What Makes Insurance Companies Voluntarily Share Proprietary Customer Information? Guofang Huang∗ February 11, 2012

Abstract This paper provides a novel game-theoretic explanation for why voluntary information-sharing arrangements in the insurance and banking industry can be self-enforcing—namely, why the information exchanges, organized as independent for-profit corporations, can count on insurance companies and banks to continue to report their private customer information to them. JEL: D82 (Asymmetric and Private Information), G14 (Information and Market Efficiency), G21 (Banks), G22 (Insurance Companies), L13(Oligopoly and Other Imperfect Markets) Key Words: information sharing, adverse selection, insurance market ∗

Guofang Huang: Yale School of Management, 135 Prospect St, New Haven, CT 06511 (email: [email protected]). I am deeply indebted to my advisor, Joe Harrington, for his encouragement and detailed comments on the previous drafts of this paper. I also thank Marc Remer and K. Sudhir for helpful discussions. Any remaining errors are my own.

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1

Introduction

It is well-known that hidden information possessed by one side of the market could induce adverse selection, which in turn lead to inefficient market outcomes in equilibrium. The problem is, in fact, ubiquitous. Practices and outcomes in many important markets—such as credit markets, insurance markets, labor markets, etc—are all heavily shaped by the problem. Numerous papers in the economic theory literature have been devoted to analyzing and designing approaches to deal with the adverse-selection problem.1 A developing empirical literature includes efforts to test for the existence of adverse selection and quantify the efficiency loss associated with it, often using data from insurance markets.2 However, part of the hidden information is often revealed during repeated interactions between market participants.3 For example, employers gradually learn more about some hidden attributes of their employees. Banks gradually learn more about the risks of lending to their clients. And insurers gradually learn more about the risk types of their insurees.4 If such useful, but often private and proprietary, information were disseminated freely to all the market participants, the efficiency loss caused by hidden information would be greatly reduced. However, a bit surprisingly, this is what actually has happened in many markets. In the U.S., consumer credit reporting agencies, such as Equifax, Experian and Transunion, collect and distribute proprietary information about consumer creditworthiness. In the insurance market, the CLUE (Comprehensive Loss Underwriting Exchange), run by Choicepoint Inc., and the A-PLUS (Automobile Property Loss Underwriting Services), operated by ISO Ltd, collect and supply many kinds of insurance claim history data. ChoicePoint also collects employee information and sells it to employers for pre1

Classical examples include Stiglitz and Weiss (1981) for the credit market, and Rothschild and Stiglitz (1976) for the insurance market. 2 Chiappori and Salani´e (2000) and Finkelstein and Poterba (2004) are examples of this literature. 3 The accumulation of such information is called asymmetric learning in the literature. 4 See Cohen (2011) for evidence of such asymmetric learning using data from the Israeli auto insurance market.

3 employment screening. The extent to which the hidden-information problem has been relieved by such institutions can be seen from the staggering amount of information that they have collected and distributed. For example, ChoicePoint’s database of personal information contains names, social security numbers, credit reports, insurance claim histories, and many other sensitive data. In 2005, this database contained data on 220 million people, and these data were sold to an estimated 100,000 clients.5 There are two important features to the various information exchanges mentioned above. The first is that every bit of customer information in the exchanges’ databases is contributed voluntarily by participating companies that own the information.6 This is puzzling because keeping such proprietary information private enables firms to make higher profits from some old customers. For example, Sharpe (1990) and Nilssen (2000) demonstrate that such an informational advantage allow firms to charge relatively high prices to their old high-quality clients without turning them away to competitors. Cohen (2011) provides empirical evidence showing that Israeli auto insurance companies are able to make higher profits from their old low-risk customers because they have informational advantage over their competitors with regard to these customers.7 Another important feature of the information-sharing institutions is reciprocity. For example, the insurance information exchanges are accessible only by the member companies that also contribute claim-history data to the exchanges. The banks that supply credit-history data to the exchanges can access the exchange database at a much lower cost. Many factors, no doubt, have contributed to the development of these information-sharing institutions. My goal in this paper is to shed light on one novel force absent from the literature. I show that such information-sharing institutions can endogenously arise in an environment with hidden information and asymmetric learning. I demonstrate the idea through an extended two-period oligopoly pricing model in the context of the insurance market. 5

These figures are cited from Magnusson (2005). The information exchanges do not own the contributed information in their database. See the next section for more details about these information exchanges. 7 There is no similar information-sharing institutions in the Israeli auto insurance market. 6

4 I assume in my model that there are two types of customers: high-risk and low-risk. Firms first choose whether to share information and set prices simultaneously in period 1, and then they set prices in period 2. Firms observe their own customers’ types after transactions with them in period 1, and then the information is shared exclusively among firms that choose to share information. Firms’ pricing behavior in period 2 depends on their information-sharing policy. On the one hand, if firms decide to keep their customer information private, they can use it to extract more surplus from their old low-risk customers in period 2 because, with inferior information about these customers’ types, other firms will not compete too aggressively on price. On the other hand, agreeing to share the information with other firms commits firms to lower prices for the low-risk customers in period 2 because competition drives down the prices after the information asymmetry between firms is eliminated. In contrast, the high-risk customers would always be charged their marginal cost, with or without information sharing. Assuming that consumers have rational expectation and that the information to be shared is complete8 , I show that a firm unilaterally withdrawing from an information-sharing agreement necessarily attracts less low-risk customers; and if the cost of sharing information is low enough, there exists a Subgame Perfect Equilibrium (SPE) in which all firms share information. There are two approaches in the literature to explain information sharing in general, including sharing private customer information and sharing private demand and cost information. The relatively simpler approach goes by the following logic, quoted here from Vives (2006), “If firms are able to enter into industry-wide agreements, they want to share information if pooling (information) increases expected profits.”Researchers taking this approach look for conditions that would guarantee a profit increase under an industry-wide information-sharing agreement. This approach is intuitive and often easy to implement. One problem with this approach is that the information-sharing agreement may not be self-enforcing under the identified conditions. The second approach lets individual firms make their information-sharing decisions 8

That is, firms reveal all their private information without adding noise to it.

5 simultaneously and independently. This approach is often harder to pursue, as the information-sharing decisions have to be considered alongside other strategic interactions. A few articles in the literature have investigated the possible mechanisms that may have led to the voluntary exchange of customer information among firms.9 The earliest I am aware of is by Pagano and Jappelli (1993). In their static model, geographically isolated banks extend loans to heterogeneous customers. The banks’ incentive to share information is to get more precise information about the types (risk) of the exogenous “immigrants.”They consider two types of market structures. When the banks are scattered monopolists (no competition between banks), the trade-off when considering sharing information is simply between more information about the types of “immigrants” and the cost of sharing information.10 When under the kind of local competition the authors assume, the trade-off is between better information about immigrants and increased competition from adjacent firms plus the cost of sharing information. So, if sharing information makes competition too intense, information sharing would not arise. In short, the information sharing in their model is motivated by the exogenous migration that moves the old customers of one bank to another bank. Another important paper is Padilla and Pagano (1997). The authors set up a two-period model of the credit market in which banks extend loans to entrepreneurs of two types, high- and low- ability, each period through oneperiod non-contingent contracts. In their model, there are many “towns,” in each of which there is one bank and a continuum [0, 1] of entrepreneurs. Each bank knows ex ante the types of entrepreneurs in its own town, but not of those from other towns. They assume that before the start of the two periods, each bank first decides whether or not to share information about entrepreneur types in period 2 . Then each entrepreneur chooses her effort level once and 9

There is a large literature that discusses the incentives and implications of sharing private information about demand and product cost in the oligopoly market. For a review of that literature, see K¨ uhn and Vives (1995) and Raith (1996). 10 Throughout the article, by “the cost of sharing information ” I mean the cost of the labor, equipment, etc. needed to organize a centralized information-sharing system.

6 for all for the next two periods, given her knowledge of all banks’ informationsharing decisions and her rational expectations of the interest rates to be offered in the next two periods. Subsequently, banks choose interest rates for period 1, and then for period 2. And each entrepreneur chooses a bank to borrow from in each period after banks set interest rates. The returns on the low-ability entrepreneurs’ projects are always zero, while the returns on the high-ability entrepreneurs’ projects in both periods are strictly positive if successful, and the probabilities of a high-ability entrepreneur’s projects being successful are equal to the effort level chosen by her ex ante. In addition, the authors assume that the parameters in their model are such that the banks are monopolists in their own towns in period 1. So, without information sharing, the high-ability entrepreneurs’ effort would always be zero as they expect that the return on their projects, if successful, would be entirely expropriated by the local banks via high interest rates. Thus, without information sharing, the credit market would close and the profit of the banks would always be zero.11 Agreeing to share information in the second period provides a way for banks to commit to lower interest rates in period 2. When the entrepreneurs’ effort cost function satisfies certain conditions, they could be induced to choose high enough effort levels due to the expected lower period-2 interest rate. Thus, banks make expected profits of zero in period 2 but strictly positive profits in period 1. Hence, given that the conditions on the parameters and functional forms are satisfied, each bank’s total profit is higher under information sharing than without information sharing.1213 11 Local banks never extend credit to local low-ability entrepreneurs, as the returns on their projects are always zero. 12 Pandilla and Pagano do not provide an explanation within their two-period model for why information-sharing agreements are sustainable when banks act independently. They argue informally that when the two-period models are repeated infinitely, an informationsharing agreement with a membership rule in the spirit of the grim trigger strategies would be able to provide the incentives for the banks to share information. Specifically, the membership rules would state that banks can keep their membership only if they continue to truthfully share their private information; otherwise, they will be deprived of their membership permanently. 13 In the above model of Padilla and Pagano (1997), the banks’ higher profit under information sharing depends crucially on the structure built on two key assumptions. The first is that the effort level chosen by high-ability entrepreneurs ex ante determines the

7 My explanation differs from those provided by the previous work in two significant ways. First, the forces driving information sharing in my model are inherent in the basic dynamic oligopoly pricing framework: An insurer is willing to share information only because doing otherwise would induce consumer adverse selection and produce less profit. In contrast, the explanations provided by the above articles rely on sharing information generating some direct benefits.14 Second, my model allows global competition among all firms and endogenizes firms’ information-sharing decisions. In Pagano and Jappelli (1993)’s model, the competition is either ruled out or restricted to be among “neighboring” banks. Padilla and Pagano (1997) allow competition only in period 2. They do not provide equilibria to support information sharing within their two-period model, but they argue that an extended infinitely repeated game model could justify the sustainability of information sharing. On the empirical side, Cohen (2011) shows that insurance companies do possess considerable market power over some customers due to an information advantage. Using a customer-level panel data from the Israeli auto insurance market, she shows that insurance companies tend to obtain a much higher markup from their low-risk customers than from their high-risk customers, and such markups increase with the low-risk customers’ experience with their insurers.15 Cohen’s work supports the hypothesis that market power develexpected returns in both periods; the other is that each bank knows the types of their local entrepreneurs ex ante, while information sharing is feasible only in period 2. The first assumption may be restrictive because: the banks cannot commit to a two-period-long contract, whereas entrepreneurs can commit to an effort level for two periods/projects. The second assumption is vulnerable to the criticism that the initial informational advantage is left unexplained: it comes as free lunch, and the benefit of it could have been dissipated by competition beforehand. Also the assumption that the information is available before period 1 but can only be exchanged in period 2 somewhat artificially created the benefits of sharing information in period 1. 14 For example, Pagano and Jappelli (1993) assume that customers migrate between towns exogenously, and, thus, information sharing becomes valuable because local banks do not have information about the creditworthiness of new immigrants from other towns. In Padilla and Pagano (1997)’s model, sharing information commits banks to lower future interest rates, which, in turn, provides some incentives for the entrepreneurs to choose higher effort ex ante. This, together with some other assumptions, creates the direct benefit (i.e., higher period-1 profit) of sharing information for the banks. 15 There is no institution established to share customer information in the Israeli auto

8 oped through asymmetric learning is fundamental for industries such as auto insurance. Such empirical evidence makes me believe that my basic framework is on solid footing. The rest of the paper is organized as follows. Section 2 briefly introduces the institutions set up for financial firms to share their customer information. The model is presented in section 3. Section 4 provides the proof of the main result. Section 5 discusses the generality and robustness of the results. Section 6 concludes.

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Industry Background

This section focuses on the current information-sharing arrangements in the United States insurance market. Pagano and Jappelli (1993) provide some discussions of features of information sharing in the credit market. There are currently two insurance-information exchanges16 in the U.S. One is the CLUE (Comprehensive Loss Underwriting Exchange), run by Choicepoint, Inc. The other is A-Plus (Automobile-Property Loss Underwriting Services), operated by the Insurance Service Office, Ltd. An insuranceinformation exchange, organized as an independent company, stores and compiles the customer information contributed by member insurers and supplies the raw and processed information exclusively to the member insurers. The information exchange does not own the information database. Its business is to “maintain the databases that contain information provided and used by insurance underwriters”, and to provide services based on these databases. The data-supplying agreements between the insurance companies and the information exchanges are generally short-term agreements. However, Choicepoint, for example, believe that the insurers will keep providing data for the database since they “depend upon the aggregated information in such databases to conduct their business operations.”17 insurance market. 16 Here, by “insurance-information exchange,” I specifically mean the information broker that collects and supplies insurance claim history data. 17 See the 2006 Annual Report of Choicepoint, Inc.

9 Both of the above two information exchanges cover most of the insurers in the U.S. “More than 95 (90) percent of insurers writing automobile (property) coverage provide claim data to the CLUE Personal Auto (Property) database.”18 “More than 1,440 insurance companies19 contribute reports of property claims to the A-PLUS database,”and “more than 980 auto insurers20 contribute to the A-PLUS database.”21 However, as a comparison, there is no similar kind of information exchange in the Israeli auto insurance market studied by Cohen (2011). To get a rough idea of the cost of sharing information, we can look at how much insurance companies pay information exchanges for related information services. For example, according to the 2006 Annual Report of Choicepoint, Inc., its revenue from insurance services amounted to around 454 million dollars, a major part of which derived from services based on the CLUE database. We may roughly treat the 454 million dollars as the total cost of sharing information, which was shared by around one thousand insurers in the U.S. In comparison, from 2001 to 2010, the average annual net income of the entire auto insurance industry was around 17.7 billion dollars,22 which is around 39 times the above total cost of sharing information. The claim history information that member insurers report to the information exchange is very detailed. It includes information that identifies individual customers and vehicles (homes), policy numbers, and claim information such as date of loss, type of loss and amounts paid. However, as Pagano and 18

Cited from the following Choicepoint webpages: http://www.choicepoint.com/products/clue auto.html?l2=underwriting&bc=bus&sb=b, and http://www.choicepoint.com/products/clue property.html?l2=underwriting&bc=bus&sb=b 19 Representing 95 percent of the industry by premium volume. 20 Representing nearly 93 percent of U.S. private passenger premium volume. 21 Cited from the ISO web page: http://www.iso.com/Products/A-PLUS/A-PLUS-theAutomobile-Property-Loss-Underwriting-Service.html 22 This is a crude number calculated based on Allstate’s net income reports and its market share. The annual net incomes of insurance companies vary a lot from year to year. The average annual net income from the auto insurance segment from 2001 to 2010 was about 1968 million dollars for Allstate. Allstate’s market share of the auto insurance market was about 11% in 2006 (source: A.M. Best). So, the average annual net income of the entire auto insurance industry is about 17,712 million dollars.

10 Jappelli (1993) point out, the customer information shared among financial companies varies in content. For example, in the United States, Britain, and Japan, lenders exchange both black and white information, while in Australia and Finland, credit bureaus provide only black information.23 The Fair Credit Reporting Act (FCRA), which regulates the practices of reporting and distributing consumer information, explicitly requires such reports be accurate and complete. The information exchange in both the insurance and the credit market is covered by FCRA. Consumers can access their own consumer reports and dispute the accuracy and completeness of the reports with the consumer rating agencies. When a consumer challenges a report, the involved consumer rating agency is required to investigate the problems raised by the consumer and make corresponding corrections when appropriate. So, with these regulations already in place, the issue of whether the firms will truthfully report their private information about their customers does not seem to be a serious problem. There are also public records from various sources—including federal, state, and local government agencies—that contain information about the risk associated with each consumer. However, these public records reveals only a small part of the customer information that is shared among the insurance underwriters. For example, relatively small accidents may not be reported to the police or recorded by DMV; by law, only crashes involving property damage, personal injury or death must be reported to the police; furthermore, law enforcement officers are required to forward a written crash report to DMV only when a traffic crash results in injury, the death of any person, or total property damage in excess of $1,000. Obviously, the crash information maintained by the DMV will contain only part of the information that is available in the claim history database. So, the additional information acquired exclusively through the information exchanges is important. 23

Black (negative) information concerns only defaults, while white (positive) information includes the credit history and current debt exposure.

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3 3.1

Basic Model Model Setup

In the following I set up a two-period oligopoly pricing model with asymmetric learning in the context of the insurance market.24 My focus will be on the forces that induce information sharing in equilibrium and the accompanying implications for competition. After setting up my model, I discuss some modeling issues in the next subsection. I use the words “firms” and “insurers” interchangeably to refer to insurance companies supplying insurance policies; and I use “customers,” “consumers” and “clients” interchangeably to refer to consumers buying insurance policies. I assume that there is only one type of contract available so as to abstract from the contract-design issues. First, assume that there are M ≥ 3 identical insurers and a continuum of N customers in the market.25 The customers are of two types: H and L. A customer’s type is constant over time. There are two periods, and in each period, the H type customer has probability 1 of incurring a loss of 1, whereas the L type customer always loses nothing.26 A proportion λ ∈ (0, 1) of the N customers are H type. Consumers always know their own types, but firms do not observe the types of their customers ex ante. The customers’ claim histories in period 1 are observed only by their own insurers. In each period, insurers offer only one type of non-contingent contract that provides full insurance. Every customer will buy one and only one insurance policy each period.27 Customers have rational expectations about the insurers’ dynamic pricing behavior. Both the insurers and customers are risk-neutral and infinitely patient (discount factor equals one). For information sharing, I assume that if a group of insurers agree to share 24

The model can be adapted to reflect the credit market without great difficulty. We will see that the problem is interesting only when M ≥ 3. 26 This assumption makes the analysis more transparent. I show in the extension section that all our results remain the same when the two loss probabilities are strictly between 0 and 1. 27 This assumption can be interpreted as mandatory purchase, as is the case for auto insurance. 25

12 information, they share their complete private information among themselves. To cover the fixed cost of sharing information, the insurers must have some N form of market power. I assume that each insurer in the market has b M captive/loyal customers, with b ∈ (0, 1), who “enjoy extra value” v1 > 0 for choosing the insurer in period 1; that is, bN customers have extra value v1 for one and only one of the M insurers’ products in period 1, while the other (1 − b) N customers are “free” so that they simply compare the premiums when choosing policies. In period 2, the extra value v2 = 0.28 To resolve ties, the captive consumers always choose the insurance company that they are loyal to when there are ties in the values of offers;29 for the other consumers, the demand is divided equally among companies that offer the same lowest premium. In addition, I assume that an insurer cannot distinguish its captive customers from other customers. The two-period extensive form game proceeds in the following way. Period 1. Stage 1. The insurers simultaneously decide whether or not to sign a multilateral agreement to share customer information (excluding the premiums to be charged) through an information exchange, and, in the meantime the insurers set their first-period rate r1 . The membership of the exchange is public information. Stage 2. New customers choose insurers based on the rates offered and the insurers’ membership statuses. Stage 3. Losses occur, and claims are paid by insurers. Period 2. Stage 4. Complete information about the claim histories is shared among the information-exchange members. 28

This assumption is made to simplify the analysis. As we will see, If v2 > 0, there exists no pure-strategy equilibrium for period 2 subgames, which makes the analysis of the period 1 reduced-form game much more complex. 29 As will become clear later, this way of resolving ties plays two roles. First, it ensures the existence of pure-strategy equilibrium for the period-2 subgames. Second, it guarantees that the payoff functions of the period-1 reduced-form game is upper-semicontinuous, which is a condition used in the proof of the existence of equilibrium for the period 1 reduced form game.

13 Stage 5. Each firm j sets rates r2j (L) and r2j (H) for its L and H type old customers. Only the premium schemes, but not the rates for specific customers, are observable to the other insurers. Stage 6. Firms simultaneously set rates r˜2j (L) and r˜2j (H) for other firms’ L and H type old customers if they have complete information about them, and otherwise, they set r˜j (B) for other insurers’ customers.30 Stage 7. Customers choose insurers by premiums. Stage 8. Losses occur, and claims are paid by insurers. A firm’s strategy in the game can be written as (the information-sharing policy, r1 , r2 (L), r2 (H), r˜2 (L), r˜2 (H), r˜2 (B)). The cost of information sharing for each participating insurer is C (M1 ), where M1 is the number of insurers that enter the information-sharing agreement. Firms not in the informationsharing agreement cannot access the database at any cost.

3.2

Discussion of Some Modeling Issues

One modeling issue is about the synchronized decision of information sharing and premiums in period 1. This setup is adopted to simplify the analysis of the equilibriums of the game. An alternative setup is letting the insurers simultaneously make the information-sharing decision first and then choose premiums for period 1, given the knowledge of the membership of the information-sharing agreement. However, this alternative setup would make the investigation of equilibriums more complex, as it is essentially a three-period extensive-form game and would entail analyzing more subgames and an asymmetric subgame.31 The same forces supporting information sharing in equilibrium are at 30

The argument “B” in r˜j (B) denotes that the insurers cannot see (i.e., blind) the types of customers that they have no information about. 31 The asymmetric two-period pricing subgame follows one insurer choosing not to share information while all other insurers choose to share information. However, with the twoperiod game structure, I can analyze the impact of the decision to not share information on the deviating insurer’s payoff without having to solve an asymmetric pricing game, as it only arises as a deviation from a symmetric equilibrium strategy (in which everyone chooses to share information and charge the same premium in period 1).

14 work in both the two-period model and the three-period model.32 The sequential move structure in period 2 seems natural to me. The customers first get a quote for renewing the contract with their current insurers before they receive competing offers from other insurers. The sequential move structure for period 2 also makes the analysis easier. It is straightforward to find pure-strategy equilibria for the sequential-move subgames, which makes it easier to compute the payoff function for the reduced-form game in period 1. Alternative model setups, such as the customers’ types not being revealed completely via one-period transaction and the customers not knowing their types a priori, are treated in Section 5.

4

Basic Results

For the analysis of the above 2-period game, I use the following notations for a series of break-even premiums. Let rl = 0, rh = 1, rλ = λrh + (1 − λ) rl , which are, respectively, the break-even rates for the L type customers, the H type customers and the whole customer population. I will keep using the notations of (rh , rλ , rl ) instead of substituting in their current numerical values in the following analysis. When I generalize the loss probabilities of the two types of consumers, we will see that I only need replace the numerical values for them for the more general model. In this section, I investigate the conditions under which information sharing can arise in equilibrium. I focus on symmetric equilibria in my analysis. The candidate symmetric equilibrium strategy I propose is “to share information (abbreviated as SI.) and use mixed strategy F ∗ in period 1, and use subgame equilibrium strategy r2∗ in period 2.”33 Figure 1 is an illustration of the 2period game. The choice of “Not to share information” is abbreviated as N SI 32

In both the two- and three-period model, the deviation to not share information leads to a disproportionately small share of the demand coming from low-risk customers, which makes deviation unattractive in both models. 33 We can take F ∗ and r2∗ just as generic notations for strategies for now. The exact forms for them will be given later.

15 in Figure 1.

Game with the Choice of n Figure 1.
The 2-Period Pricing o n Information Sharing o 0 N SI, r d , (SI, F ∗ )∀j6=d L99 (SI, F ∗ )∀j∈M 99K (SI, r)d , (SI, F ∗ )∀j6=d ↓ ↓ ↓ Subgame 3 Subgame 1 Subgame 2 Note: subscript j denotes a generic insurer, and d denotes the defecting insurer. I use backward induction to solve the 2-period game. Note that I need only check deviations to pure strategies when showing that a candidate strategy profile is a Nash Equilibrium. As shown in Figure 1, I need to analyze two types of deviations in period 1 from the proposed equilibrium strategy and three types of period-2 subgames. The first type of subgame (“subgame 1” in Figure 1) follows every insurer choosing to share information and using the same mixed strategy F ∗ in period 1. The second type of subgame (“subgame 2” in Figure 1) follows every insurer choosing to share information and all insurers using the same mixed strategy F ∗ except for the single defecting insurer d charging r in period 1.34 The third type of subgame (“subgame 3” in Figure 1) follows all insurers choosing to share information and using the same mixed strategy F ∗ except for the one defecting insurer d choosing not to share infor0 mation and to charge r in period 1.35 To investigate the conditions ensuring that the proposed strategy profile is a SPE, I need to check whether, expecting everyone else to choose (SI, F ∗ ) in period 1 and use subgame equilibrium strategies in period 2, an insurer can increase its profit by deviating from the 0
proposed strategy (SI, F ∗ ) to either (SI, r) or Do N ot SI, r . To do so, I first find the equilibrium strategy r2∗ for every type of subgame in period 2. With r2∗ , I compute the payoff functions for the reduced-form game in period 1. Then I go on to show that a symmetric equilibrium pricing strategy F ∗ for period 1 always exists, assuming that all insurers choose to share information. Finally, I show my main result by providing the conditions under which sharI denote this defector’s strategy in period 1 as (F ∗ ,r)d .  0 35 I denote this defector’s strategy as Do N ot SI, r . 34

d

16 ing information and using F ∗ is the symmetric period-1 equilibrium strategy in an SPE. The following is a summary of the intermediate results I derived to prove the existence of the information-sharing SPE. Proposition 1 presents the equilibrium strategies for subgames 1 and 2. It allows us to solve backwards for the payoff functions and the candidate equilibrium pricing strategy (F ∗ ) in period 1. Proposition 2 points out that, given the derived payoff functions, there exists no pure-strategy equilibrium for period 1. Proposition 3 follows up by showing that the existence of a symmetric mixed-strategy equilibrium (F ∗ ) is guaranteed for period 1, which then implies that deviation to (SI, r) from (SI, F ∗ ) cannot be profitable. Propositions 4 and 5 characterize the support of the equilibrium mixed strategy. Proposition 6 gives the analytical solution for the mixed-strategy equilibrium for the duopoly case. The analytical solution gives us some idea of the comparative statics properties of the general mixedstrategy equilibria involving more than two firms.36 Proposition 7 describes the equilibrium for the third type of subgame (“Subgame 3” in Figure 1). 0
Lemma 1 shows that deviating to Do N ot SI, r induces customer adverse selection. My main result is stated in Theorem 2, which shows that sharing complete information can arise in an SPE as long as the cost of sharing the information is low enough. The first proposition gives the equilibrium for the period 2 subgames I need to be concerned with when all insurers choose to share information in period ∗ ∗ (L) and r˜2j (H) to 1—i.e., subgames 1 and 2 in Figure 1. Recall that we use r˜2j denote the premiums that an insurer sets for the L and H type old customers of the other insurers. Proposition 1 Suppose that M ≥ 3, and all insurers choose to share information in period 1—i.e. M1 = M . Then, for any period-2 subgame, the following strategy profile is a Nash Equilibrium: ∗ ∗ r2j (L) = rl , r2j (H) = rh , ∗ ∗ r˜2j (L) = rl , r˜2j (H) = rh 36

We don’t have an analytical solution for F ∗ for the case of more than two firms.

17 for all j ∈ M1 .37
∗ ∗ Proof. With perfect information, r˜2j (L) = rl , r˜2j (H) = rh is the equilib ∗ ∗ rium strategy for Bertrand competition. Given r˜2j ˜2j , firm j’s 0 (L) , r 0 (H) ∀j 0 ∗ profit is always zero no matter how it changes its premiums. So, (r2j (L) = rl ,  ∗ ∗ ∗ r2j (H) = rh ) is a trivial optimal strategy for its stage given r˜2j ˜2j . 0 (L) , r 0 (H) ∀j 0 6=j

Remark 1 If I let insurers simultaneously set price schemes in period 2, the subgame equilibrium would be the same as above for the first and second type of subgames. It is the third type of subgame for which there exists no pure-strategy equilibrium if I assume a simultaneous move structure. In the following, I refer to the strategy profile for period 2 described in the above proposition simply as r2∗ . With the equilibrium strategy for the first and second type of period-2 subgames in hand, we can get the period 1 payoffs as functions of the vector of premiums set by the insurers in period 1, assuming that all insurers choose to share information. Obviously, the profits for all insurers in period 2 are zero in subgame 1 and subgame 2. So, the total payoff for each insurer is equal to its first period profit. One prominent feature of the payoff functions for the reduced-form game in period 1 is that they are not continuous in the premiums. The next proposition shows that a mixed-strategy pricing equilibrium for period 1 is the best we can hope for. Proposition 2 There exists no SPE in which all firms share information and choose pure strategies in period-1 prices. Proof. Suppose there exists an equilibrium in which all firms choose to share information and set period-1 premiums as {rj }j∈M . First note there would be no consumer adverse selection in this case, and each insurer can at least guarantee itself a payoff of zero by charging any rate above rλ . Let r0 ≡ minj∈M {rj }. We must have # {j|rj = r0 } > 1, as otherwise the insurer charging r0 can 37

Note that I abused the notation by letting M1 also be the set of insurers that choose to share information.

18 increase its payoff by increasing its period-1 premium by a small amount. Furthermore, we must have r0 ≥ rλ , as otherwise the payoffs of some insurers would be strictly negative. Now suppose that r0 = rλ ; then, the payoffs of the insurers charging r0 would be zero, but each of them could deviate to v from its captive customers. Finally, suppose r = rλ + v and get a payoff of bN M that r0 > rλ . An insurer charging r0 can increase its payoff by charging instead r0 − ε for small enough ε. Therefore, there exists no pure-strategy equilibrium for the reduced-form game in period 1. In light of the above result, I look for pricing equilibria with mixed strategies for period 1.38 As we’ll see, analytical solutions for the equilibrium mixed strategies are not available for cases with at least three insurers in the market. Yet, in my analysis of the forces supporting information sharing in equilibrium, I will use the equilibrium mixed strategy for period 1. To ensure that my efforts are not in vain, I first prove in the next proposition that such equilibria exist. As the payoff functions of the insurers in period 1 are not continuous in premiums, I cannot apply the existence result of Glicksberg (1952). Here I invoke a theorem from Dasgupta and Maskin (1986), which provides conditions that guarantee the existence of a mixed-strategy equilibrium for strategic form games with some specific types of discontinuity in payoff functions that are typical in many economic games. For convenience, I replicate the Theorem 6 in Dasgupta and Maskin (1986) here, following their notations. Let [(Ai , Ui ); i = 1, ..., M ] be a game, where the action space of the ith player Ai ⊆ R1 is a closed interval. For each pair of agents, i, j ∈ {1, ..., M }, let D (i) be a positive integer, and for each integer d, with 1 ≤ d ≤ D (i), let fijd : R1 → R1 be a one-to-one, continuous function. 38

Here is an example of mixed pricing strategy from real life. Stores sometimes give discounts to online shoppers. The discount rates are normally highlighted on the main page of the stores’ websites. Interestingly, however, different customers might see different discount rates at the same website, at the exact same time; and a customer may quickly see different discount rates if she repeatedly visits the websites without leaving traces of her old visits.

19 For each i ∈ {1, ..., M }, denote A∗ (i) = {(a1 , ..., aM ) ∈ A|∃j 6= i, ∃d, 1 ≤ d ≤ D(i) such that aj = fijd (ai )}. (1) Note that A∗ (i) ⊆ RM is of (Lebesgue) measure zero. It covers the set of discontinuous points in many games in economics, including the period-1 reducedform game in my model. Let the set of discontinuities of Ui (a) be denoted as A∗∗ (i), and suppose A∗∗ (i) ⊆ A∗ (i). For ∀ai ∈ Ai define A∗−i (ai ) = ∗∗ {a−i ∈ A−i | (ai , a−i ) ∈ A∗ (i)}, and A∗∗ −i (ai ) = {a−i ∈ A−i | (ai , a−i ) ∈ A (i)}. ∗ ∗∗ Let A∗i (i) (resp. A∗∗ i (i)) be the projection of A (i) (resp. A (i)) onto Ai . The following is a definition and a property of the payoff function that will be used in the cited theorem. Definition 1 Ui (ai , a−i ) is weakly lower semi-continuous in ai if ∀¯ ai ∈ A∗∗ i (i), ∗∗ ai ), ∃λ ∈ [0, 1] such that for all a−i ∈ A−i (¯ λ lim inf Ui (ai , a−i ) + (1 − λ) lim inf Ui (ai , a−i ) ≥ Ui (¯ ai , a−i ) −

ai →¯ ai

+

ai →¯ a

Definition 2 (Property (α)): ∀¯ ai ∈ A∗∗ (i), ∃λ ∈ [0, 1] such that for all a−i ∈ A∗∗ ai ) −i (¯ ai , a−i ) λ lim inf Ui (ai , a−i ) + (1 − λ) lim inf Ui (ai , a−i ) ≥ Ui (¯ −

ai ai →¯

+

ai →¯ a

where the inequality is strict if a−i = (¯ a , ..., a ¯ ). | i {z i} M −1 times

Theorem 1 (Theorem 6 in Dasgupta and Maskin (1986)) Let A¯ ⊆ R1 (i = 
 ¯ Ui ; i = 1, ..., M be a sym1, ..., M ) be non-empty and compact, and let A, ¯ ... × A¯ → R1 is continuous except on a subset metric game, where ∀i, Ui : A | × {z }

M times P A∗∗ (i) of A∗ (i), where A∗ (i) is defined by (1). Suppose that i Ui (a) is upper semi-continuous and ∀i Ui (ai , a−i ) is bounded and satisfies Property (α). Then, there exists a symmetric mixed-strategy equilibrium (µ∗ , ..., µ∗ ) with the ∗ property that ∀i and ∀¯ ai ∈ A∗∗ ai }) = 0. i (i), µ ({¯

20 Now, let F be a cumulative distribution function defined on [rl , rh ], and f and F¯ be the corresponding density function and survival function. π is the expected profit function. The next proposition verifies the conditions in Theorem 1 to prove the existence of a symmetric mixed-strategy equilibrium for the reduced-form game in period 1. Proposition 3 Suppose that all insurers choose to share information. Then, there exists a Nash Equilibrium in symmetric atomless mixed strategies for the period-1 reduced-form game. Proof. In period 1, the reduced-form game is essentially a Bertrand price competition with product differentiation. Let ωi ≡ (ri − rλ ), which is the profit an insurer charging ri in period 1 can expect to make from one of its customers. For (ri , r−i ) ∈ [rl , rh ]M , the payoff function of firm i in period 1 can be written as   0, if ri > minj6=i {rj } + v     bN ωi , if minj6=i {rj } < ri ≤ minj6=i {rj } + v nM  o πi (ri , r−i ) = bN 1 bN  + (1 − b) N + # {j|r > r + v} ωi , if ri = minj6=i {rj } j i  #{j|rj =ri } M   M    bN + (1 − b) N + # {j|rj > ri + v} bN ωi , if ri < minj6=i {rj } M M where #X denotes the cardinality of set X. The above payoff function of firm i is continuous except on a subset A∗∗ (i) of the following continuous manifold of dimension less than M : n o M d A (i) = (r1 , ..., rM ) ∈ [0, 1] |∃j 6= i, and ∃d, 1 ≤ d ≤ 3 such that rj = hij (ri ) ∗

where h1ij (ri ) = ri , h2ij (ri ) = ri + v, and h3ij (ri ) = ri − v. Obviously, πi is bounded for all i. When ri approaches any point of discontinuity from the left, πi always jumps down or stays constant. Therefore, πi is (left) weakly lower semicontinuous in ri . In addition, it is easy to check that Σi πi (ri , r−i ) is upper semicontinuous by noticing that the sharing rule assumed for ties in values maximizes the aggregate profit in such cases. Fur-

21 thermore, we have lim inf r →¯ πi (ri , r−i ) > πi (¯ ri , r−i ), for r−i = (¯ r , ..., r¯ ) and − i ri | i {z i} M −1 times

all r¯i ∈ A∗∗ i (i), as the demand is equally shared among insurers when there is a tie. Hence, the payoff function satisfies the property (α). Thus, Theorem 6 in Dasgupta and Maskin (1986) ensures that there exists a symmetric mixedstrategy equilibrium for the game of [[rl , rh ] , πi (ri , r−i ) ; i = 1, ...M ], and the equilibrium mixed strategy is atomless at points of discontinuity. As the equilibrium mixed strategy cannot have atoms at any point where πi (ri , r−i ) is continuous either, the equilibrium mixed strategy is atomless everywhere. For the rest of the analysis, I use F ∗ to denote a given equilibrium mixed strategy for the reduced-form game in period 1. Though the analytical solutions for equilibrium mixed strategies for period 1 are not available for the cases of M > 2, I provide some characterization of the support of the equilibrium mixed strategies,39 and use the analytical equilibrium strategy I derive for M = 2 to get additional information on the properties of the mixed-strategy equilibria. Proposition 4 For F to be a symmetric equilibrium mixed strategy, we must have f (r) = 0 for r ≤ rλ . Proof. Suppose that f (˜ r) > 0 for some r˜ < rλ . Then, π (F ) = π (˜ r) < 0. However, each insurer can guarantee itself nonnegative payoff by charging any premium larger than rλ . Thus, f (r) = 0 for r < rλ . Furthermore, if f (rλ ) > 0, then π (F ) = π (rλ ) = 0, whereas π (rλ + v) > 0, given that other insurers stick to F . Thus, we must have f (r) = 0 for r ≤ rλ for F to be a symmetric equilibrium mixed strategy. So, we should look for equilibrium mixed strategies with the support of period 1 premiums contained in (rλ , rh ]. The next proposition shows that the length of the support of the symmetric equilibrium mixed strategy is bounded below by v. Proposition 5 Let D be the support of a symmetric equilibrium mixed strategy F ∗ , r¯ ≡ sup D, r ≡ inf D. Then, we must have r¯ − r ≥ v. 39

One of the conditions I need for information sharing to arise in equilibrium is related to the upper bound of the support of the equilibrium mixed strategy.

22 Proof. Suppose that r¯ − r < v. A firm’s demand would be bN — i.e., all M its captive customers—if it charges r¯ while all other insurers use strategy F ∗ . However, a firm’s demand would still be bN if it charges r¯ + ε for small M enough ε while all other firms use strategy F ∗ . Therefore, π (¯ r) < π (¯ r + ε) as bN bN r − rλ ) and π (¯ r + ε) = M (¯ r + ε − rλ ), which contradicts the fact π (¯ r) = M (¯ that r¯ = sup D. The next proposition shows that for the case of M = 2 the equilibrium mixed strategy for period 1 has r¯ − r = v and can be solved analytically. Although the case of M = 2 is of little interest for deriving the main results, its analytical solution gives us some idea of the possible comparative statics properties of the mixed strategy equilibrium for cases of M > 2. Proposition 6 Suppose that all insurers choose to share information. Then √ with M = 2, given that b < 6−22 5 , the following is a equilibrium mixed strategy h  i M1−1 1 π∗ b for the period-1 reduced-form game: F ∗ (r) = 1 − 1−b − for N (r−rλ ) M
bv bv , r¯ = r + v, and π ∗ = bN + (1 − b) N M (1−b) r ∈ [r, r¯], where r = rλ + M (1−b) M is the equilibrium profit. For M > 2, we have r¯ − r > v for all equilibrium mixed strategies for the period-1 reduced-form game. Proof. By inspection, we know that the above F ∗ (r) is strictly increasing in r) = 1 and F ∗ (r) = 0. So, F ∗ (r) r on [r, r¯]. And it is easy to check that F ∗ (¯ is, indeed, a cumulative distribution function (cdf) on [r, r¯]. In the following, I show that F ∗ is the equilibrium mixed strategy for M = 2 by construction. Note that, unless specifically clarified, the following analysis holds for both M = 2 and M > 2, assuming r¯ = r + v. Suppose that r¯ = r + v, and let F (r) be some cdf on [r, r¯]. Let π (r) be the profit of an insurer when it charges rate r ∈ [r, r¯] while all other insurers use the mixed strategy F . Then, we have  π (r) =


M −1 bN + (1 − b) N F¯ (r) (r − rλ ) , for r ∈ [r, r¯] M

23 In order for F to be an equilibrium mixed strategy, it is necessary that π (¯ r) = π (r)

(2)

and, π (¯ r) = π (r) for all r ∈ [r, r¯]

(3)


bv bv , and π (¯ r) = π ∗ ≡ bN + (1 − b) N M (1−b) . From equation (2), we get r = rλ + M (1−b) M  
M −1 π(¯ r) 1 From equation (3), we get F¯ (r) = 1−b − Mb . N (r−rλ ) To finish proving that F = 1− F¯ is a symmetric equilibrium mixed strategy, in the following I show that π (r) ≤ π (¯ r), for all r > r¯ and r < r. For r ≥ r¯, we have π (r) =


M −1 bN ¯ F (r − v) (r − rλ ) M

We only need to be concerned with r ∈ (¯ r, r¯ + v), as the demand is zero and π (r) = 0 for r ≥ r¯ + v. We have π (r) < π (¯ r) for r ∈ (¯ r, r¯ + v) because   bN dπ (r) vπ ∗ b = − − dr (1 − b) M N (r − v − rλ )2 M < 0 for all r ∈ (¯ r, r¯ + v) For r ≤ r, the firm charging r would always get all its bN captive customers M and all the (1 − b) N contestable customers. In addition, the firm would get bN (M − 1)F¯ (r + v) captive customers of the other M − 1 firms. So, we have M for r ≤ r π (r) = (r − rλ ) [

bN bN + (1 − b) N + (M − 1)F¯ (r + v)] M M

As π (r) < π (r − v), for r < r − v, we only need to consider π (r) for r ∈ [r − v, r]. For M = 2, we have dπ (r) bN b = (1 − ) − (1 − b)N , for r ∈ [r − v, r] dr 2 2(1 − b)

24 √

So, for r ∈ [r − v, r], dπ(r) > 0, if b < 6−22 5 . Hence, π (r) ≤ π (r), for dr √ r ∈ [r − v, r], given that b < 6−22 5 . Therefore, for M = 2, F ∗ given in the proposition is, indeed, an equilibrium mixed strategy for the period-1 reduced-form game, given that b is relatively small. However for M > 2, we have π (r − ε) > π (r), for ε small enough. This ¯ → −∞ as r → r, since dF (r+v) → −∞, as r → r. Thus, for is because dπ(r) r r M > 2, we must have r¯ − r > v for the support of any symmetric equilibrium mixed strategy for period 1. Remark 2 The comparative statics properties of the above solution for M = 2 is consistent with intuition. In particular, the equilibrium profit π ∗ is increasing in v and b, both of which increase the insurance  companies’ market power. The b ∗ v, is an increasing upper bound of the support of F , r¯ = rλ + 1 + M (1−b) function of v. In addition, r¯ → rλ as v → 0. My conjecture is that, for any given M > 2, the least upper bound of the support of any given equilibrium mixed strategy also converges to rλ as v → 0. We now know that for M > 2, we must have r¯ − r > v for the period-1 reduced-form game with all insurers choosing to share information. However, with r¯ − r > v, a closed-form solution for F ∗ is unlikely. Given r¯ − r > v, the demand of an insurer charging rate r, denoted as G (r), when the other insurers use a given equilibrium mixed strategy F ∗ is G (r) =


M −1
M −1 bN ¯ ∗ bN F (r − v) + F¯ ∗ (r) (1 − b) N + (M − 1)F¯ (r + v) M M

In the above demand function, the first term measures the demand of the in
M −1 surer’s bN captive customers that the insurer gets with probability F¯ ∗ (r − v) ; M the second term measures the demand of the (1 − b) N contestable customers
M −1 that the insurer gets with probability F¯ ∗ (r) ; the third term measures the demand of the captive customers of the other M − 1 insurers that the insurer gets with probability bN (M − 1)F¯ (r + v). Given the demand function M G (r), the insurer’s corresponding profit function is then π (r) = G (r) (r − rλ ),

25 where “r − rλ ” is the profit an insurer can expect to make from a customer. Now, suppose that we know the value of r¯ − r, the set of equations in (2) and (3) is very unlikely to have closed-form solutions for F¯ ∗ (r) anymore because of the the higher-order terms of F¯ ∗ (r − v) and F¯ ∗ (r) in the insurers’ demand functions. To continue to derive my main results, I now only need to check if choosing 0
Do N ot SI, r can increase the defecting insurer’s profit relative to (SI, F ∗ ). To do so, I first characterize the equilibrium strategy in the third type of subgame40 (“subgame 3” in Figure 1) and then provide conditions that ensure the existence of information sharing in equilibrium. Hereafter I always mean 0
“the insurer deviating to Do N ot SI, r ” by “the defecting insurer”. In the following, I use s to denote the equilibrium proportion of the H type customers in the defecting insurer’s demand, subscript d to denote the defecting insurer, and r˜2j (B) to denote the premium an insurer j sets for customers it has no information about.41 I will say that the defecting insurer faces consumer adverse selection if s ∈ (λ, 1], recalling that λ is the share of H type customers in the population. The following proposition characterizes the equilibrium strategy for the third type of subgame for any given s.42 40

Recall that the third type of period-2 subgame follows all insurers using the strategy (SI, F ∗ ), except that the defecting insurer uses the strategy of (N ot to SI, r). 41 For example, the defecting insurer receives no information about the old customers of all other insurers. 42 This result relies on the assumption that in stage 5 of period 2, only the pricing scheme of an insurer but not its prices for specific old customers are observable to the competitors. More-careful motivations for the assumption is in order here. The assumption may seem strong from a practical point of view: If an old customer receives a low quote, then she may, in practice, reveal her low type to the competing insurers by showing them the quote. However, this kind of behavior does not seem to exist in practice, and it is inconsistent with the empirical evidence found in markets in which insurers do not share information (cf. Cohen (2011)). More importantly, whether the assumption really is a strong assumption should eventually depends on whether there exists a low-cost mechanism that allows the insurers to charge distinct prices to their old customers without losing them to their competitors. If such a mechanism exists, then the insurers can, and most likely would, resort to such a mechanism if they found out that their distinct quotes to their customers have been actively used as proof of customer types to seek better competing offers. The following simple mechanism might be one of such mechanisms. In the mechanism, only the premium schedule, but not customer specific prices, would be revealed to the old customers. The old customers then would be invited to buy insurance at the price that they qualify for according to their

26 Proposition 7 Suppose that M ≥ 3 and that s is the proportion of the H type customers in the demand of the defecting insurer in period 1. The following strategy profile is an equilibrium for the third type of subgame (subgame 3 in Figure 1). r2d (L) = srh + (1 − s) rl ,

r2d (H) = rh ,

For all j 6= d, r2j (L) = rl ,

r2j (H) = rh ,

r˜2j (L) = rl , ( r˜2j (B) =

r˜2d (B) = rh

r˜2j (H) = rh , rh if r2d (L) ≤ srh + (1 − s) rl srh + (1 − s) rl if r2d (L) > srh + (1 − s) rl

Proof. As I assume that M ≥ 3, with the defecting insurer choosing to not share information, there are still at least two insurers choosing to share information in the third type of subgame (subgame 3 in Figure 1). Given that there are at least two insurers sharing information, we must have r2j (L) = rl , r2j (H) = rh , r˜2j (L) = rl , and r˜2j (H) = rh , for all j 6= d, in equilibrium, for the exact same reason I used to prove that r2∗ is an equilibrium strategy profile for the first and second type of subgames.43 So, we only need to show that the addition of the above r2d (L), r2d (H), r˜2d (B) and r˜2j (B) makes up an equilibrium for the third type of subgame. Note that the defecting insurer does not have information about the types of the other insurers’ old customers. Given that r2j (L) = rl and r2j (H) = rh in the proposition, the defecting insurer would attract from the other insurers only the H type old customers if it set r˜2d (B) in the range of (rl , rh ), and it would attract from the other insurers all their H type old customers and part of their L type old customers if it sets r˜2d (B) = rl . In both cases, the defecting insurer loses money in dealing with the switching customers. Thus, r˜2d (B) = rh is the defector’s optimal strategy given r2j (L) and r2j (H). own claim histories. Now, the customer-specific premiums and types can only be revealed by actually purchasing the insurance. 43 So, M ≥ 3 is necessary to ensure that, without the defecting insurer, there are still at least two insurers that can share information with each other, so that they are still able to commit to charge the low price rl in period 2.

27 Further note that the information about the types of the deviating insurer’s old customers is not shared. As in any equilibrium insurers expect r2d (H) ≥ rh , I claim that r˜2j (B) = rh is the only rate that lets the competing insurers break even when r2d (L) ≤ srh +(1 − s) rl . The reason is as follows. If r2d (L) ≤ srh + (1 − s) rl , charging any rate in the range of (srh + (1 − s) rl , rh ) would attract only all the H type old customers of the defecting insurer, of which rh is the break-even rate. Charging exactly srh + (1 − s) rl would attract from the defecting insurer all its H type old customers but, at most, only part of its L type old customers.44 Thus, the break-even rate for these switching customers is strictly larger than srh + (1 − s) rl because the proportion of the H type among the switching customers is strictly larger than s. Charging any rate below srh + (1 − s) rl , the competing insurers would attract from the defecting insurer all its H type old customers and all or part of its L type old customers, but the break-even rate for these switching customers is at least srh + (1 − s) rl . Therefore rh is the optimal rate if r2d (L) ≤ srh + (1 − s) rl . It is easy to see that srh + (1 − s) rl is the break-even rate for the customers switching from the defecting insurer if r2d (L) > srh + (1 − s) rl . Lastly, given the r˜2j (B) in the proposition, it is easy to see that (r2d (L) = srh + (1 − s) rl , r2d (H) = rh ) is the defector’s dominant strategy for the stage. The above subgame equilibrium strategies are contingent on s, the share of H type customers in the defecting insurer’s demand. Now suppose that in period 1, all insurers use the pricing strategy F ∗ , except that the defecting insurer sets its premium at r. We now need to find the s that is consistent with consumer rational expectation for any given r. Expecting the above period-2 premium schemes, the period-1 demand of the defecting insurer from the L type customers would be (1 − λ) G (r + s (rh − rl )), as the defecting insurer is expected to charge s (rh − rl ) more than the other insurers to the low-risk customers in period 2;45 meanwhile the period-1 demand from H 44

This is generally true unless we assume that all the demand goes to the insurers competing with the defecting insurer when there is a tie in premiums. 45 Recall that G (r) is the total period-1 demand of an insurer charging rate r when the other insurers follow mixed strategy F ∗ and all the insurers choose to share information.

28 type customers would be still be λG (r).46 Let the equilibrium share of H type customers in the defecting insurer’s demand be denoted as Q (r, s) for a given r and an expected s. Then, we have Q (r, s) =

λG (r) λG (r) + (1 − λ) G (r + s (rh − rl ))

where the numerator and the denominator on the right-hand side are, respectively, the defecting insurer’s demand from the H type customers and its total demand in period 1. The assumption that consumers have rational expectations implies: s = Q (r, s) (4) That is, the expected share of H type customers of the defecting insurer, s, must equal the equilibrium share of H type customers in the defecting insurer’s demand, Q (r, s). Thus, given the defecting insurer’s period-1 premium r, s is determined endogenously by equation (4) in equilibrium. In the following Lemma, I show by analyzing equation (4) that when the information to be shared is complete, the defecting insurer almost always faces customer adverse selection. I use S (r) to denote the set of solutions to equation (4) for a given r. Lemma 1 For the solutions to equation (4), we have λ ≤ S (r) ≤ 1 for all r. Furthermore, let rˆ ≡ r − v − λ (rh − rl ) and rˆˆ ≡ r¯ + v − (rh − rl ), and assume that v < rh − rl . Then we have: a) λ ∈ S (r) for all r ≤ rˆ; b) 1 ∈ S (r) for all r ≥ rˆˆ; and c) λ < S (r) < 1 for r > rˆ and r < rˆˆ. Proof. As it is possible that there are multiple solutions to equation (4), S (r) should be treated as a correspondence. Recall that G (r) = N if r ≤ r − v, and that G (r) = 0 if r ≥ r¯ + v. So, we only need to consider the solution to the above equation for r < r¯ + v. For r < r¯ + v, as G (r + s (rh − rl )) ≤ G (r), we have Q (r, s) ≥ λ for all s ∈ [0, 1]. And, obviously, Q (r, s) ≤ 1 for all s ∈ [0, 1]. So, there is no customer selection in such a case. G (r) is decreasing in r. 46 All firms charge rh to their H type old customers in the subgame equilibrium in period 2.

29 Thus, λ ≤ S (r) ≤ 1 for all r < r¯ + v unless S (r) is an empty set. However, as Q (r, λ) − λ ≥ 0, and Q (r, 1) − 1 ≤ 0, and Q (r, s) − s is a continuous function in s, it follows that ∃ s∗ ∈ [λ, 1] s.t. s∗ = Q (r, s∗ ). So, S (r) is not an empty set, and λ ≤ S (r) ≤ 1 for all r < r¯ + v. As G(r) = N and G(r + λ(rh − rl )) = N for all r ≤ rˆ, we have λ ∈ S(r) for all r ≤ rˆ. And since G(r + (rh − rl )) = 0 for all r ≥ rˆˆ, 1 ∈ S(r) for all r ≥ rˆˆ. Furthermore, for rˆ < r < rˆˆ, we have G(r) > 0, G(r + (rh − rl )) > 0 and G(r) > G(r + λ(rh − rl )). Hence, λ < S(r) < 1, for rˆ < r < rˆˆ. Remark 3 As will become clear after the proof of the next theorem, the only period-1 premiums we need to consider for the defecting insurer are those in [ˆ r, rˆˆ]. Therefore, the defecting insurer will effectively almost always face consumer adverse selection. Remark 4 The assumption of v < rh −rl is essentially an innocuous assumption, as conceptually v should be relatively small. I will suppress the dependence of s on r hereafter for simpler notation. Given s as a solution to equation (4), we now have the expression for the twoperiod total payoff πd (r) for the defecting insurer when it sets its premium at r in period 1: πd (r) = λG (r) · (r + rh − 2rh ) + (1 − λ) G (r + s (rh − rl )) · (r + srh + (1 − s) rl − 2rl ) = λG (r) · (r − rh ) + (1 − λ) G (r + s (rh − rl )) · (r + srh − (1 + s) rl ) The first term above is the total profit from H type customers, and the second term is the total profit from the L type customers. Now, I am ready to show the main result that information sharing can arise endogenously in equilibrium as long as the cost of sharing information is not too high. As I have derived previously, when all insurers choose to share information, the demand of an insurer charging fixed rate r while the other insurers apply F ∗ is G (r) =


M −1
M −1 bN ¯ ∗ bN F (r − v) + F¯ ∗ (r) (1 − b) N + (M − 1)F¯ (r + v) M M

30 I will use this demand function in the proof of the next theorem. I use π ∗ to denote each insurer’s expected profit when all insurers choose to share information and follow the equilibrium pricing strategy F ∗ (supported on [r, r¯]) in period 1 and r2∗ in period 2; and I use π (r) to denote the profit of an insurer in the exact same situation, except that it charges rate r in period 1. We know that π ∗ = π (r), for any r ∈ [r, r¯], as F ∗ is an equilibrium mixed strategy. And, hereafter, I use r2∗ to denote the equilibrium strategies for all the three types of subgames relevant to my analysis. Theorem 2 Suppose that r¯ < rh − v & M ≥ 3. Then, it is an SPE for all insurers to choose to share complete customer information and use an equilibrium mixed strategy F ∗ in period 1 and r2∗ in period 2 if the cost of sharing information is low enough, more specifically, if 

∗

∗

min π , π −

max

r∈{r|r−v≤r+s(rh −rl )≤¯ r+v}

 πd (r) ≥ C (M )

where s is determined by the following equation s= and

λG (r) . λG (r) + (1 − λ) G (r + s (rh − rl ))

 min π ∗ , π ∗ −

 max

r∈{r|r−v≤r+s(rh −rl )≤¯ r+v}

πd (r)

>0

Proof. Recall that we have the following expression for π (r) : π (r) = G (r) (r − rλ ) For the defecting insurer (deviating to (Do N ot SI, r)), we have derived its payoff as a function of its period-1 premium r: πd (r) = λG (r) · (r − rh ) + (1 − λ) G (r + s (rh − rl )) · (r + srh − (1 + s) rl ) It is easy to check that the above two formulas hold for all r. My goal is to find the conditions under which unilateral deviation to not sharing information

31 reduces the defecting insurer’s profit, i.e., π ∗ − C (M ) ≥ maxr πd (r). Towards the goal, in the following, I first show that π ∗ > maxr πd (r). a) For r ∈ {r|r + s (rh − rl ) > r¯ + v}. As G (r) = 0 iff r ≥ r¯ + v, with the assumption of r¯ + v < rh , we have that G (r) (r − rh ) ≤ 0 and G (r + s (rh − rl )) = 0. Thus, πd (r) ≤ 0, for r ∈ {r|r + s (rh − rl ) > r¯ + v}. b) For r ∈ {r|r − v ≤ r + s (rh − rl ) ≤ r¯ + v}. Note that we have π (r + s (rh − rl )) = λG (r + s (rh − rl )) (r + s (rh − rl ) − rh ) + (1 − λ) G (r + s (rh − rl )) (r + srh − (1 + s) rl ) which is the payoff of an insurer charging r + s (rh − rl ) in period 1 when the other insurers apply mixed strategy F ∗ and all insurers choose to share information. Notice that π ∗ ≥ π (r + s (rh − rl )), for all r, as F ∗ is an equilibrium mixed strategy. Therefore, we have the following comparison of π ∗ and πd (r) for any r ∈ {r|r − v ≤ r + s (rh − rl ) ≤ r¯ + v}, π ∗ − πd (r) ≥ π (r + s (rh − rl )) − πd (r) = λ {G (r + s (rh − rl )) [r + s (rh − rl ) − rh ] − G (r) (r − rh )} = λ {G (r + s (rh − rl )) s (rh − rl ) + [G (r) − G (r + s (rh − rl ))] (rh − r)} > 0 The last strict inequality follows because rh − rl > 0, (rh − r) > 0 for all r ∈ {r|r − v ≤ r + s (rh − rl ) ≤ r¯ + v}, and (

= 0 if r + s (rh − rl ) = r¯ + v > 0 if r ∈ {r|r − v ≤ r + s (rh − rl ) < r¯ + v} ( > 0 if r ∈ {r|r − v < r + s (rh − rl ) ≤ r¯ + v} G (r) − G (r + s (rh − rl )) = 0 if r + s (rh − rl ) = r − v G (r + s (rh − rl ))

32 Since π (r + s (rh − rl )) − πd (r) is a continuous function in r on the compact set of {r|r − v ≤ r + s (rh − rl ) ≤ r¯ + v}, it obtains its minimum on the set, and obviously the minimum is strictly positive. c) For r ∈ {r|r + s (rh − rl ) < r − v}, we have πd (r) ≤ πd (r − v − λ (rh − rl )), where (r − v − λ (rh − rl )) ∈ {r|r − v ≤ r + s (rh − rl ) ≤ r¯ + v}. This is because, for all r ∈ {r|r + s (rh − rl ) < r − v}, we have47 πd (r) = λN (r − rh ) + (1 − λ) N (r + srh − (1 + s) rl ) < λN (r − v − s (rh − rl ) − rh ) + (1 − λ) N (r − v − rl ) ≤ λN (r − v − λ (rh − rl ) − rh ) + (1 − λ) N (r − v − rl ) = πd (r − v − λ (rh − rl )) where the weak inequality on the third line follows because s ≥ λ, and the last equality follows because λ ∈ S (r − v − λ (rh − rl )). Summarizing the above analysis, we have π ∗ > maxπd (r) , & r  maxπd (r) = max 0, r

max

r+v} r∈{r|r−v≤r+s(rh −rl )≤¯

 πd (r)

Therefore, π ∗ − max πd (r)  r = min π ∗ , π ∗ −

max

r∈{r|r−v≤r+s(rh −rl )≤¯ r+v}

 πd (r)

Therefore, a sufficient condition for deviating to (Do N ot SI, r) being not profitable is π ∗ − max πd (r) ≥ C (M ) (5) r

This condition would be satisfied as long as the cost of sharing information, C (M ), is not too large. Therefore, given that r¯ < rh − v, I have proved that 47

Recall G (r) = N for all r < r − v.

33 there exists an SPE with endogenous information sharing if condition (5) is satisfied. Remark 5 As I suggested earlier, information sharing allows insurers to commit themselves to not exploiting “informationally captured” customers. The force driving insurers to make such a commitment is the rational expectations of the L type customers, as they know that they would be charged srh +(1 − s) rl if they chose an insurer that opts to not share information, while they can get their insurance at the price of rl from the other insurers that share information. This logic is captured in the model as the period-1 demand from the L type customers decreases from “(1 − λ) G (r)” to “(1 − λ) G (r + s (rh − rl ))” if an insurer switches from sharing to not sharing information. Meanwhile, the demand from the H type customers in period 1 stays the same because the H type will always be charged the same rate rh in period 2. In short, it is the consumer adverse selection faced by the deviating insurer that makes quitting sharing information unilaterally unappealing. Remark 6 The condition M ≥ 3 ensures that, with one insurer defects to not share information, there are still at least two insurers who share information and compete with each other and will charge the low price rl in period 2. Otherwise, if M = 2, with one insurer defecting to not share information, the one insurer left would have no one to share information with and, thus, would not be able to commit to charge rl in period 2 anymore. Then, my proof above no longer holds, because there would be no difference between the period2 premiums for the L type customers offered by the defecting insurer and by the other insurer since both of them would be charging λrh + (1 − λ) rl to their L type old customers. Remark 7 The condition r¯ < rh − v is expected to be satisfied if v is small enough. In the case of period-1 mixed-strategy equilibrium for M = 2, we saw that the analytical solution for r¯ converges to rλ as v goes to zero. My conjecture is that the equilibriums for the case of M ≥ 3 have similar property. Through the above analysis, I have shown that: a) information sharing works as a device that allow insurers to commit to lower premiums in the

34 future; and b) when the information to be shared is complete, defecting to not sharing information induces consumer adverse selection and leads to lower profit, which is the force that sustain information sharing in equilibrium. My analysis implies that insurers can voluntarily enter into an industrywide agreement to share customer information. And even if the market is initially in a equilibrium without information sharing, it can move to the information-sharing equilibrium with a push from profit-seeking entrepreneurs. Thus, if the benefits associated with exchanging customer information are worth pursuing, my results imply that it is not necessary for the government to step in to create some government-operated or -sponsored organization to broker such information.

5

Extensions and Robustness Checks

In this section, I investigate how the above results are robust to relaxing two particular assumptions I made in the model. The first one is that insurers are able to discover their customers’ exact types through a one-period transaction; and the second is that, originally, customers have complete information and, thus, better information than insurers about their own types. In the following, I first show that the first assumption is essentially without loss of generality. I then discuss the alternative assumption that the customers are originally endowed with the same (i.e., no) information as insurers about their own types and would obtain the same information as the insurers during the process of transactions. Under this alternative assumption about the information structure of the model, the root cause that induces information sharing is the same, but the specific mechanism that supports information in equilibria is different.

5.1

Partial Revelation of Customer Types

In my basic model, I assumed that the L type customers never incur any loss, while the H type customers always incur a loss of 1. Thus, the consumers’ types would be fully revealed to their insurance companies through just one

35 period of transaction. In general, we expect losses to occur with less-extreme probabilities, and consumers’ types to be revealed only partially through their transaction histories. Yet, none of my previous results would change if we incorporate the more general setup into my model. In this more general setup, we can essentially duplicate my previous analysis and results by replacing the revealed consumer types by the observation of period 1 losses, and, accordingly, replacing the expected break-even premiums for each type with the expected break-even premiums conditional on the realizations of losses. More specifically, suppose that L and H type customers incur a loss of 1 with probability p and q, respectively, and 0 < p < q < 1. Let It be a Bernoulli random variable indicating whether, in period t, a consumer has a loss.48 Then, the posterior break-even rates conditional on claim history can be modified accordingly. Let rl = Pr (I2 = 1|I1 = 0), rh = Pr (I2 = 1|I1 = 1), and rλ = Pr (I1 = 1) = λp + (1 − λ) q, and it is easy to check that we have, as before, rl < rλ < rh . With these key variables redefined, we can essentially duplicate all the results that I have derived above. For example, the period 2 subgame equilibriums and the payoff functions for the period-1 reduced-form game would be exactly the same except that the numerical values underlying the notations are modified. Of course, here, the information being shared would be the claim history data instead of the actual risk types of the customers.

5.2

Alternative Information Structure

Now, I replace the original assumption about the information structure with the assumption that customers are originally endowed with the same information and accumulate the same information as their insurers in the process of transaction. That is, the customers originally do not know their own types, but can update their belief about their own types after the observation of period-1 losses. First, there would be no customer selection based on types in period 1—i.e., 48

It = 1 if a consumer has a loss in period t.

36 s ≡ λ—in any circumstance since types are now not known to the customers either. So, by Proposition 7, the equilibrium of the third type of subgame in period 2 is: r2d (L) = λrh + (1 − λ) rl ,

r2d (H) = rh ,

r˜2d (B) = rh

For all j 6= d, r2j (L) = rl , r2j (H) = rh , r˜2j (L) = rl , r˜2j (H) = rh , ( rh if r2d (L) ≤ λrh + (1 − λ) rl r˜2j (B) = λrh + (1 − λ) rl if r2d (L) > λrh + (1 − λ) rl That is, the defecting insurer would always charge its L type old customers λrh + (1 − λ) rl , while the other insurers would charge L type customers rl . In this model, the customers expect that the defecting insurer’s period-2 premium for L type customers would be higher than other insurers’ by λ (rh − rl ), and a priori, a customer would turn out to be the L type with probability 1 − λ. The H type customers would always be charged rh in period 2. Thus, customers would expect to be charged an extra λ (1 − λ) (rh − rl ) had they chosen the defecting insurer in period 1. The candidate equilibrium strategy profile is the same as before. Recall that G (r) is the total period-1 demand of an insurer who shares information but charges r when all other insurers follow the proposed symmetric equilibrium strategy in period 1. So, the defecting insurer’s payoff, when it charges r in period 1, would be πd (r) = G (r + λ (1 − λ) (rh − rl )) [λ (r − rh ) + (1 − λ) (r − rl ) + (1 − λ) (λrh + (1 − λ) rl − rl )] = G (r + λ (1 − λ) (rh − rl )) [r − (λrh + (1 − λ) rl ) + λ (1 − λ) (rh − rl )] which is simply the sum of its profit from period 1 and its profit from period

37 2. Meanwhile, for any r ∈ [r, r¯] we have π (r + λ (1 − λ) (rh − rl )) = G (r + λ (1 − λ) (rh − rl )) [r + λ (1 − λ) (rh − rl ) − (λrh + (1 − λ) rl )] Thus, we have πd (r) = π (r + λ (1 − λ) (rh − rl )). As π ∗ ≥ π (r + λ (1 − λ) (rh − rl )), for all r, we have π ∗ ≥ maxr πd (r). In fact, π ∗ = maxr πd (r), as we can always find an r such that r + λ (1 − λ) (rh − rl ) is in the support of F ∗ . Therefore I conclude that there exists an SPE in which insurers share information if the cost of sharing information is zero. Compared with the result I obtained with the original information structure, the new result differs in three respects. First is the assumption that r¯ < rh − v is not needed in this proof. Second, under the new information structure, the defecting insurer faces shrunken demand and, consequently, lower payoff, but no customer adverse selection. Third, the new result also requires that information sharing is costless.

6

Concluding Remarks

In this paper, I present a novel explanation for why firms might share information. The information to be shared is firms’ old customers’ types, which determines the cost of serving the customers. When the information to be shared is complete, information sharing emerges in equilibrium as long as the cost of sharing information is low enough. In effect, sharing such information commits firms to more intense competition and, thus, lowers price in period 2. All firms sharing information constitutes an equilibrium because unilateral defection simply implies that the defecting insurer would charge a higher price for its L type old customers in period 2 and, thus, would get less demand from the low-risk customers in period 1 and, consequently, lower profits. The adverse selection faced by the defecting firm makes defection unprofitable because only sales to the low-risk customers brings in positive profit.

38

Appendix 1. The proof of the nonexistence of pure-strategy Nash Equilibrium for the simultaneous move game for the third type of subgame in period 2. Proof. The key reason for the nonexistence result is that we cannot find an appropriate combination of fixed r2d (L) and r˜2j (B) that is consistent with Nash Equilibrium. First, note that we must have r2d (H) ≥ rh in any-pure strategy equilibrium. Suppose that r2d (L) = srh + (1 − s) rl . Then, the response for the other competing firms requires that r˜2j (B) = rh , ∀j 6= d49 , but to which r2d (L) = srh + (1 − s) rl is not an optimal response for the defecting insurer. If, instead, r2d (L) > srh + (1 − s) rl , the response from the other competing firms would be r˜2j (B) = srh + (1 − s) rl , ∀j 6= d, as srh + (1 − s) rl is the break-even rate when all the old customers of the defecting insurer switch. However, r2d (L) > srh +(1 − s) rl is not the defecting insurer’s optimal response to r˜2j (B) = srh + (1 − s) rl . Lastly, suppose that r2d (L) < srh + (1 − s) rl . The response from the other competing firms would be r˜2j (B) = rh . However, r2d (L) < srh + (1 − s) rl , again, is not the defecting insurer’s optimal response to r2j (B) = rh . Therefore, there is no pure-strategy Nash Equilibrium if the third type of subgame is a simultaneous-move game.

49 If any competing firm m set r˜2j (B) < rh , it would always lose money in dealing with the switching customers from the defecting insurer.

39

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