Optimal Signals in Bayesian Persuasion Mechanisms with Ranking

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1 Optimal Signals in Bayesian Persuasion Mechanisms with Ranking Maxim Ivanov y McMaster University August 2015 (First version: October 2013) Abstract This paper develops a novel approach based on the majorization theory and positive-dependence stochastic orders to solving Bayesian persuasion mechanisms in which an action of the receiver (the mechanism) depends on the posterior mean of an increasing utility function of the state and the order of this mean in the sequence of possible means. For such mechanisms, we provide necessary and su cient conditions of the optimality of monotone partitional signal structures. Our characterization results provide solutions to a variety of Bayesian persuasion applications: auctions, optimal selling mechanisms, monopoly with xed and endogenous prices, and principal-agent model of delegation with information acquisition. JEL classi cation: C72, D81, D82, D83 Keywords: information; persuasion; stochastic orders; positive dependence; majorization; mechanism design 1 Introduction The main focus of this paper is to develop a tractable approach to derive optimal signal structures, i.e., transformations of the prior information (the state) into the observed information (the signal), in Bayesian persuasion mechanisms. These mechanisms include three players, an the sender, the mediator, and the receiver whose preferences depend on an unknown state and the receiver s decision. The players information about the state is I am very grateful to Xianwen Shi for an insightful discussion which inspired this work. I am also thankful to Ricardo Alonso, Sourav Bhattacharya, Seungjin Han, Sergei Izmalkov, Emir Kamenica, René Kirkegaard, Nicolas Klein, Vijay Krishna, Hao Li, Tymo y Mylovanov, Michael Peters, Je rey Racine, Sergei Severinov, Joel Sobel, and audiences of CETC 2014, University of Western Ontario, University of British Columbia, and University of Pittsburgh for helpful comments and questions. All mistakes are mine. The paper has been previously circulated under the title Partitional Signals in Persuasion. y Department of Economics, McMaster University, 1280 Main Street Hamilton, ON, Canada L8S 4M4. mivanov@mcmaster.ca. Phone: (905) x Fax: (905)

2 initially limited by a common prior distribution. At the beginning of the game, the sender publicly selects a signal structure. Then, the mediator privately observes a realization of the signal generated by the signal structure. Given this information, the mediator updates his posterior belief about the state by the Bayes rule and sends a message to the receiver. The receiver also updates his posterior belief by the Bayes rule and makes a decision, which a ects payo s of all players. The main focus of this paper is solving the problem of the sender, which is selecting an ex-ante optimal signal structure, i.e., the one that maximizes her ex-ante payo given strategies of the mediator and the receiver. We consider scenarios in which the mediator and the receiver can be either strategic or non-strategic players. For example, if the mediator (non-strategically) commits to transmit all obtained information to the receiver truthfully, then such a setup is equivalent to the model of Bayesian persuasion by Kamenica and Gentzkow (2011). If the receiver commits to make a decision as a function of the posterior beliefs and the signal realization, this scenario corresponds to the mechanism design with imperfectly informed players. Examples of such setups are the optimal selling mechanism by Bergemann and Pesendorfer (2007) or Holmström s (1977) model of delegation with an imperfectly informed sender. In this broad class of mechanisms, we restrict attention to setups in which the sender s ex-ante payo can be decomposed as an average of an ordered sequence of interim-payo functions indexed by signal realizations, which depend on: 1) the posterior mean value of an increasing function of the state (hereafter, a posterior mean); and 2) the s lowest rank of the posterior mean in the generated sequence of means. 1 This rank has also a meaning of a signal realization. As we show below, this speci cation includes a variety of economic applications, including the aforementioned works. The main results of the paper are as follows. First, we show that for a sequence of posterior means generated by an arbitrary signal structure there exists a monotone partitional signal structure that generates a sequence of posterior means, which dominates the original sequence by the weighted majorization (or g majorization). 2 The majorization is a special case of the convex stochastic order, or the second-order stochastic dominance, such that the elements of the majorizing sequence are more dispersed than those of the original sequence while the weights (i.e., probabilities) of associated elements of both sequences are identical. Thus, the ranking of distributions of posterior means by the weighted majorization preserves the ranking of sender s ex-ante payo s if the ex-ante payo is a weighted Schur convex function of the distribution of posterior means. In the light of our rst result, this means that if the sender s ex-ante payo is weighted Schur convex, then monotone partitional signal structures provide the maximum ex-ante payo among all signal structures. Second, we characterize su cient conditions on interim payo functions under which 1 In general, the sender s interim payo can be a function of other characteristics of a posterior belief. Under some conditions, however, the ex-ante payo aggregates these characteristics into the distribution over posterior means, which solely determines the sender s ex-ante payo. 2 The proof of this result combines three results from the statistical literature. First, among all signal structures with xed marginal distributions, monotone partitions generate signals which are most positively dependent on states by the positive-quadrant dependent (PQD) stochastic order. Second, we apply the result by Tchen (1980) who shows that the average of posterior means given that the signal is below some cuto is lower for signal structures which are dominant by the PQD order. Finally, for discrete distributions of posterior means, this result implies the ranking of distributions of posterior means by the weighted majorization. 2

3 sender s ex-ante payo is weighted Schur convex. In particular, the necessary condition for the weighted Schur convexity is the supermodularity of the interim payo across signals and posterior means. At the same time, Schur convexity does not require the convexity of interim payo s in posterior means. This result highlights the crucial di erence between Bayesian persuasion games studied by Kamenica and Gentzkow (2011) and Bayesian persuasion mechanisms. In particular, in Bayesian persuasion games the sender s interim payo is a function of the posterior belief only, but not the signal realization that induces the belief. 3 If this payo depends on the posterior mean of the belief only, the sender s ex-ante payo is determined as an average of a single interim-payo function over posterior means. It is Schur convex if and only if the interim payo is convex in posterior means. In contrast, in Bayesian persuasion mechanisms the ex-ante payo is an average of the sequence of interim-payo functions of posterior means ordered by signal realizations. In this case, the sender s ex-ante payo can be Schur convex if interim payo s are concave in posterior means for some or even all signals. Third, we investigate the optimality of monotone partitional signal structures for interim payo functions which are linear in posterior means but di erent across signals. This issue is not trivial because the supermodularity condition can be violated, which is necessary for the optimality of monotone partitions. As we show, however, there are optimal signal structure of the monotone partitional form, but they might include redundant signals, i.e., ones which induce identical posterior beliefs. Finally, we apply the above results to several economic models. These are: the principal-agent model of delegation with information acquisition, auctions with imperfectly informed bidders, the optimal selling mechanisms, and provision of information by the monopolist with xed and endogenous prices. In the model of auction with private values and nite number of signals, we sharpen results by Ganuza and Penalva (2010). In particular, we demonstrate the optimality of monotone partitional signal structures if the number of bidders is su ciently large. Also, we show that the expected valuation of the winning bidder is maximized by a monotone partitional signal structure. For the optimal single-object selling mechanism by Bergemann and Pesendorfer (2007), we show that the main result of their paper (Theorem 1) about the optimality of monotone partitions is solely based on the weighted majorization of distributions of posterior means. Moreover, our results show that ndings by Bergemann and Pesendorfer (2007) and Ganuza and Penalva (2010) are based on the same statistical tools. Next, we consider the model of provision of information by a monopolist. We investigate both cases of an exogenous and an endogenous prices and show that the optimal signal structure is monotone partitional. Finally, we demonstrate the optimality of monotone partitional signals in the model of optimal delegation by Holmström (1977) and Alonso and Matouschek (2008) with covert information acquisition by the agent (the sender). Monotone partitions have several natural economic interpretations. First, if we consider a signal structure as a mechanism of transforming the actual unobserved information (the state) into an observed information (the signal), then the monotone partitional character of a signal structure means that the mechanism is deterministic and 3 Kamenica and Gentzkow (2011) focus on equilibria in which the receiver s action is solely determined by his posterior belief. In Bayesian persuasion games, there could exist equilibria in which the action of the receiver depends on the signal realization also. However, these equilibria are rather speci c, since the dependence of the receiver s action on signal stems from his indi erence between actions. 3

4 monotone. That is, each state is mapped into a single signal s (), which is non-decreasing in state. Second, monotone partitions can be considered as a convex bundling of states into signals. That is, if two states are mapped into the same signal, all states between the two states are also mapped into the same signal. Finally, monotone partitional signal structures arise endogenously in various applications. A classical example are cheap talk games by Crawford and Sobel (1982), in which a perfectly informed expert uses only monotone partitional strategies for mapping states into signals sent to the receiver. Moreover, Ivanov (2010) shows that any equilibrium outcome in this game can be replicated in the game of communication with an imperfectly informed sender, which is a special case of Bayesian persuasion mechanisms. In Bayesian persuasion games, that is, setups in which the receiver cannot commit to actions and the mediator commits to convey the signal truthfully, Kamenica and Gentzkow (2011) o er a general approach, which treats the sender s problem as the problem of the constrained optimal decomposition of the prior distribution into a distribution over posterior beliefs. The optimal decomposition provides the ex-ante payo to the sender on the lowest concave envelope of the sender s interim payo (that is, the payo conditional on the receiver s posterior belief). 4 However, there are two main di culties with using this approach for nding the optimal distribution of posterior beliefs in Bayesian games. First, the space of distributions of posterior beliefs is not a simple object, especially if the prior distribution is continuous. In particular, it is a set of probability distributions over probability distributions over the state (posterior beliefs), which are interdependent due to the Bayes-plausibility constraint. As a result, optimizing over the space of Bayes-plausible posterior beliefs is a computationally di cult problem. Second, in many economic applications posterior beliefs contain redundant information. For example, a risk-neutral buyer (the receiver) is interested only in the posterior mean value of the product before making a decision about the purchase. Because the payo to the seller (the sender) depends only on the buyer s decision, all other information contained in posterior beliefs is irrelevant for players. However, the derivation of the distribution of posterior beliefs and hence, the signal structure from a distribution of posterior means is a complex problem. Due to these issues, the approach of Kamenica and Gentzkow (2011) does not always provide precise solutions of optimal signal structures. In addition, as noted above, Bayesian persuasion games do cover a variety of economic applications in which an action of the receiver depends not only on characteristics of the posterior belief, but also on its rank in the sequence of generated beliefs. Our paper o ers an alternative approach, which is qualitatively di erent from the one by Kamenica and Gentzkow (2011) and is applicable to Bayesian persuasion mechanisms in which an action of the receiver depends on both the posterior mean of an increasing utility function of the state and its lowest order in the sequence of possible means. Instead of optimizing over distributions of posterior beliefs directly, we start with the ranking of joint distributions of the state and the signal based on the positive-quadrant dependence of these variables. Then, we use this ranking in order to show that partitional signal structures achieve the upper bound in the set of generated posterior means ordered by the weighted majorization. Finally, these results imply that the optimal signal structures 4 This approach has been extended by Alonso and Camara (2013) to the setup with heterogeneous prior beliefs of the players, and by Gentzkow and Kamenica (2014) to the case of costly signal structures, where the cost is proportional to expected reduction in entropy. 4

5 are monotone partitional in the class of problems in which the sender s ex-ante payo is Schur-convex in posterior means. In this light, it is important to note that the ranking of distributions of posterior means holds not only for posterior means of states, but also for increasing utility functions of the state. This extension is important, because some common informativeness orders of distributions of posterior means are not invariant to monotone transformations of the state (Brandt et al., 2014). Regarding the economic literature, our approach attempts to generalize methods used in more structured models, which focus on characterizing optimal signal structures as mappings of the state into signals. For example, in the auction design with the endogenous precision of information released by the seller to bidders, Bergemann and Pesendorfer (2007) demonstrate the sub-optimality of non-monotone partitional signal structures in optimal actions by variations of probabilities across (adjacent) posterior means. A similar method has been used by Ivanov (2010) for a partial characterization of optimal signal structures in Crawford and Sobel s (1982) model of cheap-talk communication. In addition, Ivanov (2010) uses the majorization to prove the optimality of monotone partitions in special cases. Li and Shi (2013) use a convex order for the analysis of the optimal signal structure in dynamic price discrimination. In a recent paper, Li et al. (2015) investigate a persuasion model with a privately informed receiver such that the signal structure depends on a receiver s report about his private information. They show that the structure and the methods of the analysis of such discriminatory persuasion mechanisms are similar to those used in the literature on the optimal mechanism design by Myerson (1981). In this light, our approach generalizes the stochastic order approaches by Ivanov (2010) and Li and Shi (2013) and applies it to a broad class of Bayesian persuasion mechanisms. This is because the weighted majorization is a special case of the convex stochastic order, which excludes the dominance of sequences by the convex order with di erent weights of elements of the sequences. At the same time, considering the more restrictive stochastic order allows us to extend the class of sender s ex-ante problems to non-convex problems. The rest of the paper is structured as follows. Section 2 presents the formal model. Section 3 focuses on the ranking of signal structures. Section 5 analyzes the relationship between signal structures and sender s ex-ante payo s. Economic applications of our results are provided in Section 5. Section 6 concludes the paper. 2 Model 2.1 Information The state is distributed according to a given prior distribution F () with a density function 0 = f () and a support = ; R. The set of signals is exogenously determined by a nite signal set S n of size n. A signal structure is a collection of n integrable functions = f s ()g s2sn indexed by signals s 2 S n, where s () = Pr fsjg 2 [0; 1] ; s 2 S n is the probability of signal s conditional on state. Let n be the set of all of size n. 5 Also, if S n o S n, then f s ()g s2sn o 2 n if we put s () = 0; s =2 S n o. Then, 5 P Formally, n = ff s ()g s2sn : s () 0; s () = 1; 2 g. s2s n 5

6 the marginal probability of signal s is given by Z g s = Pr fsg = s () f () d: (1) A signal s such that g s > 0 induces the Bayesian posterior belief (the density function) s () = s () f () g s : (2) Thus, a signal structure 2 n generates a distribution over posterior beliefs fg s ; s g s2sn, which satis es the Bayesian-plausibility constraint E gs [ s ] = X s2s n g s s = 0 : (3) Denote P n the set of Bayes-plausible distributions over n posterior beliefs about. 6 In other words, determines the set of posterior beliefs = f s g s2sn and their probabilities g = fg s g s2sn, whereas a signal s randomly selects the belief s with the probability g s. Conversely, fg s ; s g s2sn 2 P n determines the underlying signal structure 2 n by (2). Posterior means. Given a signal s, which induces a posterior belief s, let! s be a posterior mean of a right-continuous increasing and integrable utility function (): Z! s = E [ () js] = () s () d: (4) Thus, a distribution of posterior beliefs fg; g 2 P n generates a distribution of posterior means fg;!g, where! = f! s g s2sn are given by (4). We call a distribution of posterior means fg;!g plausible if there exists fg; g 2 P n which induces it. Denote G n the set of plausible distributions fg;!g of size n (and, thus, of size n o n). 7 The plausibility of P fg;!g implies n g s = 1 and E gs [! s ] = E [ ()]. The converse, however, does not hold. 8 Monotone partitions. A special class of signal structures are monotone partitional ones. A signal structure o is monotone partitional if s o () = 1 s (), where f s g s2sn is a partition of into n non-overlapping non-degenerate intervals. Thus, o determines distributions fg o ; o g 2 P n and fg o ;! o g 2 G n such that o s = f (j 2 s ) ; gs o = Pr [ 2 s ] = F (sup s ) F (inf s ), and! s o = E [ () j 2 s ] ; s 2 S n. 9 Note that if () is increasing, then! o is increasing in the strong set order of f s g s2sn. Also, o 6 P That is, P n = ffg s ; s g s2sn : g s s = 0 g or, equivalently, P n = fg s ; s g s2sn, where g s ; s are s2s n determined by (1) and (2) for some 2 n. 7 That is, fg s ;! s g s2sn 2 G n if and only if g s = R s () f () d 0 and g s! s = R () f () s () d for some 2 n. 8 Suppose that is distributed uniformly on [0; 1] and v () =. Consider fg;!g, such that g = 1 2 ; 1 2, and! = f0; 1g. Then, g 1 + g 2 = 1 and g 1! 1 + g 2! 2 = E [] = 1 2. However, fg;!g is not plausible since the variance of posterior means is higher than the variance of. 9 Thus, o is de ned up to the set of boundary points of f s g s2sn of measure zero. 6

7 is called redundant monotone partitional if o s = f (j 2 s ) ; s 2 S n, where each s is a non-degenerate interval, and there is a proper subset S no of S n, such that f s g s2sno form a partition of. That is, such a signal structure includes redundant signals, which generate identical posterior beliefs. 2.2 Bayesian persuasion mechanisms This paper analyzes a class of games which we call Bayesian persuasion mechanisms (hereafter, mechanisms). A game in this class includes three players, the sender (S), the mediator (M), and the receiver (R) whose information about the state at the beginning of the game is limited to the prior distribution F (). Given this information, the sender publicly selects the signal structure, which determines the distribution over posterior beliefs fg; g 2 P n. Then, the mediator privately observes the signal s generated by and sends a (possibly random) private message m to the receiver. Finally, the receiver takes a (possibly random) action a in the action set A. The players ex-post payo functions v i (a; ) ; i 2 K = fs; M; Rg depend on state and the receiver s action a. The interim payo of player i 2 K is the payo from action a under the Bayesian posterior belief s induced by and signal s according to (2): V i (a; s ) = E s() [v R (a; )] : (5) We say that the interim interests of players i 2 K and j 2 K are identical if arg max a2a V i (a; s ) = arg max a2a V j (a; s ) for all s 2 : (6) In our setup, the mediator and the receiver may or may not have commitment power over actions. If the receiver has no commitment power and the mediator is truthful, i.e., m (s) = s for all s 2 S n, then the action rule must satisfy the receiver s interim best-response condition a ( s ) 2 arg max y2a V R (y; s ) for any s 2 : (7) If the receiver has commitment power, we represent it by the receiver s ability to base his action a on both the distribution of Bayesian beliefs fg; g and the posterior belief s induced by the mediator s report s, that is, a = a (fg; g ; s), where a (fg; g ; s) = a (fg; g ; s 0 ) for all s; s 0 2 S n, such that s = s 0: (8) Thus, if the receiver and/or the mediator have commitment power, their strategies can generally depend on the sender s choice of and, thus, fg; g. However, we do not focus on situations in which the receiver and/or the mediator discipline the sender by using grim-punishment for non-selecting a favorable (for these players) signal structure by either not passing any information (by the mediator) or taking an unfavorable action from the sender s perspective (by the receiver) For instance, Argenziano et al. (2014) show that the threat of ignoring sender s information and taking an uninformed action by the receiver in the case of a sender s deviation from a prescribed signal 7

8 Upon observing signal s the mediator builds the posterior belief s by Bayes rule (2) and sends a non-veri able message according to a messaging strategy m (s) 2 S n. We restrict attention to equilibria with the truthtelling strategy of the mediator m (s) = s for all s 2 S n ; (9) which is sustained either by mediator s commitment to truth-telling or by the Revelation principle in the case of receiver s commitment to actions or its analogue in setups without receiver s commitment. 11 Hereafter, by equilibria we imply equilibria with the truth-telling mediator s strategy (9). In particular, if the receiver has commitment power, then by the Revelation principle we can restrict attention to truth-telling equilibria in which the action rule a (fg; g ; s) satis es the mediator s interim incentive-compatibility constraints: s 2 arg max V M (a (fg; g ; m) ; s ) for all s; m 2 S n ; fg; g 2 P n, and s 2 : (10) m2s n Finally, given the action rule a (fg; g ; s), the sender selects a signal structure 2 n or, equivalently, fg; g 2 P n, which maximizes his ex-ante payo EV (g; ) = X s2s n g s V S (a (fg; g ; s) ; s ) : (11) In this broad setup, some classes of games are of particular interest. Bayesian persuasion games. Suppose that the mediator commits to the truth-telling strategy (9). Equivalently, if the mediator does not have commitment power, but his interim interests are perfectly aligned with the receiver s interests, i.e., (6) holds for M and R, the mediator has not incentives to deviate from (9). 12 Also, assume that the receiver does not have commitment power over actions, i.e., the action rule must satisfy (7). This setup corresponds to Bayesian persuasion games considered by Kamenica and Gentzkow (2011). Selling mechanisms. Suppose that the interim interests of the sender and receiver are identical, i.e., (6) holds for S and R, the receiver has commitment power, and the mediator does not have commitment power. In particular, the sender is an auctioneer wants to maximize his ex-ante revenue by auctioning a single indivisible product to j = 1; :::; K risk-neutral bidders (mediators). Mediators have independently distributed and initially unknown product valuations j ; j = 1; :::; K. The sender selects a signal structure j of each mediator about her valuation j. Because the receiver can commit to actions, he determines the action rule (8) the bidders allocation probabilities q fs j j g K j=1 and expected payments p fs j j g K j=1 on the basis of the distribution of posterior beliefs structure provides the su cient incentive for the sender to select a preferable signal structure from the receiver s perspective. 11 For example, analogues of the Revelation principle without receiver s commitment to actions are established by Ivanov (2010) for cheap-talk communication, and by Bester and Strausz (2001) for contracting for information. 12 Given any fg; g selected by the sender, the subgame is equivalent to the decision problem of a single player, M = R. Thus, any distortion of information by the mediator is suboptimal. 8

9 fg; g = fg j ; j g K j=1 and the pro le of bidders reports s = fsj g K j=1. Since the interim interests of the sender and receiver are identical, the optimal signal structure and the action rule maximize the sender s ex-ante payo (11). Such a speci cation is identical to the auction design model of Bergemann and Pesendorfer (2007). 13 Informational control in cheap-talk and delegation. Similarly to the previous settings, suppose that the interim interests of the sender and receiver are identical, but the receiver does not have commitment power, i.e., the action rule must satisfy (7). This setup is equivalent to the model of strategic communication of Crawford and Sobel (1982) with the imperfectly informed expert (mediator), which is investigated by Ivanov (2010). In this case, the mediator s signal structure is determined endogenously by the sender (or the receiver), and the optimal signal structure must maximize the sender s ex-ante payo. Also, if the receiver has commitment power, then the setup is equivalent to the principal-agent model of restricted delegation by Holmström s (1977) (see also Alonso and Matouschek, 2008; and Kovàc and Mylovanov, 2009) in which the expert (mediator) is imperfectly informed about the state. 2.3 Ranking mechanisms Consider a sub-class of Bayesian Persuasion mechanisms in which the optimal interim action, i.e., the maximizer of the interim payo V i (a; s ) of each player i 2 K without commitment is solely determined by the posterior mean! s derived from s by (4), a i ( s ) = arg max a2a V i (a; s ) = a i (! s ) for all s 2 : (12) Also, the action rule a (fg; g ; s) of the receiver with commitment only depends on the value of! s and the s-th lowest order (s) of! s in the ordered sequence! (s) s2s n of generated posterior means!: a (fg; g ; s) = a (! s ; (s)) = a! (s) ; (s) for all fg; g 2 P n and s 2 S n : (13) Finally, we assume that the ex-ante payo of the sender in an equilibrium can be decomposed into an average over an ordered sequence of functions V! (s) ; (s) : EV (g;!) = nx g (s) V! (s) ; (s) ; fg;!g 2 G n ; (14) where V (! s ; s) is upper semi-continuous in x for each s 2 S n. 14 In particular, if the sender s interim payo V S (a; s ) is a function of an action a and a posterior mean! s only, i.e., V S (a; s ) = V S (a;! s ) for all a 2 A and s 2 ; 13 As Bergemann and Pesendorfer (2007) show, for K bidders with independently distributed values j ; j = 1; :::; K, the auctioneer s problem of maximizing the total revenue can be decomposed into K independent revenues-maximization problems for individual bidders. 14 The upper-semicontinuity of V is a mild regularity condition which is needed to guarantee the existence of the maximum of EV (g;!). 9

10 then (12) and (13) are su cient for (14) to hold. This is because EV (g;!) = X s2s n g s V S (a (fg; s g ; s) ;! s ) = X s2s n g s V S (a (! s ; (s)) ;! s ) (15) = nx g (s) V S a! (s) ; (s) nx ;! (s) = g (s) V! (s) ; (s) ; fg;!g 2 G n : (s)=1 where V! (s) ; (s) = V S a! (s) ; (s) ;! (s). However, because (12) implies that the sender s interim payo V S (a; s ) can depend on other characteristics of s, (14) requires that ex-ante payo function EV is invariant to these characteristics. In general, the class of setups which satisfy (12) (14) includes many economic applications. In selling mechanisms, for example, the action rule of the receiver (the auctioneer) with commitment power initially orders generated posterior means of risk-neutral bidders (mediators) in an increasing order. Then, the receiver s action the expected probability of getting the object Q j (s) and the expected transfer T j (s) a ects the bidder j s interim payo solely via the order (s j ) of the reported posterior mean! j in the ordered sequence f! j g. Similarly, consider cheap-talk communication with s j (s j ) informational control of the mediator. If the ex-post payo functions of players are quadratic in (a; ), then the receiver s best-response (7) is solely determined by the posterior mean! s. As a result, any equilibrium satisfy (12) (14). 15 Note that (14) means that EV is symmetric with respect to permutations of signals s or, equivalently, pairs fg s ;! s g: (s)=1 EV (g;!) = V (g 1 ; :::; g n ;! 1 ; :::;! n ) for all permutations of fg;!g : (16) Since EV is symmetric to permutations of fg s ;! s g and the player s ex-post payo s and actions do not directly depend on s, without loss of generality we can restrict attention to signal sets S n = [n] = f1; :::; ng which index values of! in an increasing order, i.e.,! = f! s g n is weakly increasing. Let G+ n G n be the set of distributions fg;!g with weakly increasing! of size n. Then, the ex-ante payo of the sender can be written as EV = EV (g;!) = nx g s V S (! s ; s) ; fg;!g 2 G n + : (17) A signal structure 2 n is optimal if it generates a distribution of posterior means fg ;! g 2 G + n which maximizes the sender s ex-ante payo (17), EV (g ;! ) = max fg;!g2g + n nx g s V S (! s ; s) : (18) Thus, the Bayesian sender is interested in an ex-ante optimal decomposition of the prior mean! 0 = E [ ()] into a plausible distribution of posterior means fg;!g by selecting signal structure. 15 Also, this class of setup include situations with boundedly rational players. For example, the action rule can replicate the optimal strategy of a partially naive receiver whose decisions is a ected by both the signal realization s and the posterior mean! s induced by it. 10

11 Allowing for the receiver s action depend on fg;!g because of commitment extends the set of Bayesian persuasion games introduced by Kamenica and Gentzkow (2011). In these games, the receiver cannot commit to actions, and his action in any equilibrium is solely determined by the posterior belief s, that is, a ( s ) = a ( s 0) for all s and s 0 such that s = s 0. Hence, the sender s interim payo V S ( s ) = E s() [v S (a ( s ) ; )] is a P function of a single variable s, and the ex-ante payo EV = n g s V S ( s ) is an average of the interim payo function V S ( s ) across s. Thus, if V S ( s ) in an equilibrium depends on the posterior mean only, V S ( s ) = V S (! s ) for all s 2, then the ex-ante payo P EV = n g s V S (! s ) becomes a special case of (17). 3 Informativeness ordering of signal structures In order to solve the sender s problem (18), we use the informativeness ranking of signal structures and then translate this ranking into the ranking of distributions of posterior means fg;!g. For these purposes, we use three statistical tools: the positive quadrant-dependent stochastic orders (Tchen, 1980; and Joe, 1997), the comonotonicity of distributions (Puccetti and Scarsini, 2010), and the weighted majorization (Marshall, Olkin, and Arnold, 2011). First, the comonotonicity of distributions means the perfect positive dependence between multiple random variables, which means that they can be represented as increasing functions of a single random variable. In particular, the joint distribution H (s; ) is called comonotonic, if H (s; ) = min fg (s) ; F ()g for all s 2 S and 2 ; (19) where S is ordered, and G (s) and F () are marginal distributions of and s, respectively. Second, the positive quadrant dependent (PQD) order (which is also called the concordance order by Joe, 1997), ranks joint distributions H (s; ) in the Fréchet class M(G; F ), which includes all bivariate distributions with xed marginal distributions F () and G (s). Namely, we say that H o 2 M(G; F ) dominates H 2 M(G; F ) by the PQD order if 16 H o (s; ) H (s; ) for all s 2 S and 2 : Third, the weighted (or g ) majorization is a special case of the single-dimensional convex stochastic order, or second-order stochastic dominance, for discrete distributions such that the values of the majorizing distribution are more dispersed than those of the original distribution while the probabilities of associated values of both distributions are identical. In particular, let fg;!g and fg;! o g be two discrete distributions with n values, 16 For bivariate distributions, Müller and Scarsini (2000, Theorem 2.5) show that the PQD order is identical to the supermodular stochastic order de ned as follows. We say that n dimensional random variable X o dominates X by the supermodular order if E [ (X o )] E [ (X)] for all supermodular functions : R n! R, i.e., such that ' (x ^ y) + ' (x _ y) ' (x) + ' (y). 11

12 where! and! o are increasing. If kx g s! s o kx g s! s ; k = 1; :::; n 1, and nx g s! s o = nx g s! s ; then we say! o g majorizes!, or! o g!. Thus, if fg;!g and fg;! o g are generated by signal structures H (s; ) and H o (s; ), respectively, from the same Fréchet class M(G; F ), P then E [! s] o = n P g s! s o = n g s! s = E [! o (s)] = E [ ()]. Thus,! o g! is equivalent to E [E o [ () js] js k] = 1 G (k) kx g o s! o s 1 G (k) kx g s! s = E [E [ () js] js k] ; k < n: (20) With a minor abuse of notation and when the nature of the distribution is clear from the context, we also call a signal structure the joint probability distribution of (s; ) induced by, Z sx H (s; ) = k (t) df (t) : k=1 Given these preliminaries, the lemma below establishes the ranking of distributions of posterior means generated by monotone partitional and arbitrary signal structures. Lemma 1 For any signal structure H (s; ) 2 M(G; F ) there is a monotone partitional H o (s; ) 2 M(G; F ), such that E o [ () js] g E [ () js]. The lemma is proved by construction and combines three results from the statistical literature. First, for a given joint distribution H (s; ) in a Fréchet class M(G; F ), consider a monotone partitional joint distribution H o (s; ) with cuto s s = F 1 (G (s)) ; s = 1; :::; n The choice of these cuto s guarantees that G o (s) = G (s), that is, H o (s; ) is in the same Fréchet class M(G; F ). Second, by construction, the signal generated by the partitional signal structure, s () = s for 2 [ s 1 ; s ], is an increasing function of. Therefore, G (s) and F () are comonotonic, or equivalently, and s are perfectly positively correlated. The comonotonicity implies that H o (s; ) = min fg (s) ; F ()g, which is the upper bound on distributions in the Fréchet class M(G; F ) (Joe, 1997). Finally, Tchen (1980) shows that if H o (s; ) 2 M(G; F ) and H (s; ) 2 M(G; F ) are such that H o (s; ) H (s; ) for all s and, and () is right-continuous increasing and integrable, then E [E o [ () js] js x] E [E [ () js] js x] for all x; (21) which implies that! o g majorizes!. Two comments are necessary. First, since the initial signal structure H (s; ) and the monotone partitional H o (s; ) are in the same Fréchet class and H o (s; ) H (s; ) for 17 The comonotonicity of s and follows from the fact that signal generated by the partitional signal structure, s () = s for 2 [ s 1 ; s ], is an increasing function of. 12

13 all (s; ), then (21) implies that the g majorization ranking of the associated sequences of posterior means,! o g!, is invariant to right-continuous increasing transformations () of the state. This invariance of the g majorization ranking of the generated posterior means is a valuable property of the PQD order, which is generally not preserved by other informativeness orders. For example, the supermodular precision and the integral precision introduced by Ganuza and Penalva (2010) are not invariant to strictly increasing transformations of (see Brandt et al., 2014). Second, it is clear from this lemma that the PQD order cannot be used directly to rank joint distributions F (a; ) over actions and states in Bayesian persuasion games and mechanisms. Because the receiver is a Bayesian player, he is interested in the meaning of a signal rather than the signal itself. Thus, the sender s ex-ante payo can be expressed as an average of the ex-post payo over actions and states, where actions depend on the signal structure : Z EV = v (a; ) df (a; j) ; A where F (a; j) = Z dh (s; t) : fs;t:a( s();s)a;tg However, in Bayesian persuasion games it is su cient to consider signal sets and equilibria that admit the obedience constraint (Proposition 1 in Kamenica and Gentzkow, 2011), a ( s ) = s for all s 2 S; (22) i.e., each signal recommends the action, and the receiver always follows the recommendation. Imposing constraint (22) on the signal set S means that two signal structures and o can induce di erent ordered action sets S and S o though the probabilities of associated actions (that is, actions with the same index in the action set) are identical, g a = g a o. That is, the constraint (22) violates the necessary condition of identical marginal distributions for ranking distributions F (a; ) by the PQD order. Equivalently, imposing the obedience constraint implies that the standard normalization of signal structures by transforming signal a into ^a = G (a), which results in the identical (uniform) marginal distribution of the modi ed signal ^a, is not applicable. Finally, because Bayesian persuasion games form a subclass of Bayesian persuasion mechanisms, this argument can be extended to mechanisms as well. 4 Ranking of ex-ante payo s Consider EV (g;!) of the form (16) which is di erentiable in! and satis es the condition (g;!) (g;!) (! s! t ) 0 for all! and s; t = 1; :::; n: (23) g s g t These functions form a set of weighted (or g ) Schur-convex functions. The following 13

14 proposition by Cheng (1977) demonstrates the relationship between nite sequences in R n (in our case, sequences of posterior means) ranked by the g majorization and values of a function of these sequences. Proposition 1 (Cheng, 1977) If EV (g;!) is di erentiable in! and satis es (23), then! o g! implies EV (g;! o ) EV (g;!) if and only if (23) holds. The key feature of (weighted) Schur-convex functions that plays a crucial role in our analysis is that a Schur-convex function is not necessarily convex, and a convex function is not necessarily Schur-convex. However, a convex function is weighted Schur-convex if P and only if it is symmetric with respect to fg s ;! s g pairs. Thus, EV (!; g) = n g s V (! s ) is weighted Schur-convex if and only if V (!) is convex. However, in Bayesian persuasion mechanisms with the sender s ex-ante payo of the form (14), EV is an average of an ordered family of functions fv (x; s)g n. In this case, condition (23) is equivalent to V 0 x (x; s) V 0 y (y; k) for all x y and s > k; (24) that is, the marginal payo V 0 x (x; s) from a higher posterior mean is isotone in (x; s). In order to see the equivalence between (23) and (24), note rst that it is su cient to consider fg;!g 2 G + n by (17). Thus, we can consider (23) only for! s! k and s > k. Then, (23) is satis ed if and only if (!; g) (!; g) = V! 0 g s g s (! s ; s) V! 0 k (! k ; k) 0 for! s! k and s > k: k If x = y, then (24) is equivalent to the supermodularity of V (x; s). Thus, the supermodularity of V (x; s) is a necessary condition for the preservation of the ranking of ex-ante payo functions of sequences of! o and! ordered by the g majorization. Also, if V (x; s) = V (x; k) = V (x) for all x; s and k, then (24) is equivalent to the convexity of V (x). Together, the supermodularity of V (x; s) and the convexity of V (x; s) in x for all s are su cient conditions for (24) to hold, since they imply Vx 0 (x; s) Vy 0 (y; s) Vy 0 (y; k) for x y and s > k. However, if V (x; s) 6= V (x; k), then the convexity of V (x; s) in x is not required. Moreover, V (x; s) can be concave in x for all s. The following lemma provides su cient conditions for (24) to hold. Lemma 2 Condition (24) holds if: (A) V (x; s) is supermodular and in each pair of functions fv (x; s + 1) ; V (x; s)g ; s = 1; :::; n 1 at least one function is convex in x; or (B) min x Vx 0 (x; s + 1) max x Vx 0 (x; s) for all s = 1; :::; n 1. The intuition behind Lemma 2 is straightforward. Suppose rst that V (x; s) is convex in x for all s and consider! = fy; xg, where x > y. Then, the marginal gains in EV from a larger spread in fy; xg due to the convexity of V (x; s), that is, the increasing di erence V (x + ; s) V (y ; s) in, are ampli ed by the supermodularity of V (x; s), that is, the increasing di erence V (x; s) V (y; s) in s. As a result, the overall e ect of the spread! o = fx + ; y g of! on EV is positive. In addition, if the supemodularity e ect is strong enough, it can still outweigh the losses in EV caused by the non-convexity of V (x; s) in x. Hence, the ex-ante higher payo can be achieved by decomposing the prior 14

15 distribution into monotone partitions if the interim payo functions V (x; s) are concave in posterior means x for all s. This is a stark di erence with Bayesian persuasion games (see Corollary 2 in Kamenica and Gentzkow, 2011) in which the necessary condition for an ex-ante bene cial decomposition of the prior distribution into posterior ones is the local convexity of the interim payo V ( s ) in the posterior belief s. 18 Conversely, if V (x; s) is concave in x for all s and V (x; s) is submodular, then the spread in! decreases EV (g;!). Hence, the optimal sequence of posterior means is constant,! o = fe []g n. This also implies that if V (! s; s) satis es (24), then introducing the cost function of the signal structure C (! s ; s) which is concave in! s and submodular in (! s ; s) preserves the optimality of monotone partitional signal structures. An implication of Lemmas 1 and 2 is the optimality of monotone partitional signal structures in mechanisms in which the sender s interim payo s V! (s) ; (s) = a (s)! (s) + b (s) are linear in posterior means!(s) and actions a! (s) ; (s) = a (s) ; b (s). Proposition 2 Consider a mechanism with action rule! (s) ; (s) = a (s) ; b (s) s2s n such that EV (g;!) = P a (s)! (s) + b (s) ; fg;!g 2 G n. Then, there is a mechanism s2s n g (s) with redundant monotone partitional o and the same action rule, such that EV (g;! o ) EV (g;!). The proof is constructive and involves two key steps. The rst step is necessary if the rst component a s in the action rule s = fa s ; b s g is not weakly increasing. In this case, we construct an iterated sequence of ironing mechanisms. In the rst iteration, consider a pair of actions k and k+1 with decreasing rst components, that is, a k+1 < a k. Denote S the subsets of signals which induce an action with value, i.e., S = fs 2 S n j s = g. Then, consider a mechanism with the initial action rule and the modi ed signal structure 1 that irons posterior means! s generated by signals in S k and S k+1 by collapsing them into a single posterior mean! s 1 = E! s js 2 S k [ S k+1 for all s 2 Sk [ S k+1, but preserves the initial probabilities of signals fg s g n. Because a function V (! s; s) = a s! s + b s is linear in! s and a k+1 < a k, condition (24) is reversed for s 2 S k [ S k+1 and! s. Also, by construction a modi ed sequence of posterior means! 1 is g majorized by the initial sequence!. As a result, Lemma 1 and Proposition 1 imply that the sender s ex-ante payo in the mechanism with the modi ed signal structure is higher. In addition, the linearity of V (! s ; s) in a s! s means that a mechanism with an ironed sequence! 1 is payo -equivalent to a mechanism with the initial sequence!, but a modi ed ironing action rule fsg 1 n. This auxiliary hypothetical rule collapses actions k and k+1 into a single average action k 1 = 1 k+1 = E a s js 2 S ak [ S ak+1 for all s 2 Sk [ S k+1, and keeps other actions unchanged. In the second iteration, if the rst component fa 1 sg n of the ironing action rule fsg 1 n is non-increasing, then we modify the signal structure 1 by ironing posterior means! s generated by signals s 2 S 1 k1 [ S 1, where a1 k1 +1 k 1 +1 < a1 k 1. The iterative process repeats T n 1 times until the sequence a T s action rule s T n of the T mechanism becomes non-decreasing. n in the ironing 18 In this case, the concave closure ^V ( s ) of V ( s ) is strictly higher than V ( 0 ) at 0, which is the necessary and su cient condition for decomposing the prior belief 0 into a distribution over posterior beliefs s. 15

16 In the second step, we construct a mechanism with a redundant monotone partitional signal structure o and an initial action rule s, which is ex-ante payo superior to the T mechanism. Let a T n o s be a strictly increasing subsequence of k a T n k=1 s, which includes all values of a T n s. Denote go k the probability of inducing an ironing action s T with value s T k, i.e., gk o = P g s. Then, consider a redundant monotone partitional s2s T sk signal structure o, which partitions the state space into n o = n T intervals f k g no k=1, such that Pr f 2 k g = gk o. Also, conditional on 2 k, o randomizes among signals s 2 S T sk with probabilities gs. By construction, each signal s 2 S gk o n is generated with the original probability g s, and the sequence of posterior means f! sg o n generated by o g majorizes the sequence! s T n generated by T in the T mechanism. Because a T n s is increasing, condition (24) holds for the modi ed interim-payo function V T (x; s) = a T s x + b T s. Therefore, Lemma 1 and Proposition 1 imply that the sender s ex-ante payo in a mechanism with the signal structure o and the modi ed action rule T n s is higher then the payo with the initial signal structure and the action rule T n. Finally, by the linearity of V T s (x; s) in a T s x and the construction of s T n, it is equivalent to a higher sender s ex-ante payo in the mechanism with the signal structure o and the initial action rule f s g n compared to the payo in the initial mechanism. The main implication of this section is that if the sender s ex-ante payo is a weighted sequence of ranked functions of!, which can be represented in a form (17) which satis es (24), then the set of optimal signal structures can be reduced to those of a simple interval form. The section below demonstrates how our results can be used in order to either solve problems or generalize the existing solutions in several applications of persuasion games and mechanisms. 5 Applications 5.1 Auctions Auctions with imperfectly informed bidders is another subclass of Bayesian persuasion mechanisms in which the seller s ex-ante revenue has the form (14). In the context of auctions, Ganuza and Penalva (2010) introduce two informativeness orders, the supermodular precision and the integral precision. In particular, a signal structure H o (s; ) is more supermodular precise than H (s; ) if for any pair of quantiles q and q 0 such that 0 < q < q 0 < 1, E o jg o 1 (q 0 ) E o jg o 1 (q) E jg 1 (q 0 ) E jg 1 (q) ; (25) where G (s) and G o (s) are the marginal distributions of s associated with H (s; ) and H o (s; ), respectively. Also, H o (s; ) is more integral precise than H (s; ) if Z q 0 E o jg o Z q 1 (t) dt E jg 1 (t) dt for all q 2 (0; 1) (26) 0 16

17 By noting that E [E [js] js G 1 (q)] = 1 q R q 0 E [jg 1 (t)] dt, (26) can be written as E E o [js] js G o 1 (q) E E [js] js G 1 (q) for all q 2 (0; 1) : (27) Because the supermodular precision and the integral precision are not invariant to monotone transformations of (Brandt et al., 2014), we put () =. Then, in a Fréchet class of signal structures M(G; F ) with discrete G (s), the integral precision is equivalent to the g majorization of posterior means, which directly follows from comparing (20) and (27) for () = and G o (s) = G (s). As a result, the g majorization of posterior means is a strictly weaker criterion than the supermodular precision. In particular, if H o (s; ) is more supermodular precise than H (s; ), then (25) means that the distance between any pair of posterior means generated by the more informative signal structure is larger,! o s 0!o s! s 0! s for all s and s 0 > s: However, if H o (s; ) 2 M(G; F ) is monotone partitional and thus, generates the sequence of posterior means! o which g majorizes! generated by any H (s; ) 2 M(G; F ), then! o is not necessarily more dispersed than! pairwise (except for extreme pairs! n o! 1 o! n! 1 ). Thus, the supermodular precision is equivalent to the g majorization of posterior means if and only if n = 2. Ganuza and Penalva (2010) consider the following setup. An auctioneer sells an indivisible object he values at zero, to N ex-ante identical risk-neutral bidders with private valuations i ; i = 1; :::; N. For each bidder i = 1; :::; N, i is independently drawn from [0; 1] according to a common distribution F (). Initially, bidders expected valuations of the object are equal to = E [ i ]. The ex-post utility of bidder i from winning the auction is u i ( i ; t i ) = i t i, where t i is a monetary payment. Prior to the auction, the auctioneer can supply information to bidders by incurring costs. By publicly paying 0, the auctioneer will generate information in the form of bidders private signals fs i gn i=1. Thus, determines the common precision of the bidders private signals, where each signal s i provides information only about i. After releasing the information by the auctioneer, each bidder updates her posterior valuation of the object! (s i ) = E [ i js i ]. Finally, the object is sold via the second-price sealed-bid auction. As Ganuza and Penalva (2010) note, given the marginal distribution G (s) of original signals s i ; i = 1; :::; N, by applying transformation ^si = G (s i ) we can consider signal structures with identical (uniform) marginal distributions G (s) for all. In other words, we can restrict attention to signal structures H (s; ) in the same Fréchet class M(G; F ). However, since both the integral and supermodular precision orders are de ned in terms of distributions of posterior means, it is not clear which criteria on signal structures induce these orders. This question has been addressed by Brandt et al. (2014), who show that the PQD order of signal structures H (s; ) implies the integral precise order and is invariant to strictly increasing and di erential transformations () of the state. In addition, our Lemma 1 shows that for a nite number of bidders signals, the monotone partitional signal structures achieve the upper bound on the informativeness of H (s; ) 2 M(G; F ) ranked by the g majorization of posterior means, and hence by the integral precision Also, the g majorization order of posterior means does not require the strict monotonicity and 17