Eco514--Game Theory gnawing game Marciano siniscalchi January 10, 2000 Introduction Signaling games are used to model the following situation: Player 1, the Sender, receives some private information 0 ee and sends a message m E M to Player 2, the Receiver. The latter, in turn, observes m but not 0, and chooses a response E R Players' payoffs depend on 6, m and r What could be simpler? Yet, there is a huge number of economically interesting games that fit nicely within this framework: Spence's job market signaling model is the leading example, but applications abound in IO(limit pricing, disclo finance(security design) and political economics Also, the analysis of signaling games is not entirely straightforward. The point is in general all (reasonable) Nash equilibria are also sequential, trembling-hand perfect etc; that is, they satisfy all sorts of backward-induction criteria However, many equilibria appear to be intuitively unreasonable. Moreover, tI to the particularly simple dynamic and informational structure, it is not hard to generalize intuitions concerning specific models to the whole class of signaling games As a result, a sizable literature on refinements for these games has developed. W shall only look at the most important(and most successful) notions: the intuitive criterion of Cho and Kreps, and divinity-like ideas a la Banks and Sobel Beer-Quiche and the Intuitive Criterion Consider the game of Figure 1 I told you the story behind this game in class, so I'm not going to bother you with it again. Let me just point out the essential features: either type of the Sender prefers to avoid a fight, even if this requires not enjoying her favorite breakfast(the incremental payoff from avoiding a fight is 2, whereas the preferred breakfast yields an
Eco514—Game Theory Signaling Games Marciano Siniscalchi January 10, 2000 Introduction Signaling games are used to model the following situation: Player 1, the Sender, receives some private information θ ∈ Θ and sends a message m ∈ M to Player 2, the Receiver. The latter, in turn, observes m but not θ, and chooses a response r ∈ R. Players’ payoffs depend on θ, m and r. What could be simpler? Yet, there is a huge number of economically interesting games that fit nicely within this framework: Spence’s job market signaling model is the leading example, but applications abound in IO (limit pricing, disclosure...), finance (security design) and political economics. Also, the analysis of signaling games is not entirely straightforward. The point is, in general all (reasonable) Nash equilibria are also sequential, trembling-hand perfect, etc.; that is, they satisfy all sorts of backward-induction criteria. However, many equilibria appear to be intuitively unreasonable. Moreover, thanks to the particularly simple dynamic and informational structure, it is not hard to generalize intuitions concerning specific models to the whole class of signaling games. As a result, a sizable literature on refinements for these games has developed. We shall only look at the most important (and most successful) notions: the intuitive criterion of Cho and Kreps, and divinity-like ideas `a la Banks and Sobel. Beer-Quiche and the Intuitive Criterion Consider the game of Figure 1. I told you the story behind this game in class, so I’m not going to bother you with it again. Let me just point out the essential features: either type of the Sender prefers to avoid a fight, even if this requires not enjoying her favorite breakfast (the incremental payoff from avoiding a fight is 2, whereas the preferred breakfast yields an 1
W F 2.0 {1} 3,0 B 2.1 Figure 1: Beer-Quiche incremental payoff of 1); also, the Receiver strictly prefers fighting the weak Sender, and not fighting the strong one That all Nash equilibria in this game are pooling is not surprising at all. However there are two classes of pooling equilibria: one in which both types of the Sender order Beer for breakfast, under the threat that Quiche will induce a Fight, and one in which the exact opposite is true-both types order Quiche, under the threat of a Fight if they deviate to Beer. Let us indicate the pure-strategy equilibria by(B, B NF)(the Weak type orders Beer, the Strong type orders Beer, and the receiver play n if he observes Beer and F if he observes Quiche)and( Q, Q, FN) Why exactly do we find the second type of equilibrium unreasonable? A moments reflection is sufficient to conclude that the reasonableness of the equilibrium hinges on the reasonableness of the Receiver's response, which in turn depends on his off- equilibrium beliefs Now, for the Receiver to be willing to choose F after observing B, it must be that he is sufficiently confident that the Sender is Weak. Hence, we might recast our question as follows: Is it reasonable for the receiver to believe that the Sender is Weak after observing Beer Suppose we changed the game so uI B, N, w)=0: then we could simply invoke a dominance argument. That is, we could say In the modified game a Weak sender gets 0 by choosing B; this is strictly less than she can get by choosing Q, regardless of the Receiver's strategy That is, B is a strictly dominated message for the Weak Sender. Hence, if the Receiver thinks that both types of the Sender are rational, he cannot place positive probability on the Weak Sender having chosen B. Thus, the out-of-equilibrium beliefs required to support the equilibrium(Q,Q, FN are unreasonable, and so is the equilibrium itself Note that this argument has a forward induction favor: we consider the out-of- equilibrium choice of B as being intentional, and we attempt to interpret it assuming
❜ w {.1} ❍❍ r✛ B 0,1 ❍❨ F ✟✟ 2,0 ✟✙ N Q ✲ r✟ F✟✟✯ 1,1 ❍ N ❍❍❥ 3,0 ❜ s {.9} ❍❍ r✛ B 1,0 ❍❨ F ✟✟ 3,1 ✟✙ N Q ✲ r✟ F✟✟✯ 0,0 ❍ N ❍❍❥ 2,1 Figure 1: Beer-Quiche. incremental payoff of 1); also, the Receiver strictly prefers fighting the weak Sender, and not fighting the strong one. That all Nash equilibria in this game are pooling is not surprising at all. However, there are two classes of pooling equilibria: one in which both types of the Sender order Beer for breakfast, under the threat that Quiche will induce a Fight, and one in which the exact opposite is true—both types order Quiche, under the threat of a Fight if they deviate to Beer. Let us indicate the pure-strategy equilibria by (B, B, NF) (the Weak type orders Beer, the Strong type orders Beer, and the receiver plays N if he observes Beer and F if he observes Quiche) and (Q, Q, FN). Why exactly do we find the second type of equilibrium unreasonable? A moment’s reflection is sufficient to conclude that the reasonableness of the equilibrium hinges on the reasonableness of the Receiver’s response, which in turn depends on his off- equilibrium beliefs. Now, for the Receiver to be willing to choose F after observing B, it must be that he is sufficiently confident that the Sender is Weak. Hence, we might recast our question as follows: Is it reasonable for the Receiver to believe that the Sender is Weak after observing Beer? Suppose we changed the game so u1(B,N,w) = 0: then we could simply invoke a dominance argument. That is, we could say: In the modified game a Weak sender gets 0 by choosing B; this is strictly less than she can get by choosing Q, regardless of the Receiver’s strategy. That is, B is a strictly dominated message for the Weak Sender. Hence, if the Receiver thinks that both types of the Sender are rational, he cannot place positive probability on the Weak Sender having chosen B. Thus, the out-of-equilibrium beliefs required to support the equilibrium (Q,Q,FN) are unreasonable, and so is the equilibrium itself. Note that this argument has a forward induction flavor: we consider the out-ofequilibrium choice of B as being intentional, and we attempt to interpret it assuming 2
that the Sender is rational However, in this game u1(B, N, w)=2, so B is not strictly dominated by Q for the Weak Sender. For instance, if the Sender expects the Receiver to play NF, contrary to the equilibrium prescription, then it's perfectly OK for her to choose B Still, a slightly modified version of the dominance idea works here. One way to motivate it is to note that, if we want to take the notion of equilibrium at least a little seriously, we should not allow the Weak Sender to expect the receiver to play anything other than what the equilibrium prescribes, as long as no deviation has occurred. In particular, the Weak Sender should believe that the Receiver will respond to Q with N Note well that we do not need to assume that the Receiver will respond to the out-of-equilibrium message B with F, as prescribed by the equilibrium. After all, the whole point is that we are not so sure this response is reasonable! What we do wish to maintain, though, is the assumption that, on the equilibrium path, players behave as prescribed by the equilibrium But this suffices to eliminate b for the Weak Sender Why should the Weak Sender choose a message which, even under the most optimistic circumstances (i.e. even if the Receiver responds with N B), leaves her with a payoff of 2, which is strictly less than what she gets in equilibrium? On the other hand, the Strong Sender is getting 2 in quilibrium, and if the Receiver was convinced that only a Strong Sender might choose B, she could potentially get a payoff of 3 To sum up, the Weak Sender can only lose(compared with the equilib- rium outcome) by sending B, whereas the Strong Sender has a positive incentive to send B (relative to the equilibrium outcome. But the out- of-equilibrium beliefs necessary to support( Q, Q, FN) do not reflect this so the equilibrium is unreasonable The Intuitive Criterion formalizes these intuitions, with an additional twist. Sup- pose that, following each message, the Receiver had another response available-to donate $10,000 out of his own pockets to the Sender. Then the preceding argument would not work, because even the Weak Sender might have an incentive to deviate to B-if she somehow expected the Receiver to donate $10,000 to her We cannot accept this as a reasonable justification for the choice of B: after all donating $10,000 is(conditionally) strictly dominated for the Receiver, i. e. it is never a best response. Thus, we restrict the beliefs of the Weak Sender to best replies of the receiver, and the argument goes through as before. The resulting test is called equilibrium domination, for obvious reasons
that the Sender is rational. However, in this game u1(B,N,w) = 2, so B is not strictly dominated by Q for the Weak Sender. For instance, if the Sender expects the Receiver to play NF, contrary to the equilibrium prescription, then it’s perfectly OK for her to choose B. Still, a slightly modified version of the dominance idea works here. One way to motivate it is to note that, if we want to take the notion of equilibrium at least a little seriously, we should not allow the Weak Sender to expect the Receiver to play anything other than what the equilibrium prescribes, as long as no deviation has occurred. In particular, the Weak Sender should believe that the Receiver will respond to Q with N. Note well that we do not need to assume that the Receiver will respond to the out-of-equilibrium message B with F, as prescribed by the equilibrium. After all, the whole point is that we are not so sure this response is reasonable! What we do wish to maintain, though, is the assumption that, on the equilibrium path, players behave as prescribed by the equilibrium. But this suffices to eliminate B for the Weak Sender: Why should the Weak Sender choose a message which, even under the most optimistic circumstances (i.e. even if the Receiver responds with N to B), leaves her with a payoff of 2, which is strictly less than what she gets in equilibrium? On the other hand, the Strong Sender is getting 2 in equilibrium, and if the Receiver was convinced that only a Strong Sender might choose B, she could potentially get a payoff of 3. To sum up, the Weak Sender can only lose (compared with the equilibrium outcome) by sending B, whereas the Strong Sender has a positive incentive to send B (relative to the equilibrium outcome.) But the outof-equilibrium beliefs necessary to support (Q, Q, FN) do not reflect this, so the equilibrium is unreasonable. The Intuitive Criterion formalizes these intuitions, with an additional twist. Suppose that, following each message, the Receiver had another response available—to donate $10,000 out of his own pockets to the Sender. Then the preceding argument would not work, because even the Weak Sender might have an incentive to deviate to B—if she somehow expected the Receiver to donate $10,000 to her. We cannot accept this as a reasonable justification for the choice of B: after all, donating $10,000 is (conditionally) strictly dominated for the Receiver, i.e. it is never a best response. Thus, we restrict the beliefs of the Weak Sender to best replies of the Receiver, and the argument goes through as before. The resulting test is called equilibrium domination, for obvious reasons. 3
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 responses 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 typedependent, 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
Second, one may think about constructing more stringent tests, according to the following intuition. After all, if types in e(m) cannot gain by sending the message m, then the Receiver should take this into account; this much is incorporated in the above alternative formalization of the Intuitive Criterion test, but not in the definition of R(m Thus. we can iterate our definitions Fe(n)=BB((m),m),(m)={∈m0):m(m and so on(what would be the third step? This leads to the Iterated Intuitive Crite D1, D2, Divinity and friends his material goes beyond what we covered in class, but i thought you might find it Useful. Check Cho and Kreps's original paper for further info The Intuitive Criterion is not a panache, however. Consider once again the game of Figure 1, but change the payoffs by letting u1(Q, N, w)=2. Then the IC does not eliminate the(Q, Q, FN)equilibrium, because ui(w)=2= maxrERI u(m, r,w) However, consider the following argument The Weak Sender never gains by sending the message B. Also, the only case in which she is indifferent is when the Receiver responds with n On the other hand, if the Receiver responds to B with N, the Strong Sender gains by deviating In other words, for any response that makes the Weak Sender indiffer ent between deviating and playing as per the candidate equilibrium, the Strong Sender has a positive incentive to deviate. This suggests that the latter should be more likely to be the deviator This idea leads to a class of refinements known as D1, D2 and Divinity. First fixing a candidate equilibrium as above, for any type 0 Ee and out-of-equilibrium message m, define the set of mired responses of the Receiver that provide type 0 with a strict incentive to deviate to m as D(m)={y∈MBR2(O,m):1()<a1(m,r,0)y(r)} r∈R where MBR2(e, M) is defined analogously to BR2(0, M) (note that the former is not the convex hull of the latter-think about it! ) Also, define the set of mixed responses
Second, one may think about constructing more stringent tests, according to the following intuition. After all, if types in Θ¯ 1 (m) cannot gain by sending the message m, then the Receiver should take this into account; this much is incorporated in the above alternative formalization of the Intuitive Criterion test, but not in the definition of R1 (m). Thus, we can iterate our definitions: R 2 (m) = BR2(Θ\Θ¯ 1 (m), m), Θ¯ 2 (m) = {θ ∈ Θ\Θ¯ 1 (m) : u ∗ 1 (θ) > max r∈R2(m) u1(m, r, θ)} and so on (what would be the third step?) This leads to the Iterated Intuitive Criterion. D1, D2, Divinity and Friends [This material goes beyond what we covered in class, but I thought you might find it useful. Check Cho and Kreps’s original paper for further info.] The Intuitive Criterion is not a panache, however. Consider once again the game of Figure 1, but change the payoffs by letting u1(Q,N,w) = 2. Then the IC does not eliminate the (Q, Q, FN) equilibrium, because u ∗ 1 (w) = 2 = maxr∈R1 u 1 (m, r,w). However, consider the following argument. The Weak Sender never gains by sending the message B. Also, the only case in which she is indifferent is when the Receiver responds with N. On the other hand, if the Receiver responds to B with N, the Strong Sender gains by deviating. In other words, for any response that makes the Weak Sender indifferent between deviating and playing as per the candidate equilibrium, the Strong Sender has a positive incentive to deviate. This suggests that the latter should be more likely to be the deviator. This idea leads to a class of refinements known as D1, D2 and Divinity. First, fixing a candidate equilibrium as above, for any type θ ∈ Θ and out-of-equilibrium message m, define the set of mixed responses of the Receiver that provide type θ with a strict incentive to deviate to m as Dθ(m) = {ϕ ∈ MBR2(Θ, m) : u ∗ 1 (θ) < X r∈R u1(m, r, θ)ϕ(r)} where MBR2(Θ, M) is defined analogously to BR2(Θ, M) (note that the former is not the convex hull of the latter—think about it!) Also, define the set of mixed responses 5