Signaling games A pure strategy for the sender is a function m(ti) specifying which message will be chosen for each type that nature might draw A pure strategy for the receiver is a function a(mi) specifying which action will be chosen for each message that the sender might send
Signaling Games • A pure strategy for the Sender – is a function 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 a(mj ) specifying which action will be chosen for each message that the Sender might send. 11
Signaling Games: EXample Sender a Receiver Nature Receiver Sender Tt,, t2, M=(m,m2), A=a,a2], and Prob(t,=p 12
Signaling Games: Example a1 a2 a1 a2 m1 m2 m1 m2 a1 a2 a2 a1 Nature p 1-p t1 t2 Sender Sender Receiver Receiver T={t1 ,t2 }, M={m1 ,m2 }, A={a1 ,a2 }, and Prob(t1 )=p 12
Signaling games Sender's strategy Strategy 1: m(t=m, and m(t2)=m, (pooling strategy) Strategy 2: m(t=m, and m(t2)=m2(separating strategy) Strategy 3: m(t=m2 and m(t2) =m, (separating strategy) Strategy 4: m(t=m2 and m(t2)=m2(pooling strategy) Receiver's strategy Strategy 1: a(m1=a1 and a(m2 =a,(pooling strategy) Strategy 2: a(m)=a, and a(m2 =a2(separating strategy) Strategy 3: a(m1=a2 and a(m2)=a,(separating strategy) Strategy 4 a(m1)=a2 and a(m2 =a2(pooling strategy) 13
Signaling Games • Sender’s strategy – Strategy 1: m(t1 )=m1 and m(t2 )=m1 (pooling strategy) – Strategy 2: m(t1 )=m1 and m(t2 )=m2 (separating strategy) – Strategy 3: m(t1 )=m2 and m(t2 )=m1 (separating strategy) – Strategy 4: m(t1 )=m2 and m(t2 )=m2 (pooling strategy) • Receiver’s strategy – Strategy 1: a(m1 )=a1 and a(m2 )=a1 (pooling strategy) – Strategy 2: a(m1 )=a1 and a(m2 )=a2 (separating strategy) – Strategy 3: a(m1 )=a2 and a(m2 )=a1 (separating strategy) – Strategy 4: a(m1 )=a2 and a(m2 )=a2 (pooling strategy) 13
Signaling games Signaling Requirement 1: After observing any message m; from M, the Receiver must have a belief about which types could have sent m Denote this belief by the probability distribution u(timi, whereu(timi>=0 for each t, in T, and y (1|m;) t∈T 14
Signaling Games • Signaling Requirement 1: After observing any message mj from M, the Receiver must have a belief about which types could have sent mj . Denote this belief by the probability distribution μ(t i |mj ), whereμ(t i |mj )>=0 for each t i in T, and ( | ) 1 i i j t T t m = 14
Signaling games Signaling Requirement 2R: For each m; in M,the Receiver's action a*(mi) must maximize the receiver's expected utility, given the belief u(tm) about which types could have sent m That is a(mi solves max u(tImUr(i, m;,ak) Signaling Requirement 2S: For each t, in T, the Senders message m*(ti) must maximize the Senders utility, given the receiver's strategy a (mi). That is m*() solves max Us(i, m;, a(mi)) m∈M 15
Signaling Games • Signaling Requirement 2R: For each mj in M, the Receiver’s action a*(mj ) must maximize the Receiver’s expected utility, given the belief μ(t i |mj ) about which types could have sent mj . That is a*(mj ) solves • 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*(mj ). That is m*(t i ) solves max ( | ) ( , , ) k i i j R i j k a A t T t m U t m a * max ( , , ( )) j S i j j m M U t m a m 15