Chapter 6 Signaling games and refinements 6.1 Adverse selection The term adverse selection comes originally from insurance applications. An insurance contract may attract high-risk individuals, with the result that the pool of insured customers may be riskier than the population at large Adverse selection is now used generically to describe situations of asymmet ric information, particularly market settings in which some individuals have private information about their characteristics and where the individuals actions may reveal some or all of that information to other individuals. The earliest work on adverse selection(Akerlof(1969), Spence(1973)made use of the competitive equilibrium framework. Later we shall want to make use of this framework, but to start with we will use a simple game-theoretic framework to develop the basic insights A signaling game is played between a single informed agent and two or more risk-neutral uninformed agents. The informed agent undertakes a risk venture which he then sells to the uninformed agents. The agent's private information is represented by his type t E T, where T is a finite set. The probability of the informed agent's type is v(t). The agent chooses an action a E A, where A is a finite set. The agent's action is publically observed. The extensive form game has three stages Nature chooses the informed agent's type t
Chapter 6 Signaling games and refinements 6.1 Adverse selection The term adverse selection comes originally from insurance applications. An insurance contract may attract high-risk individuals, with the result that the pool of insured customers may be riskier than the population at large. Adverse selection is now used generically to describe situations of asymmetric information, particularly market settings in which some individuals have private information about their characteristics and where the individuals’ actions may reveal some or all of that information to other individuals. The earliest work on adverse selection (Akerlof (1969), Spence (1973)) made use of the competitive equilibrium framework. Later we shall want to make use of this framework, but to start with we will use a simple game-theoretic framework to develop the basic insights. A signaling game is played between a single informed agent and two or more risk-neutral uninformed agents. The informed agent undertakes a risky venture which he then sells to the uninformed agents. The agent’s private information is represented by his type t ∈ T, where T is a finite set. The probability of the informed agent’s type is ν(t). The agent chooses an action a ∈ A, where A is a finite set. The agent’s action is publically observed. The extensive form game has three stages: • Nature chooses the informed agent’s type t. 1
CHAPTER 6. SIGNALING GAMES AND REFINEMENTS The informed agent chooses an action a(as a function of his type t The uninformed agents observe the action a and bid in Bertrand fashion for shares in the venture The payoff to the informed agent is u(a, p, t), where a is his action, p is the price of the venture, and t is his type. Because of the assumption of Bertrand competition, the equilibrium price of the venture will equal its expected value. The true value of the venture is v(a, t). The expected value depends on the beliefs of the uninformed agents, which are represented by probability assessment u, where u(a, t)is the probability of type t given the observed action a. Then the equilibrium price is p(a)=∑(n,)(a,1 The informed agent chooses his action a to maximize u(a,p(a),t)=u(a,2v(a, t)u(a, t),t) conditional on his true type t, taking the price function p( as given The probability assessment u satisfies Bayes'rule wherever this is ap- plicable. Let T(a) denote the set of types that choose a in equilibrium. If t∈Ta)then (a,t)= () ∑∈rav(s) 6.1.1 The IPO example This example is based on Leland and Pyle(1978). A risk averse entrepreneur wants to diversify his risk by selling shares in his startup company. The entrepreneur's type t is the probability of success. The company will be worth V if it succeeds and 0 if it fails. The entrepreneur has to decide how much ownership to retain in the company. The more he holds, the better the signal he sends to the market. The entrepreneur's payoff is given by tU(aV+(1-a)p(a)+(1-t)U(1-a)p(a) here U( is a VNM utility function, a is the fraction of the firm held by the entrepreneur, and p(a) is the price(market value)of the firm when the
2 CHAPTER 6. SIGNALING GAMES AND REFINEMENTS • The informed agent chooses an action a (as a function of his type t). • The uninformed agents observe the action a and bid in Bertrand fashion for shares in the venture. The payoff to the informed agent is u(a, p, t), where a is his action, p is the price of the venture, and t is his type. Because of the assumption of Bertrand competition, the equilibrium price of the venture will equal its expected value. The true value of the venture is v(a, t). The expected value depends on the beliefs of the uninformed agents, which are represented by a probability assessment µ, where µ(a, t) is the probability of type t given the observed action a. Then the equilibrium price is p(a) = X t v(a, t)µ(a, t). The informed agent chooses his action a to maximize u(a, p(a), t) = u ³ a,X t v(a, t)µ(a, t), t´ , conditional on his true type t, taking the price function p(·) as given. The probability assessment µ satisfies Bayes’ rule wherever this is applicable. Let T(a) denote the set of types that choose a in equilibrium. If t ∈ T(a) then µ(a, t) = ν(t) P s∈T(a) ν(s) . 6.1.1 The IPO example This example is based on Leland and Pyle (1978). A risk averse entrepreneur wants to diversify his risk by selling shares in his startup company. The entrepreneur’s type t is the probability of success. The company will be worth V if it succeeds and 0 if it fails. The entrepreneur has to decide how much ownership to retain in the company. The more he holds, the better the signal he sends to the market. The entrepreneur’s payoff is given by tU(aV + (1 − a)p(a)) + (1 − t)U((1 − a)p(a)) where U(·) is a VNM utility function, a is the fraction of the firm held by the entrepreneur, and p(a) is the price (market value) of the firm when the
6.1. ADVERSE SELECTION fraction held by the entrepreneur is a. The market value of the firm is defined p(a)=∑r(a,t) Pooling equilibrium Suppose that all types hold a fraction ap of the firm. The equilibrium price p(an)=∑以t Each type t chooses a to maximize u(a, p(a), t), which requires that +(1-a)p(a) t)U(1-a)p(a)≤ tU(apV+(1-app(ap))+(1-tU((1-app(ap)) for all a and t. The easiest way to support this equilibrium is to assume that for every a≠a H(,)=0t≠tm where tmin is the smallest value of t. Then the equilibrium condition reduces tU(av+(1-a)tminV)+(1-t)U((1-a)tminV)< for every a and t, wheret=>v(t)t Separating equilibrium Let a be an increasing function of t and define the beliefs u by putting if a=a(t) (a,t) if t=tmin, a fa-(T) 0 otherwise Define p( in the usual way. Then the strategy a is optimal if tu(av+(1-ap(a))+(1-tU((I-apla< tU(a(t)y+(1-a(t)t)+(1-t)U((1-a(t)tV) or every a and t
6.1. ADVERSE SELECTION 3 fraction held by the entrepreneur is a. The market value of the firm is defined by p(a) = X t µ(a, t)tV. Pooling equilibrium Suppose that all types hold a fraction ap of the firm. The equilibrium price is p(ap) = X t ν(t)tV. Each type t chooses a to maximize u(a, p(a), t), which requires that tU(aV + (1 − a)p(a)) + (1 − t)U((1 − a)p(a)) ≤ tU(apV + (1 − ap)p(ap)) + (1 − t)U((1 − ap)p(ap)) for all a and t. The easiest way to support this equilibrium is to assume that for every a 6= ap µ(a, t) = ½ 0 t 6= tmin 1 t = tmin where tmin is the smallest value of t. Then the equilibrium condition reduces to tU(aV + (1 − a)tminV ) + (1 − t)U((1 − a)tminV ) ≤ tU(apV + (1 − ap)tV¯ ) + (1 − t)U((1 − ap)tV¯ ) for every a and t, where t ¯= P t ν(t)t. Separating equilibrium Let α be an increasing function of t and define the beliefs µ by putting µ(a, t) = 1 if a = α(t) 1 if t = tmin,a /∈ α−1(T) 0 otherwise. Define p(·) in the usual way. Then the strategy α is optimal if tU(aV + (1 − a)p(a)) + (1 − t)U((1 − a)p(a)) ≤ tU(α(t)V + (1 − α(t))tV ) + (1 − t)U((1 − α(t))tV ) for every a and t
CHAPTER 6. SIGNALING GAMES AND REFINEMENTS 6.2 Equilibrium Definition 1 A perfect Bayesian equilibrium is an ordered pair(a, u)satis fying the following conditions )a()∈ arg maxaeA(a,p(a),t),t∈T; (ii)u(a, t)=v(t)/tea-1(av(t)if a=a(t) where p(a)=2tu(a, tv(a, t) Equilibrium beliefs must satisfy Bayes'rule wherever possible(condition (ii). This is the"Bayesian"part of the definition. Uninformed agents re- spond optimally to every action and not just to the actions chosen in equi- riun (condition(i)). This is the "perfect "part of the definition. However, because optimality is defined relative to beliefs and beliefs are more or less arbitrary for actions not chosen in equilibrium, PbE is a weak equilibrium oncept and there are many equilibria a mixed strate a function g:T→△(A), where△(A)={r A-R+>T(a)=1 and o(a, t)is the probability of choosing a when he type is t. A mpletely mixed if a(a, t)>0 (a, t). A perturbation of the game is a completely mixed strategy y and a number 0<n< 1. In the (n, y-perturbed game, a strategy o is replaced by the strategy(1-n)o +m. Equivalently, the informed agent is required to choose a strategy o subject to the constraint o > mr Definition 2 For any(u, y-perturbation of the game, o is a (Nash) equi librium of the perturbed game if )o(t)∈ arg maxo>∑ao(a,t)u(a,p(a),t),t∈T; (i)(a,t)=(a,t)/∑:o(a,t),(a,t); (iii)p(a)=tula, tu(a, t) A perfect equilibrium is the limit of a sequence (o",u)) as n-0 where on is a(Nash) equilibrium of the (n, y) perturbed game and u'" is the uniquely determined probability assessment 6.2.1 The IPO example The equilibria described in Section 6.1.1 are, in fact, perfect equilibria
4 CHAPTER 6. SIGNALING GAMES AND REFINEMENTS 6.2 Equilibrium Definition 1 A perfect Bayesian equilibrium is an ordered pair (α, µ) satisfying the following conditions: (i) α(t) ∈ arg maxa∈A u(a, p(a), t), ∀t ∈ T; (ii) µ(a, t) = ν(t)/ P t∈α−1(a) ν(t) if a = α(t); where p(a) = P t µ(a, t)v(a, t). Equilibrium beliefs must satisfy Bayes’ rule wherever possible (condition (ii)). This is the “Bayesian” part of the definition. Uninformed agents respond optimally to every action and not just to the actions chosen in equilibrium (condition (i)). This is the “perfect” part of the definition. However, because optimality is defined relative to beliefs and beliefs are more or less arbitrary for actions not chosen in equilibrium, PBE is a weak equilibrium concept and there are many equilibria. A mixed strategy is a function σ : T → ∆(A), where ∆(A) = {π : A → R+| Pπ(a)=1} and σ(a, t) is the probability of choosing a when the type is t. A strategy σ is completely mixed if σ(a, t) > 0 for every (a, t). A perturbation of the game is a completely mixed strategy γ and a number 0 <η< 1. In the (η, γ)-perturbed game, a strategy σ is replaced by the strategy (1 − η)σ + ηγ. Equivalently, the informed agent is required to choose a strategy σ subject to the constraint σ ≥ ηγ. Definition 2 For any (η, γ)-perturbation of the game, σ is a (Nash) equilibrium of the perturbed game if (i) σ(t) ∈ arg maxσ≥ηγ P a σ(a, t)u(a, p(a), t), ∀t ∈ T; (ii) µ(a, t) = σ(a, t)/ P t σ(a, t), ∀(a, t); (iii) p(a) = P t µ(a, t)v(a, t). A perfect equilibrium is the limit of a sequence {(ση, µη)} as η → 0 where ση is a (Nash) equilibrium of the (η, γ) perturbed game and µη is the uniquely determined probability assessment. 6.2.1 The IPO example The equilibria described in Section 6.1.1 are, in fact, perfect equilibria
6.3. STABILITY 6. 3 Stability The notion of perfect equilibrium refines the notion of Nash equilibrium by requiring that off-the-equilibrium-path beliefs be generated by some feasible strategy. In this sense, they are not completely arbitrary. However, perfec tion does not reduce the set of off-the-equilibrium-path beliefs very much nor does it reduce the set of equilibria very much A stronger refinement is the notion of strategic stability. What we are interested in is the actions chosen in equilibrium and the associated prices These are determined by the equilibrium strategy o(note that if> o(a, t)> 0 then the price p(a) is uniquely determined by o and condition(ii)). Hence it is meaningful to refer to o as the equilibrium outcome Definition 3 An equilibrium outcome a is called a unique stable outcome if, for any completely mi.ced strategy y and any a>0, there ecists a number no such that for any n< no there exists an equilibrium of the(n, m)-perturbed game whose outcome is E-close toa 6.3.1 The IPo example again Strategic stability is a test of robustness. In characterizing robustness, there is often a critical perturbation that destabilizes all but the stable outcome. In this example, it is easy to see that the perturbation that is likely to destabilize an equilibrium outcome a will put most of its weight on tmax, the highest value of t, thus raising the price off the equilibrium path and encouraging agents to deviate from the equilibrium strategy Re eferences Akerlof, George(1970)."The Market for 'Lemons: Quality Uncertainty and the Market Mechanism, "Quarterly Journal of economics Banks, Jeffrey and Joel Sobel."Equilibrium Selection in Signaling Games Econometrica 55(1987)647-61 H. Bester, "Screening vs Rationing in Credit Markets with Imperfect Infor mation, "American Economic Review 75(1985)850-855
6.3. STABILITY 5 6.3 Stability The notion of perfect equilibrium refines the notion of Nash equilibrium by requiring that off-the-equilibrium-path beliefs be generated by some feasible strategy. In this sense, they are not completely arbitrary. However, perfection does not reduce the set of off-the-equilibrium-path beliefs very much, nor does it reduce the set of equilibria very much. A stronger refinement is the notion of strategic stability. What we are interested in is the actions chosen in equilibrium and the associated prices. These are determined by the equilibrium strategy σ (note that if P t σ(a, t) > 0 then the price p(a) is uniquely determined by σ and condition (ii)). Hence, it is meaningful to refer to σ as the equilibrium outcome. Definition 3 An equilibrium outcome x is called a unique stable outcome if, for any completely mixed strategy γ and any ε > 0, there exists a number η0 such that for any η<η0 there exists an equilibrium of the (η, γ)-perturbed game whose outcome is ε-close to x. 6.3.1 The IPO example again Strategic stability is a test of robustness. In characterizing robustness, there is often a critical perturbation that destabilizes all but the stable outcome. In this example, it is easy to see that the perturbation that is likely to destabilize an equilibrium outcome σ will put most of its weight on tmax, the highest value of t, thus raising the price off the equilibrium path and encouraging agents to deviate from the equilibrium strategy. References Akerlof, George (1970). “The Market for ‘Lemons’: Quality Uncertainty and the Market Mechanism,” Quarterly Journal of Economics. Banks, Jeffrey and Joel Sobel. “Equilibrium Selection in Signaling Games,” Econometrica 55 (1987) 647-61. H. Bester, “Screening vs Rationing in Credit Markets with Imperfect Information,” American Economic Review 75 (1985) 850-855