Signaling games Signaling Requirement 3: For each m in M, If there exists t in T such that m*(t=m;, then the Receiver's belief at the information set corresponding to m; must follow from Bayes rule and the senders strategy (4|m) p(ti) Definition: A pure-strategy perfect Bayesian equilibrium is a signaling game is a pair of strategies m" () and a(mj) and a belief u(timi) satisfying Signaling Requirements (1)、(2R),(2S),and(3) 16
Signaling Games • Signaling Requirement 3: For each mj in M, If there exists t i in T such that m*(t i )=mj , then the Receiver’s belief at the information set corresponding to mj must follow from Bayes’ rule and the Sender’s strategy: • Definition: A pure-strategy perfect Bayesian equilibrium is a signaling game is a pair of strategies m*(t i ) and a*(mj ) and a belief μ(t i |mj ) satisfying Signaling Requirements (1),(2R),(2S), and (3) ( ) ( | ) ( ) i i i j i t T p t t m p t = 16
Signaling Games: EXample Payoff of Receiver 1.3 Sender 2,1 Payoff of Sender p L R 40d 0.5 0.0 Nature Receiver Receiver 2,4 U 0.5 [ R 0,1 Sender 1,.2 Tt, t2, M=L,R) A=u,d, and Prob(t =0.5 17
Signaling Games: Example u d u d L R L R u d d u Nature 0.5 0.5 t1 t2 Sender Sender Receiver Receiver T={t1 ,t2 },M={L,R},A={u,d}, and Prob(t1 )=0.5 1,3 4,0 2,4 0,1 2,1 0,0 1,0 1,2 [p] [1-p] [q] [1-q] Payoff of Sender Payoff of Receiver 17
Signaling games Type t1 Type t2 Approach the receiver's optimal strategy (a(L/,ap ,derive Given the Sender's strategy(m(), m(t2)) Signal L Signal R Given receiver's strategy, check whether the Sender's strategy is optimal 18
Signaling Games • Approach – Given the Sender’s strategy (m(t1 ), m(t2 )), derive the receiver’s optimal strategy (a(L), a(R)) – Given receiver’s strategy, check whether the Sender’s strategy is optimal Type t1 Type t2 Signal L Signal R 18
Signaling games The Sender's strategy Strategy 1: m(t)=L and m(t2=L(pooling The Receivers strategy: a(L)=u and a(r)=d, p=0.5, <=2/3(PBE If a(r=u, then type t, sender will deviate to play r Condition for a(R)=d is g 0+(1-9).2>=g 1+(1-9).0 Strategy 2: m(t=L and m(t2=R(separating) The Receiver's strategy: p=1, 90, a(L)=u and a(r)=d No equilibrium(the type t2 Sender will deviate to play l) Strategy 3: m(t)=R and m(t2)=L (separating) The Receiver's strategy: p=0, q=1, a (L=u and a(R)=U, (PBE) Strategy 4: m(t)=R and m(t2)=R(pooling) The Receiver's strategy: 90.5, a( R)=d, for all p, a(L)=u 0.5×0+0.5×2(payd>0.5X1+0.5×0(payu No equilibrium the type t, Sender will deviate to play l) 19
Signaling Games The Sender’s strategy – Strategy 1: m(t1 )=L and m(t2 )=L (pooling) • The Receiver’s strategy: a(L)=u and a(R)=d, p=0.5, q<=2/3 (PBE) • If a(R)=u, then type t1Sender will deviate to play R • Condition for a(R)=d is q.0+(1-q).2>=q.1+(1-q).0 – Strategy 2: m(t1 )=L and m(t2 )=R (separating) • The Receiver’s strategy: p=1, q=0, a(L)=u and a(R)=d • No equilibrium (the type t2 Sender will deviate to play L ) – Strategy 3: m(t1 )=R and m(t2 )=L (separating) • The Receiver’s strategy: p=0,q=1, a(L)=u and a(R)=u, (PBE) – Strategy 4: m(t1 )=R and m(t2 )=R (pooling) • The Receiver’s strategy: q=0.5, a(R)=d , for all p, a(L)=u 0.5x0+0.5x2 (play d)>0.5x1+0.5x0 (play u) • No equilibrium (the type t1 Sender will deviate to play L) 19