Consider the following definition: let BR2(e, m) the set of conditional best re- sponses of the Receiver after the partial history(m), for m E M, and subject to the constraint that ui(m)(e)=1 for ece. That is, BR2(6,m)={r∈R:1(m)∈△()s.t.p1(m)() ∑[2(m,r,6)-2(m,r,)m(mn)()≥0W∈fF Now fix a sequential equilibrium (o1(0))bee, o2(m))meM); denote by ui(e) the equilibrium payoff for the Sender and, for any out-of-equilibrium message m, let R(m)=BR2(,m),6(m)={∈6:u1()>max,u1(m,r,)} The interpretation should be clear: R (m)is the set of best replies to m, and 0(m) is the set of types that lose by deviating to m relative to the equilibrium. That is, we first delete message-response pairs, then we delete message-type pairs. The reason for the superscripts will be clear momentarily. The candidate equilibrium passes the Intuitive Criterion if it can be supported by out-of-equilibrium beliefs which assign zero probability to types in 0(): that is, if for every out-of-equilibrium m E M such that e(m)+0(see below), we can find u1(m)E A(0) such that p1(m)(0(m))=0 and o2(m) is sequentially optimal after given u1(m) Two observations are in order. First, if 0(m)=e for some out-of-equilibrium message m, then we need not worry about the Receiver's beliefs: intuitively, if m is qually bad"for all types, there are no inferences the receiver can make about 0 after observing m. Second, note that the candidate equilibrium fails the Intuitive Criterion if at least one type 0 has a positive incentive to deviate to m whenever the Receiver's beliefs are constrained as above. Formally, the candidate equilibrium fails the Intuitive Criterion iff 1(6) Imin u1(m,r,6) ∈BR Observe that necessarily 0 g 0(m), so 0(m)+0 I conclude with two additional observations. First, Cho and Kreps allow for more general signaling games in which the messages available to the Sender may be type- dependent, and the responses available to the Receiver may be message-dependent But the analysis is identical IRecall the notation for Bayesian extensive games with observable actions: Hi(m)EA(e)is the belief, shared by all players, about Player 1s type, after history (m) 2ur()=∑n∈M∑reu1(m,r,0)1(0)(m)o2(m)()Consider the following definition: let BR2(Θ0 , m) the set of conditional best re￾sponses of the Receiver after the partial history (m), for m ∈ M, and subject to the constraint that µ1(m)(Θ0 ) = 1 for Θ0 ⊂ Θ.1 That is, BR2(Θ0 , m) = {r ∈ R : ∃µ1(m) ∈ ∆(Θ) s.t. µ1(m)(Θ0 X ) = 1 and θ 0∈Θ0 [u2(m, r, θ0 ) − u2(m, r0 , θ0 )]µ1(m)(θ 0 ) ≥ 0 ∀r 0 ∈ R} Now fix a sequential equilibrium ((σ1(θ))θ∈Θ,(σ2(m))m∈M); denote by u ∗ 1 (θ) the equilibrium payoff for the Sender2 , and, for any out-of-equilibrium message m, let: R 1 (m) = BR2(Θ, m), Θ¯ 1 (m) = {θ ∈ Θ : u ∗ 1 (θ) > max r∈R1(m) u1(m, r, θ)} The interpretation should be clear: R1 (m) is the set of best replies to m, and Θ¯ 1 (m) is the set of types that lose by deviating to m relative to the equilibrium. That is, we first delete message-response pairs, then we delete message-type pairs. The reason for the superscripts will be clear momentarily. The candidate equilibrium passes the Intuitive Criterion if it can be supported by out-of-equilibrium beliefs which assign zero probability to types in Θ¯ 1 (·): that is, if, for every out-of-equilibrium m ∈ M such that Θ¯ 1 (m) 6= Θ (see below), we can find µ1(m) ∈ ∆(Θ) such that µ1(m)(Θ¯ 1 (m)) = 0 and σ2(m) is sequentially optimal after history (m) given µ1(m). Two observations are in order. First, if Θ¯ 1 (m) = Θ for some out-of-equilibrium message m, then we need not worry about the Receiver’s beliefs: intuitively, if m is equally “bad” for all types, there are no inferences the Receiver can make about θ after observing m. Second, note that the candidate equilibrium fails the Intuitive Criterion if at least one type θ has a positive incentive to deviate to m whenever the Receiver’s beliefs are constrained as above. Formally, the candidate equilibrium fails the Intuitive Criterion iff u ∗ 1 (θ) < min r∈BR2(Θ\Θ¯ 1(m),m) u1(m, r, θ) Observe that necessarily θ 6∈ Θ¯ 1 (m), so Θ¯ 1 (m) 6= Θ. I conclude with two additional observations. First, Cho and Kreps allow for more general signaling games in which the messages available to the Sender may be type￾dependent, and the responses available to the Receiver may be message-dependent. But the analysis is identical. 1Recall the notation for Bayesian extensive games with observable actions: µ1(m) ∈ ∆(Θ) is the belief, shared by all players, about Player 1’s type, after history (m). 2u ∗ 1 (θ) = P m∈M P r∈R u1(m, r, θ)σ1(θ)(m)σ2(m)(r). 4