Graduate Microeconomics II Lecture 5: Cheap Talk. Patrick Legros

Transcription

1 Graduate Microeconomics II Lecture 5: Cheap Talk Patrick Legros 1 / 35

2 Outline Cheap talk 2 / 35

3 Outline Cheap talk Crawford-Sobel Welfare 3 / 35

4 Outline Cheap talk Crawford-Sobel Welfare Partially Verifiable Information 4 / 35

5 Outline Cheap talk Crawford-Sobel Welfare Partially Verifiable Information Multiple Senders 5 / 35

6 Cheap talk and information transmission Similar to signaling games except that the signal is not costly for the sender. Hence if all types have the same order of preferences over actions chosen by the receiver. (as the sender in the beer-quiche prefers not to have a fight), signals cannot be informative. A necessary condition for signals to be informative in cheap talk games is therefore that different types have different orders of preferences over actions chosen by the receiver. 6 / 35

7 Map One sender (Crawford-Sobel 1982) Inefficiency, partition information equilibria prevent full aggregation of information. Efficiency with partially verifiable information Many senders: on the benefits of competition Battaglini (2002): Generic efficiency with multiple dimensions 7 / 35

8 Crawford-Sobel (1982) Simplified model One agent (sender) with private information about the state of the world θ (the set of states is an interval Θ = [0, 1].) The prior density on Θ is uniform. One agent (receiver) who has to choose a decision y (a real number) If y is chosen in state θ, the outcome is x = y θ. The sender and the receiver have preferences over outcomes: U S (x) = (x u) 2 and U R (y) = x 2 ; u and 0 are the ideal points for the sender and the receiver respectively. 8 / 35

9 Since only the sender knows the state, it is more convenient to work with the state dependent utility functions U S (y, θ) = (y θ u) 2 U R (y, θ) = (y θ) 2 Note that U i 12 > 0 and Ui 11 < 0. As u = 0,R and S have common interest : y (θ) = θ As u 0,divergent interest: S likes y = u + θ y (θ). 9 / 35

10 Key assumption: the receiver cannot commit to a decision rule. no commitment: makes it different from mechanism design. message is not costly to send ( cheap talk ): makes it different from the signaling literature, e.g., Spence labor market example. Communication: the sender can use messages m M to communicate with the receiver. Assume that M is rich enough in the sense that there exists a map Θ M that is injective (for any two different states there exist two different messages). 10 / 35

11 Crawford-Sobel Game The sender chooses a state dependent message strategy σ (m, θ), that is a distribution (mixed strategy) over M. Let M (σ, θ) = {m : σ (m, θ) > 0} be the support of σ in state θ and let M (σ) = θ Θ M (σ, θ). The receiver chooses a message dependent decision rule y (m). Since V is concave, the receiver will never use a mixed strategy. Use Bayesian Nash equilibrium: there exist a belief structure µ (θ m) that assigns a probability distribution on Θ as a function of the received message m such that 11 / 35

12 If m M (σ), µ must be consistent with Bayes law, that is µ (θ, m) = σ (m, θ) ( ). ˆθ σ m, ˆθ d ˆθ The message strategy is optimal given y, that is σ (m, θ) > 0 y (m) arg max ˆm (y ( ˆm) θ u)2. The decision rule is optimal given σ and given µ y (m) arg max (y θ) 2 µ (θ, m) dθ. y 12 / 35

13 Crawford-Sobel Examples of equilibria Babbling : the strategy of the sender is state independent, σ (m, θ) = σ (m) for all θ. Then Bayes law implies that for all m, µ (θ, m) = 1 (the initial distribution - assumed to be uniform on [0, 1]). Fully revealing : the sender uses a pure strategy that is 1-1: σ (θ) = m, and θ ˆθ implies σ (θ) σ(ˆθ). In this case, the posterior belief is for m M (σ) : µ (θ, m) = 1 when σ (θ) = m. 13 / 35

14 For fully revealing equilibria, the receiver can attain the first best. Note that if there is a fully revealing equilibrium, there are plenty of others: the content of the message is not important, what matters is that the map from state to message is 1-1. Partially revealing : for each m M (σ), the set {θ : σ (m, θ) > 0} is not a singleton but is not the full set Θ either. Hence the message transmits some information about the state. Since the content of the message does not really matter, there is no loss of generality in supposing that the set of messages is a copy of Θ. 14 / 35

15 Crawford-Sobel Properties of an Equilibrium 1. The equilibrium message strategy is (essentially) partitional: there exists k cutoff values θ k, k = 0,..., K such that for each k [0, K], θ (θ k, θ k+1 ), σ (m, θ) is the uniform distribution on [θ k, θ k+1 ] state θ 0 = 0 1 θ θ 2 θ3 4 θ θ θ 5 = 1 message σ( i θ) support(, 15 / 35

16 2. The maximum number of elements of a partition equilibrium is a function of the degree of conflict between R and S; here the ideal point u is an index of conflict. 3. If N (u) is the maximum number of elements, for any N N (u), there exists an equilibrium with N elements. 4. Under a monotonicity condition can show that the N + 1 equilibrium is more informative than the N equilibrium (finer partition). 16 / 35

17 Crawford-Sobel Some Observations In equilibrium, to each θ corresponds at most two possible decisions taken by R. Consider the set of messages that yield R to choose y N (y) = {m : y (m) = y} Say that y is induced by θ if m N (y) is in the support of the message strategy of S σ (m, θ) dm > 0. m N(y) Let Y be the set of all induced decisions. Immediate that if y is induced by θ that U S (y, θ) maxŷ Y U S (ŷ, θ) [revealed preferences] 17 / 35

18 By concavity U S 11 < 0 : US (y, θ) is maximum on Y for at most two values. Thus θ can induce at most two decisions in any equilibrium. For any y and ŷ in Y, y ŷ u 18 / 35

19 In the parametric example, the best state contingent decisions are y S (θ) = θ + u and y R = θ Let two equilibrium decisions y and ŷ where y > ŷ. Let θ induce y and ˆθ induce ŷ. If θ = ˆθ, then U S (y, θ) U S (ŷ, θ) = 0, and choose θ = θ. If θ ˆθ, by revealed preferences U S (y, θ) U S ( (ŷ, θ) ) U S ŷ, ˆθ ( ) U S y, ˆθ ( ) U S (y, θ) U S (ŷ, θ) 0 U S y, ˆθ ( ) U S ŷ, ˆθ with one strict inequality. By continuity there exists θ such that U S ( y, θ ) U S ( ŷ, θ ) = 0. Since U 12 > 0, θ > ˆθ and θ [ˆθ, θ]. 19 / 35

20 Since U 11 < 0, and moreover ŷ < y S ( θ ) < y all θ > θ prefer y to ŷ all θ < θ prefer ŷ to y. ŷ S y ( θ) = θ + u y θ < θ θ < θ 0 ŷ θ u ŷ θ u u y θ u y θ u x= y θ yˆ preferred to y by all θ < θ 20 / 35

21 It follows that the beliefs of R when he chooses ŷ have support N (ŷ) [0, θ) and when he chooses y have support N (y) ( θ, 1]. Since y > ŷ, and U12 R > 0, UR (y, θ) U R (ŷ, θ) is increasing ) in θ. Using continuity, U12 R > 0, UR 11 < 0 and θ (ˆθ, θ, we conclude that y R ( θ ) = θ [ŷ, y]. Now, y R ( θ ) y S ( θ ) = u; therefore for any two induced actions y ŷ, y ŷ u. Since u is greater than 0, the set of actions in Y must be finite. Since when y > ŷ we have shown that N (y) > N (ŷ) (set notation), it must be the case that messages that induce y and ŷ respectively generate supports for the beliefs µ that are disjoint; since this is true for any two pairs in Y, the generated supports must form a partition of Θ. 21 / 35

22 Hence any equilibrium message strategy that does not have the partitional property can be redefined as a partitional strategy. In practice, to conclude the proof, need to show that there exists cutoffs θ k, k = 0,..., K with θ 0 = 0, θ K+1 = 1, θ k < θ k+1 such that when the message strategy is to use a uniform distribution on [θ k, θ k+1 ] when the type is θ [θ k, θ k+1 ], that the best response properties hold. Note that the belief structure is trivial in this case: if 1 receives message m [θ k, θ k+1 ], µ (θ, m) = θ k+1 θ k when θ [θ k, θ k+1 ], and µ (θ, m) = 0 otherwise. 22 / 35

23 Let y k be the decision taken when the message is in (θ k, θ k+1 ). If m (θ k, θ k+1 ) the receiver will choose θk+1 y k = arg max y θ k U R (y, θ) dθ. (1) Since by sending a message θ k + ε the sender can generate decision y k and by sending θ k ε he can generate decision y k 1, the sender must be indifferent between y k and y k 1 at θ k, or U S (y k, θ k ) = U S (y k 1, θ k ). (2) Given the indifference relation (2), U12 S < 0 implies that all θ > θ k prefer y k to y k 1 and all θ < θ k 1 prefer y k 1 to y k. The question of an equilibrium boils down to finding cutoff values θ k for with conditions (1) and (2) hold. 23 / 35

24 In the example the conditions are: the receiver chooses the average state in the interval: y k = θ k + θ k+1 2 with y K = y K 1. For the sender, since θ k is increasing in k, y k θ k > y k 1 θ k ; hence can have (y k θ k u) 2 = (y k 1 θ k u) 2 only if y k θ k u > 0 > y k 1 θ k u, and (2) boils down to y k + y k 1 = 2θ k + 2u θ k+1 = 2θ k θ k 1 + 4u θ k = θ 1 k + 2k (k 1) u We need θ K = 1, hence 2K (K 1) u < 1, or K u ; the right hand side is strictly less than 2 when u 1 4. In this case K = 1 and the unique equilibrium is babbling! (3) 24 / 35

25 Given the boundary condition θ K+1 = 1, we have and therefore θ 1 = 1 2K (K 1) u K θ k = k + 2uk (k K), k = 0,..., K. K As long as K u, there exists a partitional equilibrium with K intervals. 25 / 35

26 Crawford-Sobel Welfare From an ex-ante perspective, both R and S value information transmission. In equilibrium EU S = K 1 θk+1 (y k θ u) 2 dθ and EU R = K 1 k=0 θk+1 θ k k=0 θ k (y k θ) 2 dθ. Since y k = θ k+θ k+1 2 is the average state in the interval [θ k, θ k+1 ], θ k+1 θ k (y k θ) 2 dθ θ k+1 θ k is the within variance and K 1 θk+1 k=0 θ k (y k θ) 2 dθ is the expected variance σ 2 (K). Hence, EU S = ( σ 2 (K) + u 2) EU R = σ 2 (K), Both R and S prefer (from an ex-ante point of view) equilibria in which K is the largest since then the variance σ 2 (K) is the smallest. 26 / 35

27 Partially Verifiable Information In Crawford-Sobel, there are no constraints on the type of information that the sender can transmit. In particular, the set of feasible messages is independent of the state of the world. Information is soft. Often, there are natural possibilities to verify a claim. Someone announcing that he has 10 dollars in his pocket should be able to show 10 dollars. Note that if he has 15 dollars, he may still decide to announce that he has only 10 dollars, but he cannot pretend that he has 20 dollars. Accounting rules allow firms to release verifiable information about profits in many ways; for a given financial state announced profits can vary depending on the use that is made of accounting principles. However there are limits to this and partial verifiability. 27 / 35

28 Consider a simple restriction on the set of messages: if the state is θ, then M (θ) = [0, θ], that is the sender can understate the state but not overstate it. In this case, for any value of u (the index of conflict between S and R), it is possible to achieve full revelation and the first best for R. Suppose that u > 0, that is that the sender prefers decision y S (θ) = θ + u > y R (θ) in any state. Trick : pessimistic belief about the state in the sense that the receiver takes the message that is sent at face value: if S sends m, R believes that the state is m. In this case, in state θ, if S sends m, the receiver chooses m. Since S is constrained to send m θ, his payoff (given the beliefs of the receiver) is U S (m, θ) = (m θ u) 2 By concavity of U S and the fact that m θ < y S (θ), it is immediate that U S (m, θ) is maximized at m = θ. 28 / 35

29 This is a special form of what is called the unraveling theorem : if there is a bound on messages that is state dependent and monotone in the state, skeptical beliefs from the receiver (always believing that the state is the worth possible state consistent with the announced message) induces revelation of information by the sender. More general formulations (announcements can be intervals but must be truthful in the sense that the true state belongs to the interval): Milgrom-Roberts 1986, Shin 2003, Green-Laffont (1986,87) consider general restrictions in a mechanism design framework. 29 / 35

30 Multiple Senders Literature on multiple referrals Milgrom-Roberts 1986, Battaglini 2002, Ottaviani-Sorensen 2000, Garinaco-Santos 2003,... Remember if only i sends information, the maximum transmission of information is N (u i ), decreasing in u i. Note here a significant cost of no-commitment. If R could commit to a decision rule as a function of the messages received, he could induce full revelation (mechanism design). Competition helps for information revelation but still residual inefficiencies. Most work done with one dimensional states and decisions. 30 / 35

31 Multidimensions Limits to competition with one dimension Two senders with ideal points u i ; wlog assume quadratic utility functions Ui S (y, θ) = (y θ u i ) 2 for the senders U R = (y θ) 2 for the receiver. Assume that u 1 0 u 2 and that Θ = [ θ, θ ], θ > 0. Battaglini 2002 A necessary and sufficient condition for the existence of a fully revealing equilibrium is that u 2 u 1 θ. Therefore, as long as the ideal points of the two senders are not too far apart, there exists a fully revealing equilibrium. An example of equilibrium construction when the condition of the proposition is true. y (m 1, m 2 ) = (m 1 + m 2 )/2 if m 1 m 2 θ θ if m 1 > m 2 and m 1 < 2u 2 θ if m 1 > m 2 and m 1 2u 2 θ. The beliefs are such that µ (θ; m 1, m 2 ) = y (m 1, m 2 ). 31 / 35

32 Suppose that 2 is truthful in state θ. If 1 is truthful, the decision is θ and 1 s payoff is U1 S = u2 1. Suppose that m 1 < θ; then the decision is m 1+θ 2 and the outcome is m 1 θ 2 < 0; therefore U1 S < u2 1. If m 1 > θ, it must be that m 1 2θ θ (the case m 1 < 2m 2 θ is inconsistent with m 1 > θ and θ θ), therefore the decision is θ and the outcome is θ θ < 0, hence U1 S < u2 1. This shows that 1 does not want to deviate from truth-telling Note that belief structure is extreme 32 / 35

33 Multiple dimensions Generic revelation of information Battaglini (2002) considers multiple dimensions for states and decisions. Turns out that there exist generically fully revealing equilibria. State space is Θ R 2. Two agents (senders) with ideal points u 1 and u 2 observe θ and transmit a message m Θ to a receiver with ideal point of 0. Decisions are y R 2 and the outcome is x = y θ where θ is the true state of the world. If ideal point is u = ( u 1, u 2), preference over decision y in state θ is (y (1) θ (1) u (1)) 2 (y (2) θ (2) u (2)) 2. proposition Suppose that u 1 and u 2 are not on the same ray from the origin, then there exists a fully revealing equilibrium. 33 / 35

34 Idea: For fully revealing messages, the policy is y = θ. Since u 1 and u 2 are not on the same ray from the origin, the tangent I 1 to the indifference curve of sender 1 at x = 0 and the tangent I 2 to the indifference curve of sender 2 at x = 0 form a cone. For any line parallel to I i, there exists a number x that identifies uniquely this line (since there exists a unique vector (0, x) belonging to this line); let I i (x) be the line parallel to I i passing through (0, x). Then for any x 1 and x 2 and y, I 1 (x 2 ) I 2 (x 1 ) exists and is unique. Messages are for i to announce a number x i. The decision is to choose y I 1 (x 2 ) I 2 (x 1 ). That is agent 2 announces the line parallel to I 1 going through θ and agent 1 announces the line parallel to I 2 going through θ. If 2 is truthful in state θ, that is announces the number x 2 such that θ I 1 (x 2 ), then by announcing ˆx 1 x 1, sender 1 generates outcomes on I 1 (0) which are by construction worse than the outcome under truth telling. The argument generalizes to higher dimensions. 34 / 35

35 Sequential transmission of information Endogenous communication system Loss due to no-commitment in one vs. many senders, uni vs. multi-dimensions? Noisy information: value of duplication of transmission of information? 35 / 35