Introduction to Game Theory

Transcription

1 Introduction to Game Theory Part 3. Dynamic games of incomplete information Chapter 2. Signaling Games Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

2 Topics covered Perfect Bayesian equilibrium in signaling games Job-market signaling Corporate investment and capital structure Monetary policy V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

3 Perfect Bayesian equilibrium in signaling games A signaling game is a dynamic game of incomplete information involving two players: A Sender (S) A Receiver (R) The timing of the game is 1 Nature draws a type t i for the Sender from a finite set of feasible types T = {t 1,..., t I } according to a probability distribution p Prob(T ) with full support, i.e., p(t i ) > 0 for every t i 2 The Sender observes t i and then chooses a message m j from a finite set of feasible messages M = {m 1,..., m J } 3 The Receiver observes m j but not t i and then chooses an action a k from a finite set of actions A = {a 1,..., a K } 4 Payoffs are given by U S (t i, m j, a k ) and U R (t i, m j, a k ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

4 Perfect Bayesian equilibrium in signaling games In many applications, the sets T, M and A are intervals on the real line, rather than finite sets One may allow the set of feasible messages to depend on the type Nature draws One may allow the set of feasible actions to depend on the message the Senders chooses In Spence s (QJE 1973) model of job-market signaling the Sender is the worker the Receiver is the market of prospective employers the type is the worker s productive ability the message is the worker s education choice the action is the wage paid by the market V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

5 Perfect Bayesian equilibrium in signaling games In Myers and Majluf s (JFE 1984) model of corporate investment and capital structure the Sender is a firm needing capital to finance a new project the Receiver is a potential investor the type is the profitability of the firm s existing assets the message is the firm s offer of an equity stake in return for financing the action is the investor s decision about whether to invest A signaling game may be embedded within a richer game there could be an action by the Receiver before the Sender chooses the message in step 2 there could be an action by the Sender after (or while) the Receiver chooses the action in step 3 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

6 Perfect Bayesian equilibrium in signaling games In Vicker s (1986) model of monetary policy the Federal Reserve has private information about its willingness to accept inflation in order to increase employment the Sender is the Federal Reserve the Receiver is the market of employers the type is the Fed s willingness to accept inflation in order to increase employment the message is the Fed s choice of first-period inflation the action is the employers expectation of second-period inflation the employers expectation of first-period inflation precedes the signaling game the Fed s choice of second-period inflation follows it V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

7 Perfect Bayesian equilibrium in signaling games We consider an extensive form representation of a simple case: T = {t 1, t 2 }, M = {m 1, m 2 }, A = {a 1, a 2 } and Prob{t 1 } = p V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

8 Perfect Bayesian equilibrium in signaling games A player s strategy is a complete plan of action: a strategy specifies a feasible action in every contingency in which the player might be called upon to act In a signaling game: a pure strategy for the Sender is a function ti m(t i ) specifying which message will be chosen for each type that Nature might draw a pure strategy for the Receiver is a function m j ã(m j ) specifying which action will be chosen for each message that the Sender might send V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

9 Perfect Bayesian equilibrium in signaling games In the simple signaling depicted before, the Sender and the Receiver both have four pure strategies The Sender s strategy m is said to be a pooling strategy if each type sends the same message, i.e., if m is constant a separating strategy if each type sends a different message, i..e, m is injective a partially pooling (or semi-separating) if it is neither pooling nor separating V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

10 Perfect Bayesian equilibrium in signaling games We translate the informal statements of Requirements 1 through 3 into a formal definition of a perfect Bayesian equilibrium in a signaling game Requirement 1 is trivial when applied to the Sender since his choice occurs at a singleton information set The Receiver, in contrast, chooses an action after observing the Sender s message but without knowing the Sender s type There is one information set for each message the Sender might choose Each such information set has one node for each type Nature might have drawn V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

11 Perfect Bayesian equilibrium in signaling games Signaling Requirement 1 After observing any message m j from M, the Receiver must have a belief about which types could have sent m j Denote this belief by the probability distribution µ( m j ) Prob(T ) Signaling Requirement 2R For each m j in M, the Receiver s action a (m j ) must maximize the Receiver s expected utility, given the belief µ( m j ) That is, a (m j ) solves max a A µ(t m j )U R (t, m j, a) t T V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

12 Perfect Bayesian equilibrium in signaling game Requirement 2 also applies to the Sender, but the Sender has complete information Signaling Requirement 2S For each t i in T, the Sender s message m (t i ) must maximize the Sender s utility, given the Receiver s strategy a (m j ) That is, m (t i ) solves max U S(t i, m, a (m)) m M V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

13 Perfect Bayesian equilibrium in signaling game Given the Sender s strategy t i m (t i ), let T j denote the set of types that send the message m j T j {t i T : m (t i ) = m j } or equivalently 1 T j = [m ] 1 (m j ) The signal t i is a member of the set T j if m (t i ) = m j Given a message m j, if T j is non-empty then the information set corresponding to the message m j is on the equilibrium path otherwise, mj is not sent (at equilibrium) by any type and so the corresponding information set is off the equilibrium path 1 Rigorously, we should write [m ] 1 {m j}. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

14 Perfect Bayesian equilibrium in signaling game For messages on the equilibrium path, one should apply Requirement 3 to the Receiver s strategy Signaling Requirement 3 For each m j in M, if there exists t i in T such that m (t i ) = m j, then the Receiver s belief at the information set corresponding to m j must follow from Bayes rule and the Sender s strategy: µ(t i m j ) = p(t i [m ] 1 (m j )) = p(t i ) τ i T j p(τ i ) V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

15 Perfect Bayesian equilibrium in signaling game Definition A pure-strategy perfect Bayesian equilibrium in a signaling game is a pair of strategies (m, a ) where m : t i m (t i ) a : m j a (m j ) a family of beliefs (µ( m j )) mj M with each µ( m j ) Prob(T ) satisfying Signaling Requirements (1), (2R), (2S), and (3) Requirement 4 is vacuous in a signaling game If the Sender s strategy is pooling or separating then we call the equilibrium pooling or separating, respectively V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

16 Perfect Bayesian equilibrium in signaling game Consider the following example of a simple signaling game V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

17 A simple signaling game Each type is equally likely to be drawn by Nature The Receiver belief µ( L) at information set L is denoted (p, 1 p) The Receiver belief µ( R) at information set L is denoted (q, 1 q) There are four possible pure-strategy perfect Bayesian equilibria in this two-type, two-message game 1 Pooling on L 2 Pooling on R 3 Separating with t 1 playing L and t 2 playing R 4 Separating with t 2 playing L and t 1 playing R V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

18 A simple signaling game: pooling on L Suppose there is an equilibrium (m, a, µ) in which the Sender s strategy is { m L if t = t1 (t) = L if t = t 2 Then the Receiver s information set corresponding to L is on the equilibrium path So the Receiver s belief (p, 1 p) at this information set is determined by Bayes rule and the Sender s strategy This implies that µ(t 1 L) p = = 0.5 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

19 A simple signaling game: pooling on L Given this belief µ, the Receiver s best response following L is to play u The Sender s type t 1 earns payoff of 1 and the Sender s type t 2 earns payoff of 2 To determine whether both Sender types are willing to choose L, we need to specify how the Receiver would react to R If the Receiver s response to R is u, i.e., a (R) = u then type t 1 s payoff from playing R is 2, which exceeds t 1 s payoff of 1 from playing L But if the Receiver s response to R is d, i.e., a (R) = d then t 1 and t 2 earn payoffs of 0 and 1 from playing R, whereas they earn 1 and 2 from playing L To get the pooling equilibrium on L, the Receiver s response to R must be d, i.e., a (R) = d V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

20 A simple signaling game: pooling on L One have to check that a (R) = d is an optimal action with respect to the Receiver s belief at the information set corresponding to R Observe that E µ( R) [U R (, R, d)] = q 0 + (1 q) 2 = 2(1 q) and E µ( R) [U R (, R, u)] = q 1 + (1 q) 0 = q Playing d is optimal for the Receiver for any q 2/3 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

21 A simple signaling game: pooling on L The pair of strategies (m, a ) defined by m (t) = L, t {t 1, t 2 } { a u if m = L (m) = d if m = R and the beliefs m µ( m) defined by { (0.5, 0.5) if m = L µ( m) = (q, 1 q) if m = R form a pure-strategy perfect Bayesian equilibrium if q 2/3. V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

22 A simple signaling game: pooling on R Suppose the Sender s strategy is m (t) = R for any t in T Then q = 0.5 and the Receiver s best response is a (R) = d Thus the contingent payoffs for the Sender are U S (t 1, R, d) = 0 and U S (t 2, R, d) = 1 But t 1 can earn 1 by playing L, since the Receiver s best response to L is u for any value of p There is no equilibrium in which the Sender plays m (t) = R for any t in T V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

23 A simple signaling game: Separating with m (t 1 ) = L Suppose the Sender s strategy m is defined by { m L if t = t1 (t) = R if t = t 2 Both of the Receiver s information sets are on the equilibrium path So both beliefs are determined by Bayes rule and the Sender s strategy p = 1 and q = 0 The Receiver s best responses to these beliefs are { a u if m = L (m) = d if m = R V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

24 A simple signaling game: Separating with m (t 1 ) = L It remains to check whether the Sender s strategy is optimal given the Receiver s strategy a It is not: if type t 2 deviates by playing L rather than R, then the Receiver s responds with u, earning t 2 a payoff of 2, which exceeds t 2 s payoff of 1 from playing R V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

25 A simple signaling game: Separating with m (t 1 ) = R Suppose the Sender s strategy m is defined by { m R if t = t1 (t) = L if t = t 2 Both of the Receiver s information sets are on the equilibrium path So both beliefs are determined by Bayes rule and the Sender s strategy p = 0 and q = 1 The Receiver s best response to these beliefs is a (m) = u, m {L, R} Both types t 1 and t 2 earn payoffs of 2 V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

26 A simple signaling game: Separating with m (t 1 ) = R If t 1 were to deviate by playing L, then the Receiver would react with u t 1 s payoff would then be 1, so there is no incentive for t 1 to deviate from playing R If t 2 were to deviate by playing R, then the Receiver would react with u t 2 s payoff would then be 1, so there is no incentive for t 2 to deviate from playing L V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27

27 A simple signaling game: Separating with m (t 1 ) = R The pair of strategies (m, a ) defined by { m R if t = t1 (t) = L if t = t 2 a (m) = u, m {L, R} and the beliefs m µ( m) defined by { (0, 1) if m = L µ( m) = (1, 0) if m = R form a separating pure-strategy perfect Bayesian equilibrium V. Filipe Martins-da-Rocha (FGV) Introduction to Game Theory November, / 27