Communication with Self-Interested Experts Part II: Models of Cheap Talk

Transcription

1 Communication with Self-Interested Experts Part II: Models of Cheap Talk Margaret Meyer Nuffield College, Oxford 2013 Cheap Talk Models 1 / 27

2 Setting: Decision-maker (P) receives advice from an advisor (A) who possesses useful information but whose preferences over possible decisions don t match those of P. After receiving advice, P makes a decision. Examples: 1 manager consulting supervisor about an employee 2 headquarters consulting a division manager about an investment project 3 investor evaluating information disclosed by a firm s mgt. 4 politician listening to a lobbyist representing an interest group 5 customer asking a salesman about a product s quality. Maintained assumptions: Decision-maker cannot commit in advance as to how will use advisor s advice and hence cannot offer monetary transfers contingent on the advice. Contrast mechanism design. NB: Signaling and signal-jamming models can also be used to study communication, but in these, emphasis is on effort expended as well as on information transmitted. Cheap Talk Models 2 / 27

3 Cheap-Talk Models of Strategic Communication In cheap-talk models, the privately-informed advisor (A) can costlessly make any report to the decision-maker (P) about the state of the world. seminal paper: Crawford and Sobel, Etrica, widely applied to study expert advice (e.g. lobbying, legislative committee recommendations, financial advice to consumers) in settings where legal remedies for misleading advice are costly or unavailable For cheap talk to be informative with fully rational and self-interested players, and a single agent, there are 3 necessary conditions: i) A s (ordinal) preferences over P s decisions vary with the state of the world. ii) P prefers different decisions in different states of the world. iii) P s preferences are not completely opposed to A s. Remarks: Allowing for bounded strategic rationality, Crawford (AER, 03) shows that even in a zero-sum game, cheap talk can be used for strategic advantage. Some experiments also find a greater role for cheap talk than predicted by eqm. analysis with rational, self-interested players. Cheap Talk Models 3 / 27

4 Crawford and Sobel model (uniform-quadratic special case) u P (V, x) = (V x) 2 P s ideal: V P (x) = x u A (V, x) = (V (x + b)) 2 A s ideal: V A (x) = x + b b measures degree of divergence of preferences; state of the world x U[0, 1] Timing: A learns x and then sends a costless but unverifiable report m to P; P updates beliefs about x and then chooses V. There exists a fully-pooling PBE, in which A s message reveals nothing about x. Such a babbling eqm. always exists in any cheap talk game. For any b > 0, a fully revealing PBE does not exist if P expected A s message to be fully revealing, A would wish to deviate and mislead P. Crawford+Sobel show that all PBE are equiv. to partially pooling eqa of the form m 1 x [0, x 1 ), m 2 x [x 1, x 2 ),..., m n x [x n 1, 1], where, for some n, n distinct messages (m 1, m 2,..., m n ) are sent in eqm. in eqm., some communication occurs, but it is coarse. Thus, Cheap Talk Models 4 / 27

5 Example: PBE with two distinct messages: m = L if x [0, x 1 ) and m = H if x [x 1, 1] 0 L x 1 = 1 2 2b H 1 The eqm. value of x 1 is the value of x at which A is indifferent between reporting L and H, given that P expects A to use the strategy above. P s optimal response to L is V = x1 2, and his optimal response to H is V = (1+x1) 2, [ so A is indifferent ] between reporting L and H at x = x 1 if x 1 + b = 1 2 x (1+x1) 2. Eqm. value of x 1 = 1 2 2b, so a two-message eqm. exists iff b < 1 4. The interval of values for which A reports H is larger than the interval for which he reports L. Cheap Talk Models 5 / 27

6 Example: PBE with two distinct messages: m = L if x [0, x 1 ) and m = H if x [x 1, 1] 0 L x 1 = 1 2 2b H 1 The interval of values for which A reports H is larger than the interval for which he reports L. Even given the restriction that A makes only two reports, P would be better off if A used the cutoff x 1 = 1 2. (A would also be better off (ex ante) if he could commit to using this cutoff.) But such a reporting rule would not constitute a PBE, because if P expected A to use it, A would strictly prefer to deviate: if A saw x = 1 2, he would strictly prefer to report H. The greater coarseness of the H report is the endogenous cost borne by A (but also by P) which ensures that A reports H only for those states of the world where P expects him to do so. Same reasoning shows that in all eqa, partition elements increase in size as the state x increases. Cheap Talk Models 6 / 27

7 Properties of equilibria in Crawford-Sobel model (uniform-quadratic case) Given b, define n (b) as maximum number of intervals that can arise in eqm: partially pooling PBE s exist n n (b) n (b) in b more communication can occur when preferences more closely aligned lim b 0 n (b) = and lim b 0 P s expected loss (rel. to full infor.) = 0 b > 0, n (b) < implies that for all b > 0, full revelation is impossible in a PBE n (b) 1 b Welfare analysis: 1 Given b, P s and A s ex ante expected payoffs are both in n. 2 Given n, P s payoff is in b. In this analysis of cheap talk with a one-dimensional state space, the maximal amount of information communicated, n (b), and its value to the principal, both increase as the divergence in preferences (measured by the distance between P s and A s ideal points) decreases. Cheap Talk Models 7 / 27

8 Communication vs. Delegation Dessein (REStud, 2002) asks whether a principal in the Crawford-Sobel environment might prefer (assuming it were possible) to delegate the decision to the agent, rather than communicate with him. Under delegation, A chooses, for all x [0, 1], his own ideal point, that is, V = x + b. Thus for all x, A s choice is exactly b units away from P s full-infor. optimum. Clearly, for b very large, P prefers, rather than delegate, to choose on the basis of only his prior information. And for b very large, no communication can occur in eqm. On the other hand, as b 0, both communication and delegation approach the full-infor. outcome. Dessein shows that whenever preferences are sufficiently aligned that some communication can occur in eqm. (b < 1 4 ), delegation yields higher expected payoff for P. Cheap Talk Models 8 / 27

9 Communication vs. Delegation Dessein shows that whenever preferences are sufficiently aligned that some communication can occur in eqm. (b < 1 4 ), delegation yields higher expected payoff for P. Eqm with communication yields a partition of [0, 1] in which partition elements vary in size, increasing with x. This variation in size is per se harmful to P: Although n (b) increases as b decreases, the variation in size limits the rate of increase of n (b). Defining Ā(b) 1 n (b) eqm), Dessein shows as the average interval length (in the most informative lim b 0 Ā(b) b =. Implication: As b 0, communication performs infinitely badly relative to delegation. Cheap Talk Models 9 / 27

10 Communication with Multiple Experts Consider first models with two agents, both of whom perfectly observe the one-dimensional state: e.g. Gilligan and Krehbiel (AJPS, 89), Krishna and Morgan (QJE, 01 and APSR, 01), Battaglini (Etrica, 02) By playing off the agents against each other, responding harshly to any discrepancy in their reports, P may be able to achieve his full-infor. optimum. Battaglini, for ex., considers the case where the agents biases have opposing signs, and the agents report simultaneously. A necessary and sufficient condition for existence of a fully revealing eqm is that the sum of the magnitudes of the agents biases not be too large relative to the support of the unknown state. As in the Crawford-Sobel model, the magnitudes of preference divergences (btw. P and A s) are the crucial determinant of how much information is communicated in eqm. Cheap Talk Models 10 / 27

11 Communication with 2 experts, who both observe 1-dimensional state By playing off the agents against each other, responding harshly to any discrepancy in their reports, P may be able to achieve his full-infor. optimum. But Battaglini argues that in many cases, the fully revealing eqa. are not plausible (not robust in a precise sense), because P s out-of-equilibrium beliefs are such that very small discrepancies in agents reports result in P choosing extreme punishments. Small discrepancies in reports might be due to unanticipated mistakes by A s, so P might want to respond by choosing an action close to both reports. Cheap Talk Models 11 / 27

12 Communication with 2 experts, multidimensional state/action space Battaglini shows that the possibility for full revelation of the state in a robust PBE is significantly greater when the state space and action space has two (or more) dimensions. Two agents, both of whom perfectly observe the state x (x 1, x 2 ) and report simultaneously to P P s action is V (V 1, V 2 ) R 2 u P (V, x) = (V 1 x 1 ) 2 (V 2 x 2 ) 2, so P s ideal is V P (x) = x u i (V, x) = (V 1 (x 1 + b i 1 ))2 (V 2 (x 2 + b i 2 ))2, for i = 1, 2, so A i s ideal is V i (x) = x + b i The vectors b 1 = (b1 1, b1 2 ) and b2 = (b1 2, b2 2 ) represent how much, and in which direction, the agents preferences diverge from the principal s. Battaglini shows that whenever the vectors b 1 and b 2 are linearly independent, and thus regardless of their magnitudes, there exists a fully revealing, robust equilibrium. Cheap Talk Models 12 / 27

13 Communication with 2 experts, multidimensional state/action space Battaglini shows that whenever the vectors b 1 and b 2 are linearly independent, and thus regardless of their magnitudes, there exists a fully revealing, robust equilibrium. The result is most easily understood in the special case where b 1 and b 2 are orthogonal to each other: Let x i represent the efficient level of investment in division i and V i represent the actual level of investment in division i. Assume that there is no upper limit on total investment. Let b 1 = (k, 0) and b 2 = (0, m), where k and m are constants. Ai s prefs. differ from P s only w.r.t. the i th dimension, i.e. the choice of V i. There is a fully revealing eqm in which P asks agent i how much he should invest in division j, i = 1, 2, j i. In this special case with b 1 and b 2 orthogonal, it is a dominant strategy for each agent to tell the truth. Cheap Talk Models 13 / 27

14 Battaglini s eqm construction when b 1 and b 2 are orthogonal: P asks agent each agent how much he should invest in other division, and it is a dominant strategy for each to tell the truth. V 2, x 2 A 2 s report in PBE and DSE A 1 s report in PBE and DSE b 2 b 1 V 1, x 1 Cheap Talk Models 14 / 27

15 Battaglini s eqm construction in general: Even for b 1, b 2 not orthogonal, P can still achieve full revelation as a Bayesian Nash eqm. Essentially, P limits each agent s influence to a one-dimensional subspace over which the preferences of P and that agent do not diverge. V 2, x 2 A 2 s report in PBE b 2 A 1 s report in PBE b 1 V 1, x 1 Message from Battaglini s analysis: When state and action space are multidimensional, fully revealing eqa exist as long as there is some difference (however small) in the directions of the agents biases; magnitudes of the biases are not relevant. Cheap Talk Models 15 / 27

16 But assumptions underlying Battaglini s (2002) analysis are very strong: 1 Both agents are perfectly informed about both components of the state. In investment ex., Ai might be more informed about own division. 2 There are no constraints on the action space that force a tradeoff between the two dimensions. In investment ex., a budget constraint might limit total investment. 3 The principal is perfectly informed about the directions of the agents bias vectors. Battaglini s construction fails if P is even slightly wrong about the directions of these biases. Battaglini (Adv. Theor. Econ., 2004) relaxes Assn. 1 and shows: Full revelation of agents information never arises in eqm. But eqm. outcomes approach the full-information benchmark as i) the noise in the agents signals disappears or as ii) the number of agents increases. Cheap Talk Models 16 / 27

17 Bounds on set of feasible actions: When does a FRE exist? Ambrus + Takahashi (TE, 2008) and Meyer, Moreno, + Nafziger (W.P., 2013) relax Assn. 2, analyzing when bounds on the action space prevent existence of a fully revealing eqm (FRE). Suppose support of distrib. of state x is R 2, but P s action, V, must lie in Y R 2. In a fully revealing eqm (FRE), for each state x, P learns from the agents his best feasible action. Assume now that each agent makes a report about all dimensions of the best feasible action for P. It is wlog to focus on truthful FRE, i.e. eqa in which each agent reports P s best feasible action truthfully. In a truthful FRE, if agents reports agree, P chooses the action matching their reports. Cheap Talk Models 17 / 27

18 Bounds on set of feasible actions: When does a FRE exist? Key question is: For each possible pair of discrepant reports by the agents, (x 1, x 2 ) with x 1 x 2, is there a feasible punishment V for P that would deter each A from deviating so as to generate this pair of reports? That is, does there exist V (x 1, x 2 ) Y such that, 1 in state x 1, A 2 weakly prefers to report x 1 (thus inducing P to choose x 1 ) to deviating to reporting x 2 (thus inducing P to choose V (x 1, x 2 )); and 2 in state x 2, A 1 weakly prefers to report x 2 (thus inducing P to choose x 2 ) to deviating to reporting x 1 (thus inducing P to choose V (x 1, x 2 ))? Note that, given x 1 x 2, P knows that a deviation has occurred but does not know which A was responsible, so same punishment V must work for both cases. Cheap Talk Models 18 / 27

19 Bounds on set of feasible actions: When does a FRE exist? Meyer, Moreno, and Nafziger (2013) restrict out-of-eqm. beliefs so small deviations from eqm. (or small mistakes) cause only small punishments, i.e. they focus on robust FRE. V (x, x ) b 2 V (x, x ) b 1 x x x Y B(x, ɛ) Cheap Talk Models 19 / 27

20 Bounds on set of feasible actions: When does a FRE exist? Meyer et al seek robust FRE that exist independently of the magnitudes of the biases. For given, finite magnitude of bias vector, an agent s indifference curves are circles. As magnitude of bias vector becomes arbitrarily large, an agent s indifference curve through a given point approaches a straight line. x x tb x + b b x Cheap Talk Models 20 / 27

21 Bounds on set of feasible actions: When does a FRE exist? Proposition (Meyer et al): Assume that the set Y of feasible actions is convex. When biases can be arbitrarily large, there exists a robust FRE whenever a FRE exists. That is, when punishments must work no matter how large the biases, then for any small deviation, a small feasible punishment exists whenever a large feasible punishment does. V (x, x ) V Y b 2 b 1 x x x Cheap Talk Models 21 / 27

22 When does a robust FRE exist when set of feasible actions Y R 2? We seek a simple, local geometric condition that is necessary and sufficient. Fr(Y ) frontier (boundary) of Y and n Y (x) inward-pointing normal vector to Fr(Y ) at a smooth point x Fr(Y ). Given b 1, b 2 linearly indep., C(b 1, b 2 ) open convex cone spanned by b 1, b 2. Proposition (Meyer et al): Given Y R 2 closed and convex and b 1, b 2 R 2 linearly independent, the following statements are equivalent: (i) There exists a robust FRE for biases (b 1, b 2 ). (ii) There exists a (robust) FRE for all biases (t 1 b 1, t 2 b 2 ) with t 1, t 2 0. (iii) For every smooth point x Fr(Y ), n Y (x) / C(b 1, b 2 ). Y b 2 n b 1 n b 2 b 1 Cheap Talk Models 22 / 27

23 Illustration: allocating goods vs. allocating bads Allocating goods subject to upper limit on total: For every smooth point x Fr(Y ), n Y (x) / C(b 1, b 2 ), so a robust FRE independent of the magnitudes of the biases. Y x b 2 b 1 n x V x (x, x ) n b 2 b 1 x V (x, x ) Cheap Talk Models 23 / 27

24 Illustration: allocating goods vs. allocating bads Allocating bads subject upper limit on total: x Fr(Y ) s.t. n Y (x) C(b 1, b 2 ), so a FRE. Y x b n 2 b 1 V (x, x ) x Cheap Talk Models 24 / 27

25 Recall the strong assumptions underlying Battaglini s (2002) analysis: 1 Both agents are perfectly informed about both components of the state. 2 There are no constraints on the action space that force a tradeoff between the two dimensions. 3 The principal is perfectly informed about the directions of the agents bias vectors. Meyer et al s approach, in contrast to Battaglini s, can easily handle uncertainty on the principal s part about the directions of the agents biases: what matters for existence of a robust FRE are the least aligned possible realizations of the biases, (b, b) Y x b b b 2 b 1 x x {b,b} x x {b 1,b 2 } x Cheap Talk Models 25 / 27

26 Dynamic Models of Communication with Self-Interested Experts Suppose that self-interested, privately informed experts are consulted repeatedly. Does concern for their reputation help to discipline such experts? Applications: policy advisors; financial market insiders; managers; referees for job candidates Usually studied with models in which decision-makers learn about experts intrinsic characteristics. What characteristics of experts might decision-makers want to learn about? How closely their preferences are aligned with those of decision-maker Sobel, REStud, 85; Benabou + Laroque, QJE, 92; Morris, JPE, 01; Ely + Valimaki, QJE, 03; Avery + Meyer, W.P., 12 Quality of their information Scharfstein + Stein, AER, 90; Prendergast + Stole, JPE, 96; Ottaviani + Sorensen, JET, 06, and Rand, 06; Prat, AER, 05; Grubb, JEMS, 11 Cheap Talk Models 26 / 27

27 Dynamic Models of Communication with Self-Interested Experts On what information are experts reputations based? Is more information necessarily better for disciplining the experts? No. Prat shows that an expert s incentives can be worse when both his action and its outcome are observed, compared to when only its outcome is observed. Morris, Ely+Valimaki, and Avery+Meyer show that an expert s incentives, and the decision-maker s welfare, can be worse when the decision-maker can observe the expert s previous recommendations as well as his current one, compared to when only the current recommendation can be observed. Cheap Talk Models 27 / 27