Andriy Zapechelnyuk

University of Pennsylvania†

Queen Mary, University of London‡

June 3, 2012

Abstract We provide a sufficient condition under which an uniformed principal can infer any information that is common knowledge among two experts, regardless of the structure of the parties’ beliefs. The condition requires that the bias of each expert is less than the radius of the smallest ball containing the action space. If the principal does not know the structure of the experts’ biases and is concerned with the worstcase scenario, this condition is also necessary. If this condition is satisfied, there exists an ex-post incentive compatible decision rule that implements an optimal action for the principal conditional on the smallest event commonly known by the experts. This rule is simple: it is fully determined by the action space and is independent of other details of the environment. JEL classification: D74, D82 Keywords: Communication, multidimensional cheap talk, multi-sender cheap talk, principal-expert model, common knowledge, robustness.

∗

The authors would like to express their gratitude to Attila Ambrus for very helpful comments. Email : [email protected] ‡ Corresponding author: School of Economics and Finance, Queen Mary, University of London, Mile End Road, London E1 4NS, UK. E-mail: [email protected]. †

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Introduction

In this note, we ask under what conditions an uniformed principal can infer any information that is common knowledge among two experts. We allow for general decision rules that select an action contingent on the agents’ reports and show that there exists a decision rule which makes it optimal for the experts to reveal the commonly known information if the supremum of each agent’s bias relative to the principal’s optimal action is less than the radius of the smallest ball that contains the action space. This condition becomes necessary if the principal does not know the structure of the experts’ biases and is concerned with the worst-case scenario.1 The rule constructed in this paper is simple and depends only on the boundaries of the action space, but is independent of other details of the environment, such as beliefs, the direction of biases, and the relationship between the biases and the agents’ information. It implements an optimal action for the decision maker given the commonly known event by the experts if their reports about this event agree and implements a constant stochastic action that is independent of the nature of disagreement otherwise. If the experts commonly know the optimal action for the principal, this rule is first best and implements the optimal action in each state. This note is related to the cheap-talk literature with two experts who commonly know the principal’s optimal action that has focused on establishing conditions under which the decision maker can achieve the first best (Krishna and Morgan (2001b), Battaglini (2002), Ambrus and Takahashi (2008)).2 In cheap talk, the decision maker has no commitment power and must take an action that is sequentially rational given her beliefs. In this paper, there is no such restriction. Thus, our full commitment environment serves as a natural benchmark for the various modifications of environments studied in the existing literature. Naturally, our condition for the first best under full commitment is related to but weaker than those in cheap talk environments (Krishna and Morgan (2001a), Battaglini (2002), and Ambrus and Takahashi (2008)).3 The key difference is that our condition 1

Frankel (2011) studies an optimal robust decision rule in environments with one expert, where the principal has limited information about the agent’s preferences. 2 Crawford and Sobel (1982) is the seminal reference on cheap talk communication with one expert. For models of cheap talk communication with two experts see also Gilligan and Krehbiel (1989), Krishna and Morgan (2001a), Battaglini (2004), Levy and Razin (2007), Ambrus and Lu (2010), Li (2008, 2010), See also Li and Suen (2009) for a survey of work on decision making in committees. 3 In a seminal paper, Battaglini (2002) demonstrates in generic cheap talk environments with unbounded multidimensional action spaces and common knowledge of state among the agents, there exists a fully revealing equilibrium that implements the first best. Clearly, the first best is also implementable

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bounds each agent’s bias independently, whereas this is not so in cheap talk. For instance, if the state space is unidimensional, Proposition 1 in Battaglini (2002) establishes that a necessary and sufficient condition for a fully revealing cheap talk equilibrium is that the sum of the absolute values of the agents’ biases is less than half of the measure of the action space. Under full commitment, the first best is implementable if and only if each of the agent’s biases, rather than their sum, is bounded by that value. In cheap talk, the inability of the decision maker to commit to a decision rule causes punishments in fully revealing equilibria to depend non-trivially on the experts’ reports (Krishna and Morgan 2001a, Krishna and Morgan 2001b, Battaglini 2002, Ambrus and Takahashi 2008).4 In the decision rule in this paper, the punishment for disagreement is independent of the agents’ reports and is fully determined by the boundaries of the action space available to the decision maker. In that sense, the optimal rule is simple and constant on a large set of environments. The rule in this paper is ex-post incentive compatible and provides a lower bound on the decision maker’s payoff in an optimal decision rule. Full investigation of optimal decision rules in Bayesian environments with noise is not the focus of this note.5 However, the construction presented here can be made robust to small mistakes in experts’ observations, in the sense of satisfying the property of “diagonal continuity” of Ambrus and Takahashi (2008). This paper is a natural continuation of the work on optimal delegation to one agent that was initiated by Holmstr¨om (1977, 1984) for unidimensional action spaces6 and that if the decision maker has more commitment power. In this note, it is assumed that the action space is bounded; our condition is trivially satisfied for unbounded action spaces. 4 Furthermore, as pointed out by Battaglini (2002), these cheap talk equilibria contain implausible outof-equilibrium beliefs. By contrast, there is no issue of out-of-equilibrium beliefs in our model because the decision maker can commit to her actions. 5 Enviroments with noise have been considered in Austen-Smith (1993), Wolinsky (2002), Battaglini (2004), Levy and Razin (2007), and Ambrus and Lu (2010). In particular, in a model with multiple agents, a multidimensional environment, and noisy signals, Battaglini (2004) shows that minimal commitment power is sufficient for the first best outcome to become feasible in the limit as the number of agents increases. Ambrus and Lu (2010) construct fully revealing equilibria that are robust to a small amount of noise in environments in which the state space is sufficiently large relative to the size of the agents’ biases. 6 See Holmstr¨ om (1977, 1984), Melumad and Shibano (1991), Dessein (2002), Martimort and Semenov (2006), Alonso and Matouschek (2008), Martimort and Semenov (2008), Goltsman, H¨orner, Pavlov and Squintani (2009), Kovac and Mylovanov (2009), Amador and Bagwell (2010). Armstrong and Vickers (2008), Koessler and Martimort (2012), Li and Li (2009), and Lim (2009) who study optimal decision rules in environments which are related, but not identical to the model of Holmstr¨om (1977, 1984). Optimal decision rules for environments in which a decision maker can commit to monetary payments are characterized in Baron (2000), Krishna and Morgan (2008), Bester and Kr¨ahmer (2008), Raith (2008), and Ambrus and Egorov (2009).

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was recently studied in multidimensional action spaces by Koessler and Martimort (2012) and Frankel (2011, 2012). The problem of optimal decision rules for two experts has also been studied in Martimort and Semenov (2008). They focus on experts who are biased in the same direction and consider dominant strategy implementation. By contrast with our results, Martimort and Semenov (2008) demonstrate impossibility of the first best outcome and show that a sufficiently high bias renders the experts not valuable for the decision maker. The optimal delegation rules that are robust to details of the environments are studied in Frankel (2011); the principal is assumed to know the distribution of states but has limited information about the agent’s preferences. In this paper, incentives for the experts to report their commonly known information truthfully are provided by punishment of disagreements by a stochastic action. This feature of design, together with the assumption of common knowledge of the principal’s optimal action by the experts, also appears in Mylovanov and Zapechelnyuk (2012) and Zapechelnyuk (2012). Mylovanov and Zapechelnyuk’s (2012) model is limited to a unidimensional action space and deals with the case of large and opposing biases, where the first best implementation is generally impossible. The model of Zapechelnyuk (2012) allows multiple experts to collude in their information disclosure strategies, which leads to a very different result that the first best is achievable if and only if the principal’s optimal action is Pareto undominated for the experts. This condition is generally violated in the environment of the current paper, with two experts and multidimensional action space.

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The Model

There are two informed experts, i = 1, 2, and an uninformed principal. The set of actions available to the principal is Y , a compact subset of Rd , d ≥ 1. The principal’s optimal action x, called state, belongs to a set of states X ⊂ Y . Each expert i = 1, 2 is endowed with an information partition Pi of X and knows that x belongs to Xi ∈ Pi . Denote by ˆ the smallest subset of X that is common knowledge for the experts.7 X Let || · || denote a norm on Rd . Let rY be the radius of the smallest ball that contains Y and without loss of generality set the center of this ball to be at the origin. Expert i’s most preferred action at state x is given by x+bi (x), where bi (x) is i’s bias.8 Each expert minimizes his loss function given by a convex transformation of the squared 7 8

ˆ is the element of the meet P1 ∧ P2 that contains x. That is, X We do not assume that x + bi (x) ∈ Y .

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distance between action y ∈ Y and i’s most preferred action: Li (x, y) = hi ky − (x + bi (x))k2 , i = 1, 2, where hi : R+ → R+ is a weakly convex function with hi (0) = 0. Let ∆(Y ) denote the set of probability measures on Y (stochastic actions). ConvenR tionally, we extend the definition of Li to X ×∆(Y ) by Li (x, λ) = Y Li (x, y)λ(dy), x ∈ X, λ ∈ ∆(Y ). A decision rule is a measurable function ˆ1, X ˆ 2 ) 7→ µ(X ˆ1, X ˆ 2 ), µ : X 2 → ∆(Y ), (X where X = P1 ∧ P2 is the set of subsets of X that can be commonly known by the ˆ1, X ˆ 2 ) is a stochastic action that is contingent on the experts’ reports experts and µ(X ˆ1, X ˆ 2 ) ∈ X 2 . A decision rule induces a game, in which the experts simultaneously (X ˆ1, X ˆ 2 ∈ X and the outcome µ(X ˆ1, X ˆ 2 ) is implemented. make reports X ˆ ∈ X, Decision rule µ is common-knowledge-revealing if for all X ˆ X) ˆ is a stochastic action with support on X, ˆ and (C1 ) µ(X, ˆ1, X ˆ 2 )) for all x, y ∈ X ˆ and all X ˆ1, X ˆ 2 ∈ X with X ˆ1 = ˆ2. (C2 ) Li (x, y) ≤ Li (x, µ(X 6 X Condition (C1 ) requires that if the experts’ reports agree, the decision rule implements an action in the reported set, while condition (C2 ) requires that each expert prefers any action in the commonly known set to any action that can possibly be implemented if the experts’ reports disagree. These conditions imply that truthtelling is ex-post equilibrium in the common-knowledge-revealing rule if the agents know the rule. Furthermore, truthtelling is optimal even if the agents do not know the rule but know that it will not implement actions outside of the commonly known set. Our objective is to construct a common-knowledge-revealing rule. As an example, consider action set Y illustrated by Fig. 1, with the grid representing the experts’ common knowledge partition X (we assume X = Y ). If state x is as shown on Fig. 1, then ˆ In this it is common knowledge for the experts that x is in the shaded rectangle, X. environment, how can the principal motivate the experts to reveal their common knowledge at every possible state? ˆ ∈ X and µ∗ (X ˆ1, X ˆ 2 ) = λ∗ Consider decision rule µ∗ that satisfies (C1 ) for all X ˆ 1 6= X ˆ 2 , where stochastic action λ∗ is a solution of: whenever X Z

||y|| λ(dy),

max λ∈∆(Y )

Z

2

Y

s.t.

yλ(dy) = 0. Y

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(1)

B

b2 (x)

A

x

C

b1 (x) D Fig. 1.

That is, any disagreement between the experts results into a stochastic action that has the maximum variance among all random variables on Y with zero expectation. To construct λ∗ geometrically, one draws the smallest ball that contains Y and assigns positive weights on extreme actions in Y that lie on the boundary of this ball, such that the weighted average of those actions is at the center of the ball. So, for the example on Fig. 1, one will assign positive weights on vertices A, B, and C (note that vertex D is strictly inside the smallest ball, so it is assigned zero weight). Theorem 1 Decision rule µ∗ is common-knowledge-revealing if ||bi (x)|| ≤ rY for all x ∈ X and i = 1, 2. ˆ ∈ X . For all x ∈ X, all y ∈ Y , and each i = 1, 2, Proof. We need to prove (C2 ) for all X by monotonicity9 of hi we have Li (x, y) = hi ||y − (x + bi (x))||2 ≤ hi ||y||2 + ||x + bi (x)||2 . Next, by assumption, the smallest ball that contains Y is centered at the origin and has R radius rY . Consequently, Li (x, y) ≤ hi (rY2 + ||x + bi (x)||2 ). Also, Y ||z||2 λ∗ (dz) = rY2 , since λ∗ that solves (1) must assign positive mass only on subsets of Y that have distance rY from the origin (in the example depicted on Fig. 1, λ∗ will assign positive mass only 9

The assumptions on hi imply that it is increasing.

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R on vertices A and B and C). Also, by (1), Y zλ∗ (dz) = 0. Using the above and Jensen’s inequality, we obtain for all x ∈ X, all y ∈ Y , and each i = 1, 2 ∗

Z

2

Z

∗

2 ∗

hi ||z − (x + bi (x))|| λ (dz) ≥ hi ||z − (x + bi (x))|| λ (dz) Y Z n o ∗ 2 2 = hi ||z|| − 2z · (x + bi (x)) + ||x + bi (x)|| λ (dz) Y Z Z 2 ∗ ∗ 2 = hi ||z|| λ (dz) − 2(x + bi (x)) · zλ (dz) + ||x + bi (x)||

Li (x, λ ) =

Y

Y

= hi

rY2

Y 2

+ ||x + bi (x)||

≥ Li (x, y),

and (C2 ) follows immediately. Note that common-knowledge-revealing rule µ∗ is robust to details of the preferences and the information structure of the experts. The only information relevant for construction of µ∗ is the principal’s set of actions Y and the upper bounds on the experts’ biases. The stochastic punishment action λ∗ is constant with respect to state space X ⊂ Y , information structure P1 and P2 of the experts, as well as parameters of their preferences, hi and bi , i = 1, 2; to verify the sufficient condition in Theorem 1 one needs to know only the radius of Y and the upper bounds on the experts’ biases. If the principal does not know the biases of the agents, but nevertheless would like the ensure that a common-knowledge-revealing rule exists, the condition in Theorem 1 becomes necessary. Theorem 2 Assume that the agents’ biases are constant. Then, there exists a pair of directions of biases such that a common-knowledge-revealing rule exists if and only if ||bi || ≤ rY for each i = 1, 2. Proof. We show that for any action set Y , we can find directions of the biases such that if ||bi || > rY for some i = 1, 2, then a common-knowledge-revealing rule does not exist. As illustrated by Fig. 2, consider action xˆ in the support of stochastic action λ∗ that x and ||b1 || > rX . Observe that, since satisfies (1) and assume that b1 satisfies ||bb11 || = −ˆ ||b1 || > rX , the set of actions that expert 1 prefers to action y = xˆ is inside the larger dashed circle, so she strictly prefers every action in Y \{ˆ x} to xˆ. Hence, to provide the ˆ incentive for 1 to report the element X of X that is commonly known by the agents in xˆ ˆ 1 , X) ˆ = xˆ for all X ˆ 1 ∈ X . If X ˆ is not a singleton, (C2 ) is truthfully, one must choose µ(X ˆ and x 6= xˆ. Let now X ˆ = {ˆ violated for any x ∈ X x}. Then, however, in any state expert ˆ2 = X ˆ and obtain action y = xˆ. Provided the directions of the biases are 2 can report X 6

b2 b1 x′

x ˆ

Fig. 2.

different enough, there exists x0 6= xˆ such that expert 2 (whose set of actions preferred to x0 is depicted by the smaller dashed circle) strictly prefers xˆ to x0 . Consequently, (C2 ) is violated for expert 2 at x0 . An implication of this theorem that in the environments in some natural environments in which the experts commonly know the optimal action for the principal, the bound on the experts’ biases becomes a necessary and sufficient condition for existence of a decision rule that implements the principal’s optimal action in each state. In particular, assume that the set of actions Y is the ball with radius rY , X = Y , and the experts commonly know the optimal action for the principal, that is, Pi = X, i = 1, 2. If biases are state-independent and not collinear, i.e., bi (x) ≡ bi , and ||bb11 || 6= ||bb22 || , then, a commonknowledge-revealing rule exists if and only if ||bi || ≤ rY for each i = 1, 2. (The proof of this result is essentially identical to that of Theorem 2.)10 If the experts’ information partitions are not identical, there may exist decision rules that extract more information from the experts than their common knowledge. Hence, given the principal’s preferences and beliefs about x, the common-knowledge-revealing rule imposes the lower bound on the principal’s expected payoff that can be possibly achieved. The revelation principle justifies our focus on truthtelling equilibria. Nevertheless, 10

Mylovanov and Zapechelnyuk (2010) proves this result for a unidimensional action space.

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common-knowledge-revealing rules typically permit multiple equilibria, providing the experts with an incentive to collude on other strategies. Zapechelnyuk (2012) addresses the question of collusive behavior of the experts, who commonly know the state and are able to make binding agreements, and shows that the principal can implement her most preferred action if and only if the principal’s optimal action is Pareto undominated for the experts. In this paper, this condition holds only if the expert’s biases are exactly opposing and is generically violated in multi-dimensional action spaces.

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