Abstract
We survey recent developments in the economic analysis of insurance fraud. This chapter first sets out the two main approaches to insurance fraud that have been developed in the literature, namely the costly state verification and the costly state falsification. Under costly state verification, the insurer can verify claims at some cost. Claims’ verification may be deterministic or random, and it can be conditioned on fraud signals perceived by insurers. Under costly state falsification, the policyholder expends resources for the building-up of his claim not to be detected. We also consider the effects of adverse selection, in a context where insurers cannot distinguish honest policyholders from potential defrauders, as well as the consequences of credibility constraints on antifraud policies. Finally, we focus attention on the risk of collusion between policyholders and insurance agents or service providers.
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Notes
- 1.
See the chapter by Georges Dionne in this book on empirical evidence about insurance fraud.
- 2.
Note that a claimant is not fraudulent if he relies in good faith on an erroneous valuation of an apparently competent third party—see Clarke (1997). However, insurance may affect fraud in markets for credence goods, i.e., markets where producers may provide unnecessary services to consumers who are never sure about the extent of the services they actually need. See Darby and Karni (1973) on the definition of credence goods and Dionne (1984) on the effects of insurance on the possibilities of fraud in markets for credence goods.
- 3.
- 4.
Dominance is in a Pareto-sense with respect to the expected utility of the policyholder and to the expected profit of the insurer.
- 5.
If both payments were equal, then it would be welfare improving not to audit the corresponding level of loss in the verification region and simultaneously to decrease the premium. Note that Lemma 1 could be presented as a consequence of the Revelation Principle (see Myerson 1979).
- 6.
In fact, the policyholder would never increase the damage if and only if t(x) − x were nonincreasing over \([0,\overline{x}]\). Given that t(x) is nondecreasing (see Lemma 2), this no-manipulability condition implies that t(x) should be continuous. Note that extra damages can be made either deliberately by the policyholder (arson is a good example) or, thanks to a middleman, such as a car repairer or a health-care provider. In such cases, gathering verifiable information about intentional overpayment may be too time consuming to the insurer. See Bourgeon and Picard (1999) on corporate fire insurance when there is a risk of arson.
- 7.
On collusion between physicians and workers, see the analysis of workers’ compensations by Dionne and St-Michel (1991) and Dionne et al. (1995). See Derrig et al. (1994) on empirical evidence about the effect of the presence of an attorney on the probability of reaching the monetary threshold that restricts the eligibility to file a tort claim in the Massachusetts no-fault automobile insurance system. In the Tort system, Cummins and Tennyson (1992) describe the costs to motorists experiencing minor accidents of colluding with lawyers and physicians as the price of a lottery ticket. The lottery winnings are the motorist’s share of a general damage award.
- 8.
The CARA assumption eliminates wealth effects from incentive constraints.
- 9.
The payment R(. ) is net of standard audit cost c a.
- 10.
- 11.
Picard (2000) shows that allowing for audit cost manipulation (i.e., e > 0) at equilibrium is a weakly dominated strategy for the insurer.
- 12.
See also Maggi and Rodriguez-Clare (1995).
- 13.
- 14.
When β is negative, the optimal coverage schedule is equivalent to a deductible \(m = -\beta /\alpha\) with a coinsurance provision for larger losses, i.e., \(T(y(x)) = \mathrm{Sup}\{0,\alpha (y - m)\}\).
- 15.
Crocker and Morgan assume that the insurer can observe whether a loss occurred or not. Hence, there may be falsification only if x > 0.
- 16.
There are some minor differences between the Crocker–Morgan setting and ours. They are not mentioned for the sake of brevity.
- 17.
The second-order condition for incentive compatibility requires y(x) to be monotonically increasing. If the solution to the less constrained problem satisfies this monotonicity condition, then the optimal allocation is characterized as in Proposition 6. See Crocker and Morgan (1997) for a numerical example. If this is not the case, then the optimal allocation entails bunching on (at least) an interval \(({x}^{{\prime}},{x}^{{\prime\prime}}) \subset [0,\overline{x}]\), i.e., \(y(x) =\hat{ y},t(x) =\hat{ t}\) for all x in (x ′, x ′ ′). In such a case, the coverage schedule T(y) that sustains the optimal allocation is not differentiable at \(y =\hat{ y}\).
- 18.
The Revelation Principle does not apply anymore if the maximal penalty also depends on the claim \(\hat{x}\). In such a case, there may be false report at equilibrium.
- 19.
Under this interpretation, it may be more natural to assume that the policyholder should pay the penalty B in addition to the premium P, since the latter is usually paid at the beginning of the time period during which the insurance policy is enforced. In fact, both assumptions are equivalent when the policyholder is affected by a liquidity constraint. Indeed, in such a case, it would be optimal to fix the insurance premium P at the largest possible level (say \(P = \overline{P}\)) and to compensate adequately the policyholder by providing large insurance payments t N and t A unless a fraudulent claim is detected by audit. This strategy provides the highest penalty in case of fraud, without affecting equilibrium net payments t N − P and t A − P. If the law of insurance contracts specifies a penalty \(\hat{B}\) to be paid in case of fraudulent claim, we have \(P - t_{\mathrm{A}}(x,\hat{x}) \leq \overline{P} +\hat{ B}\) which corresponds to (13.25) with \(B(x) \equiv \overline{P} +\hat{ B}\).
- 20.
In a more realistic setting, there would be several reasons for which imposing maximal penalties on defrauders may not be optimal. In particular, audit may be imperfect so that innocent individuals may be falsely accused. Furthermore, a policyholder may overestimate his damages in good faith. Lastly, very large fines may create incentives for policyholders caught cheating to bribe the auditor to overlook their violation.
- 21.
Deterministic auditing may be considered as a particular case of random auditing where p(x) = 1 if x ∈ M and p(x) = 0 if x ∈ M c, and Lemma 1 may be obtained as a consequence of the incentive compatibility conditions (13.28). If \(x,\hat{x} \in {M}^{\mathrm{c}}\), (13.28) gives \(t_{\mathrm{N}}(x) \geq t_{\mathrm{N}}(\hat{x})\). Interverting x and \(\hat{x}\) gives \(t_{\mathrm{N}}(\hat{x}) \geq t_{\mathrm{N}}(x)\). We thus have t N(x) = t 0 for all x in M c. If x ∈ M and x ∈ M c, (13.28) gives \(t_{\mathrm{A}}(x) \geq t_{\mathrm{N}}(\hat{x}) = t_{0}\). If t A(x) = t 0 for x ∈ [a, b] ⊂ M, then it is possible to choose p(x) = 0 if x ∈ [a, b], and to decrease P, the other elements of the optimal contract being unchanged. The policyholder’s expected utility would increase, which is a contradiction. Hence t A(x) > t 0 if x ∈ M.
- 22.
See Fagart and Picard (1999).
- 23.
Technically, this rules out the possibility of taking up the differential approach initially developed by Guesnerie and Laffont (1984) and widely used in the literature on incentives contracts under adverse selection.
- 24.
Let \(\overline{U}(x) = [1 - p(x)]U(W - P - x + t_{\mathrm{N}}(x)) + p(x)U(W - P - x + t_{\mathrm{A}}(x))\) be the expected utility of a policyholder who has incurred a loss x. Using p(m) = 0 shows that \(\overline{U}(x)\) is continuous at x = m.
- 25.
This asymmetric information problem may be mitigated in a repeated relationship framework.
- 26.
See also Boyer (1999) for a similar model.
- 27.
For the sake of simplicity, we assume that no award is paid to the insurer when an opportunist is caught cheating. The fine B is entirely paid to the government.
- 28.
α = 0 is an optimal strategy for opportunists when \(p =\tilde{ p}(t,P)\) and it is the only optimal strategy if \(p =\tilde{ p}(t,P)+\varepsilon,\varepsilon >0\).
- 29.
We assume that (t c, P c) is a singleton.
- 30.
Proposition 10 shows that a pooling contract is offered at equilibrium: there does not exist any separating equilibrium where honest and opportunist individuals would choose different contracts. This result is also obtain by Boyer (1999) in a similar framework.
- 31.
Opportunists cannot benefit from separating and (t c, P c) is the best pooling contract for honest individuals.
- 32.
We have \(\tilde{\sigma }= 1\) if all honest policyholders choose \((\tilde{t},\tilde{P})\) and \(\tilde{\sigma }= \frac{2\theta } {\theta +1}\) if \((\tilde{t},\tilde{P})\) and (t c, P c) are equivalent for honest policyholders.
- 33.
Poverty may also affect morality. In particular, moral standards may decrease when the economic situation worsens. Dionne and Wang (2013) analyze the empirical relationship between opportunistic fraud and the business cycle in the Taiwan automobile theft insurance market. They show that fraud is stimulated during periods of recession and mitigated during periods of expansion.
- 34.
See also Dean (2004) on the perception of the ethicality of insurance claim fraud.
- 35.
See also Strutton et al. (1994) on how consumers may justify inappropriate behavior in market settings.
- 36.
The perceived unfairness factor is comprised of items related to the perception of unfair business practice, for instance, because the insurer is overcharging or because ACB is nothing but retaliation against some inadequate practice or because of weak business performance. Other factors are labeled evaluation (loading variables relating to the easiness to engage in ACB or to the general attitude toward ACB), social participation (with variables representing the social external encouragement to ACB), and consequence (measuring the extent to which the outcomes of ACB are seen as beneficial or harmful).
- 37.
See Bourgeon and Picard (2012) for a model where policyholder’s moral standards depend on the attitude of insurers who may nitpick claims and sometimes deny them if possible.
- 38.
See Andreoni et al. (1998) for a survey on tax compliance.
- 39.
Cummins and Tennyson (1994) analyze liability claims fraud within a model without Stackelberg advantage for insurers: each insurer chooses his fraud control level to minimize the costs induced by fraudulent claims.
- 40.
We assume t > c and we neglect the case c = c1 (t,σ). See Picard (1996)for details.
- 41.
As shown by Boyer (1999), when the probability of auditing is strictly positive at equilibrium (which occurs when θ is large enough), then the amount of fraud (1 −δ) \({\theta \alpha }^{\mathrm{n}}({t}^{\mathrm{n}},{P}^{\mathrm{n}},\theta ) =\delta c/({t}^{\mathrm{n}} - c)\) does not depend on θ. Note that t n does not (locally) depend on θ when c < c 1(t n, θ).
- 42.
We assume that (t n, P n) is a singleton.
- 43.
It can be shown that t n > L when there is some audit at equilibrium, that is, when θ > θ ∗. Boyer (2004) establishes this result in a slightly different model. Intuitively, increasing t over L maintains the audit incentives at the right level for a lower fraud rate π, because we should have π t = c for \(p =\tilde{ p}(t,P) \in (0, 1)\) to be an optimal choice of insurers. In the neighborhood of t = L, an increase in t only induces second-order risk-sharing effects, and ultimately that will be favorable to the insured.
- 44.
As before, the optimal contract maximizes the expected utility of honest policyholders under the constraint \(P \geq \hat{ {C}}^{\mathrm{c}}(t,P,\theta )\), where θ still denotes the proportion of opportunist individuals in the population. If the optimal contract without fraud signal is such that \(\delta [t +\tilde{ p}(t,P)c\frac{q_{2}^{\mathrm{n}}} {q_{2}^{\mathrm{f}}} ] < t[\delta +\theta (1-\delta )] <\delta [t +\tilde{ p}(t,P)c],\) then auditing claims is optimal only if the insurer can condition his decision on the fraud signal.
- 45.
As in Dionne et al. (2009), s may be a k-dimensional signal, with k the number of fraud indicators (or red flags) observed by the insurer. Fraud indicators cannot be controlled by defrauders and they may make the insurer more suspicious about fraud. For instance, when all indicators are binary, then ℓ = 2k and s may be written as a vector of dimension k with components 0 or 1: component j is equal to 1 when indicator j is “on,” and it is equal to 0 when it is “off.”
- 46.
Of course if q i n = 0 and q i f > 0, then it is optimal to trigger an audit when s = s i because the claim is definitely fraudulent in that case.
- 47.
In the present model, insurers fully deter fraud when they can commit to their auditing strategy and the proportion of opportunist individuals is large enough. This is no longer true when there is a continuum of types for individuals. Dionne et al. (2009) consider such a model, with a continuum of individuals and moral costs that may be more or less important. In their model, there is a positive rate of fraud even if insurers can commit to their audit strategy. π a would then correspond to the equilibrium fraud rate, which is positive, but lower than the equilibrium fraud rate under the no-commitment hypothesis.
- 48.
If ℓ = 2 and deterring fraud is optimal, then we have i ∗ = 2 if \(q_{2}^{\mathrm{f}} \geq \tilde{ p}(t,P)\) and i ∗ = 1 if \(q_{2}^{\mathrm{f}} <\tilde{ p}(t,P)\).
- 49.
This case has been studied by Schiller (2006).
- 50.
Moral hazard is ruled out because there is no significant effect of replacement cost endorsements on partial thefts (i.e., thefts where only a part of the car is stolen: hubcaps, wheels, radio, etc.) although the same self-protection activities affect the claims distribution of total and partial thefts. Dionne and Gagné (2002) also rule out adverse selection because the effect is significant for only 1 year of ownership and not for all years.
- 51.
Bourgeon and Picard (1999) also consider stochastic mechanisms in which the restoration of damaged assets is an option given by the insurance contract to the insurer but not always carried out at equilibrium. The (randomly exercised) restoration option is used as a screening device: larger indemnity payments require larger probabilities of restoration, which prevents firms with low economic losses from building up their claims.
- 52.
- 53.
The choice of distribution system affects the cost to the insurers of elicitating additional promotional effort of their sales force. For instance, exclusive representation prevents the agents from diverting potential customers to other insurers who pay larger commissions. Likewise giving independent agents ownership of policy expirations provides incentives for agents to expend effort to attract and retain customers—see Kim et al. (1996).
- 54.
Modelling promotional effort in an independent agency system would be more complex since, in such a system, the agent’s decisions are simultaneously affected by several insurers.
- 55.
Rejesus et al. (2004) use indicators of anomalous outcomes. Some of them are applicable to the three types of agents (e.g., the indemnity/premium ratio); others are specific to agents (e.g., the fraction of policies with loss in the total number of policies sold by the agent) or to adjusters (e.g., the indemnity per claim for the adjuster divided by average adjusted claims in the county).
- 56.
See also Brundin and Salanié (1997).
- 57.
Bourgeon et al. (2008) also consider the case of common affiliation in which insurers choose the same provider as their unique referral, and the case of asymmetric affiliation in which one insurer is affiliated with one single provider while customers of the other insurer are free to call in the provider they prefer.
- 58.
Indeed, under nonexclusive affiliation, if there is only one insurance company (the one that has detected collusion) that excludes the defrauder from its network or that switches to collusion-proof contracts, then insureds will move to its competitor and the malevolent provider will not be affected.
- 59.
Alger and Ma (2003) do not obtain the same result when the insurer can use menus of contracts.
- 60.
- 61.
This proof follows Bond and Crocker (1997).
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Picard, P. (2013). Economic Analysis of Insurance Fraud. In: Dionne, G. (eds) Handbook of Insurance. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0155-1_13
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