Evidence and Skepticism in Verifiable Disclosure Games

Transcription

1 Evidence and Skepticism in Verifiable Disclosure Games Daniel Rappoport Columbia University February 2, 2017 Abstract A shared feature of communication ames with verifiable evidence is that the receiver will be skeptical followin any non-disclosure: he will tend to believe that the messae comes from an informed sender who is withholdin unfavorable evidence. It then follows that when the receiver is more skeptical he will choose a less preferable action for the sender. This paper asks which evidence distributions induce more skepticism on the behalf of the receiver? Our first step in answerin this question is to provide an explicit characterization of the receiver optimal equilibrium payoffs in a eneral model. We then define the more evidence relation between type distributions: a distribution has more evidence than another if types with larer messae sets are more probable in a monotone likelihood ratio sense. Usin our characterization, we show that when the sender has more evidence, the equilibrium utility for all sender types decreases, i.e. the receiver becomes more skeptical followin any messae. We also show that the more evidence relation completely characterizes when the receiver is more skeptical, in that no other relation on type distributions allows for uniform sender utility comparisons. Keywords: Verifiable Disclosure, Hard Information, Monotone Likelihood Ratio Property, Comparative Statics I am deeply indebted to Navin Kartik and Andrea Prat for their continued uidance. I thank Charles Anelucci, Ashna Arora, Yeon-Koo Che, Jacob Leshno, Pietro Ortoleva, Valentin Somma, and the audience at the Columbia micro-economic theory colloquium for their comments. all errors are my own.

2 1. Introduction In many economic contexts, hard evidence is the only viable means of informative communication: a politician will say anythin to et elected, a car salesman will never admit to sellin a lemon, and a job applicant will never say that he is untalented. In these situations the bias of the informed party is so extreme that any cheap talk communication would be frauht with fabrication. Fortunately, certain types of communication- hard evidence - are immune to such fabrication: The upstandin politician can release non-scandalous tax returns, the car salesman with a workin car can offer a test drive, and the ood job applicant can present her A + transcript. In each case, hard evidence is informative because its availability in communication is directly related to (or is correlated with) the decision maker s payoff. More enerally, verifiable disclosure ames involve an informed aent whose set of available messaes depends on his private information. These ames have received extensive attention since they were introduced to the literature in Milrom & Roberts (1986), Grossman (1981), and Dye (1985). However, many of these papers focus on the specific evidence framework in Dye (1985) in which informed aents can either truthfully reveal or completely withhold their private information. In reality, evidence structures can be more complicated, e.. multidimensional and correlated, wherein aents selectively disclose the most favorable pieces of evidence available and omit the unfavorable ones. Also, recent models such as Acharya et al. (2011), Dziuda (2011), and Guttman et al. (2014) present a rane of seeminly different comparative statics results concernin chanes in the distribution of evidence. The main contributions of this paper are (i) a tractable characterization of equilibrium in a unifyin framework and (ii) a eneral result comparin sender equilibrium payoffs for different evidence structures that subsumes the above comparative statics results. The model in this paper encompasses a lare class of verifiable disclosure ames. An informed sender communicates with an uninformed receiver in order to influence his action choice. While the receiver s preferences over actions depend on the private information or type of the sender, the sender independently prefers hiher actions. Followin Hart et al. (2017) (Henceforth HKP), and Ben-Porath et al. (2017), we model the structure of hard evidence as a partial order: we say that type t dominates type s accordin to the disclosure order, or t d s, if type t has all the evidence necessary to masquerade as s. Importantly, there is no assumed relationship between whether a type is hih value (commands a favorable best response from the receiver) and whether that type is dominant accordin to the disclosure order (has a lare feasible messae set). We focus on the receiver optimal 1

3 equilibrium which is selected by the refinement in HKP 1. Solvin for equilibria in eneral disclosure ames can be difficult. We exploit the observation that any equilibrium induces a partition of sender types into payoff equivalence classes. Our first task is to characterize when any set of types can form such a payoff equivalence class, or poolable set, in the receiver optimal equilibrium. Intuitively, poolable sets involve low value types that are more dominant in the disclosure order pretendin to be hih value types that are less dominant in the disclosure order. Proposition 2 formalizes this intuition: Poolable sets are those for which all lower contour subsets accordin to the disclosure order are hiher value than the set as a whole 2. When this is satisfied we say that the receiver s best response is downward biased on this set. Thus any partition such that the receiver s best response is downward biased on its parts constitutes an equilibrium partition 3. This observation underlies two novel solution methods for receiver optimal equilibria: (i) an alorithm to find the equilibrium partition; and (ii) an explicit expression for the equilibrium action for each type. Crucially, our analysis bypasses the issue of the which messaes the sender uses in equilibrium and thereby the complicated derivation of mixed strateies. Armed with this characterization, we seek eneral comparative statics results. A shared feature of verifiable disclosure ames is that non-disclosures or partial disclosures are met with a deree of skepticism: The receiver will believe that these omissions come from types that are strateically withholdin unfavorable information. Intuitively then, when the sender has more evidence the receiver will be more skeptical followin any messae and consequently choose a less favorable action for the sender. Many existin models find seeminly different comparative statics results that exemplify this effect: Grubb (2011) and Acharya et al. (2011) consider uniformly increasin the probability that types receive verifiable evidence, Guttman et al. (2014) consider introducin additional evidence types, and Dziuda (2011) considers decreasin the probability of honest types that can only reveal truthfully; all these studies find that the equilibrium action for every type decreases. Our results identify the unifyin feature behind these seeminly different comparisons 1 A rowin literature (Sher (2011), Glazer & Rubinstein (2004), and most recently Ben-Porath et al. (2017)) has shown that the optimal mechanism solution in which the receiver can commit ex-ante to his best response can be implemented as an equilibrium without receiver commitment. HKP identify the truth leanin refinement that selects this equilibrium. 2 A lower contour subset S of a partially ordered set (X, ) is all the elements dominated by elements in S, i.e. {s : s S, s s }. 3 Some additional constraints on the partition are required and made precise in Proposition 1. Broadly these are (i) the action obtained by types in a iven part is the receiver s optimal action in response to that part, and (ii) the partition is an interval partition with respect to the disclosure order in that no type from a lower payoff part dominates any type from a hiher payoff part. 2

4 that lead to the same conclusion. We introduce the more evidence relation which compares prior distributions over types. For two distributions f and over the type space, we say that f has more evidence than if whenever t d t, the likelihood ratio f is reater at t than at t. In other words, f represents a monotone likelihood ratio shift up the disclosure order (to types with larer messae sets) from. Theorem 1 shows that increased skepticism in the receiver is characterized by the sender havin more evidence. That is, (i) if f has more evidence than then any receiver takes a lower action for every type (and messae) under f than under and (ii) if f does not have more evidence than then there exists a receiver and a type t, such that t obtains a strictly hiher action under f than under. Due to our characterization of poolable sets, the key insiht in establishin this result is that the downward biased property ensures that the best response action decreases when the sender has more evidence. This is not implied by existin results concernin monotone likelihood ratio shifts in the distribution. To see this, consider a measurable function ε : [0, 1] [0, 1] and the associated conditional expectation function Ef ε (S) E[ε(x) x S, x f] for some probability density f on [0, 1] and subset S [0, 1] 4. Assume that the disclosure order is the usual order on [0, 1], and the receiver s best response to any subset is iven by Ef ε(s)5. Then the receiver s best response is downward biased on [0, 1] if Ef ε ([0, y]) Ef ε ([0, 1]) for all y [0, 1], i.e. all lower contour subets have hiher value than the set itself. Also f has more evidence than if f monotone likelihood ratio dominates, i.e. if f(x) is (x) increasin in x. The fact that Ef ε is downward biased on [0, 1] will mean that in equilibrium all types pool toether. Thus, verifyin Theorem 1 requires showin that the expectation of ε under f is lower than that under. Notice that Ef ε bein downward biased is implied by but is weaker than the condition that ε is decreasin in x. This means that we cannot use the off the shelf result that the expectation of a decreasin function reduced under a monotone likelihood ratio shift in the distribution. Instead, we take a eneral approach and find a property-proposition 7- that holds for all conditional expectation functions under monotone likelihood ratio shifts in the distribution. Our result provides the followin corollary in this example: Either the expectation of ε under f is hiher than that under or there exists y [0, 1] such that Ef ε([0, y]) < Eε f ([y, 1]). This is sufficient to imply that the former holds for downward biased functions 6. In its eneral form, our result also implies 4 In the paper we consider a partial order and only finite sets, but the continuum provides a more convenient illustration here, and it is straihtforward to show that our results extend to this case. 5 This is true if the receiver s utility is quadratic loss i.e. if U R (a, t) = (a ε(t)) 2, where ε(t) represents the optimal action for each type. 6 Note that if Ef ε([0, y]) < Eε f ([y, 1]), then Eε f ([0, y]) < Eε f ([0, 1]) < Eε f ([y, 1]) which contradicts the fact that 3

5 classic results from the literature such as that in Athey (2002). Finally, we show how our results apply and eneralize examples from the literature. First, we use our results to quickly solve the classic Dye evidence model from Dye (1985) and Jun & Kwon (1988). Next we confirm that the comparative statics results in this model by Acharya et al. (2011) and Guttman et al. (2014) are specific examples of our comparative statics result-theorem 1. Next we move to the more complicated multidimensional framework of Dziuda (2011) and aain quickly solve for equilibrium usin our characterization results. Finally we use Theorem 1 aain to eneralize a key result in this model to all verifiable disclosure ames. That is, if we increase the probability of honest types - types that must declare truthfully - we increase the equilibrium utility of every type. The paper proceeds as follows. Subsection 1.1 previews our model, characterization approach, and comparative statics result in a simple economic example. Subsection 1.2 discusses the related literature. Section 2 lays out the model and equilibrium concept, as well as listin well known examples that fit our framework. Section 3 presents our characterization results. Section 4 introduces the more evidence relation, presents our main comparative statics result, and details the novel methodoloy behind it. Section 5 presents some applications of our results to the literature and eneralizations of their results. Finally Section 6 concludes. Unless noted otherwise, all proofs are in the appendix Preview of Important Concepts Consider an entrepreneur (sender) who instructs his enineers to run a beta test for a new software before its launch. The test can result in four different outcomes. The software could possibly outperform expectations arnerin rave reviews from its users. Alternatively the test could uncover an unknown fatal flaw in the software. For example, the users may find it inaccessible to non-enineers. Given the fatal flaw another possible outcome is that after the test the enineers could partially salvae the problem by addin a useful tutorial. Lastly, the beta test could yield no usable evidence, either because the users were not a representative sample or because faulty instructions were distributed. The entrepreneur reports to his investor (receiver), and no matter the test result he would like to create the most favorable impression for his product, and thereby induce the hihest action choice (The action could proxy for amount or duration of future fundin, or advertisin effort). However, certain results can be certified. If the product performs above expectations, or if the product has a fatal flaw, reviews can be presented that suest as much. In addition if the fatal flaw is realized but partially salvaed the entrepreneur can E ε f is downward biased. 4

6 credibly present the new tutorial. Althouh, in this case it will be apparent that the software was flawed to bein with. Finally independently of the test result, the entrepreneur can always claim that the test was unusable. There are four types of sender - no evidence (NE), above expectations (AE), fatal flaw (F F ), and Partially Salvaed (P S). A convenient representation of this problem is illustrated in the left panel of Fiure 1. The directed raph illustrates the disclosure order. Each vertex represents a type. The available messaes to each type are the set of vertices accessible via a directed path. For example, P S can declare {P S, F F, NE} but not {AE}. The investor s type dependent value for the product is displayed above each vertex. Suppose that the prior over sender types is uniform and the receiver chooses an action equal to the expected value of the product 7. v = 5 Above Expectations v = 5 AE v = 0 v = 0 No Evidence v = -6 NE v = -6 Fatal Flaw v = -4 FF v = -4 Partially Salvaed PS (a) Disclosure Order and Best Response (b) Equilibirum Strateies Fiure 1: Entrepreneur with Uniform Prior What is the receiver optimal equilibrium in this simple example? It turns out that the equilibrium involves the pure strateies represented by the dotted arrows in the riht panel of Fiure 1 8. Types in {P E, F F, NE} all claim to have no evidence, and induce the receiver to take a(ne) = The AE type truthfully reveals and obtains an action a(ae) = 59. The interpretation is that an entrepreneur will only reveal that the test produced results if 7 Formally, let the receiver s utility be iven by U R (a, t) = (a v(t)) 2. 8 We chose an example with a pure stratey receiver optimal equilibrium for simplicity. However mixed stratey equilibrium are very common in eneral verifiable disclosure ames, and so our method of findin the equilibrium partition proves to be far simpler. 9 There are many ways to set the actions for the off path declarations F F and P S. In the main text we do this accordin to the truth leanin refinement by HKP. In this case the refinement dictates that a(f F ) and a(p S) are 6 and 4 respectively. 5

7 said results are positive and will claim the test was faulty otherwise. The receiver anticipates this, and is skeptical upon receivin NE, i.e. he forms a lower expectation of the value of the product than if he were certain that the test were faulty. Notice that the above equilibrium can also be seen as a partition into sender payoff equivalence classes. This partition is (P 1, P 2 ) where P 1 = {NE, F F, P S} and P 2 = {AE}. As a convention we label the hiher payoff parts with hiher indices. Also note that the types from P 1 cannot masquerade as types from {P 2 }, i.e. P 2 is not lower in the disclosure order than P 1. Most importantly, note that the receiver s expected belief is downward biased on each of P 1 and P 2 10 : lower contour subsets- those with less evidence- in P 2 would induce a hiher receiver expectation than P 2 as a whole. To verify this, note that the expected values of NE and {NE, F F } (the two (strict) lower contour subsets) are 0 and 3 respectively which is reater than 10. In Section 3, we show that these properties are 3 eneral to any equilibrium partition. In addition, this equilibrium is prior dependent. Now suppose that tests are of hiher quality and provide the entrepreneur with more evidence about his product: He is both more likely to et a test result and more likely to partially salvae the fatal flaw. Specifically, assume the test now has only a 1 probability of bein faulty, the probability of a fatal flaw 6 or performin above expectation remains unchaned at 1, and the probability of partially 4 salvain the fatal flaw increases to 1. In this example, the receiver optimal equilibrium is 3 iven by the same partition as with the uniform prior. However, the equilibrium actions for types in P 1 and P 2 are now 34 9 utility has (weakly) decreased for all types ( 34 9 comparative statics result holds in eneral. and 5 respectively. This means that the equilibrium < 10 ). In Section 4, we show that this 3 Intuitively, when the sender has more verifiable evidence the report of NE is more likely to come from F F and P S. Althouh motivated by a stron intuition, upon further inspection the result seems more complicated. Notice that the investor decreases his expected value of types in P 1 even thouh the sender has become better at salvain the fatal flaw. More formally, this is surprisin because the value of the project in P 1 is (0, 6, 4) is not decreasin in the disclosure order. Instead we have the strictly weaker property that the value is downward biased on P 1. Our methodoloy shows that this property is necessary and sufficient to deduce that the expected value of P 1 decreases whenever the sender has more evidence. 10 This is obvious for P 2 as it is a sinleton. 6

8 1.2. Related Literature For our purposes, The relevant verifiable disclosure ames can be split into three cateories. First, Milrom & Roberts (1986) and Grossman (1981) introduce a model that prevents the sender from lyin but allows the sender to be vaue, i.e. the sender can declare any subset of types to which his true type belons. These papers find that communication unravels and all types declare truthfully 11. These papers also introduced the concept that the receiver is skeptical upon hearin vaue messaes, i.e. the receiver will believe that any vaue messae comes from the lowest value type who could have possibly sent the messae. This intuition will be key in our comparative statics analysis. The second class of models initiated by Dye (1985) and Jun & Kwon (1988) include the possibility that the sender is uninformed. More specifically types either do or do not obtain verifiable information and types with verifiable information can either truthfully reveal or pretend to be uninformed. The equilibrium involves all verifiable types with value below some threshold to pretendin to be uninformed. We apply our methods to this model in Section 5. This oriinal model has seen a number of extensions. Shin (2003) and Dziuda (2011) consider multidimensional versions of this model in which the aent obtains potentially multiple pieces of verifiable evidence and can disclose any subset. Dziuda (2011) also considers uncertainty over the preferences of the sender and over whether he is honest or strateic. Guttman et al. (2014) considers a dynamic model with multidimensional evidence in which the receiver is also uncertain about when the sender has obtained evidence. Acharya et al. (2011) also considers a dynamic model in which public information continuously arrives about the value of the sender s evidence. We will refer back to these models in Section 5 when we eneralize their comparative statics results. Both of the above cateories of verifiable disclosure ames (in their static form) fit the model of this paper. The last class of disclosure ames initiated by Verrecchia (1983) involves disclosure costs and is not covered by our model. Recently, a set of results have been established showin that the receiver s utility in some equilibrium of the verifiable disclosure ame is the same as the case in which the receiver can commit to a best response before hearin the sender s messae. This equivalence was first introduced in Glazer & Rubinstein (2004) and further eneralized by Sher (2011) to a model similar to the one in this study. More recently Ben-Porath et al. (2017) have shown this equivalence in a model with multiple aents and more eneral sender preferences. HKP identify the equilibrium that achieves this equivalence throuh what they term the 11 See Mathis (2008) and Haenbach et al. (2014) for more eneral and recent results on unravelin. 7

9 truth leanin refinement. We focus on this receiver optimal equilibrium throuhout. More relatedly, Glazer & Rubinstein (2004) and Sher (2014) derive methods to find the receiver optimal equilibrium. However, their models involve a binary action choice and only two types of senders - acceptable and unacceptable. Thus our characterization methods are not similar. Althouh our comparative statics result eneralizes findins from the literature, there is no other study that examines eneral likelihood ratio shifts in the distribution of types accordin to the disclosure order. Our methodoloy involves iteratively applyin the classic result that a monotone likelihood ratio shift in the distribution lowers the expectation of a decreasin function. This result can be found in Milrom (1981) and Shaked & Shanthikumar (2007) 2. Model The settin involves a sinle sender and a sinle receiver. The sender is endowed with a type t T, which constitutes his private information. T is a finite set with T = n. Let h T be the prior over types. Throuhout the main text (excludin examples and applications) we assume that the prior has full support. In Appendix D we show that our results extend without modification to eneral prior distributions. After observin his type, the sender chooses a messae from those available to his type and communicates this to the receiver. The receiver then takes an action a A, where A is a compact convex subset of R. Next we discuss the specific assumption on preferences and messae availability Preferences The receiver s preferences depend on both the action and the sender s private information. The sender simply wants the hihest action. We therefore assume that the sender s utility is iven by U S : A R where U S is strictly increasin in a 12. Let the receiver s utility be iven by U R : A T R which is assumed to be strictly concave and differentiable in a 13. Define the receiver s unique best response to type t as v : T R iven by v(t) ar max a U R (a, t). Similarly, define V : 2 T R as V (S) ar max a E[U R (a, t) t S, t h] 14 to be the receiver s best response action conditional on the sender s type bein in the set S 12 It is without loss to normalize the sender s utility to U S (a) = a. Also we could allow U S to depend on the type as lon as it is strictly increasin in a. 13 We could make the same weaker assumptions as in HKP, i.e. that the receiver s utility over actions is sinle-peaked iven any fixed distribution over T. More specifically q T t q tu R (a, t) is strictly quasiconcave in a. 14 Note that because we consider h to have full support V (S) is defined for every S. 8

10 and distributed accordin to the prior 15. We will refer to sets of types with relatively hih (low) optimal actions, i.e. hih V, as hih (low) value. The leadin example for the receiver s utility will be quadratic loss, i.e. U R (a, t) = (a v(t)) 2 for any function v : T R. Here, v(t) specifies the optimal action for each type. Also note that in this case, V (S) = E[v(t) t S] takes the form of a conditional expectation. We next make some observations that are obvious in the quadratic loss case but that also hold in eneral. Lemma 1. For any distribution over types q T, define a (q) ar max a E[U R (a, t) t q]. 1. Consider two distributions q 1, q 2 T and λ (0, 1). min i {1,2} a (q i )) a (λq 1 + (1 λ)q 2 ) max i {1,2} a (q i ) 2. Consider q R T + and define the distribution q T as q(t) q(t) t q (t). We have that a ( q) is increasin (decreasin) in q(t) if v(t) > (<)a ( q). (1) says that the optimal action for the mixture of two distributions is between the optimal actions in response to each individual distribution. In the quadratic loss example the best response is linear, i.e. a (λq 1 +(1 λ)q 2 ) = λa (q 1 )+(1 λ)a (q 2 ), so that (1) is satisfied. (2) says that if we shift weiht to a hiher valued type, the optimal action increases Messain Technoloy The sender s utility is type independent (and invertible). If the set of messaes available to the sender were also type independent, no informative communication would be possible. The potential for informative communication in this model is instead driven by type dependence in the the sender s set of available messaes. We make the assumption that the messae space is the type space. The set of feasible messaes available to each type is iven by the correspondence M : T 2 T : type t can send any messae in M(t). We make the followin assumptions on this correspondence, t M(t), t (Reflexivity) (1) s M(t) = M(s) M(t) (T ransitivity) (2) 15 Later on we make this prior dependence explicit, but as we consider a fixed prior until Section 4, we initially omit the dependence of V on h. 9

11 These assumptions induce a partial order d on T iven by inclusion of feasible messae sets, i.e. t d s if M(s) M(t), or in other words if t can pretend to be s 16. M(t) is also the lower contour set of t in this partial order, i.e. M(t) {s : t s}. Abusin notation, for S T we refer to M(S) s S M(s) as the lower contour set of S. This is the set of types that can be declared by some type in S. Similarly for any subset S T, we will refer to the upper contour set of S as B(S) {s T : t S, s d t}. B(S) is the set of types that can declare some type in S. For notational simplicity we omit the dependence of M and B on the disclosure order. When we refer to different partially ordered sets (X, ), the lower and upper contour correspondences will be denoted by M and B respectively. Notice that due to (1), the above messae structure always permits aents to report truthfully, i.e. report a messae with the same label as their own type. The assumptions on the messae structure are not entirely without loss of enerality. The more eneral framework would be one in which the messae space is some arbitrary set G, and the feasible messaes for each type is iven by some correspondence E : T G. When can we replace the more eneral messae space with the assumptions above and maintain the same set of equilibrium payoffs? This question has been addressed in the verifiable disclosure literature. If the messae structure satisfies a property called normality then we can represent it usin the assumptions above 17. As non-normal messae structures are beyond the scope of this study we focus on the assumptions above 18. We emphasize two key eneralities of this model. First, the disclosure order is arbitrary. And second, there is no assumed relationship between the disclosure order and the receivers best response function - v. We next list some well known examples of disclosure orders from the literature that fit our framework Examples Dye Evidence Assume that T = {t 1,..., t n 1, t }. The disclosure order is iven by M(t i ) = {t i, t } i < n and M(t ) = {t }. The interpretation is that the types {t 1,..., t n 1 } are evi- 16 These assumptions actually identify a preorder on T, however because sender preferences are type independent, we can without loss consider the quotient type space iven by d, and the associated partial order. Throuhout, we simply refer to (T, d ) as this partially ordered quotient space. A partially ordered set is a pair (S, ) such that is a binary relation over S that is transitive, reflexive, and antisymmetric. A preordered set is a pair (S, ) such that is a binary relation over S that is transitive and reflexive, but not necessarily anti-symmetric. 17 A messae structure is normal if for all types t, there exists e t, such that for any t, t T, e t E(t ) = E(t) E(t ). In words, normality means that for each type there is a distinuishin messae, such that if one type has the power to distinuish himself from another, he can do so usin this distinuishin messae. 18 For further details on normality see Bull & Watson (2004), Green & Laffont (1986), or Kartik & Tercieux (2012). For an of equilibrium characterization in a ame without normality, see Sher (2014) and Rubinstein & Glazer (2006). 10

12 dence types, while t is the uninformed type. The evidence types can credibly communicate their type to the receiver, or they can pretend to be uninformed. The uninformed type has no hard evidence, and so does not have the ability to separate from these would be masqueraders 19. This model was first introduced by Dye (1985), and has been widely used in the verifiable disclosure literature, e.. by Grubb (2011), Acharya et al. (2011), and Bhattacharya & Mukherjee (2013). In subsubsection we quickly solve this model usin the methods from Section 3, and then show that many comparative statics results are specific examples of Theorem 1. Multidimensional Evidence The aent draws an inteer n from some {0, 1,..., k}. The aent then draws a sample of size n from some distribution f X where X is a finite subset. Thus each type of the aent is a set of size n, i.e t = {x 1,.., x n }, and the type space is T = {t 2 X : t k}, i.e. all subsets with fewer elements than k. The disclosure order is iven by M(t) = {t : t t} t T. The interpretation is that each type can report any combination of the evidence in his possession. Note that the dye evidence framework is a specific case of this model in which n is drawn from {0, 1}, X = {t 1,..., t n 1 }, and t =. Multidimensional evidence models have also been employed in the verifiable disclosure literature by Guttman et al. (2014), Dziuda (2011), and Shin (2003). In Subsection 5.2 we show how to solve and present comparative statics for the multidimensional evidence model in Dziuda (2011). Vaueness All aents obtain a sinle piece of verifiable evidence x X drawn from h X, where X is some finite set. The type space is all non-empty subsets of X, T = 2 X \. The disclosure order is iven by M(t) = {t T, t t } t T. The interpretation is that each positive probability type t can credibly reveal himself or be vaue. For example, if X = {0, 1}, type {0} can either truthfully report {0}, or be vaue and report {0, 1}, however she cannot lie and report {1} 20. Messae structures like the one above were first introduced by Grossman (1981) and Milrom & Roberts (1986). Honest Types Consider a payoff relevant type space T endowed with disclosure order d. In addition to the payoff relevant type, aents can either be strateic, s, or honest, h. 19 Because of the interpretation that the evidence types have information while the uninformed type does not, it is usually assumed that v(t ) = V ({t 1,..., t n 1 }. 20 Notice that we are explicitly usin zero probability types to model additional messain options for positive probability types. Any non-sinleton type t is zero probability, but represents a common feasible messae for two types t 1, t 2 t. In Appendix D we show that the receiver optimal equilibrium of this ame is the same as if we were to consider only the positive probability types, i.e. T X. 11

13 The receiver s utility U R : A T R only depends on the payoff relevant type, and not on whether the aent is strateic. The total type space and disclosure order are iven by (T, d ) defined as follows: T = T {s, h}, and t T, M(t, s) = M d {s, h} M(t, h) = (t, h) The interpretation is that honest types must report truthfully and therefore have a sinle available messae. Strateic types can report accordin to some arbitrary disclosure order. In subsubsection 5.2.3, we use the results of Section 4 to show that as the equilibrium utility of all types increases with the probability of honest types. Complete Order and Empty Order Althouh less commonly used in the literature, the cases where the disclosure order is complete or empty repeatedly serve as illustrative examples throuhout this study. A completely disclosure ordered type space is iven by (T, d ), with T = {t 1,..., t n } and i j t i d t j. That is, types with hiher indices can report all types with lower indices. An empty disclosure ordered type space (T, d ) is iven by t d t t = t. That is each type is forced to truthfully reveal his type Strateies, Equilibria, and Preliminaries A feasbile stratey for the sender is σ : T T where Supp(σ t ) M(t) t 21. A pure stratey for the receiver is a : T A which specifies an action choice in response to each messae 22. A Bayes Nash equilibrium is strateies for the sender and receiver such that, σ t (s) > 0 = s ar max s M(t) a(s ) and a(s) = ar max a E[U R (a, t) σ, s] s Supp(σ). We omit the implications of perfect bayesian equilibria since we focus on a stroner refinement. We focus on the receiver optimal equilibrium. A number of studies (Ben-Porath et al. (2017), Sher (2011), Glazer & Rubinstein (2004)) have found that commitment is of no value in this disclosure ame. This means that there exists an equilibrium in which the receiver s utility is equivalent to that when he can commit to a stratey before receivin the sender s messae. HKP show that this (receiver optimal) equilibrium is found throuh their truth leanin equilibrium refinement below. A pair of strateies, σ and a is truth leanin if they 21 σ t (s) refers to the probability that type t declares type s. 22 Because the receiver s utility is strictly concave, it is without loss to focus on pure strateies 12

14 satisfy the followin conditions, t ar max s M(t) a(s ) = σ t (t) = 1 σ t (s) = 0 t T = a(s) = v(s) As HKP state, truth leanin says that if the sender is indifferent between truthful revelation and some other report, he truthfully reveals with probability one. On the receiver s side, upon observin an off path messae, the receiver assumes it is a truthful messae. HKP prove existence, but throuh a fixed point arument. In contrast, we will show existence throuh an explicit construction. Althouh it is not the only refinement that ensures that the equilibrium is receiver optimal, focusin on truth leanin strateies will round the analysis. Before movin on to the novel part of our analysis, we recall one result from HKP concernin truth leanin equilibria. For any iven equilibrium denote the payoff to the sender of type t by the function π : T R. Throuhout the paper, we will sometimes refer to π as the sender payoff vector Lemma 2. If (σ, a) constitute a truth leanin equilibrium, then for every t T exactly one of the followin holds, σ t (t) = 1, and π(t) = a(t) v(t) (3) σ s (t) = 0 s, and π(t) > v(t) = a(t) (4) Lemma 2 says that in any truth leanin equilibrium, we can split the sender s types into two roups - those who lie and those who tell the truth. The types that tell the truth obtain a lower action than their type commands, while the types that lie obtain a hiher action than their type commands. As we will focus solely on truth leanin equilibria or receiver optimal equilibrium, we will often henceforth refer to both as equilibria. In addition, when we refer to best responses, or feasible strateies of either the sender or the receiver, we implicitly refer to best responses and feasible strateies that satisfy the above refinement. 3. Equilibrium Characterization In this section we fully characterize the equilibrium, and provide two simple ways to calculate it. We also show how these results easily establish uniqueness of the sender equilibrium payoff vector. 13

15 3.1. Equilibria as Partitions We bein by noticin that an equilibrium is associated with a partition of the type space, P = {P 1, P 2,..., P m }, such that any type in a iven part obtains the same action. More specifically, let the set of available sender payoffs in an equilibrium be {π 1 <... < π m }, and let P i {t : π(t) = π i }. Because each type in P i obtains the same payoff, and the sender s payoff is strictly increasin in the action taken, all types in P i also obtain the same action with probability 1. We call this partition into payoff or action equivalence classes the equilibrium partition 23. In addition, for each type t, the set of types that t declares with positive probability must also be in the same payoff equivalence class (or part of the equilibrium partition) as t. In this sense the stratey of types in each part of the equilibrum partition is self-contained, i.e. t P i = Supp(σ t ) P i. This means there exists a sender stratey for the types in each P i, such that the best response of the receiver is to choose the same action for all on path declarations in P 24 i. Usin Lemma 1 we can deduce that this action must be equal to V (P i ). That is, the receiver best responds to the prior belief conditioned on t P i. This means that the types in P i are poolin, in the sense that the receiver treats each type in P i the same and as if he only knew that the type were in P 25 i. If such strateies exist we call them poolin strateies and say that P i is poolable. We make this notion precise in the followin definition. Definition 1. A subset S T is poolable if there exists a feasible sender stratey for types in S, σ S : S (S) and a receiver best response a : S R such that s Supp(σ S ) a(s) = V (S), and s / Supp(σ S ) a(s) < V (S). Focusin on the concept of poolable sets is the first step in abstractin from complicated mixed strateies, and thereby makin the disclosure ame more tractable. From the above discussion, we know that the payoff equivalence classes of an equilibrium (P 1,..., P m ) must each be poolable sets. How does the disclosure order, d, relate to the payoff equivalence classes of an equilibrium? As the equilibrium action is hiher for hiher parts, it must be that no type in t P j 23 The fact that any equilibrium partitions sender types into action equivalence classes holds for all equilibria and not just those satisfyin the truth leanin refinement. 24 In addition, the action for off path declarations must be less than π i. 25 This does not mean that iven a declaration in P i the receiver s belief about the type is the same as the prior conditioned on t P i. Nor does it mean that the belief is the same followin each on path declaration in P i. The only conclusion is that the set of beliefs followin on path declarations in P i induce the receiver to take the same action as that of the prior conditioned on t P i. 14

16 has the ability to declare any type in s P k when j < k. Otherwise, t would deviate to the stratey of s and obtain a strictly hiher payoff. We call a partition that satisfies this property an interval partition. We formalize this concept in the context of arbitrary partial orders. Definition 2. Let (X, ) be a partially ordered set. An interval partition of (X, ) is P = (P 1,..., P m ) such that, M (P k ) P j = j > k In other words, an interval partition is an ordered partition, such that no element from a lower part dominates any element from a hiher part accordin to the oriinal partial order. Remark 1. We use the term interval partition, because any interval partition P = (P 1,..., P m ) of the reals (R, ), has that each part is an interval, i.e. P i = [a, b] i with (extended) real numbers a b. In a partial order, the set of interval partitions is larer and has less structure. For example, if the partial order is empty, then every partition is an interval partition. It turns out that this is the final requirement for an equilibrium partition. We summarize the previous discussion below in our preliminary characterization of the equilibrium partition. Proposition 1. Let π : T R with equivalence classes s {π(s)} = {π 1 <... < π m } and let P i {s : π(s) = π i }. π is an equilibrium sender payoff vector P i is a poolable set i (5) (P 1,..., P m ) is an interval partition of (T, d ) (6) π i = V (P i ) i (7) This result ives us a partial road map to findin equilibrium payoffs 26, however it leaves a fundamental question unanswered. Given a sender payoff vector π, (6) and (7) are relatively easy to check. However condition (5) is more opaque. The next section characterizes poolable sets in terms of the primitives of the model. 26 Before movin on we note which parts of Proposition 1 depend on the refinement under consideration. The only aspect that is specific to truth leanin equilibria is our definition of poolable sets. This definition requires that the strateies that ensure poolin satisfy the refinement. If we were to remove these conditions from the definition of poolable sets, Proposition 1 would still characterize equilibria without chane. 15

17 3.2. Downward Biased Functions and Poolable Sets A natural intuition is that poolable sets will involve types of lower value pretendin to be types of hiher value. This says that a poolable set will involve hiher value types that are less dominant in the disclosure order and lower value types that are more dominant in the disclosure order. With the correct formalization presented below, this intuition turns out to be correct 27. Definition 3. The set function H : 2 X R is downward biased on (X, ) if H(M ( X)) H(X) X X (8) A function is downward biased when all lower contour sets have hiher value than the set as a whole. If the best response function V is downward biased on (S, d ), we have that V (M( S) S) V (S) S S (9) Therefore V is downward biased on (S, d ) if less dominant subsets accordin to the disclosure order of S command a hiher optimal action than the set S as a whole. The definition requires that the comparison in (8) hold in a very stron sense: all lower contour subsets must be of hiher value than S. We will see in the next result that downward biased is the key concept in characterizin poolable sets. It is worth notin that as a consequence of Lemma 1, an equivalent upper-contour version of (9) can be stated as follows V (B( S) S) V (S) S S (10) That is, upper contour subsets of S-more dominant subsets of S accordin to the disclosure order-have lower value than the set as a whole. Althouh for simplicity, we write that condition (8) must hold for all S S, this formulation involves sinificant redundancy because many subsets will have the same lower contour set within S. This means that checkin this condition is easier than the above formulation suests. This is apparent in the followin two examples. Example 1. Consider that the disclosure order is the empty order, i.e. t d t t = t. In this case, any subset of S is a lower contour subset. Thus V is downward biased on (S, d ) if and only if v(s) = V (S) s S, i.e. the optimal action is constant across types. 27 We state this definition for eneral partial orders and set functions because it will be useful in Section 4. 16

18 Example 2. Consider that the receiver s utility is quadratic loss and let the disclosure order be complete on S = (s 1,..., s m ) where s i d s j i j. If V is downward biased on (S, d ) we have that i = 1,..., m, V ({s 1,..., s i }) V (S). A more specific example is illustrated in Fiure 2. In this case S (s 1,..., s 8 ), (v(s 1 ),..., v(s 8 )) = (6, 4, 5, 3, 4, 2, 3, 1), and h is the uniform distribution. In this case V is downward biased on (S, d ) 28. This demonstrates that on a completely ordered set, V is downward biased on (S, d ) is strictly weaker than v is decreasin on (S, d ). v(s i ) i Fiure 2: V is downward biased on (S, d ) Remark 2. These two extreme examples illustrate the more eneral property that weakenin the disclosure order makes the condition that V is downward biased on (S, d ) more restrictive. More formally, consider two disclosure orders, d and d on S, such that d is a refinement of d, i.e. t, t S t d t = t d t. If V is downward biased on (S, d ), then V is also downward biased on (S, d ). We now present the characterization of poolable sets. Proposition 2. A set S is poolable V is downward biased on (S, d ). 28 As a verification, note that V ({s 1 }) = 6, V ({s 1, s 2 }) = 5, V ({s 1, s 2, s 3 }) = 5, V ({s 1, s 2, s 3, s 4 }) = 9/2, V ({s 1, s 2, s 3, s 4, s 5 }) = 22/5, V ({s 1, s 2, s 3, s 4, s 5, s 6 }) = 4, V ({s 1, s 2, s 3, s 4, s 5, s 6, s 7 }) = 27/7, V ({s 1, s 2, s 3, s 4, s 5, s 6, s 7, s 8 }) = V (S) = 7/2. 17

19 The arument that V bein downward biased on (S, d ) is sufficient for S bein poolable is complex as it requires construction of the poolin strateies. A sketch of the arument for necessity is as follows. Consider that S is poolable with some poolin strateies σ and a. In contradiction with Proposition 2, say that there exists some S S, with M M( S) S and V ( M) < V (S). By feasibility, the support of the strateies for the types in M must be contained in M, i.e s M Supp(σ s ) M. In addition, from Lemma 2, we know that other types t / M that declare messaes in M must have value less than the action they obtain, i.e. v(t) < π(t) = V (S). But this means that the types that declare messaes in M are either, (i) contained M and have lower averae value than V (S) by assumption, or (ii) are not in M and also have lower value than V (S) by Lemma 2. This implies by Lemma 1 that the receiver must take an action less than V (S) followin some declaration in M, contradictin that S is poolable. The above result is a powerful tool in abstractin from the complex equilibrium strateies that make poolin possible. Indeed, armed with this result, the remainder of this paper does not make any further reference to sender strateies. In addition, verifyin that V is downward biased on (S, d ) is far simpler than constructin a poolin stratey for the sender Solvin for Equilibrium Now that we have characterized poolable sets and equilibrium, we operationalize this understandin to find the equilibrium partition and sender payoff vector. We bein by showin uniqueness. Then we introduce a lemma that allows us to easily identify poolable subsets. Usin this lemma iteratively, we provide an alorithm that constructs the equilibrium partition. Finally we provide an explicit expression for the sender s equilibrium payoff vector. Corollary 1. The equilibrium partition and thereby sender payoff vector is unique. The arument for Corollary 1 is illustrative of how Proposition 1 and Proposition 2 work toether to produce stron implications. To see this, consider aain that T = (t 1,..., t n ) with a complete disclosure order iven by t i d t j i j. Suppose that there exists two equilibrium partitions (bipartitions for simplicity), P = (P 1, P 2 ) and Q = (Q 1, Q 2 ) both satisfyin Proposition 1. Accordinly, V (P 2 ) > V (P 1 ) and V (Q 2 ) > V (Q 1 ). If P Q, i.e. the equilibrium partition is not unique, then P 1 Q 1. Because both are interval partitions, it is without loss of enerality to suppose that P 2 Q 2 and Q 1 P 1. Also note that Q 1 P 2 is both a lower contour subset of Q 2 and an upper contour subset of P 1, i.e. M(P 1 Q 2 ) Q 2 = B(P 1 Q 2 ) P 1 = P 1 Q 2, and B(Q 1 P 2 ) P 2 = Q 1 P 2. This example is illustrated in 18

20 Fiure 3. Thus because Q 2 and P 1 are poolable, and because of Proposition 2, we have V (Q 2 ) V (P 1 Q 2 ) V (P 1 Q 2 ) V (P 1 \ Q 2 ) = V (Q 1 ) These imply that V (Q 2 ) V (Q 1 ), a contradiction. Q 1 P 1 Q 2 Q 2 Hiher Disclosure Order P 1 P 2 (T, d ) Fiure 3: The Equilibrium Partition is Unique We now move to the construction of the equilibrium. The first step is the followin lemma which selects poolable subsets. Lemma 3. For any subset S T, let J ar max S S V (B( S) S). Ŝ J B(Ŝ) S is poolable. Lemma 3 will imply that any maximal valued upper contour subset is poolable. The arument for this result is straihtforward. Say that B(Ŝ) S ˆB were not poolable. By usin the upper contour version of the definition of downward biased in (10), there exists R ˆB such that V (B(R) ˆB) > V ( ˆB). But this contradicts the maximality of ˆB. The ability to find poolable sets throuh maximization is extremely useful in findin the equilibrium partition. Consider applyin the above result iteratively as follows. Bein with the entire type set T and use Lemma 3 to find a poolable set P m. Now, remove P m and repeat the process. More specifically, apply Lemma 3 to T \ P m to find another poolable set P m 1. Next consider T \ (P m P m 1 ) to find poolable P m 2, and so on until every type is in some P i. This alorithm, which we call partition into poolable sets, enerates the equilibrium partition described in Proposition 1. We formalize this result below. Proposition 3. The output of Partition into Poolable Sets (P 1,..., P m ) is the equilibrium partition, with sender payoff vector iven by t P i π(t) = V (P i ) The indexin in the output partition is reversed from that of Proposition 1. The hihest value part is actually P 1 and the action is decreasin in the index. This is why we correct the indexin in the last step of the alorithm. 19

21 Alorithm 1 Partition into Poolable Sets Initialize: Partially ordered type space (T, d ) 1: S 1 = T 2: i = 1 3: while S i do 4: P i = ar max Si S i V (B( S i ) S i ) 5: P i = ( P P B(P )) S i. 6: i = i + 1 7: S i = S i 1 \ P i 1 8: Relabel P j P i j 1 j i 1 Alorithm 1 produces a partition of T into disjoint sets (P 1, P 2,..., P m ). First, each P i poolable by Lemma 3. Because of the iterative maximization, types in hiher index parts obtain strictly hiher actions. Also note that at each stae we remove an upper contour subset of the remainin set of elements. This means that the output is an interval partition. In summary, the requirements of Proposition 1 are satisfied and the output is the equilibrium partition. The above demonstrates a simple way to find the equilibrium partition in a eneral disclosure ame. The alorithm concludes in at most T iterations, so it is fairly quick. This method is so simple in fact that we can o further in characterizin the equilibrium. In the followin result, we present an explicit expression for the sender payoff vector. Proposition 4. Let π : T R be the equilibrium payoff vector. π(t) = min {S a:t S a} max V (M(S a) B(S b )) (11) {S b :t S b } The expression in (11) exactly corresponds to the equilibrium utility (and thereby obtained action) of the sender of type t. This depends on the choice of two sets S a and S b. For any such choices let P (S a, S b ) M(S a ) B(S b ). Notice that for any feasible choices of S a and S b, t P (S a, S b ) and so the above problem is well defined. The above result implies that if S a, S b solve the problem in (11), then P (S a, S b ) is exactly the poolable part of the equilibrium partition that contains type t. It is also worth notin that we could rewrite the equaltion in (11) as, π(t) = max {S b :t S b } min V (M(S a) B(S b )) {S a:t S a} 20