Communication with Detectable Deceit

Transcription

1 Communication with Detectable Deceit Preliminary Wioletta Dziuda 1 Kellogg School of Management Managerial Economics and Decision Sciences wdziuda@kellogg.northwestern.edu February 2012 ( rst draft: January 2011) 1 This project is a fruit of my conversations with Markus Brunnermeier, for which I express my gratitude.

2 Abstract I analyze a model of communication in which players have no common interest, messages are cheap talk, but the deceit is detectable with positive probability. The sender observes his type and sends a message to the receiver. If he lies, his message becomes inconsistent with positive probability. In any informative equilibrium, the intermediate types of the sender reveal themselves, while the extreme types of the sender pool. The amount of information revealed increases with the detectability of lies. In the extreme cases, the model encompasses cheap talk (no information revelation) and veri able evidence (full information revelation). I extend the model to a T period game and show that there exists an equilibrium in which the probability of informative communication is positive in each period. In each period, only the highest types reveal the truth, and the threshold for truthtelling decreases with time. The receiver su ers from multiple rounds of communication when the probability of lie detection is high enough and bene ts otherwise. Hence, when lie detectability is high, the receiver would like to commit to not giving second chances.

3 1 Introduction Communication oftentimes takes a form of storytelling. Hardly ever a simple "yes" or "no" su ces for an answer, and a supporting story is expected. This observation has important consequences for lie detectability. Since the memory of a self-experienced event is likely to be vivid, an account of an event that has occurred is likely to be internally and externally consistent, but an account based on fabrication may not. To illustrate this point, consider the following example. Imagine that yesterday you invited your friend Alice to see a movie, but she declined, claiming to have an important social engagement. Suppose that you only care whether your friend Alice was truly busy, in which case you will remain her friend, or simply does not enjoy your company, in which case you will end the friendship. There are many stories consistent with Alice being busy, and each story may involve many details. She could have been at Stained Glass with a job market candidate talking about the supremacy of Italian over French wines, or she could have spent the evening with her high-school friend Bob, who happened to be visiting Chicago for his sister s wedding. When Alice is telling the truth, she simply needs to recount her activities from yesterday. If she is lying, however, she needs to create a story. But creating a consistent story is not easy. For example, Alice may claim to have spent the entire evening bar hopping with Bob, but later she may accidentally reveal that she watched an episode of Grey s Anatomy that aired that night. In such a case, you know that these two events are mutually exclusive, and you most likely conclude that Alice is lying. Psychologists have long recognized that when communication involves storytelling, lies may be detectable even in the absence of hard evidence. The Criterion-Based Content Analysis (CBCA), for example, which is used extensively in European courts, has been developed on the premise that an account derived from the memory of a self-experienced event will di er in content and quality from an account based on fabrication or imagination. 1 CBCA 1 See Vrij (2005). CBCA is designed to decipher truthful from false memories in children and is a part of the Statement Validity Analysis (SVA). SVA consists of structured interviews, systematic analysis of verbal statements and statement validity checklist. CBCA refers to the quality analysis of the content, while SVA refers to the overall diagnostic procedure. An excellent overview of the CBCA can be found in Vrij (2009). 1

4 is used, for example, in sexual abuse trials in which the accuser and the accused agree that the sexual act occurred, but they disagree on whether it was consensual. In such cases, there is little physical evidence, but the courts routinely provide rulings, relying mainly on the testimony of the interested parties. 2 In this paper, I build a stylized model that incorporates this basic insight from the psychological literature. Two players, a sender and a receiver, engage in one round of communication. The sender observes the state of nature (his type) and sends a cheap-talk message to the receiver. If he tells the truth, the receiver observes his message. If he attempts to lie, his message either reaches the receiver unaltered, or appears to the receiver as inconsistent. The receiver observes only the message and takes an action. I assume that the players have no common interest: the receiver wants to take an action that matches the state, while the sender wants her to take the highest possible action. That is, I am considering a setting in which there would be no communication if the talk were purely cheap. I show that in any informative equilibria, the sender types can be divided into three groups. The rst group consists of the highest types. These types tell the truth, but their messages are taken by the receiver with a grain of salt, and result in actions lower than these sender s types. The second group consists of the lowest types. These types lie, pretending to be the highest types. Hence, when observing a high message, the receiver does not know whether the state is high and the sender is telling the truth, or whether the state is low and the sender is successful at lying. As a result, she takes an action that is lower than the messages would imply if taken at face value. When the receiver observes an inconsistent message, however, she knows that the sender is lying. In such a case she infers that the state is low, but she does not learn the state perfectly. The third group of sender types consists of intermediate types. These types are low enough not to be mimicked by liars, but high enough to prefer to tell the truth than to risk that their deceit is detected and that they end up pooled with the lowest types. Hence, in any partially revealing equilibrium, the receiver 2 Formally, one could model this as a game between the accuser and the judge, in which the judge wants to take an action that matches the state of the world, and the accuser wants to be acquitted. Such a game is analyzed in this paper. 2

5 can perfectly distinguish between intermediate types. The incentives for truthtelling arise in the model endogenously. In equilibrium, all detected liars induce the same action. This makes lying costly for higher types. They prefer to tell the truth, even if that does not induce the highest possible action, to pretending that the state is higher but risking at the same time being pooled with the lowest types. Perhaps intuitively, more information is revealed when lies are likely to be detected. If the deception is more likely to be discovered, the sender lies in fewer states. As a result, an inconsistent message induces a lower action, which decreases the incentives to lie even further. As the probability of lie detection approaches one, almost all information is revealed in equilibrium. Note that this is not an immediate result of high lie detectability, as the detection technology only allows to identify the instance of lying but not the true state. Information revelation is a consequence of players strategic considerations. On the other hand, as the probability of lie detection approaches zero, almost no information is revealed in equilibrium. Hence, for the case of no common interest, the model can reconcile the polar predictions of the two canonical models of communication: no information transmission in cheap-talk models and complete unraveling in veri able information models. The informative equilibria of this game are not renegotiation proof. To see this, note that in equilibrium the types of the sender that lie and are detected, induce the same action. Hence, after lying proves unsuccessful, but before the receiver takes an action, the highest of these types have an incentive to send another message in which they reveal the truth. Such a message would be bene cial to the receiver as it would help her to di erentiate between di erent types further. Hence, ex post both players may bene t from the second round of communication. Multiple rounds of communication can be interpreted as the receiver giving the liars second chances. To address a question whether giving second chances bene ts the receiver, I analyze a T period extension of the model. I show that there exist equilibria in which there is informative communication in every period. In each period, all types below a certain threshold lie, and this threshold decreases over time. Receiver takes action after the rst consistent message, but this action decreases 3

6 over time. I show that the receiver does not necessarily bene t from multiple rounds of communication. 3 Whether she does, depends on the probability of lie detection. Giving second chances has two competing e ects. On the one hand, an additional round of communication allows the receiver to discriminate more precisely between the types who lie in the initial periods. On the other hand, anticipating second chances, more types of the sender lie in the initial rounds. I show that if lies are hard to detect, many types of the sender lie in the rst period. In such a case, discriminating among the liars in the subsequent periods is quite bene cial to the receiver, and the rst e ect dominates. On the other hand, if lying is di cult, few types of the sender lie; hence, the rst e ect is small. As a result, the receiver prefers to commit herself to not giving second chances in order to decrease the incentive to lie for the intermediate types. This project is a part of a larger research agenda that tries to incorporate the psychological insights on storytelling, lying, and detection into the economic literature on communication. As such, it does not attempt to model these processes in detail, but focuses instead on the impact of lie detectability on information transmission in standard communication settings. A fully- edged model of storytelling and lie detectability is beyond the scope of this paper, but is by no means less interesting, and is left for future research. To illustrate better how the assumptions of my model can follow from the psychological ndings, however, I discuss a basic outline of such a model in Section 2.1. The paper is organized as follows. Section 1.1 contains an overview of the related literature. Section 2 presents the model. Section 3 describes the equilibria. Section 3.3 analyzes how the informative equilibrium changes as the lie detectability and the prior distribution of types changes. In Section 4, I analyze a T period game. Section 5 concludes. 3 In the current setting, the sender is risk neutral, and hence indi erent between any number of rounds of communication. 4

7 1.1 Literature overview Most of the literature on strategic information transmission falls into one of two categories. One approach, pioneered by Crawford and Sobel (1981) and Green and Stokey (2009), assumes that messages exchanged between the communicating parties have no direct impact on these parties payo s. In other words, lying is costless. On the other side of the spectrum, there is literature on veri able disclosure, pioneered by Grossman (1981), Milgrom (1981), and Milgrom and Roberts (1986), which assumes that the informed party can provide only statements that are consistent with the truth. Hence, the informed party can attempt to manipulate the uninformed party solely via selective disclosure of evidence. There are many actual communication situations that fall into one of these categories. In casual conversations in which punishments for lying are not enforceable and there is no hard evidence, cheap talk is a tting assumption. In legal trials and nancial disclosure, on the other hand, veri able evidence is likely to exist. However, these models do not seem to exhaust the set of possibilities. This paper ts best the line of research on communication with costly misrepresentation. Ottaviani and Squintani (2006) and Chen (2007) assume that the receiver may be strategic or naive. Since the naive receiver takes the messages at face value, in ating the message too much may be costly. This transforms the cheap-talk game into costly signaling. This transformation, however, does not allow for in uential communication when, like in my model, there is no common interest between the players. 4 Kartik, Ottaviani and Squintani (2007) and Kartik (2009) assume that lying carries an exogenous cost. Kartik, Ottaviani and Squintani (2007) analyze an unbounded state space and obtain full information revelation. Every type of the sender in ates his message, but only until the bene t from in ating equals the cost. Since the cost of each message is type dependent, di erent types achieve this point at di erent messages. The paper closest to mine is Kartik (2009). Like me, Kartik (2009) analyzes a bounded state space. He obtains language 4 Chen (2011) additionally assumes that the sender may be honest or strategic, which implicitly introduces a cost of lying that varies across types. This is su cient for in uential communication even when there is no common interest between players. 5

8 in ation and full information revelation of low types, but high types pool at the highest message. Unlike in these models, in this paper the cost of lying is determined endogenously. I assume that lying is costless per se, but since lying can be detected, in equilibrium it becomes costly for high types. However, it is not costly for all types, as the lowest types strictly bene t from lying. An interesting feature of the model of Kartik (2009) is that it encompasses the cheap-talk and veri able disclosure literatures using a single cost parameter. My model shows that these two literatures can be also reconciled using the detectability of lies. One of the motivations for an exogenous cost of lying assumed in the cited papers comes from a supposition that people have intrinsic aversion to lying. Some experimental work seems to support this conjecture; see, for example, Gneezy (2005), Hurkens and Kartik (2009), and Sánchez-Pagés and Vorsatz (2009). Although this is probably true in certain environments, Vrij (2008) argues that many people experience no guilt or fear associated with lying; hence, assuming that lying is costless is a tting assumption for many interactions. Subrahmanyam (2005) builds a model of fraud in which the sender can distort the probability of the receiver observing a correct signal about the state. The fact of this distortion is revealed ex post, and unlike in my work, Subrahmanyam (2005) allows for an exogenous ex-post punishment of any distortion. My paper is also related to the literature on veri able disclosure with uncertain evidence. Dye (1985) analyzes a model in which the sender may or may not receive hard evidence about the payo -relevant state, and the fact of receiving it is private knowledge. Dye (1985) shows that in this setting unraveling fails: the informed sender reveals the evidence when the state is good, and withholds it otherwise. My model can be embedded in this framework with an additional assumption that the sender can fabricate evidence, but this fabrication is not guaranteed to be successful. 5 Finally, several papers look at multiple-rounds of communication. Aumann and Hart (2003) and Krishna and Morgan (2004) show that multiple stages of communication may be 5 See also Verrecchia (2001) and Shin (2003). Shin (1994a), Shin (1994b), and Dziuda (2011) analyze di erent settings in which the receiver is uncertain about what kind of information the sender has. 6

9 bene cial if they introduce noise that relaxes the incentive compatibility constraint of the sender. Es½o and Fong (2010) analyze a multiple-sender multiple-stage game and show that the threat of delay can induce the senders to reveal the state. In those papers the presence of multiple rounds facilitates information transmission, but unlike in my model the payo relevant information is revealed only once. Golosov et al. (2011) show that multiple rounds of communication bene t the receiver if the decision maker makes multiple decisions over time. 6 I show that the receiver may bene t from multiple rounds of communication even if the sender is risk neutral and does not discount the future. Moreover, the current paper shows that multiple rounds of communication may actually harm the receiver: she may prefer to commit to one round of communication if lying is highly detectable. 2 Model In this section I present the one-period game. In Section 4, the model will be extended to allow for multi-stage communication. There are two players: a sender and a receiver. The state space is = [0; 1] ; with a typical element denoted by : Let f () be the prior over ; which is common knowledge, and F () be the corresponding distribution function. The receiver s action space is A = [0; 1], with a typical element denoted by a: The sender s message space is M = [0; 1] : The game proceeds as follows. After observing the state, the sender sends a message m 2 M. The message is then transformed by a communication technology that will be speci ed later. The receiver observes the transformed message and chooses an action a 2 A: The receiver s payo is u R (; a) = (a ) 2 ; and the sender s payo is u S (a) = a: That is, the receiver wants to choose an action that matches the state, while the sender wants the receiver to take the highest possible action. 7 A sender who observes will be called a sender of type : I will say 6 Carrasco and Fuchs (2009) show that in a setting in which both players have private information and make decisions jointly, an optimal mechanism can be implemented by a protocol with multiple rounds of communication. Hörner and Skrzypacz (2010) show that the seller can bene t from gradual information revelation in a setting with no con ict of interest but payments for information. 7 Most of the results extend easily to a strictly increasing u S (d) ; and u R (; d) that for each a admits a unique max d u R (; d) and is supermodular in (; d) : Those assumptions are likely to be relaxed in the subsequent drafts of this paper. 7

10 that the sender tells the truth if m = ; otherwise the sender lies. Before the sender s message reaches the receiver, it is transformed using the following communication technology. If the sender tells the truth, the receiver observes m: If the sender lies, the receiver observes either m or m inc ; where the latter can be interpreted as a statement "message is inconsistent". Formally, let (m; ) be the message observed by the receiver if the state is and the sender sends m. Then (m; ) = 8 >< >: m inc with probability p if m 6= ; m otherwise. (1) The message m inc is called an inconsistent message. The remaining messages are called consistent. Since m inc signals lying, the parameter p measures the detectability of lies. If p = 1; lies are perfectly detectable, and if p = 0; lies are indistinguishable from the truth. 8 The strategy of the sender is a probability function (j) :! M: The strategy of the receiver is a function a () : M [ m inc! A: 9 Let b (j) be the probabilistic belief of the receiver over upon observing a message : In what follows, E [] denotes the expectation derived with respect to the prior, and E [j] denotes the expectation derived with respect to the beliefs b (j) : I look for Perfect Bayesian Equilibria equilibria. Formally, a Perfect Bayesian Equilibrium (later referred to as simply an equilibrium) is characterized by measurable (mj) ; b (j) ; and receiver s mapping a (), such that: 1. If m 2 M is in the support of (j) ; then m 2 arg max m2m a (m) I (m=) + (1 p) a (m) + pa m inc I (m6=) ; 2. For any ; the receiver chooses a () such that a () = E [j] ; 8 The sender is not allowed to send an inconsistent message strategically. Allowing for such messages, would alter some arguments in the proofs, but the main results would hold. 9 Assuming that the receiver is limitted to playing pure strategies is without loss of generality. 8

11 3. The belief b (j) is derived using Bayes rule from the strategy of each expert s type and the prior distribution over ; whenever possible. The symbol I () denotes an indicator function. One technical comment is in place. Given that there is a continuum of possible messages, in equilibrium some of the messages might be sent with probability 0; in which case the beliefs of the receiver b (j) and hence a () are not pinned down by Bayes rule. However, b (j) and a () must satisfy Bayes rule on any set of strictly positive measure. Hence, some statements in the propositions will be true up to the sets of messages of measure zero. As it is common in cheap-talk models, the game admits a plethora of equilibria. To restrict their set, I focus on equilibria with the property that out-of-equilibrium consistent messages are believed. Formally, if there exists a message m 2 M such that no type sends this message in equilibrium, then upon seeing = m; the receiver believes that = m; and takes action a (m) = m: In the appendix, I show that such re nement is equivalent to D1 Criterion of Cho and Kreps (1987) In particular, I focus on equilibria in which some information is revealed with positive probability. De nition 1 An equilibrium is informative if there exist two messages and 0 that are on the equilibrium path and a () 6= a ( 0 ) : Any equilibrium not satisfying these conditions is uninformative. A few comments are in order. First, the possibility of inconsistent message m inc re ects the observation that a lying player runs a risk of producing a story that is internally inconsistent. Since there is only one such message, upon seeing an inconsistent message, the receiver does not know what message the sender sent. Clearly, this is not a truly realistic assumption: 10 See also Banks and Sobel (1987). 11 In standard cheap-talk games re nements that restrict out-of-equilibrium beliefs tend not to have a bite. The reason for this is that since all messages are costless, for any proposed equilibrium with some unused messages, one can construct an equilibrium with the same outcome in which all messages are used. The same is not true in the current model. If some equilibrium prescribes a type to reveal the truth, then altering the equilibrium by requiring this type to send some previously out-of-equilibrium message is not innocuous, as those messages may turn out inconsistent, altering the outcome of the game. 9

12 even if a story contains contradictory details, in general it is possible to understand what claim the speaker tried to make. As it will become clearer after the equilibrium analysis, however, all equilibria of this game continue to be equilibria when this assumption is relaxed. Hence, this assumption amounts to being a re nement in which the receiver treats all lies equally. Second, one can argue that it is easier to create a false account, if this account is close to the truth, in which case p in the communication technology should be a function of and m:this extension is left for future research. The model presented in this section is arguably quite stylized. Modeling carefully the process of lie detection is an interesting exercise, but it is beyond the scope of this paper and is a separate research project. Instead, the focus of this paper is on the possible implications of lie detectability on the amount of information transmitted; therefore, the details of communication are abstracted away. However, in Section 2.1 I outline a model that could lead to the proposed communication technology in hope of convincing the reader about the plausibility of the assumptions made in this paper. An anxious reader can skip Section 2.1 and proceed directly to the equilibrium analysis. 2.1 An auxiliary model of storytelling Let b denote the string of bits that come after the binary point in the binary representation of 2 : 12 For example, if = 1 4 ; then its binary representation is 0:01000 (0) ; and b = (0). Let b k denote the rst k digits of b : Let S 0 and S 1 be two abstract sets with a nite number of elements each, and assume that S 0 \ S 1 = ;: For each b ; construct a set M k of strings of length k using the elements of S 0 [ S 1 in the following way: whenever b calls for 0; the string contains an element from S 0 ; and whenever b calls for 1; the string contains an element from S 1 : Let M = [ 1 k=1 M k : The message space is M = [ M : Let C M be called a set of consistent messages. The interpretation of the above is as follows. The set corresponds to the payo -relevant states, M is the set of all stories that could correspond to the state, but some of these stories are inconsistent. That is, only a certain combinations of 12 In this setting, a binary representation is assumed to have in nitely many bits, as it is innocuous to add in nitely many zeros to the end of the binary representation of any non-integer. 10

13 elements from S 0 and S 1 are consistent, and C denotes all such combinations. The sender and the receiver are endowed with limited memory m S and m R respectively. The memory may be xed or random. When recalling the true state of nature ; the sender remembers b m S +x; and can costlessly send m ms +x = b m S +x to the receiver. When trying to make a false claim 0, the sender has to create a message m k 2 M 0: The sender knows the binary representation of 0 ; and has to choose which elements from S 0 [S 1 to use to create this representation. However, due to the limited memory the sender can check internal consistency of only m S elements of m k. Adding an extra element carries a risk of creating m k 62 C; and therefore, of revealing that the sender is lying. Note that x > 0 is supposed to re ect the observation that true events are remembered more vividly than the events never experienced before. Similarly, the receiver can verify internal consistency of only m R bits of m k : In the simplest case, m R = 1; m S < 1 and x = 1: In this case, the sender always sends m 1 ; and the lying sender faces a probability that m 1 62 C; which amounts exactly to the assumptions placed on the communication technology in the main model. 3 Equilibrium 3.1 Preliminaries The following Proposition describes all equilibria in two extreme cases of p = 0 and p = 1: Proposition 1 When p = 0; then in all equilibria, no information is revealed. When p = 1; then in all equilibria, all information is revealed. Proof. All proofs are in the Appendix. When p = 0; i.e., when lying is easy and a liar can tell any consistent story with certainty, the model becomes a standard cheap-talk model. Since there is no common interest among players, no information can be revealed in equilibrium. On the other extreme, when p = 1; lying is in nitely di cult. That is, either the sender tells the truth, or he reveals that he is lying. In this case, the game turns into a game of veri able disclosure, and the standard unravelling argument delivers full disclosure of information. Hence, using a single parameter 11

14 p; the model reconciles the polar predictions of two canonical models of information transmission. 13 When p 2 (0; 1) ; the message of the sender is not veri able, but the fact of it being consistent signals truthfulness. However, as the next proposition shows, the uninformative equilibria still exist. Proposition 2 For any p < 1; there exist uninformative equilibria. In any uninformative equilibrium, a positive measure set of sender s types with > E [] lie with positive probability. The intuition for the existence of uninformative equilibria is as follows. Assume that each type lies with probability one by randomizing uniformly over all messages that do not correspond to his type. Given this, upon observing a consistent message, the receiver must believe that it is a lie, even if the probability of a lie being consistent is small. Given that every type randomizes uniformly, the receiver cannot infer anything from any message. As a result, lying is an equilibrium response of the sender. Uninformative equilibria in this model are somewhat less appealing than the babbling equilibria in cheap-talk models, as they rely crucially on even the highest types lying: they require that all messages m > E [] are used, otherwise they would have to generate a (m) = m > E [] ; which would create an incentive to deviate. 3.2 Informative equilibria Proposition 3 characterizes all informative equilibria. Proposition 3 In any informative equilibrium, there exist two thresholds t and l > t; de ned implicitly by l = (1 p) R t 0 f () d + R 1 l f () d ; (2) F (t) (1 p) + 1 F (l) t = (1 p) l + p R t 0 f () d ; (3) F (t) such that the sender 13 Kartik (2009) bridges the gap between those two extremes by assuming that lying is costly. 12

15 a. reveals the truth when > t; b. randomizes over m 2 [l; 1] according to a probability density function h (mj) when < t: The receiver c. takes a () = when < l; d. takes a () = l when l; e. takes a(m inc ) = E [j < t] : The following picture demonstrates the equilibrium graphically. Proposition 3 says that the sender tells the truth if and only if his type is higher than t: Call t the truthtelling threshold. When the sender lies, he sends messages higher than l: Call l the lying threshold. Proposition 3 says that l > t; hence, in equilibrium the intermediate types of the sender 2 (t; l) reveal themselves. The extreme types of the sender pool by sending high messages. When these messages turn out consistent, the receiver does not know whether they come from a high type, or some low type who lies. Hence, the receiver takes an action l, which is lower than the received message would imply if taken at face value (that is, if believed to be truthful). When these messages turn out inconsistent, the receiver knows that the state is low and takes an action that maximizes her payo conditional on the sender lying. The intuition for this proposition is as follows. In equilibrium, each message is either (i) used as a lie with positive probability, (ii) sent by a truthteller only, or (iii) o the equilibrium path. If (ii) and (iii), then either by sequential rationality (in case of ii) or by assumption (in case of iii), a (m) = m: Let l be the highest possible action induced by a consistent message. Then the last observation implies that all messages m l must induce l: Moreover, all types l strictly prefer to tell the truth and induce the highest possible action l for sure. This in turn implies that only types < l can possibly be sending m < l: Hence, such messages are either out of equilibrium or are used by the truthtellers; therefore, they induce a (m) = m: 13

16 Figure 1: The horizontal axis depicts the type and the message space at the same time. The vertical axis depicts receiver s actions. Hence, for < l; the payo from truthtelling is increasing in ; while the payo from lying is type independent, which implies the existence of the truthtelling threshold t: The threshold type t must be indi erent between telling the truth and lying. This indi erence condition is described by equation 3: telling the truth induces a = t with certainty, while lying results in l if successful and in being pooled with low types if unsuccessful. The lying threshold l is determined by the receiver s inference. Since l is a message that is sent only by l; upon seeing l the receiver cannot believe that the state is higher than l; hence, a (l) l: But if a (l) < l; then the lying types of the sender would prefer to send a message slightly lower than l; as those messages are taken at face value and result in a (m) = m > a (l) : Hence, a (l) = l: And nally, a (l) must be the same for all messages used by the lying types; and by sequential rationality, it must be equal to the expected state conditional on m l; which is exactly the right-hand side of equation of 2. To complete the characterization of the informative equilibria, one needs to specify how types t lie. In equilibrium, liars are indi erent between all m l; but their strategies must induce the receiver to take a constant action a () = l for all l: There are many strategies that ful l this requirement, but perhaps a natural one is as follows. All types 14

17 t randomize according to a type-independent density function h (m) with the support on [l; 1]. If h () is appropriately increasing in m; it will o sets the fact that higher messages are associated with higher truthtelling types. The details of such a strategy are not very illuminating, and therefore are delegated to the appendix. In the equilibria of Proposition 3, some information is revealed because lying becomes costly for high types: a lying sender risks being regarded as one of the low types. However, it is important to stress that this cost arises endogenously, and depends on the pool of types that lie. The last observation suggests that the equilibrium does not have to be unique. If in equilibrium t is low, then an inconsistent message induces a low action, and this deters types higher than t from lying. However, if t increases, the receiver s action after an inconsistent message must increase as well. This increases the incentives to lie, and may therefore induce more types to lie, generating a new equilibrium with a higher truthtelling threshold and a lower lying threshold. Proposition 4, however, gives su cient conditions for uniqueness of the thresholds. Proposition 4 If F () is log-concave, then for all p; the thresholds (t; l) de ned in proposition 3 are unique on (0; 1) (0; 1). Since many of the standard distributions are log-concave, for the rest of the paper, I assume that so is F (). 14 It is instructive to compare this model with models of costly lying. Kartik (2009) assumes that sending a message that does not correspond to the true state is associated with an exogenous cost. The cost of lying allows for partial information revelation. In the model of this paper, lying is costless and the uninformative equilibria always exist. However, partially informative equilibria can be sustained, because in partially informative equilibria, lying costs arise endogenously For example, the c.d.f.s of normal, uniform, logistic, exponential, and Weibull distributions are all logconcave. See Bagnoli and Bergstrom (2005). 15 Since in Kartik (2009) the cost of lying is increasing in the distance between the lie and the true state, and in the current model the probability of lie detection is independent of the extent of the lie, comparing the equilibria between these two papers is not very revealing. 15

18 A comment on the role of noise in this model is in place. It is well-known that in the standard cheap-talk models noise may improve communication; if the sender is risk-averse, noise may relax his incentive-compatibility constraint. 16 In this model, the sender is risk neutral; hence, even though the communication technology makes communication noisy, a di erent mechanism is responsible for the possibility of information transmission. 3.3 Comparative statics Proposition 5 states an intuitive result that more information is transmitted when the deceit is likely to be discovered. Proposition 5 Assume that F () is log-concave. Then: dl dt a. dp > 0; and dp < 0; b. lim p!1 l = 1; lim p!1 t = 0; and the equilibrium converges to a fully revealing equilibrium; c. lim p!0 l = lim p!0 t = E [] ; and the equilibrium converges to an uninformative equilibrium. Proposition 5 implies that low types of the sender lose and high types of the sender bene t from the increase in the probability of lie detection. As p increases, low types are less successful at pooling with the high types; and hence, consistent claims that the state is high are met with lower skepticism. Since fewer types lie, however, an inconsistent message induces a lower action, and as a result the overall payo of liars decreases. The amount of information revealed in equilibrium depends not only on the detectability of lies, but also on the prior distribution f () : Proposition 3 states that for any f () ; the intermediate states are revealed fully. Hence, intuitively, if f () puts more weight on the intermediate states, the true state should be revealed with higher probability. This is demonstrated in the example below. 16 See Blume, Board, and Kawamura (2007) and Goltsman et al. (2009). 16

19 Example 1 Fix p, and let (t; l) be the equilibrium thresholds when f () is uniform. It is easy to show that l = 1 t: Let f (; z) = z if 2 [t + "; 1 t "] ; and f (; z) = 1 z(2t+2") 1 (2t+2") otherwise. This transformation is demonstrated in the gure below. It is easy to show that (t; l) that satisfy equations (2) and (3) for f (), satisfy them also for f (; z) : Therefore, as z! 1 1 2t 2" ; the ex-ante probability that the true state is revealed goes to 1: In the example above, the equilibrium thresholds are not a ected because f (; z) is constructed in such a way that the relative probabilities of the states in which the sender pools are the same for each z. In general, however, this does not have to be the case. And if the thresholds (t; l) move toward each other as f (; z) shifts more weight to the middle states, less information may be revealed. The proposition below shows conditions under which a more concentrated prior results in higher probability of the receiver learning the true state. Proposition 6 Let f (; z) be a family of symmetric distributions parametrized by z; and di erentiable in both arguments, with dfz d > 0 for < 1 2 : Then d (F (l; z) dz F (t; z)) > 0 a. if F (;z) f(;z) 1; or b. if F (;z) f(;z) < 1 and p p 6= 1:17 17 Condition F (;z) < 1 is satis ed if f () is increasing for < 1 : Intuitively, d(f (l;z) F (t;z)) > 0 will be f(;z) 2 dz satis ed only if f () is not increasing too fast. I conjecture that one can nd relatively weak conditions such that part (b) of Proposition 6 holds for all p. 17

20 Note that condition dfz d middle states. 18 > 0 implies that as z increases, f (; z) puts more weight on the Proposition 6 implies that shifting weight towards the center increases the probability that a state is fully revealed, if either the density function f () is single dipped, or it is single peaked, but the probability of lie detection is high enough. 4 A multi-period game From the structure of the informative equilibria outlined in Proposition 3, it is easy to see that if given a chance to engage in another round of communication, the receiver and some types of the sender would strictly prefer to do so. To see that, note that in the partially informative equilibrium, the unsuccessful liars are pooled together and induce the same action. Hence, the highest of those types would like to separate themselves from the lowest types. If given another chance to communicate, they will do so by telling the truth. Since truthtellers are more likely to be consistent than liars, the receiver may nd it optimal to listen to the second message, as this will allow her to discriminate between the lying types. In many circumstances, it is unrealistic to assume that players are exogenously constrained to one interaction. Hence, since both players may bene t ex post from another round of communication, a multi-period game may be a better description of actual situations. Following this observation, in this section I analyze a T period version of the game. I characterize an equilibrium in which in every period some information is revealed with positive probability, and compare the payo s of the receiver across the games. The game proceeds as follows. The sender learns at the beginning of the game. After that, each period consists of two stages. In the rst stage of period, the sender sends a message m 2 M; and the message gets transformed into 2 M via a communication technology speci ed below. In the second stage, the receiver either takes an action a ( ) 2 A, in which case the game ends, or she asks the sender for another message. In the latter case, the game moves to the next period. There is no discounting. For this section, I assume 18 This condition implies the second order dominance. It is neither stronger nor weaker than monotone likelihood ratio dominance. 18

21 that is distributed uniformly. In a game with multiple rounds of communication, the sender may attempt to tell the same lie in many periods. I assume that if the sender lies successfully in period, then he can repeat the same consistent message in any subsequent period. Formally, if in period the sender sends m 6= ; and there exists 0 < such that 0 = m, then = m : Otherwise, the message is transformed using the communication technology outlined in (1). And nally, in each period the sender learns whether his lie has been consistent. The assumption that a successful lie can be repeated without fail may not perfectly describe the actual process of lying. In fact, many interrogation techniques rely on the premise that a repeatedly questioned liar may accidentally change the details of his story. However, this modelling assumption is not without merit, as long as one accepts that even after a reasonably long questioning, lies are detectable with probability less than 1: 19 Clearly, any equilibrium outcome of the T period game can be achieved in an equilibrium of the (T + 1) period game by prescribing the players to play an uninformative equilibrium in = 1; and to follow the strategies from the T period-game equilibrium from = 2 onwards. In particular, a T repetition of an uninformed equilibrium is an equilibrium. I nd such equilibria uninteresting and focus instead on equilibria in which in every period some types reveal information. Proposition 7 below describes such equilibria. It states that in each period, there are two thresholds (t ; l ) ; and the behavior of the sender is similar to his behavior in the one-period game. Types higher than the truthtelling threshold reveal their type, and all other types lie using high messages. The game ends after a consistent message. After an inconsistent message, the game moves to the next stage, with lower thresholds (t +1 ; l +1 ). Formally, de ne (; p) 1 p (1 p)p p 2p+1 ; and let f p+p 2 2 4p 2 +2p 3 +p 2 p 3 g T =0 be a sequence de ned recursively by t = 1 p t+1 t+1 ; p ; with T = To see that the current game can incorporate multiple rounds of interrogation, consider the following. Let p denote the probability of lie detection after multiple rounds of communication. Assume that in each period, the sender can be interrogated multiple times as long as his message is consistent. If the message is inconsistent, the game moves to the next period. Such a game is equivalent to the current game. 19

22 Proposition 7 There exists an informative equilibrium, which is characterized by a sequence of thresholds ft ; l g T =1 ; with t and l decreasing in, such that on the equilibrium path: a. For < T; if is consistent, then the receiver ends the game; otherwise, she continues the game; b. In each period ; all types 2 (t ; t 1 ) reveal the truth, and all types < t randomize over m 2 [l ; t 1 ]; c. If l ; then a ( ) = ; if 2 [l ; t 1 ]; then a ( ) = l ; and a T m inc = t T 2 : O the equilibrium path: d. If is consistent and the receiver continues the game, then m 0 = for all 0 > ; and in each 0 > ; the receiver plays the strategy prescribed for 0 1; e. If the receiver has not deviated and > t 1 ; then a ( ) = 0; f. If the receiver has not deviated and > t 1 is still in the game at time ; she sends m 0 2 (l 0; t 0 1) in each 0 : The thresholds are de ned recursively by l = (1 p ) ( ; p) t 1 ; t = (1 p) ( ; p) t 1 ; (4) where t 0 = 1: Part (a) of Proposition 7 states that the receiver ends the game when she observes a consistent message. This is because she has nothing to gain by continuing it. If she does, she will hear the same consistent message in any subsequent period (part d) and take the same action (part d). Part (b) of Proposition 7 states that in every period the sender lies if and only if her type is low enough. When she lies, she claims to be one of the highest types remaining in the 20

23 game. Part (c) describes the receiver s actions on the equilibrium path. It is optimal for the receiver to choose a ( ) = after all consistent messages unused by liars, as long as the types consistent with these messages are still expected to be in the game. All messages that are used by liars in generate the same action l : And nally, part (e) states that if the receiver observes a message consistent with a type that should not be in the game in period ; she chooses, a = 0: This is sequentially rational, and assures that the types which are supposed to tell the truth in 0 < do not want to deviate. The lying threshold l is constructed in such a way that conditional on observing a consistent message associated with lying, a ( ) = l is optimal. The truthtelling threshold t is determined by the type who is indi erent between identifying himself and ending the game, and lying and possibly facing more communication rounds. Let us discuss now how the truthtelling and lying thresholds change as we increase the number of rounds of communication. In the proof of Proposition 7, I show that for a xed p the mapping! (1 p) (; p) has a xed point, and for any ; and +1 are on the opposite sides of this xed point. This implies that adding the second round of communication shifts the rst-period thresholds (t 1 ; l 1 ) towards each other, while adding the third round of communication shifts (t 1 ; l 1 ) away from each other. The latter move is smaller in absolute value than the former; and if T is large enough, adding an extra stage of communication a ects (t 1 ; l 1 ) only in nitesimally. The intuition for these ndings is as follows. Consider a one-period game and the type that is indi erent between lying and truthtelling in that game. Adding a round of communication to this game increases the continuation value of this type if he lies, as in the one-period game he is pooled with the lowest types, while in the two-period game, he has a possibility to distinguish himself from these types. As a result, the indi erent type in the one-period game has a strict incentive to lie in the two-period game, and hence, t 1 increases. Since, as compared to the one-period game, more types lie in the rst period of the two-period game, l 1 must decrease. If we add the third period, then by the same logic l 2 decreases, which 21

24 decreases the continuation value of the highest-type liars from the rst stage. This must decrease their incentive to lie in the rst period, and hence it must increase t 1 : Since the truthtelling and lying thresholds are not monotone in T; comparing the payo s in two games that di er in length is cumbersome. For that reason, Proposition 8 compares the payo s of the receiver as T! 1 to her payo s from the one-period game. Note that the sender is ex ante indi erent between any T; because he is risk neutral, and sequential rationality on the part of the receiver assures that her expected decision is always equal to E [] : Proposition 8 There exists ^p 2 (0; 1) such that a. if p > ^p; for T su ciently large, receiver s expected payo in the one-period game is higher than in the T period game; b. if p < ^p; for T su ciently large, receiver s expected payo is higher in the T period game. Adding more rounds of communication can be interpreted as the receiver giving a lying sender second chances. Proposition 8 says that allowing for second chances does not have to be optimal ex ante. This is because giving second chances has two e ects. On the one hand, it allows the receiver to discriminate between the types who lie in the initial rounds. This bene ts the receiver and makes second chances ex post optimal. On the other hand, expecting second chances, more intermediate types decide to lie. This is because in the oneperiod game, all types that lie are pooled together, while in the T period game, the highest types of unsuccessful liars in have a chance to reveal their type in + 1. As a result, lying is more prevalent in the multi-period game. When p is low, that is, when lying is easy, the incentives to lie in the rst period are high, and many types lie. This means that upon seeing an inconsistent message the receiver still faces a lot of uncertainty. Therefore, the rst e ect dominates, and giving second chances is bene cial. When p is high, very few types lie in the one-period game, and hence the bene t from giving second chances is low. As a result, when 22

25 p is high, the receiver prefers to commit herself to not giving any second chances in order to increase the truthtelling incentives in the rst interaction. 5 Conclusions This paper represents a step toward understanding the implications of lie detectability for the process of communication. It shows that even if lies are costless, their detectability introduces an endogenous cost, which may prevent some types of the sender from lying. As a result, some information may be revealed in equilibrium, even if there is no common interest between the players. The nature of lie detection modeled in this paper makes multiple rounds of informative communication possible. The welfare e ect of multiple rounds of communication, however, depends on lie detectability in the rst place. The receiver bene ts from multiple rounds of communication only if lying is easy, and hence prevalent. Three immediate extensions of the current model come to mind. First, it would be interesting to study a model in which the players have some common interest as in Crawford and Sobel (1982). If the common interest is strong enough, one can conjecture that the lowest types of the sender will not pool with the highest types, but will pool with some intermediate types instead. Second, one may argue that the probability of lie detection depends on the distance between the true state and the lie. And nally, an incoherent message may carry information not only about the fact that the sender has lied, but also about what message he wanted to send. As mentioned before, under this assumption the equilibria presented in this paper would remain, but new equilibria would arise. For example, the lowest types may still pool with the highest types, but the message sent by each lying type could be negatively correlated with the type. That is, the lower types of the sender could use bigger lies. As a result, the belief of the receiver upon receiving a coherent message would be increasing in the message, but her action upon this message turning incoherent would be decreasing in the message. Such a behavior would be sequentially rational and could justify the sender s behavior. 23