Chapter 7 Markets with adverse selection 7.1 A market model These notes introduce some ideas for modeling markets with adverse selec- tion. This framework was originally intended to deal with markets that cannot be easily accommodated by the standard signaling game e. g, be- cause there is two-sided adverse selection. For present purposes, however, it is enough to deal with the simplest case in which there is adverse selection on one side of the market only. The use of the competitive paradigm to analyze markets with adverse selection goes back to Spence(1973). The ideas presented here were deve oped in a series of paper 6). There are two mutually exclusive classes of individuals(agents). We can think of them as buyers and sellers, but nothing depends on this interpretation. The agents on one side of the market have private information, so we call them the informed agents The agents on the other side of the market are the uninformed. An agent's private information is represented by his type. There is a finite set of types T. Each type consists of a(non-atomic) continuum of identical agents and the measure of agents of type t is denoted by N(t)>0. There areM>0 informed agents There is a finite set of contracts e. Later the theory is extended to an infinite set). Each contract involves one agent from each side of the market If an uninformed agent exchanges a 0 contract with an informed agent of type t, the uninformed agent's payoff is u(8, t) and the informed agent's payoff v(0, t). Each agent has a reservation utility, the utility he gets if a contract is
Chapter 7 Markets with Adverse Selection 7.1 A market model These notes introduce some ideas for modeling markets with adverse selection. This framework was originally intended to deal with markets that cannot be easily accommodated by the standard signaling game e.g., because there is two-sided adverse selection. For present purposes, however, it is enough to deal with the simplest case in which there is adverse selection on one side of the market only. The use of the competitive paradigm to analyze markets with adverse selection goes back to Spence (1973). The ideas presented here were developed in a series of papers Gale (1991, 1992, 1996). There are two mutually exclusive classes of individuals (agents). We can think of them as buyers and sellers, but nothing depends on this interpretation. The agents on one side of the market have private information, so we call them the informed agents. The agents on the other side of the market are the uninformed. An agent’s private information is represented by his type. There is a finite set of types T. Each type consists of a (non-atomic) continuum of identical agents and the measure of agents of type t is denoted by N(t) > 0. There are M > 0 uninformed agents. There is a finite set of contracts Θ. (Later the theory is extended to an infinite set). Each contract involves one agent from each side of the market. If an uninformed agent exchanges a θ contract with an informed agent of type t, the uninformed agent’s payoff is u(θ, t) and the informed agent’s payoff is v(θ, t). Each agent has a reservation utility, the utility he gets if a contract is 1
CHAPTER 7. MARKETS WITH ADVERSE SELECTION not exchanged and he has to take his next best option. With an appropriate normalization of the payoff functions, the reservation utility is 0 for every ty The equilibrium choices made by the agents are described by an alloca- tion, that describes the number of agents of each type that chooses a given contract. An allocation of agents consists of a pair of functions(, g) where f:e→R+andg:×T→R+. We interpret f() as the measure of uninformed agents choosing contract 0 and g(e, t) as the measure of informed agents of type t choosing contract 8. The allocation(, g) is attainable if ∑f(0)≤M ∑9(,1)≤N(t,Mt The attainability condition contains an inequality because it is possible that some agents will choose their outside option (i.e, no trade e. When an agent selects a contract 0 he does not know the probability of rade or the type of agent from the other side of the market that he will be matched with. Let A(8, t) denote the probability that an uninformed agent choosing contract 6 will exchange the contract with a type-t agent and let u(e) denote the probability that an informed agent choosing contract 0 will exchange the contract with an uninformed agent. Note that all agents on a given side of the market have the same beliefs about their trading possibilities, that is, the same probability assessment of trading with a given type on the other side of the market The probability assessment(, u) is consistent with the allocation ( 9 A(,t) (6,t) (6) f() for any contract such that m(0)>0, where m(0) measures the long side of the market for contract 0, that is () f()∑9(0,1)
2 CHAPTER 7. MARKETS WITH ADVERSE SELECTION not exchanged and he has to take his next best option. With an appropriate normalization of the payoff functions, the reservation utility is 0 for every type. The equilibrium choices made by the agents are described by an allocation, that describes the number of agents of each type that chooses a given contract. An allocation of agents consists of a pair of functions (f,g) where f : Θ → R+ and g : Θ × T → R+. We interpret f(θ) as the measure of uninformed agents choosing contract θ and g(θ, t) as the measure of informed agents of type t choosing contract θ. The allocation (f, g) is attainable if X θ f(θ) ≤ M and X θ g(θ, t) ≤ N(t), ∀t. The attainability condition contains an inequality because it is possible that some agents will choose their outside option (i.e., no trade) When an agent selects a contract θ he does not know the probability of trade or the type of agent from the other side of the market that he will be matched with. Let λ(θ, t) denote the probability that an uninformed agent choosing contract θ will exchange the contract with a type-t agent and let µ(θ) denote the probability that an informed agent choosing contract θ will exchange the contract with an uninformed agent. Note that all agents on a given side of the market have the same beliefs about their trading possibilities, that is, the same probability assessment of trading with a given type on the other side of the market. The probability assessment (λ, µ) is consistent with the allocation (f, g) if λ(θ, t) = g(θ, t) m(θ) , ∀t µ(θ) = f(θ) m(θ) , for any contract θ such that m(θ) > 0, where m(θ) measures the long side of the market for contract θ, that is, m(θ) = max ( f(θ), X t g(θ, t) )
7. 2. STABILITY If m(0)=0 consistency is automatically satisfied Then a market equilibrium consists of an(attainable) allocation ( g and a probability assessment(, u) such that the allocation maximizes the payoff of each type, that is, f(e)>0 implies u(0,1)N(O,t)=t'= max(,t)(0,t)},v; and g(0, t)>0 implies v(6,t)p(6)=t(t) max 3u (6,t)p(6)},v6,t The equilibrium (, 9, A, u) is said to be orderly if at most one side of the market for any contract is rationed. that max A(6,+,;1()}=1 Without this requirement there exist many trivial equilibria 7. 2 Stability A perturbation is an allocation (, g) such that f()=∑9(,1)>0, Define an equilibrium for the (E, f, 9-perturbed market by replacing the allocation(, g)by(1-e(, g)+e(, g in the equilibrium conditions above Then a perfect market equilibrium(, g, A, u) is defined to be the limit of a sequence of equilibria(fg, A, us)of the (E, f, g-perturbed market as a converges to 0. Note that in a perturbed market the probability assessment is uniquely determined by the allocation and the consistency condition Call ( g) an equilibrium allocation if (f, g, A, u)is a market equilib- rium for some(A, u) and call ( g) an equilibrium allocation for the(e, f, 9)- perturbed market if(, g, A, p)is a market equilibrium of the(E, f, 9-perturbed market for some(A, p). An attainable allocation ( g) is stable if, for any
7.2. STABILITY 3 If m(θ)=0 consistency is automatically satisfied. Then a market equilibrium consists of an (attainable) allocation (f, g) and a probability assessment (λ, µ) such that the allocation maximizes the payoff of each type, that is, f(θ) > 0 implies X t u(θ, t)λ(θ, t) = u∗ = max θ (X t u(θ, t)λ(θ, t) ) , ∀θ; and g(θ, t) > 0 implies v(θ, t)µ(θ) = v∗ (t) = maxθ {v(θ, t)µ(θ)} , ∀θ, t. The equilibrium (f, g, λ, µ) is said to be orderly if at most one side of the market for any contract is rationed, that is, max (X t λ(θ, t), µ(θ) ) = 1. Without this requirement there exist many trivial equilibria. 7.2 Stability A perturbation is an allocation (f,g) such that f(θ) = X t g(θ, t) > 0, ∀θ. Define an equilibrium for the (ε, ˆf, gˆ)-perturbed market by replacing the allocation (f,g) by (1−ε)(f,g)+ε( ˆf, gˆ) in the equilibrium conditions above. Then a perfect market equilibrium (f, g, λ, µ) is defined to be the limit of a sequence of equilibria (f εgε, λε , µε) of the (ε, ˆf, gˆ)-perturbed market as ε converges to 0. Note that in a perturbed market the probability assessment is uniquely determined by the allocation and the consistency condition. Call (f, g) an equilibrium allocation if (f, g, λ, µ) is a market equilibrium for some (λ, µ) and call (f,g) an equilibrium allocation for the (ε, ˆf, gˆ)- perturbed market if (f, g, λ, µ) is a market equilibrium of the (ε, ˆf, gˆ)-perturbed market for some (λ, µ). An attainable allocation (f, g) is stable if, for any
CHAPTER 7. MARKETS WITH ADVERSE SELECTION perturbation(, g)there is a sequence of equilibrium allocations(f, g)such that(f, g)is an equilibrium allocation of the(e, f, g)-perturbed market and lin(f,92)=(f,9) Note that the probability assessments do not necessarily converge to a unique limit. It is easy to see that(, g) must be an equilibrium allocation if ( 9) is stable Let (, g) be a stable allocation and let to be a fixed but arbitrary type For any type t, let u* denote the equilibrium payoff of the uninformed and u(t) the equilibrium payoff of type t. Then there exists an equilibrium ( 9, A, u) such that for any contract 0 and any type tf to t’(t)>p()v(,切=A(O,t)=0 To see this, let(fk, n,gn)be the perturbation defined by +一 9(e, t) 1/nt≠to By choosing k and n appropriately, we can ensure that(fKm, gn) tainable allocation. By stability, there exists a sequence(f, 9, i, u)con- verging to(, g, A", ukin)as e-0, where(f, g, A, u)is an equilibrium for the(e, fkn, gkn)-perturbed game. By compactness, the sequence (kn, u has a limit point uo and it is clear that if (, g, A m, ukn )is an equilibrium then so is(f,g.A,p° Consider tf to. For each 0, if (6)v(6,t)<t’(t) then, for all e >0 sufficiently small b)v(6,t)<t2(t), where v (t) is the equilibrium payoff in(f, g, A, u). Then for tf to (,t) g(,t)1 (0)m(0)(k+r-1)/n-(k
4 CHAPTER 7. MARKETS WITH ADVERSE SELECTION perturbation ( ˆf, gˆ) there is a sequence of equilibrium allocations (f ε, gε ) such that (f ε, gε) is an equilibrium allocation of the (ε, ˆf, gˆ)-perturbed market and limε→0 (f ε , gε )=(f, g). Note that the probability assessments do not necessarily converge to a unique limit. It is easy to see that (f,g) must be an equilibrium allocation if (f, g) is stable. Let (f, g) be a stable allocation and let t0 be a fixed but arbitrary type. For any type t, let u∗ denote the equilibrium payoff of the uninformed and v∗(t) the equilibrium payoff of type t. Then there exists an equilibrium (f, g, λ, µ) such that for any contract θ and any type t 6= t0, [v∗ (t) > µ(θ)v(θ, t)] =⇒ λ(θ, t)=0. To see this, let (f k,n, gkn) be the perturbation defined by f kn(θ)=(k + |T| − 1)/n gkn(θ, t) = ½ 1/n t 6= t0 k/n t = t0. By choosing k and n appropriately, we can ensure that (f kn, gkn) is an attainable allocation. By stability, there exists a sequence (f ε, gε, λε , µε) converging to (f, g, λkn, µkn) as ε → 0, where (f ε, gε, λε , µε) is an equilibrium for the (ε, f kn, gkn)-perturbed game. By compactness, the sequence © (λkn, µknª has a limit point µ0 and it is clear that if (f, g, λkn, µkn) is an equilibrium, then so is (f, g.λ0 , µ0). Consider t 6= t0. For each θ, if µkn (θ) v(θ, t) < v∗ (t) then, for all ε > 0 sufficiently small, µε (θ) v(θ, t) < vε (t), where vε(t) is the equilibrium payoff in (f ε, gε, λε , µε). Then for t 6= t0 λε (θ, t) = gε(θ, t) mε(θ) = 1/n mε(θ) ≤ 1/n (k + |T| − 1)/n = 1 (k + |T| − 1)
7.3. ROBUSTNESS OF SEPARATING EQUILIBRIUM So, in the limit, An(e, t)=1/(k+r-1) and, taking limits as ki, n-o it follows that A(8, t)=0. Since this is true for any 0 and t, the desired property holds Note that if(f, 9, A, u)is a perfect market equilibrium then, by construc ion,(A, p)is orderly. The reason is that for any perturbation, the definition of consistency implies that every(A, p) is orderly and it remains so in the unit 7.2.1 A continuum of contracts The assumption of a finite number of contracts is convenient. It simplifies the description of an equilibrium and makes the existence of equilibrium a technically straightforward matter. For some purposes, it is more convenient to have a continuum of contracts. In particular, when it comes to charac- terizing the degree of separation in an equilibrium it is nice to be able to consider "neighboring "contracts. So let us assume that e is a subset of some finite-dimensional Euclidean space and suppose that u(, t )and v(, t) are continuously differentiable functions of 0 on some open superset of The theory can be extended from a finite subset of e to the entire space by taking limits, but for simplicity I shall assume that the definition of equi- librium and the restrictions on beliefs, derived above, can be applied directly to the limit market. With the assumption that(f, g) has a finite support the definition of equilibrium extends in the obvious way a stable allocation ( g) is defined to be an equilibrium allocation such that for any type to we can find an equilibrium probability assessment(A, u) having the properties derived in the proposition above 7. 3 Robustness of separating equilibrium Let(, g) be a stable allocation and suppose that there exists a contract Bo belonging to the interior of e such that two or more types "pool"at Bo. We can assume without loss of generality that there exists a pair tt and that 9(60,t)>0andg(6o,t)>0 Let u' and u*(t)denote the equilibrium payoffs to the uninformed and type t, respectively, for the equilibrium allocation(, g) and let To=ItE Tlv(t)=u(Bo)v(8o, t))
7.3. ROBUSTNESS OF SEPARATING EQUILIBRIUM 5 So, in the limit, λkn(θ, t)=1/(k + |T| − 1) and, taking limits as k, n → ∞, it follows that λ0 (θ, t)=0. Since this is true for any θ and t, the desired property holds. Note that if (f, g, λ, µ) is a perfect market equilibrium then, by construction, (λ, µ) is orderly. The reason is that for any perturbation, the definition of consistency implies that every (λ, µ) is orderly and it remains so in the limit. 7.2.1 A continuum of contracts The assumption of a finite number of contracts is convenient. It simplifies the description of an equilibrium and makes the existence of equilibrium a technically straightforward matter. For some purposes, it is more convenient to have a continuum of contracts. In particular, when it comes to characterizing the degree of separation in an equilibrium it is nice to be able to consider “neighboring” contracts. So let us assume that Θ is a subset of some finite-dimensional Euclidean space and suppose that u(·, t) and v(·, t) are continuously differentiable functions of θ on some open superset of Θ. The theory can be extended from a finite subset of Θ to the entire space by taking limits, but for simplicity I shall assume that the definition of equilibrium and the restrictions on beliefs, derived above, can be applied directly to the limit market. With the assumption that (f,g) has a finite support, the definition of equilibrium extends in the obvious way. A stable allocation (f,g) is defined to be an equilibrium allocation such that for any type t0 we can find an equilibrium probability assessment (λ, µ) having the properties derived in the proposition above. 7.3 Robustness of separating equilibrium Let (f, g) be a stable allocation and suppose that there exists a contract θ0 belonging to the interior of Θ such that two or more types “pool” at θ0. We can assume without loss of generality that there exists a pair t 6= t 0 and that g(θ0, t) > 0 and g(θ0, t0 ) > 0. Let u∗ and v∗(t) denote the equilibrium payoffs to the uninformed and type t, respectively, for the equilibrium allocation (f,g) and let T0 = {t ∈ T|v∗ (t) = µ(θ0)v(θ0, t)}