Nash Equilibrium in Competitive Insurance - SSRN

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Rothschild and Stiglitz (1976) show that an equilibrium may not exist in competitive in- surance markets with adverse selection. Nonetheless, their analysis does ...
Nash Equilibrium in Competitive Insurance Anastasios Dosis∗ December 16, 2016

Abstract I formalise a rather stylised insurance market with adverse selection as a standard duopoly. I formally specify demand functions and profits and prove that a Nash equilibrium in pure strategies exists if and only if the well-known Rothschild-Stiglitz allocation is efficient. JEL C LASSIFICATION : D86 K EYWORDS : Insurance, adverse selection, duopoly, contracts, Nash equilibrium

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M OTIVATION

Rothschild and Stiglitz (1976) show that an equilibrium may not exist in competitive insurance markets with adverse selection. Nonetheless, their analysis does not explicitly specify a competition game, and the arguments are, for the most part, diagrammatic and concern only two possible types. Furthermore, each company is allowed to offer only one contract. Riley (1979) and Wilson (1977) extend the result to more than two types. They also propose “reactive” equilibria. Miyazaki (1977) and Spence (1978) allow companies to offer menus of contracts and show that reactive equilibrium exists and is efficient. Notably, neither of these papers explicitly specifies a competition game. Engers and Fernandez (1987) and Hellwig (1987) propose extensive-form games that depart from “Bertrand-type” competition and show that equilibrium exists but highlight the difficulties of explicitly modeling the reactive equilibria of Wilson and Riley. Classic microeconomics textbooks such as Jehle and Reny (2011) and Mas-Colell, Whinston, and Green (1995) examine games in which companies compete by offering menus of contracts but focus on the two-type case. Netzer and Scheuer (2014) analyse an extensive-form game ∗

Department of Economics - ESSEC Business School and THEMA, 3 Avenue Bernard Hirsch, 95021 Cergy-Pontoise Cedex, France, Email: [email protected]. I am grateful to Gorkem Celik, Wilfried SandZantman and Olivier Tercieux for useful comments and to two anonymous referees and an associate editor, whose suggestions helped me to simplify the exposition and the results. I would also like to thank participants at the THEMA Economic Theory lunchtime Seminar, the Conference for Research in Economic Theory and Econometrics 2015 (Crete, Greece) and the TOM seminar at the Paris School of Economics for useful feedback. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01).

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in which companies can become inactive at a cost and show that an equilibrium may also exist or fail to exist in the two-type case. Dasgupta and Maskin (1986a,b) and Rosenthal and Weiss (1984) prove the existence of mixed-strategy equilibria in the two-type case. In this note, I formalise a rather stylised insurance market with any finite number of types as a standard duopoly and provide a step-by-step proof for the (non) existence of (pure strategy) Nash equilibrium. 2

T HE MODEL

 Consumers and Companies. There is a measure one of consumers. Each consumer belongs to one of a finite set of types θ = 1, ..., N . For simplicity, I sometimes denote P θ the θ set of types by Θ. The share of type-θ consumers in the population is λ , with θ λ = 1. There are two possible (individual) states ω = 0, 1, where ω = 1 represents the state in which a consumer suffers an accident, and ω = 0, the state in which there is no accident. Uncertainty is purely idiosyncratic, and hence, states occur independently across different consumers. Each consumer begins with endowment W and suffers a loss `, where W > ` > 0 if and only ifP the accident occurs. A consumer of type θ has probability πωθ of being in state ω, with ω πωθ = 1 for every θ. Moreover, let π01 < π02 < ... < π0N . An insurance contract is x = (p, b) ∈ X , where X = {(α, β) ∈ R2+ : α ≤ W, α − β ≤ W − `}. In insurance terms, p specifies the insurance premium and b the benefit that the consumer receives if and only if the accident occurs. A consumer of type θ has preferences represented by an expected utility function U θ (x) = π0θ u(W − p) + π1θ u(W − ` − p + b), where u is continuous, strictly increasing and strictly concave. The status quo utility of type θ is U θ = π0θ u(W ) + π1θ u(W − `). Finally, there exist two symmetric companies in the market i = 1, 2. Because I only consider symmetric companies, there is no loss of generality in assuming the existence of only two companies. If type θ buys contract x from company i, then the latter earns an expected profit equal to ζ θ (x) = p − π1θ b.  Allocations. An allocation is a vector of contracts indexed by the set of types, (xθ )θ .1 0 An allocation (xθ )θ is incentive compatible iff U θ (xθ ) ≥ U θ (xθ ) for every θ, θ0 ∈ Θ. Efficient allocations play a key role in studying the existence of an equilibrium. An efficient allocation is formally defined below. θ Definition (ii) P θ θ θ 2.1. An allocation (x )θ is efficient if and onlyθ if: (i) it is incentive compatible, θ θ x )θ that satisfies (i), (ii) and U (ˆ x )≥ θ λ π (x ) ≥ 0, and (iii) there exists no other allocation (ˆ θ θ U (x ) for every θ, with the inequality being strict for at least one θ.

Efficiency, as is defined here, is standard Pareto efficiency subject to incentive constraints. Note that, as is fairly standard in these environments, efficiency is defined with respect to the payoff of the consumers and the average resource constraint. One can establish the following result regarding the set of efficient allocations. P Lemma 2.2. If allocation (xθ )θ is efficient, then θ λθ ζ θ (xθ ) = 0. 1 An allocation defines a mapping from the type space to the set of contracts. In mechanism design jargon, an allocation is a direct revelation mechanism.

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Proof. I prove the result contraposition. Suppose that (xθ )θ is an incentive compatible P by allocation such that θ λθ ζ θ (xθ ) > 0. Consider allocation (˜ xθ )θ , where for x˜θ , u(W − p˜θ ) = u(W − pθ ) + (1 − )u(W )

(2.1)

u(W − ` − p˜θ + ˜bθ ) = u(W − ` − pθ + bθ ) + (1 − )u(W − ` + ˆb)

(2.2)

and

for ˆb > 0. Because u(·) is strictly concave, by Jensen’s inequality, for every θ ∈ Θ, W − p˜θ < (W − pθ ) + (1 − )W

(2.3)

W − ` − p˜θ + ˜bθ < (W − ` − pθ + bθ ) + (1 − )(W − ` + ˆb)

(2.4)

and

Multiplying Eq. (2.3) by π0θ and Eq. (2.4) by π1θ and summing them up yields ζ θ (˜ xθ ) > ζ θ (xθ ) − (1 − )π1θˆb

(2.5)

Multiplying Eq. (2.5) by λθ and summing over θ yields X X X λθ π1θˆb λθ ζ θ (xθ ) − (1 − ) λθ ζ θ (˜ xθ ) >  θ

θ

(2.6)

θ

Because (xθ )θ is incentive compatible by definition and due to Eqs. (2.1) and (2.2), for every  ∈ (0, 1) the following are true: 0

U θ (xθ ) ≥ U θ (xθ ) ∀ θ, θ0 U θ (˜ xθ ) = U θ (xθ ) + (1 − ) π0θ u(W ) + π1θ u(W − ` + ˆb)



∀ θ  0 0 U θ (˜ xθ ) = U θ (xθ ) + (1 − ) π0θ u(W ) + π1θ u(W − ` + ˆb) ∀ θ, θ0 Therefore, (˜ xθ )θ is incentiveP compatible. Evidently, there exist  and ˆb such that U θ (˜ xθ ) > U θ (xθ ) for every θ ∈ Θ and θ λθ ζ θ (˜ xθ ) > 0. Hence, (xθ )θ is not efficient. An allocation that plays a significant role in insurance markets with adverse selection is what is usually called the Rothschild-Stiglitz Allocation (RSA). This is identified in nearly all studies mentioned in the introduction. It maximises the payoff of every type within the set of incentive compatible allocations that make positive profits type-by-type. A formal definition of a RSA follows. Definition 2.3. An allocation (xθ )θ is an RSA if and only if: (i) it is incentive compatible, (ii) ζ θ (xθ ) ≥ 0 for every θ ∈ Θ, and (iii) there exists no other allocation (˜ xθ )θ that satisfies (i), (ii) and U θ (˜ xθ ) ≥ U θ (˜ xθ ) for every θ, with the inequality being strict for at least one θ.

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- Remarks. It is well known that with only two possible types, the RSA is efficient when the share of type-1 consumers (i.e., the high-risk consumers) in the population is sufficiently large. A similar result applies here. Note first that in the RSA, type 1’s contract is (π11 `, `) (i.e., the full-coverage contract that makes zero profits if taken by type 1 only) and all incentive constraints are binding. Therefore, every contract that is preferred by a group of types higher in the rank than type 1 over the RSA allocation is also preferred by type 1. Evidently, every such contract is loss-making if taken only by type 1, given that (π11 `, `) is the payoff-maximizing contract for type 1 that makes zero profits. If the share of type-1 consumers is sufficiently large, then every menu of contracts that is preferred by a subset of types (e.g., {1, ..., n}) necessarily makes negative profits. Hence, the RSA satisfies Definition 2.1.  Menus, Demands and Profits. Each of the two companies selects a menu of contracts. The set of possible menus for each company is X N . Let mi denote a menu for company i and m = (m1 , m2 ) a profile of menus. Based on all contracts that are available in the market, each consumer purchases a contract from one of the two companies. Let (q1θ (m), q2θ (m)), where qiθ (m) : {x : x ∈ mi } → [0, λθ ], denote a pair of measures for every m ∈ X 2N . Each of these measures represents the demand function from type-θ consumers to company i when the menus of contracts are m = (m1 , m2 ). For every θ, m and i, the following sequential rationality conditions must be satisfied: qiθ (x|m) = 0 if U θ (x)
0, if U θ (0, 0) > maxy∈m1 ∪m2 U θ (y) qiθ (x|m) = λθ , where q0θ (m) = 0, otherwise x∈mi

XX i

(2.8) Eq. (2.7) states that the demand for a contract is zero when this contract does not belong to the set of contracts that maximise the utility of type θ among all the contracts that are offered in the market (i.e., m1 ∪ m2 ∪ {(0, 0)}). Eq. (2.8) states that the measures sum to λθ ; the ex ante share of type θ. q0θ represents the share of types that does not buy any insurance. This is strictly positive if and only if no contract offered by the two companies provides a strictly higher payoff than the null contract (i.e., (0, 0)). Based on qi (m), the expected profit of firm i is written as XX Πi (mi , m−i |qi (m)) = qiθ (x|m)ζ θ (x) θ

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x∈mi

E XISTENCE OF AN EQUILIBRIUM

A formal definition of Nash equilibrium follows: ¯ = (m Definition 3.1. A Nash equilibrium consists of a profile of actions m ¯ 1, m ¯ 2 ) such that for θ θ every i m ¯ i ∈ arg max Πi (mi , m ¯ −i |qi (m)) for some (q1 (m), q2 (m)) satisfying Eqs. (2.7) and mi

(2.8) for every θ and m. 4

To study the existence of an equilibrium, I proceed in steps. The following trivial lemma facilitates the proofs. ¯ = (m Lemma 3.2. Let m ¯ 1, m ¯ 2 ) be an equilibrium in pure strategies. Then, ¯ ≥ 0 for every i, (i) Πi (m ¯ i, m ¯ −i |qi (m)) ¯ ≤ (ii) There exists j suchPthat Πj (m ¯ j, m ¯ −j |qj (m)) ¯ >0 being strict when i Πi (m ¯ i, m ¯ −i |qi (m))

P

i

¯ Πi (m ¯ i, m ¯ −i |qi (m)), with the inequality

Step 1. The first step is to show that in equilibrium all consumers of the same type purchase a contract with the same terms. This is formally stated in the following lemma: ¯ = (m Lemma 3.3. Suppose that m ¯ 1, m ¯ 2 ) is an equilibrium in pure strategies. Then, for every θ x1 , x2 ∈ arg maxx∈m¯ 1 ∪m¯ 2 ∪{(0,0)} U (x), x1 = x2 . ¯ = (m Proof. Suppose that m ¯ 1, m ¯ 2 ) is an equilibrium profile in pure strategies and there ¯ denote the exists η and x1 , x2 ∈ arg maxx∈m¯ 1 ∪m¯ 2 ∪{(0,0)} U η (x) such that x1 6= x2 . Let xθ (m) ¯ This contract is unique by contract chosen by type θ 6= η when the profile of menus is m. definition. Consider contract x¯η = (¯ pη , ¯bη ) such that P P ¯ η ¯ q (x | m) i 1 i∈{j:x2 ∈m ¯ j } qi (x2 |m) i∈{j:x1 ∈m ¯ j} U η (x1 ) + U (x2 ) (3.1) U η (¯ xη ) = η η λ λ ¯ ≥ U θ (¯ As in Lemma 2.2, U θ (xθ (m)) xη ) for every θ 6= η. The profit of contract x¯η is P P ¯ η ¯ η i∈{j:x1 ∈m ¯ j } qi (x1 |m) i∈{j:x2 ∈m ¯ j } qi (x2 |m) η η ζ (¯ x )> ζ (x ) + ζ (x2 ) 1 λη λη  ¯ θ6=η , x¯η is The aggregate profit of allocation (xθ (m)) X ¯ > λη ζ η (¯ xη ) + λθ ζ θ (xθ (m)) θ6=η

X i∈{j:x1 ∈m ¯ j}

X

X

¯ η (x1 ) + qi (x1 |m)ζ

¯ η (x2 ) + qi (x2 |m)ζ

i∈{j:x2 ∈m ¯ j}

X

¯ = λθ ζ θ (xθ (m))

θ6=η

¯ ≥0 Πi (m ¯ i, m ¯ −i |qi (m))

i

where the last inequality follows from Lemma 3.2. As in Lemma 2.2, there exists an allo¯ for every θ and cation (˜ xθ )θ such that U θ (˜ xθ ) > U θ (xθ (m)) X X X ¯ λθ ζ(˜ xθ ) > (λη ζ η (¯ xη ) + λθ ζ θ (xθ (m))) − (1 − ) λθ π1θˆb θ

θ6=η

θ

for  ∈ (0, 1) and ˆb > 0. For  and ˆb appropriately chosen, and due to Lemma 3.2, there exists j such that X ¯ ≥ Πj (m ¯ Πj ((˜ xθ )θ , m ¯ −j |qj ((˜ xθ )θ , m ¯ −j )) > Πi (m ¯ i, m ¯ −i |qi (m)) ¯ j, m ¯ −j |qj (m)) (3.2) i

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Therefore, from Eq. (3.2), ¯ Πj ((˜ xθ )θ , m ¯ −j |qj ((˜ xθ )θ , m ¯ −j )) > Πj (m ¯ j, m ¯ −j |qj (m)) and, from Definition 3.1, ¯ Πj ((˜ xθ )θ , m ¯ −j |qj ((˜ xθ )θ , m ¯ −j )) ≤ Πj (m ¯ j, m ¯ −j |qj (m)) Hence, we have a contradiction. Thanks to Lemma 3.3, we can uniquely define by (xθ (m))θ an allocation associated ¯ = (m with equilibrium m ¯ 1, m ¯ 2 ) such that: xθ (m) ∈

arg max

U θ (x)

x∈m1 ∪m2 ∪{(0,0)}

(xθ (m))θ will henceforth be called an equilibrium allocation. Step 2. The second step is to examine the efficiency properties of equilibrium allocations. We can establish the following result: ¯ θ is an equilibrium allocation, then it is efficient. Proposition 3.4. If (xθ (m)) ¯ θ is not efficient. With Proof. I prove the result by contradiction. Suppose that (xθ (m)) a straightforward extension of the argument given in the proof of Lemma 2.2, one can show that for every  ∈ (0, 1), there exists an incentive compatible allocation (xθ )θ such ¯ for every θ and that U θ (xθ ) > U θ (xθ (m)) X X X ¯ = ¯ λθ ζ θ (xθ (m)) Πi (m ¯ i, m ¯ −i |qi (m)) λθ ζ θ (xθ ) >  θ

i

θ

Therefore, consider m ˜ j = (xθ )θ . The profit of company j from this menu is X X ¯ Πj (m ˜ j, m ¯ −j |qj (m ˜ j, m ¯ −j )) = λθ ζ θ (xθ ) >  Πi (m ¯ i, m ¯ −i |qi (m))

(3.3)

i

θ

because company j attracts all types when the other company offers menu m ¯ −j . For a sufficiently large : ¯ Πj (m ˜ j, m ¯ −j |qj (m ˜ j, m ¯ −j )) > Πj (m ¯ j, m ¯ −j |qj (m)) which follows from Lemma 3.2, and ¯ Πj (m ˜ j, m ¯ −j |qj (m ˜ j, m ¯ −j )) ≤ Πj (m ¯ j, m ¯ −j |qj (m)) which follows from Definition 3.1. Therefore, we have a contradiction. Step 3. The third step is to show that in equilibrium, cross-subsidization is not possible. This is the underlying idea behind the non-existence of equilibrium in Rothschild and Stiglitz (1976). 6

¯ = (m ¯ = 0 for every θ. Lemma 3.5. If m ¯ 1, m ¯ 2 ) is a pure strategy equilibrium, then ζ θ (xθ (m)) ¯ = (m Proof. I prove the result by contradiction. Suppose that m ¯ 1, m ¯ 2 ) is an equilibrium ¯ > 0. Due to Lemma 2.2 and Lemma 3.5, Πi (m ¯ = such that, for some θ, ζ θ (xθ (m)) ¯ i, m ¯ −i |qi (m)) 0 η 0 0 for every i. Because of the single-crossing property, there exists x such that U (x ) > ¯ for every η ≥ θ and U η (x0 ) < U η (xη (m)) ¯ for every η < θ. Consider contract x˜ U η (xθ (m)) such that ¯ + (1 − )U θ (x0 ) U θ (˜ x) = U θ (xθ (m))

(3.4)

¯ + (1 − )ζ η (x0 ) ζ θ (˜ x) > ζ η (xθ (m))

(3.5)

 ∈ [0, 1]

(3.6)

Consider now company j and action m ˜ j , where m ˜ j = (˜ x, ..., x˜). For any  satisfying (3.4), (3.5) and (3.6), at least type θ buys contract x˜. For a sufficiently small , ζ η (˜ x) > 0. The profit of company j from m ˜ j when company −j offers m ¯ −j is 0 < Πi (m ˜ j, m ¯ −j |qj (m ˜ j, m ¯ −j )) ≤

N X

λη ζ η (˜ x)

(3.7)

η≥θ

where the bounds in (3.7) follow from the fact that for every x, ζ 1 (x) < ζ 2 (x) < ... < ζ Θ (x). From Definition 3.1, it is true that ¯ =0 Πi (m ˜ j, m ¯ −j |qj (m ˜ j, m ¯ −j )) ≤ Πi (m ¯ i, m ¯ −i |qi (m)) and therefore, we have a contradiction. Proposition 3.6. A Nash equilibrium in pure strategies exists if and only if the RSA is efficient. Proof. For convenience, I denote by (xθRS )θ the RSA. For the ”if” part, suppose that (xθRS )θ is efficient. I prove by contradiction that mRS = mRS = (xθRS )θ satisfies Definition 3.1. 1 2 θ Suppose that it does not. There exist j and m ˜ j 6= (xRS )θ such that for some Φ ⊆ Θ, (i) maxx∈m˜ j U θ (x) > U θ (xθRS ) ∀ θ ∈ Φ (ii) maxx∈m˜ j U θ (x) < U θ (xθRS ) ∀ θ ∈ Θ − Φ RS RS RS RS (iii) Πj (m ˜ j , mRS ˜ j , mRS −j |qj (m −i )) > Πi (mj , m−j |qj (mj , m−j ))  Consider now allocation (xθRS )θ∈Θ−Φ , (˜ xθ )θ∈Φ ) , where

x˜θ ∈ {arg max U θ (x)} ∩ {arg max ζ θ (x)} x∈m ˜j

x∈m ˜j

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The profit of this allocation is X X λθ ζ θ (xθRS )) + λθ ζ θ (˜ xθ ) > 0 θ∈Φ

θ∈Θ−Φ

which follows from (iii) 2.3. Note, however, that because of (i) and (ii),  and Definition θ θ θ (xRS )θ∈Θ−Φ , (˜ x )θ∈Φ ) dominates (xRS )θ , which contradicts that (xθRS )θ is efficient. For the ”only if” part, suppose that (xθRS )θ is not efficient. Then, for every efficient allocation (ˆ xθ )θ , there exists θ such that ζ θ (ˆ xθ ) > 0. Suppose that an equilibrium exists. From Proposition 3.5, the equilibrium allocation is efficient. This immediately contradicts Lemma 3.5. - Remarks. Note that for the existence part (i.e., the first part of Proposition 3.6) no structural assumptions are necessary. The argument straightforwardly extends to considerably more general environments. R EFERENCES Dasgupta, Partha, and Eric Maskin, “The existence of equilibrium in discontinuous economic games, I: Theory,” The Review of Economic Studies, 53 (1986a), 1–26. ——, “The existence of equilibrium in discontinuous economic games, II: Applications,” The Review of Economic Studies, 53 (1986b), 27–41. Engers, M., and L. Fernandez, “Market equilibrium with hidden knowledge and selfselection,” Econometrica, 55 (1987), 425–439. Hellwig, Martin F., “Some recent developments in the theory of competition in markets with adverse selection,” European Economic Review, 31 (1987), 319–325. Jehle, Geoffrey A, and Philip J. Reny, Advanced Microeconomic Theory, 3rd ed. (Pearson Education, 2011). Mas-Colell, Andreu, Michael Dennis Whinston, and Jerry R. Green, Microeconomic Theory, vol. 1 (New York: Oxford University Press, 1995). Miyazaki, Hajime, “The Rat Race and Internal Labor Markets,” The Bell Journal of Economics, 8 (1977), 394–418. Netzer, Nick, and Florian Scheuer, “A Game Theoretic Foundation of Competitive Equilibria with Adverse Selection,” International Economic Review, 55 (2014), 399–422. Riley, John G., “Informational Equilibrium,” Econometrica, 47 (1979), 331–359. Rosenthal, R. W., and A. Weiss, “Mixed-strategy equilibrium in a market with asymmetric information,” The Review of Economic Studies, 51 (1984), 333–342.

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Rothschild, Michael, and Joseph Stiglitz, “Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information,” The Quarterly Journal of Economics, 90 (1976), 629–649. Spence, Michael, “Product Differentiation and Performance in Insurance Markets,” Journal of Public Economics, 10 (1978), 427–447. Wilson, Charles, “A Model of Insurance Markets with Incomplete Information,” Journal of Economic Theory, 16 (1977), 167–207.

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