification and monitoring; by contrast, verifying nonreported losses is usually impractical or against the law.

There is anecdotal empirical evidence for such strategic behavior by both buyers and insurers, especially in the private passenger automobile and homeowner's insurance industries. For instance, after minor car accidents of the "fender-bender" type, it is commonplace that insurance agents advise their clients to pay for repairs themselves and not file a claim, so as to maintain their high "rating." Conversely, it is often observed that the actual malus penalties for reporting auto claims are relatively severe and would be excessive in a world with perfect reporting (i.e., they compensate for the fact that only major accidents are reported).

In this article, we study the implications of this double information-asymmetry insurance problem in a multiperiod model with the insurer learning through Bayesian updating. To focus on the ex post moral hazard aspect of the problem, we rule out adverse selection by postulating mandatory insurance (no opt-out clause) and avoid ex ante moral hazard by making the claim probabilities completely exogenous. Mandatory insurance also allows us to sidestep the issue of participation constraint as all buyers are required to nominally subscribe to the insurance market, even if they will then choose to partially self-insure. Instead, we focus on determining the optimal reporting strategies of the buyers of insurance, while assuming that insurers apply the classical Bayesian learning rule to update the premia.

Analysis of games with incomplete information dates back to the pioneering work of Harsanyi (1967, 1968a, 1968b) on Bayesian game theory. In the context of insurance with asymmetric information, recall the seminal paper of Rothschild and Stiglitz (1976) who consider an insurance model with exogenous accident probabilities, constant claim sizes, and publicly known risk preferences of the buyer. In this context, in a one-period model, Rothschild and Stiglitz find a separating equilibrium in which the insurance contract is designed to reveal information, with higher risks buying full coverage at their actuarially fair premium and lower risks buying partial coverage at their actuarially fair premium. Rothschild and Stiglitz focus on adverse selection, so that the only choice of the buyer is to select among a group of possible insurance contracts.

An extensive literature (Dionne, 2000) also considers moral hazard, where the buyer controls some variable related to his insurance losses, in contrast to adverse selection, where the unknown in the model is the static risk parameter. In a typical example, the buyer can exert (unobservable) effort that lowers his accident probability; the aim is, then, to design an insurance contract that forces the buyer to exert a first-best effort level even under the information asymmetry.

The extension of the model of Rothschild and Stiglitz (1976) to multiple periods is considered in Townsend (1982), Dionne (1983), Dionne and Lasserre (1985), and Cooper and Hayes (1987). The presence of multiple periods leads to questions of contract commitment and renegotiation. With ex post moral hazard, a typical buyer cannot commit to more than one period since his reports are intrinsically nonverifiable. Accordingly, with multiple periods, one can focus either on contracts that vary coverage (looking for separating equilibria via the insurer offering a menu of contracts) or on contracts that vary premia. Varying the premia corresponds to an experience rating or bonus-malus mechanism (Hey, 1985; Dionne and Vanasse, 1992) that is often employed in ratemaking for automobile insurance. The bonus-malus system penalizes occurrence of claims, thereby providing incentives for prudent behavior.

The second aspect of an experience rating system consists of the insurer attempting to learn the type of the buyer over time. The classical example of Bayesian learning is explicitly considered by Hosios and Peters (1989) and Watt and Vazquez (1997). Hosios and Peters consider the Rothschild and Stiglitz (1976) framework with varying coverage. In their model, low risks always act myopically, report all claims, and purchase the one-period contract; high risks choose a randomized reporting strategy. This setup leads to a stochastic game between the monopoly insurer who offers varying coverage and the high-risk buyer who strategically reports claims. In contrast to our work, the model of Hosios and Peters does not require studying the joint interaction of heterogeneous buyers, and hence, an equilibrium strategy always exists. From a different point of view, Watt and Vazquez (1997) show that with multiple periods, a Bayesian bonus-malus contract dominates the multiperiod extension of Rothschild and Stiglitz; therefore, the former is a more plausible contract structure (in other words, it is beneficial to vary premia rather than coverage). However, Watt and Vazquez assume that no underreporting is possible and, thus, treat only the case of adverse selection.

Our work is most allied with a recent paper by Robinson and Zheng (forthcoming) who consider the effect of experience rating on precautionary behavior by insurance buyers. Robinson and Zheng study a two-period model with compulsory insurance and high-and low-risk types, where the buyer faces both ex ante moral hazard (through an unobservable effort variable) and ex post moral hazard (through a possibility of not reporting a claim). In addition, Robinson and Zheng also assume that the buyers do not know their own type; moreover, the experience rating is solely based on the occurrence of a claim during period 1 (thus, there is no "prior" risk type). The authors show that even if the effort is unobservable by the insurer, the buyer will engage in precaution in order to maintain a good rating and avoid future costs, mitigating ex ante moral hazard while creating ex post moral hazard.

Starting with the same framework, we choose instead to focus solely on ex post moral hazard. Accordingly, we drop the option of the buyer to influence accidence probabilities and make the latter completely exogenous. Moreover, in contrast to Robinson and Zheng (forthcoming), we assume that the buyers are aware of their risk type and, therefore, strategically manipulate their reporting. In order to focus on underreporting, we rule out the other mentioned type of ex post moral hazard, namely, insurance fraud; see Moreno, Vazquez, and Watt (2006) for a study of effect of a bonus-malus contracts on fraud, and Dionne, Giuliano, and Picard (2009) for a Bayesian approach to fraud signals. As in Hosios and Peters (1989), we assume that the insurer uses a Bayesian experience rating that depends both on the latest claim report and on the prior of the buyer's risk type. Thus, the insurer has initial ratings assigned to each buyer, which are public knowledge, and the buyer can manipulate his experience rating by reporting or not reporting losses, modeled as a mixed game strategy (in contrast to only pure strategies considered in Robinson and Zheng, forthcoming). The explicit model of the learning mechanism of the insurer allows us to compute the optimal reporting strategies in closed form.

In order to focus on the strategic reporting behavior of the buyer, we fix the insurance market mechanisms and require full insurance so that no separating equilibrium is possible. This can be seen as a proxy for real markets in which a wide variety of risk types are typically offered a small menu of contracts. Thus, within a class of risks, there still exists significant residual variability leading to ex post moral hazard. Also, by taking the insurance market as given, we only consider the buyer's strategy. This reflects real insurance markets in which the buyer is the main "actor," and the insurer passively reacts to the buyer's strategy through the experience-rating mechanism.

In our setting, existence of equilibrium is nontrivial since it requires a compatibility between the optimal behavior for buyers of all possible risk types (in terms of resulting randomized reporting strategies) and the global experience-rating mechanism employed by the insurer across all (unknown to her) risk types of buyers and their strategies. One of the main contributions of our model is showing that this compatibility is nontrivial and can sometimes fail. Hence, it is possible that for some buyers the insurer has no consistent mechanism to update premia and ratings. Moreover, the model features abrupt phase transitions between different reporting strategies as a function of perceived risk of buyer.

Finally, we mention some recent empirical literature that attempts to test the significance of various information asymmetries in actual insurance markets around the world. Of particular note are the recent studies of Abbring, Chiappori, and Pinquet (2003), Chiappori et al. (2006), and Abbring, Chiappori, and Zavadil (2008) who consider the automobile insurance markets in France and the Netherlands, respectively. The Dutch system, in particular, incorporates a complicated bonus-malus class structure that can be viewed as a proxy for a Bayesian updating mechanism. The cited papers find conflicting evidence of ex ante and ex post moral hazard and discuss the econometric difficulties of disentangling the multiple information asymmetries. We point out that in those studies, a key emphasis is made on the claim severity, a feature that is absent in our model where all claims are of unit size.

The structure of this article is as follows. In the "Insurance Market With Ex Post Moral Hazard" section, we describe the multiperiod insurance game that we study; then, in the "Probabilistic Model" section, we present the mathematical details of our model. In the "Case Study" and "Classification of Optimal Strategies in the Case Study" sections, we study a two-period example in detail and show that an equilibrium might not exist even in this simple case. We also study the types of equilibria that can occur and show that in the first period of the two-period example, low risks either always report or never report their losses, whereas high risks report their losses with a positive probability. In other words, high risks might try to make themselves look more like low risks by hiding some of their losses, but they never completely mimic low risks by never reporting their losses. In the "Extensions of the Case Study" section, we discuss how the equilibria found in the two-period model might change in a more general setting, for example, under a nonlinear premium rule or with more than two periods. The "Summary and Conclusions" section concludes the article.


In this section, we provide an economic description of the dynamic insurance market model we consider. The precise mathematical construction is then presented in the "Probabilistic Model" section. Our notation follows Rothschild and Stiglitz (1976) and Hosios and Peters (1989).

Information Asymmetries

To focus on the stated information asymmetry, we assume that the buyer has no control over his own risk type (i.e., no ex ante moral hazard). Moreover, the buyer acts rationally in that he chooses a reporting strategy to minimize his total expected costs over a given finite time period. For simplicity, all losses are of fixed size 1, so that the issue of partial reporting is nonexistent. We also assume that the buyer can hide losses but cannot manufacture them, nor can the insurer discover losses that are not reported. In other words, a reported claim is verifiable; a nonreported one is not.

The main level of information asymmetry is that the buyer is aware of his risk type X [member of] E; in contrast, the insurer never knows X precisely (unless the buyer by his actions reveals X) but starts with a prior distribution q(0) at time 0 over possible values of X. Thus, from the insurer's point of view, X is a random variable, whose distribution is updated in response to incoming insurance information. Buyers differ only in their probability of a loss; a buyer of risk type X = x has probability of loss equal to [p.sub.x].

Since the buyer knows his type (i.e., his probability of loss), he can strategically manipulate the insurer's beliefs in the manner most advantageous to him. We denote by q(k) the insurer's beliefs at time k, with the space of values of q(k) denoted by

D = {([q.sup.1], [q.sup.2], ... ,[q.sup.d])[member of][R.sup.d][summation over (x[member of]E] [q.sup.x] = 1, 0 [less than or equal to] [q.sup.x][less than or equal to] 1}, d = [absolute value of E].

Here, [q.sup.x] is the probability that a risk type is of type x [member of] E according to the beliefs of the insurer. We write [q.sup.x](k) if we want to emphasize that the belief is held at time k, that is, at the beginning of period (k + 1). We assume that at time 0, the buyer knows his own initial rating q (0).

For a given rating q = q(k - 1) at the beginning of period k, denote by [d.sub.x](q,k) the probability that a realized loss will be reported to the insurer at the end of period k, that is, at time k. Therefore, from the insurer's point of view, the probability that a claim is filed at time k equals [[summation].sub.x[member of]E] [q.sub.x](k - 1) [p.sub.x][d.sub.x](q,k).

Remark 1: The strategic behavior of the buyers is driven by their knowledge of their true risk type X. An alternative formulation could assume that the buyers are not aware of their type either. This would require rational buyers to first filter their accident histories and only then strategically decide whether to report or not report the claim. Consequently, the buyer's prior [??](k) (which is different from the insurer's prior q(k) that is based on reporting history) will need to be taken into account during the updating mechanism leading to intractable complexity, at least with current methods.

Insurance Market

The insurance market operates as follows. The description below is based on a given fixed collective reporting strategy (across all risk types x), [??][equivalent to] {[??](q, k)} = [([d.sub.x](q, k)).sub.x[member of]E], for q [member of] D and k [member of] {1, ... ,T}, and a fixed insurer rate-making policy. Since verifying nonreported claims is impossible, the buyer cannot credibly commit to a multiple-period contract. Consequently, we only consider one-period contracts with the different periods linked by reputational considerations (insurer beliefs). The insurance contract consists of an update mechanism G that implements the experience-learning scheme and a (contingent) premium schedule f(q(k - 1), [??](x, k)) for the kth period that determines the premium payable at time k--1 based on the current rating and the heterogeneous reporting strategy that will be implemented at time k. The insurer anticipates partial nonreporting and will lower upcoming premia to reflect lower insurance costs, thus making "lying" more profitable for the buyer.

At the beginning of each period k = 1, 2, ..., T, the premium f(q (k - 1), [??] (q (k - 1), k)) is first collected. Second, the buyer either does or does not experience an insurance event (recall that all losses are of size 1). The occurrence of a loss is modeled as an independent Bernoulli trial with probability [p.sub.X] of occurring, the probability being a function of the true risk type X. If a loss occurs during period k, then the buyer has the option of (1) reporting the "claim" and receiving reimbursement at time k, or (2) settling the loss himself and reporting "no claim" at time k. If no loss occurs, the buyer reports "no claim." From the insurer's point of view, the latter outcome is identical to case (2) aforementioned. The reporting strategy of the buyer is represented by a randomization policy with parameter [d.sub.X](q(k - 1), k): conditional on the type X of the buyer and an event occurring in the kth period, the buyer performs an independent Bernoulli trial (hidden from the insurer), and with probability [d.sub.X](q(k - 1), k) reports the event (case (1)) to the insurer at time k. Thus, with probability 1 - [d.sub.X](q (k - 1), k) the event is not reported and case (2) takes place.

Let [Y.sub.k] [member of] {0,1} represent the indicator of a reported claim at the end of the kth period (time k) and denote by [F.sub.k] = [sigma]{[Y.sub.s]: s = 1,2, ... ,k} the claim information received by the insurer by the end of period k. For typographical convenience, we will write [??](k) for [??](x, k). Based on the buyer's announcement of "claim" or "no claim" (as well as the collective reporting strategy [??](k)) the insurer now updates the buyer's rating using a Markovian update mechanism G(q(k- 1), [Y.sub.k];[??](k)). More directly, we will write q+(q(k-1),[??](k)) = G(q(k-1),1;[??](k)) and [q.sup.-](q(k-1),[??](k)) = G(q(k-1),0;[??](k)) to indicate the updated ratings in the event of a "claim" ("no-claim") report. We reiterate that premia are paid at the beginning of each period and claims are settled at the end; thus, at time k (the end of the kth period), the premium f(q(k),[??](q(k),k + 1)) for period (k + 1) and any losses for period k (either self-absorbed or reimbursed by the insurer) are settled simultaneously.


After all these adjustments, the market moves to the next round. After the last round, the Tth, the game stops (i.e., no more insurance is issued). The overall mechanism is illustrated in Figure 1.

Insurance Market Equilibrium

In order to create an equilibrium, two items are now necessary:

1. The seller must ex ante commit to a verifiable and public update mechanism G regarding ratings.

2. The buyers must be able to collectively "announce" their entire reporting strategy ([??][(x,k).sub.k=1], ..., T]) at time 0, such that it is credible. That is, for any buyer of type X = x, the reporting frequency [d.sub.x](x, k) is optimal at period k assuming all other buyers implement d. In other words, deviation from the collective [??] must be suboptimal for any given buyer.

Our first assumption is easily enforced on the grounds that an insurer is regulated and, hence, faces mandatory rules regarding ratings. Alternatively, there is a competitive insurance market with public ratings that weeds out any unreliable insurers. This situation is generally true in practice where experience rating systems are publicly announced and always honored for settling claims. They also tend to be "sticky" over time, so that changes to the update mechanism are made rarely.

The requirement of collective compatibility is clearly necessary to prevent deviations from buyers (as verifying their reporting strategy is obviously impossible). The aspect of the buyers collectively "announcing" their reporting strategy is to resolve an issue of potentially several coexistent equilibria. The insurer must know the prevailing equilibrium; otherwise, she does not have a consistent way of updating her beliefs as she does not know the precise information set with which she is working. As we will show below, in some cases it is possible to prove that at most one equilibrium exists, and therefore, no public announcement is necessary by a rational buyer. With a unique equilibrium, the insurer can directly infer the only collectively compatible strategy [??]*.

Because premia are mandatory and paid at the beginning of each period, the reporting strategy is required to be ex post optimal. Thus, even though the period k premium f(q(k - 1), [??](k)) is affected by the reporting strategy [??](k) employed at time k, from the point of view of the buyer who effectively makes his reporting decision at time k, this premium is a sunk cost. Indeed, at the beginning of a period, the buyer may commit to a reporting strategy at the end of that period only if it is ex post optimal once premium is paid. This fact is most starkly manifested during the last period. At time T-1, a rational buyer may be better off self-insuring and, therefore, could claim that his premium should be zero. However, insurance is mandatory, so once premium is paid, the buyer clearly has no incentive to self-insure at time T and will report all claims. Hence, in the last period the only ex post optimal strategy is full reporting.

One key feature of our model postulates that the updating of the buyer's rating is based on a Bayesian mechanism (Hosios and Peters, 1989; Watt and Vazquez, 1997); that is, given beliefs q (k-1) in the beginning of the kth period, the announced reporting strategy d(k) for period k, and the actual report [Y.sub.k] at time k, the new insurer beliefs q(k) are computed using a relative-likelihood approach based on conditional probabilities of respective events. Namely, q(k)[??][p.sup.q(k-1)](X|[Y.sub.k],[??](k)). Thanks to this nonlinear mechanism and to the second level of information asymmetry that gives the buyer control over this updating (via the reporting strategy), the resulting insurance market exhibits a rich and interesting structure. Other update mechanisms are also possible (e.g., a non-Markovian mechanism, a moving-average mechanism such as q(k) = [beta]q(k-1) + (1 - [beta])[Y.sub.k], etc.).

We choose classical Bayesian updating because (1) the resulting q(k) is the vector of conditional probabilities for the buyer type given the reporting strategy [??], and (2) q(k) is nonlinear in the prior rating q (k-1), which results in nontrivial reporting strategies even under linear preferences, as we show later. Property (1) justifies Bayesian updating from the point of view of learning; it is well known that conditional probabilities minimize the square error loss, so that an insurer that wishes to best estimate the buyer's risk type will use this method (see Mikosch, 2004, for references). Features of Bayesian updating can be detected in some of real-world experience rating systems. For instance, in the Dutch car insurance market, Abbring, Chiappori, and Zavadil (2008) report that the updates are more dramatic for middle rating classes than for the extreme classes, as is also obtained in a Bayesian setting.

Remark 2: We argue that a straight bonus-malus system is not consistent with our asymmetric information model. Indeed, because the update in bonus-malus only depends on the previous experience rating and the latest claim information, it fails to take into account the different strategic behaviors of heterogeneous buyers. A bonus-malus model assumes that the reporting strategy of a particular buyer is independent of the strategies of the other buyers; thus, it ignores the fact that the observed probability of claims is a function of the reporting strategies across buyers. In other words, a consistent update mechanism G must be a function of all [d.sub.x](k), x [member of] E, since it should weigh the latest claim observation by its relative likelihood for each buyer type.

Markov Strategies

In general, the reporting strategy [??](k) may be a function of all the past information (in particular, the report history) rather than just a function of the latest rating q(k-1). Our Markovian assumption can be justified by assuming that the report history is not public knowledge (with verification costly or impossible), so that the only credible strategy announcements for reports in the kth period must be based on current rating q(k-1), which is public knowledge. This parallels the real-life setting of confidential credit histories that are summarized with a single public credit score for any commercial credit enquiry.

The premium rule f might also be a function of the report history. Such a situation would be ruled out in a competitive insurance market with multiple insurers at each period. If report histories are not publicly available, premia would then have to be based only on the latest rating (and implicitly on the anticipated reporting strategy). For example, in private passenger automobile insurance in the United States, insurers and other economic agents rely on a demerit or points system that acts as a proxy for risk rating. In general, our minimal requirement is that the premium paid at time k should be [F.sub.k] measurable.


In this section, we provide the mathematical details of our model.

Bayesian Learning

For simplicity, we consider the two-state case, whence X [member of][??]{L, H} so that risks are either of low or high type. Let 0 [less than or equal to] [p.sub.L] < [p.sub.H] [less than or equal to] 12 be the time-homogeneous probabilities of an insurance event for the respective risk types.

With a slight abuse of notation relative to the previous section, in this section and the next, let q(k) [equivalent to] [q.sub.H](k) = P(X = H | [F.sub.k]) be the belief of the insurer concerning whether the buyer is of high type based on the information available at time k. If the time k is irrelevant or understood, then we also write q for q(k). Also, let ([d.sub.L] (k), [d.sub.H](k)) be the reporting strategy of the buyers in the kth period conditional on q (k-1). Recall that [Y.sub.k] [member of] {0, 1} indicates whether a loss is reported at time k. Then, from the insurer's point of view the probability of a reported claim at time k equals

P([Y.sub.k] = 1) = [p.sub.H] x [d.sub.H](k) x q(k-1) + [p.sub.L] x [d.sub.L](k) x (1- q(k - 1));

therefore, by Bayes' formula the updated beliefs at the end of the kth period [q.sup.+](k) and [q.sup.-] (k) conditional on whether a claim was reported are


Again, we stress that the update mechanism, G:(q, [d.sub.L], [d.sub.H]) [member of] D x [[0,1].sup.2] [right arrow] ([q.sup.-], [q.sup.+]) [member of] [D.sup.2], depends on the announced strategy ([d.sub.L](k), [d.sub.H] (k)) since it is based on the relative likelihood of the corresponding risk types filing a claim. The Bayesian update G in (1) is nonlinear and asymmetric in the low-and high-risk-type parameters. Also, note that if [d.sub.L] = 0 (respectively, [d.sub.H] = 0), then a claim is a clear signal that the buyer is of high type, [q.sup.+] = 1 (respectively, low type, [q.sup.+] = 0); this is the only situation in which the buyer unambiguously reveals his type.

Objective of the Buyer

Let T be the finite horizon of the insurance game, and let v [less than or equal to] 1 be the discount factor for future payments. We assume that the buyer possesses time-additive risk-averse (or risk-neutral) preferences with respect to insurance costs. Let U be the convex loss function of the buyer, with U(0) = 0. Let [L.sub.k] [member of] {0, 1} be the actual realized loss in period k (as opposed to [Y.sub.k] reported to the insurer; note that [Y.sub.k] [less than or equal to] [L.sub.k]). Then, at time k (end of the kth period), conditional on his true risk type X, the buyer's risk-adjusted costs are:


As argued before, since the market terminates at time T, it is ex post optimal to report all the period T losses; thus, we assume henceforth that [L.sub.T] = [Y.sub.T], which implies that the time T costs are U(0) = 0. Recall that the buyer selects his reporting policy just before the end of the k-period, at time k-. The losses are then settled at time k, and immediately the rating is updated to q(k). Thus, for a given reporting strategy [??], at time k- the total discounted expected costs from time k through time T for a buyer of type X are


with the premium at time T understood to be 0. Note that the premium for period k, namely, f(q(k - 1), [??](q(k-1),k)), is not part of V(k,q(k-1)) since by time k--it is a sunk cost and does not enter into the optimization problem that we introduce below.

By using the time additivity of utility, (3) can be recursively written as [V.sub.X](T, q) = 0 and


The first term represents the value in the case of a report, the second term represents the cost of the self-imposed loss that occurs when an event is not reported, and the last term represents the expected value in the case of no loss. The values ([q.sup.-] (k), [q.sup.+] (k)) = G(q (k - 1), [d.sub.L] (k), [d.sub.H](k)) are obtained from (1).

A rational buyer chooses his reporting strategy to minimize expected future costs. However, if the buyer is of type X, his control is just [{[d.sub.X](q, k)}.sub.k=1, ... ,T], whereas [V.sub.X] depends on the entire collective [??]. Thus, a consistency condition is needed, namely, that


simultaneously, so that a given buyer has no rational reason to deviate from the announced collective strategy [[??].sup.*] (q, k) regardless of his true risk type. In other words, for a fixed [d.sup.*.sub.H](q, k) (respectively, [d.sup.*.sub.L] (q, k)), if the buyer is low risk (respectively, high risk), then it is optimal for him to employ the reporting strategy corresponding to [d.sup.*.sub.L](q,k) (respectively, [d.sup.*.sub.H](q,k)) at time k, and [d.sup.*.sub.L](x, k + 1), ..., [d.sup.*.sub.L](x,T) (respectively, [d.sup.*.sub.H](x, k + 1), ... , dh(x, T)) thereafter.

The insurance market will exhibit a stable equilibrium if the aforementioned collective-compatability condition (5) is satisfied for exactly one pair ([d.sup.*.sub.L](k), [d.sup.*.sub.H](k)).

In such a case we say that an equilibrium exists, and we define the value function of a buyer of type X at time k--to be


for a given rating q = q(k-1),k = 1, ..., T.


We now study a simple example in which there are two periods, the buyers are risk neutral, and the premium equals the expected discounted payout with a proportional risk load. Specifically T = 2, the loss function U is given by U(x) = x, and the time k premium f is given by

f(q(k),[??](q(k),k + 1)) = (1 + [theta])v [[p.sub.H][d.sub.H](k + 1)q(k) + [p.sub.L][d.sub.L](k + 1)(1 - q(k))], (7)

for some proportional risk load [theta][greater than or equal to] 0 and for k = 0,1. For simplicity, we assume that the discount v used by the buyer and the insurer are identical. Any difference between them could be absorbed in the factor of (1 + [theta]) by allowing 0 to take values greater than -1 instead of 0.

Since there are only two periods, in the second (last) period the buyer has no incentive to nonreport, so that trivially ([d.sup.*.sub.L] (q, 2), [d.sup.*.sub.H](q, 2)) = (1,1). In the first period, the premium paid is a function of the insurer's prior q(0) and the reporting strategy at time 1; however, at time 1, the buyer chooses his strategy to minimize future expected costs, so the premium paid at time 0 is irrelevant at that time. Note that the premium paid at time 1 equals f(q(1), [??](q(1),2))[equivalent to] f(q) = (1 + [theta])v([p.sub.Hq] + [p.sub.L](1 - q)) = (1 + [theta])v[p.sub.L] + [[alpha].sup.2]q, in which [alpha][??][square root of (1 + [theta])v([p.sub.H] - [p.sub.L]). In other words, the insurer charges a base premium (1 + [theta])v[p.sub.L] plus a surcharge proportional to the updated risk rating q = [p.sup.q(0)](X = H|[F.sub.1]) of the agent, the proportionality constant being [[alpha].sup.2].

Henceforth, we focus on the expected costs of the low and high risks at time 1. For typographical convenience, we omit time dependence and q dependence of the strategies at time 1 and write [d.sub.H] = [d.sub.H](x, 1) and [d.sub.L] = [d.sub.L] (x, 1). From (4), the expected costs at time 1 for a risk of type X = L, H equals

[V.sub.X](q)([d.sub.L],[d.sub.H]) = [p.sub.X](1 - [d.sub.X]) + (1 - [p.sub.X][d.sub.X])f([q.sup.-]) + [p.sub.X][d.sub.X] f([q.sup.+]) = [[alpha].sup.2][q.sup.-] + [[alpha].sup.2] [p.sub.X][d.sub.X]([q.sup.+] - [q.sup.-]) - [p.sub.X][d.sub.X] + [C.sub.x], (8)

in which q = q(0), the updated beliefs ([q.sup.-], [q.sup.+]) = G(q, [d.sub.L], [d.sub.H]) are defined in (1), and the constant [[bar.C].sub.X] = [p.sub.X] + (1 + [theta]) v[p.sub.L] is independent of ([d.sub.L], [d.sub.H]).

Uniqueness of Equilibrium

Our first result shows that at most one equilibrium ([d.sup.*.sub.L], [d.sup.*.sub.H]) satisfying (5) exists in this case.

Lemma 1: The function [V.sub.L] is concave in the reporting strategy [d.sub.L]; the function [V.sub.H] is convex in the reporting strategy [d.sub.H].

Proof: We directly compute


for some constants [C.sub.i][greater than or equal to] 0, which is a strictly decreasing function of [d.sub.L]. Similarly,


for some other [C.sub.i][greater than or equal to] O. Thus, [partial derivative][V.sub.H]/[partial derivative][d.sub.H] is a strictly increasing function of [d.sub.H]. Q.E.D.

The aforementioned lemma immediately implies the following key theorem.

Theorem 1: The optimal reporting strategy for low risks satisfies [d.sup.*.sub.L][member of] {0,1}. More precisely, by comparing [V.sub.L] at [d.sub.L]= 0 with the cost at [d.sub.L] = 1, we obtain that low risks report all claims, namely, [d.sup.*.sub.L] = 1, if and only if


Thus, for the low-risk types, pure reporting strategies (either always report or never report) are the only optimal strategies. On the other hand, since [V.sub.H] is convex, typically [d.sup.*.sub.H] will be strictly between zero and one, so high-risk types seek a randomized strategy.

Corollary 1: In equilibrium, the high-risk buyer reports some of his losses, that is, [d.sup.*.sub.H] > 0.

Proof: The possibility that [d.sup.*.sub.L] = 0 and [d.sup.*.sub.H] = 0 is disallowed since in that case, no claims are ever reported and the insurer has no mechanism to update beliefs. In particular, [q.sup.+] is undefined, [q.sup.-] = q, and (11) holds because the right-hand side is negative when [d.sub.H] = 0, which contradicts [d.sup.*.sub.L] = 0. So, we can assume [d.sup.*.sub.L] > 0, and by evaluating (10), we obtain


Thus, as a function of [d.sub.H], [V.sub.H] is initially decreasing and achieves its minimum at a point strictly greater than 0. Q.E.D.

This corollary relies on the assumption that [p.sub.L] > 0. See the "Limiting Cases" section for study of equilibria in the degenerate case for which [p.sub.L] = 0.

The next two lemmas study the case [d.sup.*.sub.L] = 0 and 0 < [d.sup.*.sub.H] < 1, whereby low risks self-insure during period 1 and high risks employ a mixed reporting strategy.

Lemma 2: The optimal strategy is


if and only if [alpha] > 1 and the initial rating satisfies [??] < q < [alpha] - 1/[alpha] - [p.sub.H], in which


If [??] > [alpha] - 1/[alpha] - [p.sub.H], then there are no prior q's such that [d.sup.*.sub.L] = 0 and 0 < [d.sup.*.sub.H] < 1 is optimal.

Proof: Assuming that optimal [d.sup.*.sub.H] is strictly between 0 and 1 implies that the first-order condition for [V.sub.H] must hold. Therefore,


To satisfy the constraint 0 < [[??].sub.H] < 1, one must have [alpha] > 1 and [alpha]-1/[alpha] < q < [alpha]- 1/[alpha]-[p.sub.H]. At the same time, to satisfy [d.sup.*.sub.L] = 0 at [d.sup.*.sub.H] = [[??].sub.H], we must have


It can be shown that [??] > [alpha]-1/[alpha] always, so that the lower bound for both conditions to hold is [??] < q and the lemma follows. Q.E.D.

Note that for [alpha] - 1 small enough, q > [alpha]-1/[[alpha].sub.H]; therefore, the two conditions aforementioned cannot be simultaneously satisfied, so that the scenario [d.sup.*.sub.L] = 0 and 0 < [d.sup.*.sub.H] < 1 is ruled out.

Lemma 3: If [d.sup.*.sub.L] = 0 and 0 < [d.sup.*.sub.H] < 1, then [W.sub.L] (1,q) = [p.sub.L] - [alpha] + (1 + [theta])v [p.sub.H].

Proof: From (8) with X = L, one can check that [V.sub.L] (1, q)([d.sub.L] = 0, [d.sub.H] = [[??].sub.H]) simplifies to the aforementioned expression. Q.E.D.

Perfect Reporting

We term the case ([d.sup.*.sub.L], [d.sup.*.sub.H]) = (1,1) the first-best outcome since it represents the situation for which the buyer has no incentive to withhold information and, therefore, avoids ex post moral hazard. When [d.sup.*.sub.X] ( < 1, the implicit deductibles are inefficient and are an externality of the information asymmetry; that is, the buyer pays the losses himself because the insurer cannot efficiently monitor his type. Thus, the presence of an equilibrium with [d.sup.*.sub.X] < 1 for either X = L or H is a sign of something "breaking down" in the system. Accordingly, it is important to understand the parameter values that lead to full reporting and avoid such externalities.

Lemma 4: We have that ([d.sup.*.sub.L], [d.sup.*.sub.H]) = (1,1) for any parameter values when q is sufficiently small (close to zero) or sufficiently large (close to one). Moreover, the interval of q's for which ([d.sup.*.sub.L,][d.sup.*.sub.H]) = (1,1):

* Increases in width as the risk loading [theta] decreases.

* Increases in width as the high-type riskiness [p.sub.H] decreases.

* Increases in width as the low-type riskiness [p.sub.L] increases.

Proof: In order for ([d.sup.*.sub.L], [d.sup.*.sub.H]) = (1,1), it is necessary and sufficient that (11) hold with [d.sub.H] = 1 and that (10) be nonpositive at ([d.sub.L], [d.sub.H]) = (1,1). By substituting ([d.sub.L], [d.sub.H]) = (1,1) into (11) and (10), these two conditions become, respectively,

1 [greater than or equal to] [I.sub.L], and 1 [greater than or equal to] [I.sub.H], (13)

in which

[I.sub.L] = [[alpha].sup.2]q(1-q)/1 - [p.sub.H]q[(1 - [p.sub.L]/)(1-[p.sub.H])/1 - [p.sub.L] - ([p.sub.H] - [p.sub.L])q + [p.sub.H] - [p.sub.L](1 - [p.sub.H])/[p.sub.L] + ([p.sub.H] - [p.sub.L])q], (14)


[I.sub.H] = [[alpha].sup.2][(1 - q).sup.2][(1 - [p.sub.L])2/[[1 - [p.sub.L] - ([p.sub.H] - [p.sub.L])q].sup.2] - [p.sup.2.sub.L]/[[p.sub.L] + [([p.sub.H] - [p.sub.L])q].sup.2]

It is an easy check that as q [right arrow] 0 + or q [right arrow] 1-, [I.sub.L] and [I.sub.H] converge to zero; therefore, because [I.sub.L] and [I.sub.H] are continuous functions of q, the inequalities in (13) are satisfied for q ~ 0 and q ~ 1. Q.E.D.

Moreover, since [[sigma].sup.2] = (1 + [theta])v([p.sub.H] - [p.sub.L]) is a linear multiplier in [I.sub.L] and [I.sub.H], smaller values of [theta] decrease [I.sub.L] and [I.sub.H], thereby, making the inequalities in (13) more likely. Similarly, one can show that the derivatives of [I.sub.L] and [I.sub.H] are positive with respect to [p.sub.H] and negative with respect to [p.sub.L] leading to the comparative statics stated in the lemma.

The economic intuition behind these results is that when q is very small or very large, the gain from nonreporting is minimal since it is very hard to change the beliefs of the insurer (i.e., [q.sup.-], [q.sup.+] are very close to q); therefore, full reporting is best. This is consistent with empirical observations that show that very low-risk and very high-risk buyers typically report all their accidents since they "have nothing to lose." For instance, in the Dutch auto insurance study by Abbring, Chiappori, and Zavadil (2008), the premia of the safest drivers (experience rating 20) and of the worst drivers (experience rating 1) do not increase after an accident; thus, there is essentially no gain from underreporting.

Similarly, if the risk loading [theta] is relatively small, the benefit from nonreporting (through lowering one's future premium) is not enough to outweigh the cost that comes from concealing information about one's type, so full reporting is optimal. Also, if the difference [p.sub.H] - [p.sub.L] shrinks, then there is less opportunity to distinguish high and low types and, therefore, less gain from manipulating [q.sup.[+ or -]] via nonreporting.

Our second result, which uses the aforementioned proof, shows that the buyer only has incentive to settle out-of-pocket (self-insure) when the risk loading is sufficiently large.

Corollary 2: ([d.sup.*.sub.L], [d.sup.*.sub.H]) = (1,1) for all q [member of] [0,1] whenever [[alpha].sup.2] (1 +O)v(pH -- PL) [less than or equal to] 1.

Proof: [I.sub.L] in (14) is a decreasing function of Pc and is an increasing function of [alpha].

Moreover, when [p.sub.L] = 0 and [alpha] = 1, [I.sub.L] equals 1-q/1-[p.sub.H]q[q(1-[p.sub.H])/1-[p.sub.H]q + 1], which is less than or equal to 1 because q(1-[p.sub.H])/1 - [p.sub.H]q[less than or equal to] 1 - [p.sub.H]q/1 - q - 1 = q(1 - [p.sub.H])/1 - q.

Similarly, [I.sub.H] in (15) is increasing in [p.sub.H], and by taking [p.sub.H] = 1 and [alpha] = 1, we find that

[I.sub.H] = 1 - [p.sup.2.sub.L][(1-q).sup.2]/[(q + [p.sub.L](1 - q)).sup.2][less than or equal to] 1.

Therefore, both equalities in (13) are satisfied whenever [alpha] [less than or equal to] 1, so [d.sup.*.sub.L] = [d.sup.*.sub.H] = 1 in that case.

The sufficient condition in Corollary I is consistent with the intuition that nonreporting is a form of market breakdown and should only occur if the market is sufficiently inefficient (i.e., insurance is mandatory but carries an excessive risk loading).

The set of q's for which both inequalities in (13) are satisfied can consist of two or three disjoint intervals. Figure 2 illustrates this phehomenon. We see that the second inequality in (13), namely, 1 [greater than or equal to] [I.sub.H], dominates for small q and the first inequality, namely, 1 [greater than or equal to] [I.sub.L], dominates for large q. That is, there is [] such that [d.sup.*.sub.H] = 1 and [d.sup.*.sub.L] - 1 for q [member of] [0, []], and just aforementioned q = [] we switch to [d.sup.*.sub.L] = 1 and [d.sup.*.sub.H] < 1. Similarly, there is a [bar.q] such that for q [member of] [[bar.q], 1], we again have [d.sup.*.sub.L] = 1 and [d.sup.*.sub.H] = 1, and just below q = [bar.q] we switch to [d.sup.*.sub.L] = 0 and [d.sup.*.sub.H] = 1.



Additionally, for some values of [alpha], there may be a third interval of q's in which both inequalities in (13) are satisfied. This is illustrated when [alpha] = 1.215 for q near 0.33 in Figure 2 and is also illustrated in the second panel of Figure 3 for [alpha] = 1.215. Note that the first inequality in (13) is cubic in q and the second is quadratic in q, so that although analytic expressions exist for [] and [bar.q] above, they are quite complicated. Q.E.D.


A variety of reporting strategies are possible in the simple two-period model from the previous section.

1. Full reporting: [d.sup.*.sub.L] = 1 and [d.sup.*.sub.L] = 1, that is, all buyers report all accidents. This equilibrium holds if the cost from self-imposed deductibles outweighs any gain from information manipulation. Since information sensitivity is small for q's close to 0 or 1, in particular, this outcome happens for buyers whose risk type is essentially known a priori as demonstrated in Lemma 1.

2. Obscuring high risk: [d.sup.*.sub.L] = 1 and [d.sup.*.sub.H] < 1, that is, low-risk buyers report all losses, and some of the losses generated by high risks are self-absorbed. The effect of such a policy is to reduce the information gain from observing a claim as the rate of reporting losses [p.sub.X][d.sub.X] is similar for both risks.

3. Obscured signaling: [d.sup.*.sub.L] = 0 and 0 < [d.sup.*.sub.H] < 1, that is, low-risk buyers absorb all losses, and some of the insurable events of the high-risk buyers are also not reported. The effect is that a reported claim is a clear signal that the buyer is a high risk. However, due to a large proportion of out-of-pocket settlements, losses are filed very infrequently. Such a policy may arise if the expected out-of-pocket costs are small for low-risk types relative to the gain of moving [q.sup.-].

4. Clear signaling: [d.sup.*.sub.L] = 0 and [d.sup.*.sub.H] = 1, that is, low risks report no losses, whereas high risks report all losses. Again, a reported loss is a clear signal that the buyer is a high risk. It may occur for medium values of q at which the accident cost for a high risk is sufficiently large to make under-reporting of Case 3 above suboptimal.

5. No equilibrium (No-Eqm): There is no pair ([d.sup.*.sub.L], [d.sup.*.sub.H]) that satisfies the consistency condition (5). Thus, in this case Bayesian learning essentially fails since there is no reporting strategy that is rationally optimal. In our numerical experiments, we observe that No-Eqm arises as a transition phase between Cases 2 and 3 aforementioned. In the transition from Case 2 to Case 3, both [d.sup.*.sub.L] and [d.sup.*.sub.H] change simultaneously, with the former changing discontinuously, which is impossible.

Due to the discrete nature of [d.sup.*.sub.L], as model parameters vary, the equilibrium strategy exhibits instantaneous phase transitions whence an infinitesimal change in a parameter causes a shift from one type of equilibrium to another. In particular, negligible changes in the prior q can lead to radically different equilibria. In the next subsection, we investigate these transitions more closely.

In the next lemma, we show that even though high risks might attempt to mimic low risks by settling some losses out-of-pocket, as in Cases 2 and 3, they always appear more risky to the insurer. This is intuitive since one expects that reporting a loss is a signal of riskiness, rather than its opposite, so that [q.sup.+] > q.

Lemma 5: The reporting frequency observed by the insurer pxdx maintains the risk order, that is, [p.sub.L][d.sup.*.sub.L] < pH[d.sup.*.sub.H].

Proof: We have two cases to consider: (1) [d.sup.*.sub.L] = 1, and (2) [d.sup.*.sub.L] = 0. If [d.sup.*.sub.L] = 1, then by substituting [d.sub.L] = 1 and [d.sub.H] = [p.sub.L]/[p.sub.H] into (10), we obtain

[partial derivative][V.sub.H]/[partial derivative][d.sub.H] = [p.sub.H] < 0,

which implies that [d.sup.*.sub.L] > [p.sub.L]/[p.sub.H] by convexity of [V.sub.H]. It follows that [p.sub.H][d.sup.*.sub.H] > [p.sub.L] = [p.sub.L] [d.sup.*.sub.L][d.sup.*.sub.L]

We know from Corollary i that [d.sup.*.sub.H] > 0, so that if [d.sup.*.sub.L] = 0, then we automatically have pH [d.sup.*.sub.L] > [p.sub.L][d.sup.*.sub.L].

Note that in the case for which [d.sup.*.sub.L] = 0, it is possible to have [p.sub.H][d.sup.*.sub.H] < [p.sub.L]. Indeed, by substituting [d.sub.H] = [[??].sub.H] and [d.sub.L] = 0 into (10), we obtain [partial derivative][V.sub.H]/[partial derivative][d.sub.H] = 0. Thus, [p.sub.H][d.sup.*.sub.H] = [p.sub.H][[??].sub.H] = [alpha] - [alpha]-1/q. It is possible that [alpha] - [alpha]-1/q < [p.sub.L]; for example, take [alpha] = 1.5, q = 0.5, and [p.sub.L] = 0.8. Q.E.D.

Classification of Phase Transitions According to [alpha]

In our numerical work, we find four possible types of phase transitions, three of which appear in Figure 3 (the phase labels refer to the classification in the beginning of the "Classification of Optimal Strategies in the Case Study" section):

1. [alpha] [less than or equal to] 1: Corollary I implies that the equilibrium is always Case 1, full reporting for all risk types. This also happens for [alpha] - 1 being small, for example, [alpha] = 1.05 in the top panel of Figure 3;

2. [alpha] [greater than or equal to] 1 small (e.g., [alpha] = 1.15): 1 [right arrow] 4 [right arrow] 1 (high risk always reports, no randomized strategies);

3. [alpha] [greater than or equal to] 1 moderate (e.g., [alpha] = 1.215): 1 [right arrow] 2 [right arrow] 1 [right arrow] 5 [right arrow] 3 [right arrow] 4 [right arrow] 1 (second panel);

4. [alpha] [greater than or equal to] 1 large (e.g., [alpha] = 2): 1 [right arrow] 2 [right arrow] 5 [right arrow] 3 [right arrow] 4 [right arrow] 1 (third and fourth panels).

Limiting Cases

In this section, we study a variety of limiting cases with respect to the parameters of the model. Recall that insurance is mandatory in our model, so that it is conceivable that some buyers may, in fact, be riskless [p.sub.L] = 0 or surely risky [p.sub.H] = 1. As long as 0 < [p.sub.X] < 1 for at least one type X, the insurance market (and buyers' strategies) will remain nontrivial.

[p.sub.L] = 0. In this case, low-risk types never have accidents; accordingly, [d.sup.*.sub.L] is indeterminate. Indeed, when [p.sub.L] = 0, [V.sub.L] = v(1 + [theta])[p.sub.H](1 - [p.sub.H][d.sub.H])q/(1 - [p.sub.H][d.sub.H]q), which is independent of [d.sub.L]. Although there is no unique value of [d.sub.L] that minimizes [V.sub.L] when [d.sub.H] = [d.sup.*.sub.H], we still say that an equilibrium exists with [d.sup.*.sub.H][member of] [0,1] equal to the minimizer of [V.sub.H] because the (intended) actions of the low risks are irrelevant when [p.sub.L] = 0.

It is easy to show that [d.sub.H] = [[??].sub.H] = 1-[alpha](1-q)/q[p.sub.H] minimizes [V.sub.H] when [p.sub.L] = 0. By the possible values of [[??].sub.H], we obtain the following:

1. If q [less than or equal to][alpha]-1/[alpha], then [[??].sub.H] [less than or equal to] 0, which implies that [d.sup.*.sub.H] = 0;

2. If [alpha]-1/[alpha] < q < [alpha]-1/[alpha]-[p.sub.H], then 0 < [[??].sub.H] < 1, which implies that [d.sup.*.sub.H] = [[??].sub.H];

3. If q [greater than or equal to] [alpha]-1/[alpha]-[p.sub.H], then [[??].sub.H] [greater than or equal to] 1, which implies that [d.sup.*.sub.H] = 1.


No discontinuous phase transitions are present because [V.sub.H] is continuous in q. See the illustration in the top panel of Figure 4. Note that the case [p.sub.L] = 0 is not the limit of [p.sub.L] [right arrow] 0; in particular, for [alpha] large enough and any [p.sub.L] = [member of] > 0 there exists an interval ([], [bar.q]) with length strictly bounded away from zero for which No-Eqm is the only solution. In contrast for [p.sub.L] = 0 equilibrium exists for all q [member of] [0,1].

[p.sub.L] = [p.sub.H]. In this case, [partial derivative][V.sub.L]/[partial derivative][d.sub.L] = [p.sub.L] < 0, and [partial derivative][V.sub.H]/[partial derivative][d.sub.H] = - [p.sub.H] < O. Thus, [d.sup.*.sub.L] = 1 = [d.sup.*.sub.H]. This first-best outcome makes sense because there is no reason not to report a loss due to the fact that there is no difference between the risk classes.

[p.sub.H] = 1. In this case, high-risk types are guaranteed to have claims. This accentuates ex post moral hazard because high risks have a strong incentive to hide their claims and "masquerade" as low risks. We find the following:

1. [d.sup.*.sub.L] = 0 and [d.sup.*.sub.H] = [??]H = 1-[alpha](1-q)/q [member of] (0,1) if and only if [alpha] > 1 and [bar.q] < q < 1, in which [??] is given in (12).

2. [d.sup.*.sub.L] = 1 and [d.sup.*.sub.H] = [[??].sub.H] = 1-[alpha](1-q)/q [member of] (0, 1) if and only if (11) holds at [d.sub.H] = [[??].sub.H] and [partial derivative][V.sub.H]/[partial derivative][d.sub.H] > 0 at [d.sub.L] = 1 = [d.sub.H]. The latter condition holds if and only if [alpha] > 1 and q [member of] ([p.sub.L][alpha][square root of [[alpha].sup.2]-1-[[alpha].sup.2]+(1-[p.sub.L])]/[[alpha].sup.2](1-2 [p.sub.L]-[(1- [p.sub.L]).sup.2], 1).

As [alpha] approaches 1, the lower bound on the interval for q converges to 1. The lower bound decreases with respect to [alpha], so that as the risk loading [theta] increases, it is more likely that [d.sup.*.sub.H] < 1 in this case. Also, if [[alpha].sup.2] (1 - 2[p.sub.L]) = [(1 - [p.sub.L]).sup.2], then the interval for q becomes (1-2[p.sub.L]/2(1-[p.sub.L]), 1).

To find the values of q such that (11) holds at [d.sub.H] = [[??].sub.H] requires solving a cubic inequality in q, so although there is an analytical solution for such an inequality, it is not simple.

3. [d.sup.*.sub.L] = 1 = [d.sup.*.sub.H] if and only if [alpha] [less than or equal to] 1 or both [alpha] > 1 and q [member of] 0, [p.sub.L] [alpha][square root of [[alpha].sup.2]-1-[[alpha].sup.2]+(1-[p.sub.L])/[[alpha].sup.2](1-2[p.sub.L]) -[(1 - [p.sub.L]).sup.2][union] {1}.

4. No-Eqm is also possible; see the bottom panel of Figure 4.

Note that when [p.sub.H] = 1, an equilibrium with [d.sup.*.sub.L] = 0 and [d.sup.*.sub.H] = 1 is not possible. Indeed to have it, one needs


that is, [d.sub.L] = 0 is optimal when [d.sub.H] = 1,

that is, [d.sub.H] = 1 is optimal when [d.sub.L] = 0,

which are impossible to satisfy together for any 0 [less than or equal to] q [less than or equal to] 1.

q = 1. As in the previous case, [d.sup.*.sub.L] = 1 because (11) holds when q = 1. Also, [partial derivative][V.sub.H]/[partial derivative][d.sub.H] = - [p.sub.H] < 0 when q = 1 and [d.sub.L] = 1; thus, [d.sup.*.sub.H] = 1. This result makes sense because when q = 1, the updated value [q.sup.-] = 1, so there is no possible gain from not reporting a loss.

[theta] or [alpha] large. Our final result shows that as the risk loading [theta] increases, which makes [alpha] increase, the game breaks down and no equilibrium is possible.

Lemma 6: For any fixed q [member of] (0,1) there is an [alpha] large enough such that there is no equilibrium for the two-period model.

Proof: Fix q [member of] (0,1). If an equilibrium exists, then either [d.sup.*.sub.L] = 0 or [d.sup.*.sub.L] = 1. Suppose [d.sup.*.sub.L] = 0; then, when [d.sub.L] = 0, [partial derivative][V.sub.H]/[partial derivative][d.sub.H] = [p.sub.H] + [p.sub.H][[alpha].sup.2][(1-q).sup.2]/[(1-[p.sub.H][d.sub.H]q).sup.2], which is positive for all [d.sub.H] [member of] [0, 1] for [alpha] > 1/(1 - q), from which it follows that [d.sup.*.sub.H] = 0. This result contradicts Corollary 1, so we cannot have [d.sup.*.sub.L] = 0.

Next, suppose [d.sup.*.sub.L] = 1; by looking at (10), we see that if [alpha] is large enough, then [partial derivative][V.sub.H]/[partial derivative][d.sub.H] is determined by the sign of the expression in the curly brackets. It is easy to check that that expression is negative for [d.sub.H] < [p.sub.L]/[p.sub.H] and positive for [d.sub.H] > [p.sub.L]/[p.sub.H].

Therefore, for [alpha] large, [V.sub.H] achieves its minimum at [d.sub.H] = [p.sub.L]/[p.sub.H] + [member of], in which [member of] [right arrow] 0+ as [alpha] [right arrow] [infinity]. Specifically, [alpha] and [epsilon] are related as follows:

[[alpha].sup.2] = 1/[(1 - q).sup.2][[[(1-[p.sub.L]/1-[p.sub.L]-[member of][p.sub.H]q).sup.2] - [([p.sub.L]/[p.sub.L] + [member of][p.sub.H]q).sup.2]]-.sup.1].

By substituting [d.sub.H] = [p.sub.L]/[p.sub.H] + [member of] into (11), we obtain 1 [greater than or equal to] [[alpha].sup.2]q(1-q)/1-[p.sub.L]q. (1 + O([member of])), which is a contradiction when [alpha] is large. Thus, [d.sup.*.sub.L] = 1 is also impossible. Q.E.D.

Value Functions

By solving for [d.sup.*.sub.L] and [d.sup.*.sub.H] and substituting these values into [V.sub.L] and [V.sub.H], we obtain the value functions [W.sub.L](q) and [W.sub.H](q), respectively. In the case of No-Eqm, [W.sub.L] and [W.sub.H] are undefined, but one can set [W.sub.L] = [W.sub.H] = + [infinity] there, so that individuals avoid having an updated q lie in the region of No-Eqm. This is not important in a two-period model but becomes relevant once more periods are considered.

We find that [W.sub.L] is a nondecreasing continuous function of q; so that for a low-risk type, the higher the original rating, the lower the expected cost to the buyer. Also, recall from Lemma 1 that [W.sub.L] is constant on the set {q : [d.sup.*.sub.L] = 0, [d.sup.*.sub.H] < 1}. Surprisingly, we find that [W.sub.H] is not monotone or continuous with respect to q. In particular, WH experiences a negative jump at [??] where we switch from [d.sup.*.sub.L] = 0, [d.sup.*.sub.L] = 1 to [d.sup.*.sub.L] = 1, [d.sup.*.sub.H] = 1([??] = 0.924 in Figure 5).


The analysis of the case study presented in the previous two sections provides insight into the more general settings of our model. In this section, we discuss how the aforementioned results carry over in general.

General Premium Rules and Utility Functions

First, we consider general premium rules rather than the linear function used in the "Case Study" and "Classification of Optimal Strategies in the Case Study" sections. These rules might arise due to nonlinear pricing mechanisms employed by the insurer. For instance, it is common practice that the premium paid is most elastic for buyers with uncertain risk profiles (medium q's), whereas it is relatively inelastic for buyers that belong to a given type with high probability (see, e.g., table 2 in Abbring, Chiappori, and Zavadil, 2008). Such pricing rules can be represented by an "arctangent'-shaped premium function f(q). Alternatively, insurance regulations might state that high-risk buyers should be penalized for being "careless" (in terms of ex ante moral hazard), so that f(q) would be convex in that case.


A nonlinear premium rule can be viewed as an extension of the linear case, with the risk-loading [alpha] being a function of the current rating q. Consider first a piecewise linear f(q); in that case, since the main problem (5) is optimized locally for each q, one may treat each linear interval of f([q.sup.[+ or -]]) separately and then combine the results. Therefore, the resulting reporting strategies will be a combination of the outcomes observed in the "Case Study" and "Classification of Optimal Strategies in the Case Study" sections. Of course, for some ratings q, depending on the value of [??], [q.sup.[+ or -]] will be on different segments of f, in which case a more complicated equilibrium may emerge. In particular, [V.sub.L] might no longer be concave. As a result, it is possible that 0 < [d.sub.L] < 1 is optimal; also multiple solutions of (5) may exist.

In the case study considered, the buyers were taken as risk neutral. The effect of risk-averse buyers is similar to the effect of nonlinear premium rules. The overall impact on market equilibrium is not clear. On the one hand, risk-averse buyers will be more loath to self-insure because nonreporting increases the variance of their immediate one-period payoff. On the other hand, risk aversion increases the long-term benefit of nonreporting because the buyer will be more sensitive to his experience rating.

More Than Two Periods

Nonlinear pricing rules also arise in the case of a multiperiod model with more than two periods. In that case, the next-period value function [V.sub.X](k + 1, x), as well as the next period premium f(q(k), [??](k + 1)), enter into the optimization problem. Even if f itself is linear, the nonlinear features in [??] (k + 1) (which arise as soon as nonreporting is optimal for some q's), and in [V.sub.X] make the "effective" next-period premium nonlinear in current rating. This can be seen in Figure 5, where the value functions are highly nonlinear, including points of discontinuity and intervals of constancy.

With more than two periods, the buyer has a stronger incentive to manipulate his rating early on, since he anticipates that it will affect his expected costs for several periods into the future. For instance, with T = 3 and a small [alpha] = 1.1, the solution structure is that [d.sup.*.sub.X](2) = 1 for X = L and H, but [d.sup.*.sub.X](1) belongs to a variety of equilibrium phases with the overall picture similar to the third and fourth panels of Figure 3.

A multiperiod problem also requires considering the precise interpretation of what happens in the case of No-Eqm or multiple solutions to (5). An orthodox solution is to set V to be undefined in such cases; this would rule out solutions for nearly all q after several periods, since the nondefined regions expand as more periods are added because at each step all possible updates [q.sup.[+ or -]] must be inside the defined region.

One may also consider the infinite-horizon problem, where (4) is replaced by the time-stationary version


Heuristically, one expects that W(k, q) [right arrow] H(q) as k [right arrow] [infinity]. However, given the complications arising already with two or three periods, this does not appear to be a feasible way of computing H. Also, on an infinite horizon, the learning problem of the insurer must be treated differently. Indeed, given a sufficiently large claim history and an equilibrium solution to the insurance reporting problem, the insurer can determine the risk type of the buyer to an arbitrary degree of precision (since she can infer his reporting strategy). In other words, q(k) [right arrow] [1.sub.X] as k [right arrow] [infinity] almost surely. Therefore, with an infinite horizon (and hence infinite past history), the insurer will, in fact, know the true type of the buyer with certainty.

Other Extensions

Many other cases can be considered. The buyers can belong to multiple risk types, in which case the insurer's beliefs are not one-dimensional as in case study but live in the general space D. With multiple risk types and a two-period setting, we still expect the lowest risk type either to always report or to never report losses in the first period, with the same ordering of the reporting frequency [p.sub.X][d.sub.X] that we obtained in Lemma 1.

The claim sizes [Y.sub.i] may also be varying; under the assumption that the buyer must either report the full amount or nothing at all (so that underreporting is not possible), this becomes a variation of our model in which the claim probabilities [p.sub.X] are modified by the claim size distribution [f.sub.X](Y) and reporting strategy [??](y) depends on the claim size. In that case, the update rule (1) becomes


The resulting problem is now infinite-dimensional in the buyer's strategy [d.sub.X](y), X [member of] E, y [member of] [R.sub.+], and we will study this problem in future work. Intuitively, higher claims are more costly to self-absorb; therefore, we expect a self-imposed deductible Ded(q) to emerge such that [d.sub.X](q, y) = 1 for all y > Ded(q) and [d.sub.X](q, y) < 1 for y [less than or equal to] Ded(q).

Finally, we have assumed that the insurer has zero knowledge that an underreporting has occurred and has no recourse to auditing. In practice it may be that the insurer receives some signals regarding actual loss, similar to the fraud signals recently studied in Dionne, Giuliano, and Picard (2009).


In this article, we studied an asymmetric information insurance problem with ex post moral hazard. Since the buyer has the option of not reporting losses, and since his premia are based on his experience rating, the buyer will strategically manipulate the claim information received by the insurer. Equilibrium reporting strategies depend on the joint behavior of all risk types, and upon the assumption that the insurer updates the risk ratings via Bayesian conditioning and anticipates the optimal reporting strategies when setting the premia. This Bayesian updating is crucial in our two-period model, since in the second period, the buyer has no incentive to underreport. Thus, it is imperative for the insurer to properly estimate the buyer's type in order to charge an accurate second-period premium. The same situation would persist in a multiperiod (but finite) model: as time progresses, the buyer is less likely to underreport and the insurer must be vigilant against this vanishing ex post moral hazard. The case of infinite horizon where this phenomenon would not be present is left to future work.

We find that even in a simple case study with two periods, linear risk premia, and risk-neutral (but captive) buyers, nontrivial reporting strategies might be optimal. In particular, the behavior of different risk types can be quite varied, ranging from total nonreporting, full reporting, and mixed strategies. This analysis provides probabilistic underpinnings to anecdotal evidence of ex post moral hazard in real-life private automobile and homeowner's insurance markets.

The emergence of mixed reporting strategies can be viewed as controversial for a realistic buyer of, say, car insurance. However, it is consistent with (1) unknown buyer types from the insurer's perspective and (2) strategic awareness by buyers of others' actions. Both of these features would disappear with pure strategies that are immediately discernible to the insurer and cancel any interaction between different risk types. The need for mixed strategies could be removed by allowing partial reporting of random claim sizes but would then lead to an infinite-dimensional optimization problem.

We also find that the equilibrium strategies are highly sensitive to the buyer's current risk rating and exhibit abrupt phase shifts as the risk rating varies. In particular, the fragility of equilibrium existence and corresponding phase even under mixed strategies showcases major challenges arising from ex post moral hazard under information asymmetry. This challenge is very real, as testified by the widespread evidence of strategic underreporting and complicated experience-rating schemes. It would be desirable to investigate whether such effects can be mitigated in more complex theoretical models, for example, with partially separating contract menus or further bonus-malus mechanisms.

DOI: 10.1111/j.1539-6975.2010.01366.x


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Michael Ludkovski is in the Department of Statistics and Applied Probability, University of California. Virginia R. Young is in the Department of Mathematics, University of Michigan. The authors can be contacted via e-mail:, and, respectively. We are grateful to the editor and the anonymous referees for their constructive feedback on the previous versions of the article that have greatly strengthened our presentation. We also thank Ivar Ekeland for a very useful discussion at an early stage of our research.

(1) We make the standing assumption that the insurer is a she and the buyer is a he.

(2) If [p.sub.L] = [p.sub.H] then the two risk types are the same, thereby making the problem degenerate.
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Author:Ludkovski, Michael; Young, Virginia R.
Publication:Journal of Risk and Insurance
Article Type:Report
Date:Dec 1, 2010
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