Screening 110320 JPE

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Active Screening in Insurance Markets

Shinichi Kamiya

Nanyang Business SchoolNanyang Technological University

Nanyang Avenue, Singapore 639798

Mark J. Browne

Wisconsin School of Business

University of Wisconsin-Madison975 University Avenue

Madison, Wisconsin 53706-1323

Submitted to Journal of Public EconomicsMarch 21, 2011

Corresponding author, Email: [email protected], Tel.: +65 6790 5718, Fax: +65 6791 3697


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Active Screening in Insurance Markets

Abstract

In a market characterized by asymmetric information, a party with private informa-

tion may reveal the information through its self-selection, the mechanism of which has

received considerable attention in the academic literature. The party might also reveal

the private information through a revelation test offered by an uninformed party. We

consider competitive insurance markets in which insurers induce information revelation

with self-selection mechanisms and risk classification tests. We demonstrate that con-

ditional and unconditional contracts may coexist in equilibrium when the conditional

contract offered by an insurer has an accuracy level that is superior to that of other

insurers. We also show that when multiple firms use tests that are similar in terms of

their accuracy, conditional contracts do not hold in a Nash equilibrium.

Keywords: test, screening, adverse selection, insurance

JEL Codes: D81, D82, G22

1 Introduction

Self-selection models in markets characterized by asymmetric information have received con-

siderable attention in the literature. Sorting of informed parties in these models occurs with

different types choosing different contract from menus offered by competing uninformed par-

ties. In the context of insurance markets, uninformed insurance companies compete against

each other in terms of the premium and coverage that they offer to informed individuals who

then self-select. Self-selection results in a decrease in information asymmetry. In contrast

to the attention paid to self-selecting models, the literature has paid little attention to an-

other widely observed market response to information asymmetry, risk classification through

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testing, with which uninformed parties actively screen informed parties. Underwriting in

financial markets, interviewing in job markets, and dating in the marriage market are all

examples. In the current study, we analyze equilibrium in a market in which asymmetric

information is reduced through self-selection and through testing. In practice, testing tends

to be imperfect, and noisy tests may entail pooling contracts. We consider when contracts

conditional on a test result are offered in insurance transactions and whether such conditional

contracts hold in equilibrium.

In our construct, insurers have a choice to offer either a single unconditional contract that

allows individuals to self-select or a single conditional contract that requires individuals to

take a test first that reveals whether they are high risks or low risks. Individuals have a choice

between self-selecting the unconditional contract and taking a test to purchase a conditional

contract. In the absence of the risk classification associated with a conditional contract, the

market is characterized by a Rothschild-Stiglitz (RS, hereafter) Nash equilibrium (Rothschild

and Stiglitz, 1976).

We demonstrate that individuals are willing to deviate from a self-selection contract to a

conditional contract if the test used for the conditional contract is relatively accurate. Equi-

librium conditions are investigated under several different market assumptions, and we show

that both conditional and unconditional contracts can coexist in several circumstances. In

contrast, it is also shown that conditional contracts are not sustainable as a Nash equilibrium

if multiple insurance companies employ a similar test in the market.1

The remainder of this article is organized as follows. In Section 2, a brief overview of

the literature on screening is presented. Competitive insurance markets and active-screening

are characterized in Section 3. In Section 4, we investigate equilibrium in several marketsettings where insurance companies use symmetric tests which reveal both high-risk and

low-risk individuals risk type with the same probability. To establish robustness and to

provide explicit conditions, the case of asymmetric tests which perfectly identify low-risks

1Our observation is that while insurers employ similar tests, for instance most use gender in personal

lines of coverages, the classification schemes are unique across insurers.

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is discussed in Section 5. A summary of our findings and a discussion of limitations in our

work are contained in Section 6.

2 Existing studies

Screening models in a competitive environment have been considered by many others since

the seminal work of Rothschild and Stiglitz (1976). A rich literature of competitive screening

models explores possible equilibria under various assumptions (see, for instance, Miyazaki,

1977; Riley, 1979; Spence, 1978; Wilson, 1977). Recent literature has considered refinement

of the RS model and resulting equilibria (e.g., Dubey and Geanakoplos, 2002; Martin, 2007;

Netzer and Scheuer, 2010). For instance, Netzer and Scheuer (2010) consider a model where

individuals are heterogeneous in their wealth and argue that equilibria in their model are

more consistent with empirical findings. Most studies extending the RS model have relied on

the self-selection mechanism for risk sorting. Typically efforts made by uninformed parties

to acquire knowledge regarding unobserved information are not considered.

Prior studies on information acquisition have primarily dealt with constructs where con-

sumers are unaware of their risk type ex ante and invest in learning their risk type (see,

for instance, Crocker and Snow, 1992; Doherty and Thistle, 1996; Hoy and Polborn, 2000;

Polborn, Hoy, and Sadanand, 2006). In contrast, consistent with the RS model, our analysis

assumes that individuals know their risk type, but insurance companies do not.

The assumption that test results are public information also sets this study apart from

earlier work on information acquisition by Doherty and Thistle (1996) and Hoy and Pol-

born (2000). Their work assumes that tests such as genetic tests are independent from the

purchase of an insurance contract and that insurers do not have access to the test results.

This article follows research by Browne and Kamiya (2010) which investigates the market

outcome when a noisy underwriting test is offered by the insurer. Their study focuses on

identifying equilibrium conditions when insurance companies are non-myopic and all insurers

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have access to an identical underwriting test. This study differs from theirs by focusing on

Nash competition and examining the case where each insurers test is unique.

3 Competitive Insurance Market

Homogeneous risk-neutral insurers offer one type of contract defined by coverage quantity,

q, and premium, p. Individuals live in one period with initial endowment W > 0. Terminal

wealth is uncertain with two states: W1 with a fixed loss D > 0 and W2 with no loss.

Individuals differ in their likelihood of loss, which is private information known only to the

individual. The probability that loss state W1 occurs for a high risk (H-type) is H and

that for a low risk (L-type) is L, where 0 < L < H < 1. Individuals also differ in their

observable attributes, which can be used by insurance companies to predict their risk-type.

The risk-averse individual, who has a standard risk-averse utility function (i.e., U > 0,

U < 0), applies for one contract. The L-type individuals expected utility with insurance

contract C = (p, q) is VL(C) = LU(Wp + qD)+(1L)U(Wp). The proportion of

H-type individuals and that of L-type individuals in the market are public information and

are denoted by and 1 , respectively.

The classic screening models well describe the market outcomes where there is no public

information expected to be associated with private information. However, in market trans-

actions an uninformed party tends to observe informed parties characteristics. In insurance

transactions, individual characteristics such as age, gender, and marital status may be rel-

atively easily obtained by uninformed insurance companies. It is natural in such a market

that competing insurance companies immediately attempt to process a set of observed in-

formation to predict individual risk type. If one firm successfully develops an algorithm to

predict risk type more accurately than other firms, the firm will gain a market advantage.

Consider such an algorithm as a function f : X {H-type, L-type} where X represents

a set of observable attributes. The output correctly predicts the individuals risk type with

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probability y (0, 1) but fails with probability 1 y. If a firm invents a test, it has

a choice to offer either an unconditional contract or a conditional contract that may be

purchased by individuals depending on their test outcome. Thus, a conditional contract

requires individuals to take a test first and the test result determines if they can purchase

the contract.

Consider a sequential game in which a new firm decides to enter the existing insurance

market by offering contracts. When individuals maximize their expected utility, Cournot-

Nash equilibrium identified in this game can be characterized by no contract in the equilib-

rium making a loss and no contract outside the equilibrium making a non-negative profit if

offered.

In the absence of any test, the well-known RS separating equilibrium identified as a set

of contracts (H, L) holds (see Figure 1) if and only if the population of H-type individuals

is sufficiently large relative to that of L-type individuals as pointed out by Rothschild and

Stiglitz (1976). We follow Crocker and Snow (1985) and denote the RS critical value of the

proportion of H-type individuals by RS, with which the RS equilibrium holds if and only

if the actual proportion of H-type individuals is greater than or equal to the critical value

(i.e., RS). In other words, the critical value is defined where L-type individuals are

indifferent between a pooling contract with RS and the RS separating contract L. Let M be

the pooling contract which satisfies the equality, VL(M) = VL(L), where the fair premium

rate for the pooling contract is:

M = L + RS(H L) (1)

[Insert Figure 1 Here]

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4 Insurance Market with Active Screening

To investigate the possibility that individuals demand a conditional contract which upsets

the RS separating equilibrium in the case of RS, we first consider H-type individuals

demand for a conditional contract followed by L-type individuals demand. H-type individu-

als are willing to deviate from their unconditional contract H to a conditional contract, A, if

and only if EVH(A, H) VH(H), assuming that individuals choose a conditional contract

when they are indifferent between these two contracts. The left hand side of the inequality

represents the H-type individuals expected utility attained by taking the test. If the test

misclassifies one into a L-type, contract A, defined by the L-type individuals optimal pool-

ing contract at its actuarially fair premium pA, can be purchased. Otherwise, one takes the

unconditional contract H. The light hand side of the condition is the utility corresponding

to the RS separating contract H. This condition is explicitly expressed as:

yU(W HD) + (1 y)[HU(WpA + qA D) + (1 H)U(WpA)]

U(W HD)(2)

which can be reduced to VH

(A) VH

(H). This inequality holds when L-type individuals

prefer the conditional contract A to their separating contract L given the conditional contract

is offered. That is, whenever L-type individuals deviate from the unconditional contract L to

the conditional contract A, H-type individuals also deviate from the unconditional contract

H to conditional contract A. This result confines our attention to the L-types demand.

Lemma 4.1. L-type individuals prefer the conditional contract A to their unconditional

contract L in the case of

RS

if and only if the accuracy of a test employed for theconditional contract satisfies:

(1 y)

(1 y) + y(1 ) RS (3)

Proof. The conditional contract A requires a subsidy s defined by the firms resource con-

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straint where its income is equal to the expected claim payment:

y(1 )AqA + (1 y)AqA = y(1 )LqA + (1 y)HqA (4)

where A = L+ s. The subsidy rate is proportional to coverage because the subsidy paid by

L-type individuals to misclassified individuals depends on the optimal coverage determined

by maximizing the L-types utility for contract A. Then the subsidy rate can be rewritten

as:

s =H L

1 + y(1)(1y)

. (5)

Note that the RS critical value RS

is determined where L-type individuals are indifferentbetween a pooling contract with RS and its separating contract L (i.e. VL(M) = VL(L)).

Hence the L-types demand condition, VL(A) VL(L), can be restated as VL(A) VL(M).

Since the subsidy rate ofA and that ofM are s and RS(HL), respectively, the condition

is equivalent to:

H L

1 + y(1)(1y)

RS(H L). (6)

which is rearranged, in (3), by the relationship between the fraction of H-type individuals in

the pooling contract A and the RS critical value. The minimum required accuracy of a test

decreases to 0.5 as RS.

Thus both L-type and H-type individuals no longer choose their separating contracts that

fully rely on their self-selection when a test can reduce the fraction of H-type individuals in a

pooling contract to at least the RS critical value. They instead prefer a conditional contract

that utilizes a test.

When < RS, the underlying market fully relying on self-selection is characterized by

non-existence of Nash equilibrium. Individuals prefer a pooling contract W (a Wilson pooling

contract) at the average fair premium pW, to their separating contracts. It is straightforward

to show that individuals are better off with the conditional contract, A, than the uncondi-

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tional pooling contract, W, if the test accuracy is greater than 0.5. However, this accuracy

does not necessarily guarantee that insurance companies cannot offer non-equilibrium un-

conditional contracts that attract only L-type individuals given the conditional contract A

is offered. In the next section, we identify a robust condition as a part of our equilibrium

argument in the case of < RS.

4.1 Monopoly on Conditional Contracts

We start with the case where a test f is invented by one of the competing insurance compa-

nies and only the insurance company can offer a conditional contract based on that test. In

contrast to the market discussed by Stiglitz (1977), the market analyzed here is still com-petitive in that other insurance companies can offer an unconditional contract. Therefore,

in the market where RS, H-type and L-type individuals can at least attain the RS

separating contracts, H and L, respectively.

4.1.1 Equilibrium

When individuals choose a monopolists conditional contract A, L-type individuals who are

misclassified by the test and H-type individuals who are correctly identified by the test take

unconditional contracts. Therefore, a set of contracts consisting of a conditional contract

and two unconditional RS separating contracts, (A,L,H), may hold in equilibrium when

RS. In contrast, when < RS, unconditional contract W never holds in equilibrium,

while the conditional contract A may not be upset even in that case. Equilibrium holds if

no new unconditional contract can earn a non-negative profit given the conditional contract

is offered.

Although the conditional contract A is also a pooling contract, the contract can be

sustained in equilibrium. To understand this possibility, it is important to note that an

individuals decision whether to take a conditional contract is based on its ex-ante utility

for the conditional contract A, which could result in either of two contracts: A and L for

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L-type individuals and A and H for H-type individuals when RS. This is because

L-type individuals take their unconditional separating contract L if they are misclassified

as H-types by a noisy test and H-type individuals demand their unconditional separating

contract H if a noisy test correctly identifies H-type individuals as H-types. The H-types

expected utility is strictly less than the level of utility gained solely by the contract A,

EVH(A, H) < VH(A). Analogously, L-type individuals expected utility is strictly less than

the level of utility gained solely by the contract A, EVL(A, L) < VL(A).

Thus, the individuals ex-ante utility is strictly lower than its utility gained solely from

the conditional contract A. It can be shown that the deviation of the H-type individuals

expected utility from VH(A) is positively associated with the accuracy of a test and makes it

possible that no unconditional contract that attracts only L-type individuals can be offered.

It is clear that EVH(A, H) VH(H) while EVL(A, L) VL(A) as y 1 (see Figure 1).

Lemma 4.2. A set of contracts (A,L,H) holds in equilibrium when RS if there exists

an unconditional contract L defined by VH(L) = EVH(A, H) which satisfies:

EVL(A, L) VL(L) (7)

where the contract L is offered at L-types fair premium pL. In equilibrium, a conditional

contract and unconditional contracts coexist when the test employed by the conditional

contract is relatively accurate.

Lemma 4.3. Conditional contract A holds in equilibrium when < RS if there exists an

unconditional contract L defined by VH(L) = EVH(A, W) which satisfies

EVL(A, W) VL(L) (8)

These incentive conditions may be thought problematic because unconditional pooling

contract W does not hold in equilibrium. However, the Wilson pooling contract W can be

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used to define the constraint. This is because the contract W yields lower utility for L-type

and greater utility for H-type individuals than any new potential unconditional contract

offer that attracts only L-type individuals (see contract B in Figure 2). Thus, the incentive

constraints are robust against the concern of non-existence of equilibrium for the pooling

contract W, though it is still true that the unconditional pooling contract itself does not

hold in equilibrium.


Proposition 4.1 (Active Screening Equilibrium). A conditional pooling contract may

exist in equilibrium regardless of the proportion of H-type individuals, if an insurer with a

unique underwriting test that is sufficiently accurate offers a conditional contract.

This argument can be established by the lemmas discussed above.

4.1.2 Optimal Contract

Given that an equilibrium exists, we next consider the rent earned by a monopolist, the

single best underwriter in the market, in the case where RS. We denote the profit-

maximizing contract as . Clearly, the optimal contract is obtained where the L-types

incentive constraint holds with the equality, EVL(, L) = VL(L). Otherwise, there exists a

contract (p, q

) such that

p

+ (1 )(p

q

) > p + (1 )(p q) (9)

The optimal contract may be seen diagrammatically in Figure 3. Due to the test, the

monopolists breakeven point is at A = L + s as discussed above. The contract that

maximizes profits must lie on the indifference curve VL() and on the line parallel to the

line pA. The reason is that the vertical distance between the indifference curve VL() and

the line pA represents the profit which is maximized where a line shifted upward from pA

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is tangent to the indifference curve. Let p = (L + s)q + be the tangent line where

represents a profit per insured. Thus, the profit-maximizing contract is identified as

the tangent point of the line p to the indifference curve VL(). Note that p implies

that q = qA. Therefore, the conditional contract offered to those classified as a L-type is

= (p, qA) where p = (L + s)qA + .


The profit also represents the value of a noisy test, which may be sold to other firms.

A conditional contract, however, can hold in equilibrium only when the monopolist alone

offers the conditional contract. We show later that there is no equilibrium for conditional

contracts if the same test is used by other competing insurance companies.

4.2 Competitive Market with Multiple Tests

4.2.1 Heterogeneous Tests

We investigate the impact of firm competition in developing a better test to gain a com-

petitive advantage. Our particular interest here is whether a conditional contract at the

actuarially fair premium still holds in equilibrium. Consider a simple case where firm i in-

vents a test fi : X {H-type, L-type} with probability yi where i = 1, 2. Assume that

y1 > y2 > 0. Thus, one firm has a more accurate test than the other. Firm i can offer

a conditional contract Ai for those identified as a L-type by its test at the fair premium.

Hence, firm 1 offers a conditional contract A1 = (pA1, qA1) where:

pA1 = qA1

L + H L

1 + y1(1)(1y1)

(10)

A question is whether the conditional contract A1 can be sustained when both another

conditional contract and unconditional contracts are offered. Our primary concern here is

another conditional contract, A2, potentially offered by firm 2 (see Figure 4).

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For the conditional contract A1 to be sustained against any new conditional contract A2, the

new contract must result in an immediate loss if it is offered. This condition is rephrased as:

if a conditional contract A2 attracts L-type individuals, it must attract H-type individuals

as well. That is, the following two inequalities must be satisfied simultaneously.

EVL(A2, L) EVL(A1, L) (11)

EVH(A2, H) EVH(A1, H) (12)

To analyze these conditions, it should be noted that by definition the test for contract A2

is less accurate than the test for existing contract A1. Therefore, if firm 2 offers a conditional

contract A1 with test accuracy y2, only H-type individuals deviate from A1 to A2. Thus, in

order to avoid attracting H-type individuals, firm 2 may offer a combination of coverage and

premium which at least offsets the H-types benefit gained by the less accurate test, but still

attract L-type individuals.

Proposition 4.2. A set of contracts (A1, L , H ) holds in equilibrium in a market where firms

offer conditional pooling contracts if there is no new contract A2 that satisfies both:

EVL(A2, L) EVL(A1, L) (13)

EVH(A2, H) < EVH(A1, H) (14)

As long as the accuracy of the second best test satisfies (13) and (14), a conditional contract

A1 cannot be upset. The accuracy of a test required for a conditional contract to survive in

equilibrium cannot be explicitly derived in this case. The explicit form under the assumption

of an asymmetric test is derived in the next section. This argument can be extended to the

case where more than two insurance companies offer tests with different degrees of accuracy.

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4.2.2 Homogeneous Tests

In contrast to the previous case, we now consider the case in which multiple firms employ

the same test. Each firm offers a conditional contract A to those who are identified as a

L-type. We find that the conditional contract does not hold as a part of equilibrium.

It can be shown that there exist possible profitable deviations against a new conditional

pooling contract A. Given that contract A is offered, insurance companies may offer a

conditional contract that attracts only L-type individuals but not misclassified H-type in-

dividuals (a contract such as A2 in Figure 4). Such a conditional contract always exists

as long as L-types individuals subsidize misclassified H-type individuals. The non-existence

of equilibrium for a conditional contract can be explained just as a pooling contract can-

not be sustained in the absence of any tests. Browne and Kamiya (2010) show that the

conditional contract holds in a Wilson equilibrium where insurance companies are assumed

to be non-myopic. Thus, equilibrium in a market where firms use the same test reverts to

the RS equilibrium argument. A set of unconditional separating contracts (H, L) holds in

equilibrium if RS. Otherwise, there is no equilibrium in the presence of conditional

contracts.

Efficiency of equilibrium is another major issue when contracts fully rely on individuals

self-selection. Since we discuss conditional contracts that could be introduced to the market

where self-selection contracts are offered, by definition both H-type and L-type individuals

are better off ex ante when conditional contracts exist in equilibrium. The allocation of

the utility gain between risk types is determined by the accuracy of a test in equilibrium.

The utility gain of H-types is maximized at the minimum required accuracy of the test that

sustains equilibrium and decreases as the accuracy increases. In contrast, L-types expected

utility monotonically increases with test accuracy.

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5 Asymmetric Test

The analysis in Section 4 leads to several conclusions: first, active-screening may be utilized

in a market where private information prevails; second, a pooling contract may hold in

equilibrium; third, a conditional contract cannot be a part of an equilibrium, if the most

accurate test can be used by more than one insurance company.

We next consider an asymmetric test, which imperfectly identifies H-types but can per-

fectly identify L-types. We show that our conclusion does not depend on the assumption

that both risk-types are correctly classified with the same accuracy. We also show that some

equilibrium conditions are explicitly provided in terms of the accuracy of a test. Specifically,

we consider a test that always correctly identifies L-type individuals as L-types while H-type

individuals are misclassified with the probability 1 y. It is clear that L-type individuals

will purchase a contract A if they take the test.

With the asymmetric test, the subsidy implicit in a conditional contract A is:

s = (H L)1 y

1 y, (15)

and the required accuracy of the test for the case when RS is defined as:

y 1 (RS/)

1 RS. (16)

As RS, the right hand side of (16) decreases to zero. When the market is characterized

by < RS, it is straightforward to show that L-type individuals always prefer a condi-

tional contract, regardless of its accuracy. Thus, the accuracy condition above is a sufficient

condition such that L-type individuals prefer a conditional contract A to any unconditional

contract offer regardless of the underlying market.

Market equilibrium when the conditional contract offered by the monopolist of a con-

ditional contract employs an asymmetric noisy test can be identified analogously to the

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symmetric test case. The argument is omitted here because any further simplification can-

not be made.

The insurers profit in this case can be expressed in terms of the L-types risk premium

for contract and L

, denoted as and L, respectively. With these risk premiums, the

profit-maximizing condition, VL() = VL(L), can be expressed as:

U(Wp A A(D qA) ) = U(W LD L). (17)

This implies that the profit per insured is

= (L ) sqA. (18)

Next, we investigate a market discussed in Section 4.2.1, in which two conditional con-

tracts are offered. A conditional contract A1 holds in equilibrium if a new conditional contract

A2 that attracts L-type individuals also attracts H-type individuals (see Figure 5). That is,

a contract A2 that satisfies VL(A2) VL(A1) must also hold EVH(A2, H) EVH(A1, H).


Consider a new contract offer such that H-types expected utility from a new conditional

contract A2, with which only L-type individuals are worse off, makes H-types better off by

the less accurate test y2 as before. It is straightforward to show that contract L, defined by

VL(A1) = VL(L) at L-types fair premium, maximizes L-types utility loss but still attracts

L-type individuals. Therefore, if the test employed by firm 2 is inaccurate enough to increase

H-types expected utility even for contract L

, the equilibrium condition is satisfied. Thus,

we can restate the condition as EVH(L, H) EVH(A1, H), which can also be rewritten

as:

y2 VH(L) EVH(A1, H)

VH(L) VH(H). (19)

The equilibrium condition is identified in terms of the accuracy of the second best test.

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A principle difference from the symmetric test case is observed in regards to an uncon-

ditional contract offer. When L-types are perfectly classified, no unconditional contract for

L-type individuals is offered. This also implies that, regardless of the fraction of H-type

individuals, no unconditional pooling contract is offered. Therefore, equilibrium, if it exists,

is always characterized by a set of contracts (A, H).

6 Conclusion

Competitive screening has been discussed in the absence of uninformed parties efforts to

classify informed parties. Many different forms of screening are, however, employed when

private information prevails. It is reasonable for uninformed parties to attempt to predict

informed partys private information to obtain a competitive advantage in a market. We

investigate the impact of testing on the existence of equilibrium.

This paper identifies the demand for a conditional contract and we show that individuals

deviate from their self-selection contracts to a conditional contract, which requires them to

take a test, if the test reduces the fraction of H-type individuals in a pooling contract to

lower than the RS critical value of the underlying population.

Perhaps most importantly, it is shown that a pooling conditional contract holds in equi-

librium when the most accurate test has significant competitive advantage and can attract

all individuals. In contrast, we also find that conditional contracts do not hold in equilibrium

when multiple firms utilize tests similar in terms of their accuracy.

Acknowledgements

We thank Michael Hoy and seminar participants at the 2010 Risk Theory Seminar for helpful

comments. All errors are our own.

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W2

pLEVL(A,L)

EVH(A,H)

pA

A

H L

VH(H)

045 degree

pH L

E

W1

Figure 1: Demand of Conditional Contract A

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W2

pLEVL(A,W)

EVH(A,W) A

A

H

WLB

0 45 degree

pH L

E

W1

Figure 2: Conditional Contract A in Equilibrium

pH

H

p

pA

EV (,H)

VL()

pL

A

q

D

EVL(,L)

Figure 3: Profit-maximizing Contract

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W2

pL

EVL(A1,L)

EVH(A1,H)

pA

A1A2

VH(A1)

H L

0 45 degree

E

W1

Figure 4: Multiple Conditional Contracts

W2

pLVL(A1)

EVH(A1,H)

pA

A1A2

VH(A1)

H L

L

0 45 degree

E

W1

Figure 5: Conditional Contract A with Asymmetric Tests

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