adverse selection and bilateral asymmetric information

Adverse Selection and Bilateral Asymmetric InformationAuthor(s): Fredrik AnderssonSource: Journal of Economics, Vol. 74, No. 2 (2001), pp. 173-195Published by: SpringerStable URL: http://www.jstor.org/stable/41794908 .

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voi. 74 (2001), No. 2, pp. 173195 Journal of Economics Zeitschrift für Nationalökonomie © Springer- Verlag 2001 - Printed in Austria

Adverse Selection and Bilateral Asymmetric Information

Fredrik Andersson

Received November 26, 1999; revised version received October 10, 2000

A simple principal-agent model with bilateral asymmetric information and common values is developed. The agent(s) has private information about his characteristics but does not know how these affect outcomes. The principal knows how the characteristics translate into outcomes, but does not observe the characteristics. It is shown that equilibrium contracts are simple in being de- signed not to reveal the agent's characteristics. When the agent knows how some of his characteristics affect the outcome, contracts will be differentiated with respect to precisely those characteristics. An application to the use of genetic information is considered.

Keywords: bilateral asymmetric information, adverse selection, signalling, simple contracts, genetic information.

JEL classification : D82.

1 Introduction

The author of this paper owns two private health-related insurance policies, one providing accident insurance and one providing retire- ment benefits. When buying these policies, he has not answered any questions regarding medical standing beyond making a vague decla- ration that he considers his health status to be "normal". The same is true for an overwhelming majority of such insurance policies, at least in Sweden. The fact that insurance policies are sold in this way is somewhat remarkable since it seems certain that further health-related information would have a bearing on anticipated losses if insurance companies systematically used it in actuarial statistics. Moreover, it would clearly be feasible to collect such data, and using it does not seem prohibitively expensive. The possibility of other states of affairs is



174 F. Andersson

indicated by recruitment of employees to international organisations and provision of automobile insurance policies. These are two examples where principals are known to base their hiring decisions and contracts on detailed information about the agent. In the former case there is use of health-related information, and in the latter case the information concerns characteristics of the car as well as the driver.

In this paper, we will argue that the pattern described above can be understood as a consequence of two characteristics of the author's two insurance policies. First, the policyholder has little basis for knowing the impact of particular health characteristics for payoff-relevant outcomes.1 Second, the insurance policies are supplied on markets where competition is restrained. The two other examples mentioned seem not to share these characteristics.

In this paper, we develop a simple principal-agent model with bilateral asymmetric information, and we explore the extent to which equilibrium contracts reveal the private information of the principal and the agent. We consider a hidden information principal-agent model where the agent (or agents) has private information about his characteristics, and where these characteristics determine a success probability in the undertaking of a two-outcome project which may be the object of a contractual relationship with the principal. How- ever, the principal knows, and the agent does not know, how the characteristics affect the success probability. Thus, there is bilateral asymmetric information and common values ; i.e., each party's payoff depends directly on the other party's private information. The

principal's private information concerns the stochastic technology.2 The specification is quite general, but for concreteness one may think of it as an insurance model. Our main result is a decomposition of the agent's information into two kinds: information whose relevant impact the agent knows, and information the impact of which the

agent does not know. We will show that equilibrium contracts are

1 Clearly, the policyholder sometimes has an idea about the qualitative impact of some health characteristics, but he is likely to have an extremely vague idea about magnitudes.

2 There are some conceptual problems involved in assuming that an economic agent is unaware of the possible consequences he faces, while it is less problematic to assume that he is uncertain about a probability distribution with known support, as in our case.



Adverse Selection and Bilateral Asymmetric Information 175

structured in such a way that the former kind of information is revealed, whereas the latter kind of information is not revealed.

Whereas the basic adverse-selection model has been thoroughly studied,3 there has been relatively little work done on principal-agent theory involving bilateral asymmetric information with common values. Most studies consider situations where only one party has private information, and generally this is "the same" party.4 An important exception is Maskin and Tiróle (1992) who provide a comprehensive treatment of informed-principal models. They, however, are mainly concerned with the case where the principal has all information ex ante, which in our model she does not have, but where she knows the technology as a function of agent-characteristics; they consider bilateral asymmetric information only with quasi- linear preferences. Our approach differs from that of Maskin and Tiróle (1992) also in focusing on a much more narrowly specified environment, thereby getting correspondingly more precise results.5

Our analysis shows that the principal benefits from the agent's ignorance by essentially being able to insure the agent against uncertainty about how his characteristics affect outcomes. It also shows that the principal's benefits from this will be realised in the equilibrium of the resulting signalling game where the principal initially commits to a menu of contracts, potentially revealing the relationship between characteristics and outcomes. When the agent knows how some of his characteristics affect the outcome, we show that contracts will be differentiated with respect to precisely those characteristics.6 The model

3 Examples include Akerlof (1970), bringing attention to the problem, and Mirrlees (1971), Spence (1973), Rothschild and Stiglitz (1976), and Stiglitz (1977), pointing out means of dealing with it.

4 An exception is the work on trading mechanisms pioneered by Myerson and Satterthwaite (1983) which, however, is concerned with strictly private values; i.e., cases where each agent is affected by the other agents' information only indirectly through their behavior.

5 Sadanand et al. (1989) demonstrate the optimality of probationary contracts when there is bilateral asymmetric information in a two-period worker- employer setting with risk neutral workers.

6 Taking a shortcut, we have also analysed a model where the agent is allowed to be uncertainty averse (Andersson, 1999). That model employs preferences that, following Schmeidler (1989), are consistent with the Ellsberg paradox (Ellsberg, 1961). The qualitative implications are largely the same, but uncertainty aversion in addition creates scope for over-insuring the agent.



176 F. Andersson

is then applied in discussing the usage of genetic information in health insurance.

Our main result contrasts with the case where the agent completely knows how his characteristics affect outcomes; then, separating contracts are optimal quite generally and equilibrium contracts are separating since there are no signalling issues. This has been shown, e.g., by Stiglitz (1977) in the case of adverse selection in a monopoly insurance problem, and by Maskin and Riley (1984), or Laffont (1989, chap. 10) for monopoly adverse selection models. Rothschild and Stiglitz (1976) show that essentially the same result holds on a competitive insurance market.

Once we think of the principal as insuring the agent against the uncertainty about the technology, it is easy to see why the results contrast so sharply. When the agent knows his success probability, the participation constraint forces any pooling contract to be acceptable by the agent whose willingness to pay is lowest - i.e., acceptable by the agent with the highest success probability. Starting from such a contract, however, there are gains for the principal from removing the original contract and instead offering a more expensive full insurance contract to agents with a high willingness to pay, and introducing a distorted, less than fully insuring contract for agents with a low willingness to pay. In the simplest version of our model, the willingness to pay is the same for all types of agent unless the agent is given information about his success probability. The fact that this willingness to pay is given by average beliefs together with risk aversion indicates that the principal may indeed gain by pooling all types of agents.

The paper is organised as follows. In Sect. 2, we describe the model, and in Sect. 3 we solve it for the case where the agent's private information is of no value unless combined with the principal's knowledge. In Sect. 4, we allow the information privately held by the agent to be more substantive, and show how our results combine with those for traditional hidden information models and adverse selection models. Section 5 is a somewhat informal application of the model to genetic information in health insurance; Section 6 is a conclusion.

2 The Model

To make the logic of the model clear, we start by presenting a simple example satisfying the assumptions to be detailed below.




2.1 An Example

Suppose that a monopoly insurer tries to sell health insurances to a population consisting of blue-eyeds and brown-eyeds in equal pro- portions. There are two possible relationships between eye colour and the risk of suffering illness: one where the probability of utilising health care is 0.4 for blue-eyeds and 0.6 for brown-eyeds, and one where the relationship is the opposite, the probability of utilising health care is 0.6 for blue-eyeds and 0.4 for brown-eyeds; the two relationships are equally likely ex ante. The insurer has private information about the relationship but it is verifiable. Our results show that the following contract is an equilibrium which, moreover, is unique in a precise sense. The insurer does not reveal the relationship prior to offering and signing insurance contracts with agents. The insurer offers a standard contract to all policyholders, the contract not being contingent on eye colour. The contract is fully insuring, and the premium is determined by policyholders' willingness to pay ex ante which is independent of eye colour.

Next, we will develop the model for the single agent case. It will be clear that the model applies to the case of many agents as well, however, and we will comment further on this in Sect. 3.

2.2 The Players

Consider a game between a principal, she , and an agent, he. The agent may undertake a two-outcome project that either fails or succeeds, and the success probability is q. As will be specified below, q depends on the agent's characteristics but not on his actions. Undertaking the project by himself may be unavoidable, as in many cases of insurance, im- possible, as in the case of the principal delegating something, or op- tional, as in choosing whether to borrow money for an investment. If the agent undertakes the project by himself, his payoffs are w if he is successful, and w - L if he fails; w is the sum of initial wealth and the additional, nonnegative, payoff from success, and L is the loss resulting from failure. The agent has a von Neumann-Morgenstern utility- of-wealth function, v, that is strictly increasing and strictly concave.

The principal is a risk neutral profit maximizer, and may provide insurance to the agent.

We will frame our problem as one where the principal proposes a menu of contracts to the agent in a take-it-or-leave-it fashion subject



178 F. Andersson

to the agent's reservation utility. The reservation utility depends on the agent's own assessment, q , of his success probability, q ?

ä(q) = (1 -q) -v(w-L) + q- v(w) . (1)

2.3 The Game

The game played by the principal and the agent has the following extensive form. First, the principal proposes a menu of contracts ( Pj,Dj ), where p is the price she demands for assuming the risk of losing L, and D is a deductible that she withholds in case the project fails and the loss is incurred. (One might consider more general contracts involving message games but, as we will argue in the end of Sect. 3, that would not change our results.) Second, the agent either picks a contract from the menu, or rejects all of them. Third, nature determines whether the project fails or succeeds.

Given a contract (p,D ), the payoffs at the interim stage - i.e., after the contract has been signed, but before nature has chosen the outcome - are obtained by taking expectations with respect to the outcome of the project. The principal's expected profit is n = p- (1 - q)(L - D), where q is her assessment of the success probability conditional on her information. The agent's expected utility is

u{p,D) = ('-q)-v{w-p-D)+q-v{w-p) , (2)

conditional on his accepting the contract.

2.4 Technology and Information

At the beginning of the game, nature chooses a pair (0, co) from the set 0 X Q. The agent learns 6 and the principal learns co. The pair (0, co) determines the success probability, qy of the project under- taken by the agent according to a function

/ : @ X Qi- > / Ç [0, 1] , (3)

to a subset / of the unit interval; /(•,•) is common knowledge. We will refer to 0 e ® as the agent's characteristics, and to co € Q

as the principal's information. We take 0 to be finite with n > 2

7 One might consider the possibility that the reservation utility is independent of q as well. The results would be similar, but in some cases trivial.




elements, and we take Q to be finite with m > 2 elements. Before the game begins, all parties have common beliefs over 0xQ, and these beliefs are common knowledge. The beliefs take the form of a product G X //, where G is a probability distribution over 0 with probabilities (gj)"=i, and H is a probability distribution over Q with

probabilities (hk)™=i. Denote the expected success probability with respect to G, i.e., the

expected success probability from the principal's point of view, by

Qa(œ) = ±gjnejtœ). (4) 7=1

We will assume that this expected success probability is independent of co; i.e., that the principal has no private information about the average success probability in the population of agents. This assumption is, arguably, quite natural. One way to interpret it is as saying that the average success probability is known from past experience, but that the agent does not have information that enables him to gain more specific knowledge about the distribution. In other words, the agent observes average success probabilities of earlier generations, but not the incidence of successes with respect to characteristics. The average probability may also be known and made public by e.g., a regulator (or by the principal) prior to the interaction between the principal and the agent. This assumption is, importantly, not a knife-edge assumption; our results depend on the variation in Qq across œ being small relative to the variation in the individual success probabilities.8

We assume initially that the agent's beliefs about the expected probability with respect to H ,

m Qh(0) =

^hkf{d,(0k) , (5) k= 1

are independent of 0; i.e., that the agent is completely ignorant about the average impact of the characteristics. The more general case is considered in Sect. 4.

8 If the principal would receive important enough private information about the aggregate that was independent of the type-specific (0-specific) information, she would signal the aggregate information and the results would be unchanged for the type-specific information.



1 80 F. Andersson

It is now helpful to note that QG = Qh', this follows from the fact that the agent's ex ante expected success probability is

n n m m

X! 9jQn(ej ) = S Sjhkf{ßj, <°k) = hkQo{o}k) , (6) j= 1 j= 1 k=l k= 1

and from the fact that QH is independent of 0 and Qq is independent of co by construction.

Concerning the principal's knowledge about 0, we assume to start with , that she learns 0 when the contract is signed and that 0 is verifiable. It is thus possible for the principal to offer a contract contingent on 0. We will also assume that if the principal wishes to reveal a> to the agent (and hence reveal the success probability, q- /(0, &>)), that is possible; co is verifiable as well. The reason that we make these assumptions is that they will constitute the worst case when proving our main result. We will come back to this below.

In those cases where the principal learns 0, our assuming that she does so after the signing of the contract amounts to assuming that she can commit herself to a menu of contracts prior to learning the agent's characteristics. Hence, she cares about expected profits (this is true without the assumption on timing when there are a large number of agents). The case where the principal observes 0 before proposing any contract is clearly of interest but not considered here.9 Our analysis should be thought of as conditional on the principal being able to commit.

3 Analysis

The timing makes clear that we have a variety of a signalling game. When offering a menu of contracts, the principal has private information about the function /(0, •) which she potentially reveals by the menu she chooses to offer. We will now consider the equilibria of this

9 That case is covered by the general analysis of Maskin and Tiróle (1992). We analyse it in the context of this specific model in Andersson (1997), and the results are seemingly counter-intuitive: the best risk is offered the efficient contract while all others are offered contracts with a positive deductible that in- creases as the success probability decreases. The logic is clear however, the best risk is the one for which the principal faces no incentive problem in convincing the agent about his true risk.




signalling game. Our equilibrium concept will be the perfect Bayesian equilibrium, which is defined by the following requirements:10

(i) the principal's contract offer is optimal given the agent's strategy and beliefs,

(ii) the agent's strategy is optimal given his beliefs and the principal's strategy,

(iii) beliefs are updated using Bayes' rule when applicable.

3.1 A Solution

For reasons that will become clear, we will single out one particular solution, the profit-maximizing pooling solution. This solution is pooling in two distinct respects. First, it entails the principal offering a menu consisting of one single contract which the agent accepts for all 9. Second, the contract offered is independent of <x> and the outcome thus represents a pooling equilibrium of the signalling game.

We will derive the most profitable contract that pools all agents; i.e., that offers only one pair ( p,D ). The relevant success probabilities are then the respective expectations: profits depend on the principal's expectation, Qg = J29jf(0j>co)> and the constraint depends on the agent's expectation, Qh = hkf(Q, o)k). By construction, QH = QG = Q. The contract thus solves

ma xp<D it =p - (I - Q)-(L - D) , s.t. (1 - Q)v(w - p - D) + Qv(w - p) >u(Q) .

The first-order conditions are easily seen to imply that v'(w -

p - D) = v'(w - /?), i.e. D = 0. The participation constraint then gives v(w - p) = u{Q). Note that the pooling contract is completely characterized by the premium, /?, and that it does not reveal any new information about co since the agent knows Q from the start.11

10 For the general definition, see e.g., Fudenberg and Tiróle (1991, chap. 8). For finite signalling games, perfect Bayesian equilibrium is equivalent to a sequential equilibrium , which was introduced by Kreps and Wilson (1982).

11 It may be worth noting what other solutions might look like. If co is observable and verifiable at the principal's will, she may simply announce it credibly. If 6 is contractible, the principal may propose a contingent contract. If 9 is not contractible (which we will allow below), the situation would be that of the standard adverse selection model with only the agent having private information, and the solution would be a separating contract along the lines of e.g., Stiglitz (1977).



1 82 F. Andersson

3.2 Results

We are now able to state and prove our main result. Note, however, that its full implications rely on Proposition 2 below. Moreover, in the end of this section we will provide a uniqueness result.

Proposition 1: The pooling contract described above is a perfect Bayesian equilibrium. Moreover, it gives the principal the highest profit she can attain in any equilibrium.

Proof: A non-constant contract, {pj,Dj)"=i , induces updated beliefs held by the agent, (A*)™=l over Q. For any such contract, however, there is a zero-deductible contract, ((/^,0))"=1, (possibly informa- tionally infeasible) that gives the agent the same utility, and gives the principal higher profits. We will show that such a contract can be no better than the pooling one.

Any such contract must satisfy the agent's participation constraint, and the principal's profits (but for the loss, L ) are then at most

= Ylgi{^~v~X j

, (8)

but since v is strictly concave, v~x is strictly convex and hence, by Jensen's inequality,

¿ 9j - v~] (u^J2hkf(0j, o)k)j ^ j

, (9)

with strict inequality if Ylk=ihkf(Oj,u>k) varies with 9 ; i.e., if the contract reveals any substantive information (there may exist contracts that lead to updating but keep this expectation independent of 9; they must, however, be renegotiated to D = 0 to be optimal, in which case they lead to the outcome of the profit-maximizing pooling contract). The expected loss is (1 - Q)L in both cases. Since the right- hand side of (9) is independent of co, non-revelation is superior for




all cd. This proves that the profit-maximizing pooling contract is indeed an equilibrium. It also proves that the profit-maximizing pooling contract maximizes willingness to pay, which in turn proves that the equilibrium is sustained by any out-of-equilibrium beliefs. Hence, the maximum profit is attained since all the surplus is extracted. □

We may note that there is no scope for renegotiation of the pooling contract since it is on the Pareto frontier.

3.3 Variations of Informational Assumptions

We have, so far, assumed that the principal observes 0 when the contract is signed, and that 0 and co are verifiable. It is clear, however, that the result remains exactly the same if the principal cannot observe 0, or if she can observe it imperfectly. In particular, the principal will never design the contract in such a way that it induces revelation of 0; this is because inducing such revelation is costly and the principal obtains her best possible payoff without the information. Note, however, that the principal must not observe 0 prior to signing the contract.

Although following straightforwardly, these facts demonstrate robustness in important respects, and are important enough to be stated formally.

Proposition 2: Under otherwise the same assumptions, Proposition 1 is valid no matter how well or at what cost the principal observes 0, and of whether œ is verifiable.

At this point, it is clear that the above analysis applies without any change at all to the case with many agents. Note that then, since the principal does not learn the agents' characteristics until the contract has been signed (if ever), all agents must be offered the full menu of contracts.

3.4 Interpretations

The above result may strike some readers as surprising, but it is really a rather straightforward consequence of the fact that the principal is able in effect to insure agents against the implications of their own characteristics by offering a contract that does not reveal the success



184 F. Andersson

probability. The substantive part of the argument is that the principal can exploit this in spite of the signalling logic. Note that it is a consequence of the result that contracts would look fundamentally different, being separating, if co were announced prior to contracting.

The implications for actual principal-agent relationships are important. Under the conditions stated, it will be in the principal's interest to conceal the actual determinants of the probability. For example, insurance companies will prefer not to reveal actuarial statistics. Furthermore, and at least as importantly, the contracts in question will not be made contingent on certain typical adverse selection parameters even if it would be possible and the parameters are relevant for the contract. The examples from insurance markets provided in the very introduction seem to satisfy this property. Phrased differently, we have provided a reason why contracts in some adverse selection environments should be simple. In the sense described, equilibrium contracts are incomplete.12

3.5 Uniqueness

Signalling games suffer quite generally from multiplicity of equilibria since a perfect Bayesian equilibrium puts no restrictions on beliefs off the equilibrium path. Our game is no exception. We will not go deeply into this issue, but we will provide a simple restriction on out- of-equilibrium beliefs that guarantees uniqueness. Our restriction is implied by the notion of perfect sequential equilibrium due to Grossman and Perry (1986). Grossman and Perry consider only finite games, but the restriction applies equally well to games with infinite strategy spaces. It is based on an equilibrium dominance argument similar to that of the Intuitive Criterion of Cho and Kreps (1987). The equilibrium dominance logic says, roughly, that if there is an out-of-equilibrium move and a realisation co' of the private information such that the move combined with the response to co' is strictly preferred to the equilibrium by co', while the equilibrium is

12 According to the most common notion, an incomplete contract is one leaving outcomes unspecified in some contingencies; see Hart (1995). Spier (1992) finds equilibrium contracts to be incomplete in the sense of non-insuring (not contingent on success/failure) in a model where (in our terminology) the agent proposes the contract and there are transactions costs; the result is rein- forced if the agent has private information.




strictly preferred by to", then that out-of-equilibrium move should be

conjectured not to come from a>". (For a more precise account, consult Cho and Kreps 1987, or Fudenberg and Tiróle 1991, chap. 1 1 .) Perfect sequential equilibrium imposes the additional restriction (called the GP-restriction below) that the relative probabilities of the cus gaining strictly not be updated.

In our case all types of principal strictly prefer the outcome of the profit-maximizing pooling contract to any other outcome. Hence the GP-restriction requires that if no contract leading to the profit- maximising pooling outcome is played with positive probability in

equilibrium, and if such a contract is offered out-of-equilibrium, the posterior over Q must equal the prior.

Proposition 3: Under the assumptions of Proposition 1, the outcome of the profit-maximizing pooling contract is the unique perfect Bayesian equilibrium outcome satisfying the GP-restriction.

Proof: Consider first the case where a contract producing the outcome of the profit-maximizing pooling contract is played with positive probability. Then there are two cases: (i) it is accepted for all 6, but then no contract producing another outcome can be offered in

equilibrium since the principal would make a strictly lower profit on such a contract and hence deviate; (ii) it is not accepted by all 6 but

only by a subset, J (we confine the notation to pure strategies by both parties), but then the expected success probability for those who accept, ̂ 2jej gjf(dj,a>), must be less than or equal to Qg (otherwise they would not accept); but then since Qg is the overall mean, it must hold for the rest of the 0s, Jc, that gjf(dj,o)) > Qg, and their willingness to pay falls short of that from the profit-maximizing pooling contract; total profits would be smaller. If out-of-equilibrium beliefs satisfy the GP-restriction, the principal would necessarily deviate to a contract producing the outcome of the profit-maximizing pooling contract since the agent would accept for all 0. The same argument applies if the outcome of the profit-maximizing pooling contract was not reached with positive probability. □

Let us now comment on the possibility that the principal would like to offer more general contracts than pairs (p,D ), e.g. contracts having the parties play a message game. The proof of Proposition 1



186 F. Andersson

makes clear that the profit of the profit-maximizing pooling contract is the highest attainable in equilibrium since it extracts all surplus when the agent's willingness to pay is maximal. Proposition 3 makes clear that the outcome is unique under the GP-restriction. Hence, under the qualifications of these propositions, there is no loss of generality in confining our attention to simple contracts.

4 Analysis with two Categories of Agents

We will now generalize our framework by letting the agent have more substantive private information. This extension can be viewed as an attempt to incorporate our model more completely into the framework of the adverse selection model that has been employed in various forms by, e.g. Stiglitz (1977) and in the series of papers by Laffont and Tiróle reviewed comprehensively in their 1993 book.

Above, we have assumed that the agent is characterized by a set of characteristics, 9 e 0, that are relevant for the undertaking of a project but whose effects are unknown to him. We will now assume that the agent has an additional characteristic i € {1,2}, which does affect his beliefs about his success probability. We will say that the agent belongs to category i. We will avoid the term type in order to prevent confusion, even though the categories here correspond quite closely to the types in the other models mentioned above.

We will let the agent's (correct) expectations about the probability depend precisely on his category, i.e., they depend on his characteristics only through /,

m í = 1'2- (10)

k='

Let the distribution of 6 be independent of that of i, and let n denote the probability that i = 2. The expected success probability from the principal's point of view conditional on category i is

ÔgM = ž 9jf ( I , Oj, to) . (11) 7=1

We impose the condition that this average be independent of <y category-wise; i.e., agents may have historical information about




success probabilities category-wise. Again, it is not critical that QG is constant in co for each i, but uncertainty about this average needs to be relatively small. The obvious analogue of expression (6) now immediately implies that

Qg = Qu (=ßi). i =1,2. (12)

We let Ql < Q 2; i.e., we assume that there is a relevant difference between the categories. This justifies our talking about a bad category and a good category. For technical reasons, we make the slightly stronger assumption that / is strictly increasing in /.

It is important to recognise that qlj = /(/, 0y, co) depends on the same œ for i = 1 , 2, and hence that the two categories of agent are uncertain about the same object. This reflects the notion that the realised technology, co , is common to both categories. It follows from the construction that the menu of contracts offered initially by the principal is observed by both categories, and hence that a contract aiming at separating only category-2 agents with respect to 9 necessarily reveals information to category- 1 agents too.

Reservation utilities are given by (1); note that this implies that uX<u2 for a contract with pooling within categories. If the principal were able to distinguish between the two categories costlessly, she would solve two separate problems and nothing substantive would change. However, the principal does not know to which category the agent belongs but may find out by designing a menu of contracts that separates the two categories.

Proposition 4: The principal attains the best payoff of any perfect Bayesian equilibrium by a contract that does not induce revelation of 6 (the characteristics whose consequences the agent does not know). Furthermore, the principal always separates the two categories with a menu of contracts: (/^O), (p2,/)2), where D 2 > 0; i.e., the change (compared to the case with observed categories) entails a positive deductible for the good category.

Proof: See the Appendix.

As in Sect. 3, the result applies to the case with many agents as well. It is also clear that a uniqueness result analogous to Proposition 3 holds for this case too.



188 F. Andersson

The two results provide a decomposition of the agent's characteristics in two ways, those whose impact the agent knows and those whose impact he does not know. The results are clean in the sense that for characteristics whose impact the agent does not know, we have pooling however well the principal observes the characteristics, and for characteristics whose impact the agent knows, we have separation however badly the principal observes the characteristics.

5 An Application - Revelation of Genetic Information

This section is devoted to applying the model to the use of genetic knowledge in health insurance. Although one can argue that the assumptions of the model are less than perfectly satisfied, we will argue that the model provides a useful frame of reference for thinking about the tradeoffs. The emergence of increasingly precise knowledge about genetic determinants of the incidence of various diseases makes this issue an important one. The setting should be thought of as a stylised model of medical insurance; the agents'/ policyholders' characteristics, 0 and /, correspond to more or less intimate personal traits potentially relevant for health, and the principal's/insurer's technological knowledge corresponds to knowledge about the implications of these personal traits for the likelihood of facing large health-care costs.

First, we will briefly argue that the most important assumptions made in the paper seem to be satisfied to a reasonable extent in the context of genetic knowledge and insurance. The most demanding assumption is, arguably, the independence assumption following expression (4), stating that the insurer's information, co, does not reveal information about the average risk in the population.13 In the genetic-information setting, however, this seems plausible indeed: average health risks do not depend on knowledge about which particular personal trait gives rise to an increased risk of facing a particular medical condition. Also, the assumption that the insurer can commit to a set of contracts is very weak in the insurance setting. In addition, we would argue that key parts of the information embodied in co are generally not available to third parties due to their key dependence on actuarial statistics.

13 Note that the related independence assumption for the agent of expression (5) is removed by the existence of characteristics i and the analysis of Sect. 4.




One might object that the insurer would not in reality know co since it would be costly to learn it, and since she would not want to use the information; note that this is not true for aspects of co relevant for the characteristic i. This is true, but it rather reinforces the result from the analysis: a monopoly insurer will not use such information. Importantly, however, it is not in general true in the presence of competition.

Interpreted literally, the model shows that an insurer with com- plete monopoly power will never use genetic information whose consequences are not known by the policyholder - and, as pointed out, she might therefore not bother to acquire it. This is true both in the sense that the insurer will not inform the policyholder, and in the sense that the insurer will not try to elicit information about the policyholder that would help assess the policyholder's risk. The model also shows that this is not true for characteristics whose consequences are known by the policyholder - the insurer will try to elicit information about such characteristics.

The model is a positive statement about a monopolist, and the normative significance is not immediate. However, the arrangements derived are Pareto optimal given the adverse selection problem since the monopolist, being residual claimant, seeks an efficient outcome. The presence of any competition introduces vast externalities between insurers. In particular, it introduces the potential for cream- skimming: any insurer will try to extract information about characteristics of policyholders in order to obtain a better-than-average pool of policyholders. In this quest for a better-than-average pool, all information about the agent is relevant, including information about 0. 14 The likely result of such a process is the effective revelation of personal traits, and insurers ending up differentiating contracts based on such personal traits. Thus, the insurance against effects of policyholders' own characteristics, which was proved to be beneficial above, is undermined.

This undermining is unambiguously undesirable only for characteristics 0 in the model, it is not necessarily undesirable concerning

14 Insurers may, for example, encourage "voluntary" revelation of personal traits. Holmberg (2000) finds evidence that a non-negligible fraction of em- ployers ask questions relating to family health in recruitment. This in spite of its being prohibited.



190 F. Andersson

characteristics i whose impact the agent knows. In practice the revelation of "0-type" characteristics and the revelation of "/-type" characteristics go together. Thus, it is not an unambiguous implica- tion of the model that such revelation ought to be discouraged. The model does, however, show that this process may entail some pure social waste. If it is a policy objective of the potential regulator to facilitate cross-subsidisation among policyholders, one means of achieving such an objective is limitation of the extent to which insurance contracts are individually differentiated. The model shows that the prevention of contingent contracts has additional benefits.

6 Conclusion

We have analyzed a model with bilateral asymmetric information where one party knows a parameter, and the other party knows a functional relationship. In more precise terms, we have introduced private information held by the principal about the stochastic technology of the transformation of agent-characteristics into outcomes, while we have retained the agent's having private information about some aspects of his characteristics. Although the model is rather special in some respects, the structure seems conceptually plausible and therefore important; in addition, the set-up seems to have some relevance for certain insurance environments.

When there is bilateral asymmetric information in this form, equilibrium contracts turn out to have properties quite different from those produced by models with one-sided hidden information. Our main conclusion is that equilibrium contracts of the model will be precisely the contracts that the principal would want to sell, and that these contracts will not be contingent on characteristics of the agent whose consequences he does not know. This is true even if these characteristics are relevant for the contractual relationship, and even if the principal could have made contracts contingent on them (directly or indirectly). The reason is that contracts contingent on these characteristics typically would reveal information about the technology and make the agent less uncertain, thereby removing possibilities for the principal to insure the agent against his uncertainty. The results contrast with results concerning characteristics whose consequences are known by the agent: optimal contracts differentiate with respect to such characteristics. The conclusions




hold quite generally with respect to the exact observability properties of the characteristics. The solution resulting from our specification is renegotiation proof.

The results have significant implications for agency contracts in hidden information environments. Our model predicts that contracts should be simple in the sense of not being made contingent on certain characteristics of the agent even if these are both observable and relevant for the contract; this as long as the agent does not know their consequences. A related phenomenon on which our results throw light is the so called principle of solidarity in Swedish insurance which states that insurance contracts, out of solidarity among policyholders, should not be made contingent on certain characteristics even when it is possible.15 Even though it may be partly out of solidarity, we have found another possible reason in showing that it may well leave insurers better off. It seems, furthermore, that insurers have enough means of coordination for this to apply also to observable characteristics.16

Several avenues of further research are pointed out by this paper. The most obvious one is to incorporate moral hazard in the model, which seems reasonably manageable except for one key problem - the specification of the relationship between the agent's beliefs about the distribution of outcomes and his effort. Finally, empirical work offers, as in information economics in general, an extremely rich source for further research.

Appendix

Proof of Proposition 4

We start by noting that if the principal could observe the category of agents, it follows immediately from Proposition 1 that she would offer full insurance contracts making no effort to differentiate with respect to 9. Since she does not, we must consider incentive compatibility with respect to /, and we will start by characterizing the

15 The principle of solidarity is discussed in Skogh (1980). 16 In Andersson and Skogh (1998), we show how seemingly rather innocu-

ous arrangements to sustain quality in insurance may facilitate tacit collusion quite powerfully.



192 F. Andersson

optimal contracts given that there is no differentiation with respect to 6 (noting that this step entails no signalling since the principal's information about i is a subset of the agents' information).

Let us introduce the following notation, and let primes denote ordinary derivatives evaluated at the points indicated by the indices (note that throughout, superscripts refer to category),

v' = v{w - pX - DX), Vi2 = v(w-pi),

v' = v(w - p1 - D2), v' = v(w-p1).

The problem faced by the principal is now

maxpi ¿¡J»# n = ( 1 - n) 'pl - ( 1 - Ö1 ) (L - D[ )]

+ fi'p2-(l-Q1)(L-D2))

s.t. (1 - Ql)v' +Qxv' > m1 (a1)

(1 -ó2)!?! +g2t>2 >M2 (a2)

(1 -Ql)v'+Qlv¡ > (1 -QX)v' + Q{v' (Ã1)

('-Q¿)v2 + <fvj>(l-<f)v'+Q¿vl, (Ä2)

(A.l)

where the first two constraints are the participation constraints just like above, while the last two constraints are the incentive-compatibility constraints , guaranteeing that the principal's intention to separate the two categories of agents works as intended. The first-order conditions are

A

g- f = 1 - ß - (a1 + ^)[(l - Ql)d' + ß1^1]

+ A2[(l-ßV11+ßV,1] = 0,

g=(l-Ju)(l-ß')-(a1+A1)(l-ß1K11 + A2(l-ß2)i/11=0,




r) <; P = ft - (a2 + ^ )[0 - Q1 )f;'|2 +

+ Al[(l-ß>f + ßli>?]=0,

|^= mi - ß2) - (o2 + ^2)(i -e>ř

+ i1(i-e>;2 = o.

Manipulations according to 8^ JŽ7 - 0/> Jžf /( 1 - 0*) = 0 gives for the two categories, respectively

(a1 + A})Qx {v'l -

}

+ Á2 (1 - ß2)^1 + ßV22 - j _ ^ t// = 0 , (A.2)

(a2 + ^2)ß2{t;/12-i;/22} ^

+ A1 (1 - ö1)^2 + QXv' 2 - J_ g2 ü2 (A.3)

In the absence of incentive compatibility constraints, the expressions in the first brackets are zero, and it is easy to verify that, as a consequence, at that solution, the last term of (A.2) is positive, while the last term of (A. 3) is negative. This implies that if À2 > 0, then Dx < 0 by incentive compatibility, while if Xx > 0, then D2 > 0. Now, it is easily seen (starting from the hypothetical situation where two full insurance contracts could be offered) that the second incentive compatibility constraint may bind only if m1 > w2, which is not the case for any of our specifications of reservation utilities. Hence, D2 >0.

However, once contracts are not fully insuring, Proposition 1 does no longer guarantee that there are no gains from separating agents with respect to 9. To establish that there indeed are no such gains, we again consider a separating menu of contracts, (Pj,Dj)j=l, for category 2. This menu induces beliefs (A*)7 over Q. To simplify notation, let V'j = v(w - pj - D j ) and v-ij = v(w - p¡). Since the menu of contracts is publicly observable, however, category 1 agents learn the same infor-



194 F. Andersson

mation; i.e., they also hold beliefs (hk)™. Now suppose that the principal can observe and verify 0 so that there are no incentive compatibility issues with respect to 9 - if there were, the principal would be more constrained and do no better. The constraints on the category 2 contract are then the participation constraint and the incentive compatibility with respect to category l's mimicking. (It follows from the first part of the proof that separating categories is optimal, and it is clear that the constraints will hold with equality.) Letting q'j =

X) Â*/(i, ö/i (Ok) denote the agents' updated expected probabilities, the constraints can be written

( 1 - q j>ij + 9>2/ >a + bq2j , (1 - q))v'j + q)v2] <a + bq) ,

where the right-hand sides simplify the reservation utility. Solving for the constraints gives vy - a and ity = a + b (here we exploit that / is increasing so that q' ý <?/)• Thus, the contract does not differentiate with respect to 91 Further, the resulting contracts also solves the problem pooled with respect to 9, (just take expectations in the constraint: Ey 9 $ = E, 9j E k hf(i, ej, <»*) =E* MEy QjfiU Oj , (Dk)' = Q , since the expression in brackets is independent of coi).

Since the profit of the constant category 2 contract, (p,D ), is independent of o, the constant contract is - by the same logic as in the proof of Proposition 1 - indeed the most profitable one for all co. □

Acknowledgements I have received very valuable comments and suggestions from Martin Hellwig and two anonymous referees. The paper has benefited also from suggestions from Svend Albk, Anders Borglin, Hans Carlsson, Mattias Ganslandt, Bo Larsson, Eric Maskin, Lars-Gunnar Svensson, and seminar participants at Harvard University. Financial support from the Bank of Sweden Tercentenary Foundation is gratefully acknowledged.

References

Akerlof, G. (1970): "The Market for Lemons: Qualitative Uncertainty and the Market Mechanism." Quarterly Journal of Economics 84: 488-500.

Andersson, F. (1997): "Insurance with an Informed Insurer." Notes, Depart- ment of Economics, Harvard University.

Andersson, F. (1999): Uncertainty Aversion in a Simple Insurance Model. Finnish Economic Papers 12: 16-27.




Andersson, F., and Skogh, G. (1998): "Quality, Competition and Self-Regula- tion: The Case of Insurance." Mimeo, Department of Economics, Lund University.

Cho, I. K., and Kreps, D. M. (1987): "Signalling Games and Stable Equilibria." Quarterly Journal of Economics 102: 179-221.

Fudenberg, D., and Tiróle, J. (1991): Game Theory. Cambridge, Mass.: MIT Press. Grossman, S., and Perry, M. (1986): "Perfect Sequential Equilibrium." Journal

of Economic Theory 39: 97-119. Hart, O. (1995): Firms, Contracts, and Financial Structure. New York: Oxford

University Press. Holmberg, P. (2000): "Genetisk diskriminering p+ den svenska arbetsmarkna-

den" (Genetic Discrimination on the Swedish Labour Market). Bachelor's Thesis, Department of Economics, Lund University (in Swedish).

Kreps, D. M., and Wilson, R. (1982): "Sequential Equilibria." Econometrica 50: 863-894.

Laffont, J.-J. (1989). The Economics of Uncertainty and Information. Cambridge, Mass.: MIT Press.

Laifont, J.-J., and Tiróle, J. (1993): A Theory of Incentives in Procurement and Regulation. Cambridge, Mass.: MIT Press.

Maskin, E., and Riley, J. G. (1984): Monopoly with Incomplete Information. RAND Journal of Economics 15: 171-196.

Maskin, E., and Tiróle, J. (1992): "The Principal-Agent Relationship with an Informed Principal, II: Common Values." Econometrica 60: 1-42.

Mirrlees, J. A. (1971): "An Exploration in the Theory of Optimum Income Taxation." Review of Economic Studies 38: 175-208.

Myerson, R., and Satterthwaite, M. (1983): "Efficient Mechanisms for Bilateral Trading." Journal of Economic Theory 29: 265-281.

Rothschild, M., and Stiglitz, J. (1976): "Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information." Quarterly Journal of Economics 90: 629-649.

Sadanand, A., Sadanand, V., and Marks, D. (1989): "Probationary Contracts in Agencies with Bilateral Asymmetric Information." Canadian Journal of Economics 22: 643-662.

Schmeidler, D. (1989): "Subjective Probability and Expected Utility without Additivity." Econometrica 57: 571-587.

Skogh, G. (1980): "Regiering av konsumentförsäkringsmarknaden" (Regulation of the Consumer Insurance Market). Ekonomisk Debatt 8: 603-612 (in Swedish).

Spence, A. M. (1973): "Job Market Signalling." Quarterly Journal of Economics 87: 855-874.

Spier, K. (1992): "Incomplete Contracts and Signalling." RAND Journal of Economics 23: 432-443.

Stiglitz, J. E. (1977). "Monopoly, Non-linear Pricing, and Imperfect Informa- tion: The Insurance Market." Review of Economic Studies 44: 407-430.

Address of author: Fredrik Andersson, Department of Economics, Lund University, P.O. Box 7082, S-220 07 Lund, Sweden (e-mail: fredrik.andersson@ nek.lu.se)



adverse selection and bilateral asymmetric information

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