Communication with forgetful liars

Jehiel, Philippe Article Communication with forgetful liars Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Jehiel, Philippe (2021) : Communication with forgetful liars, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 16, Iss. 2, pp. 605-638, https://doi.org/10.3982/TE4154 This Version is available at: https://hdl.handle.net/10419/253520 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/ Theoretical Economics 16 (2021), 605–638 1555-7561/20210605 Communication with forgetful liars Philippe Jehiel Paris School of Economics and University College London I consider multiround cheap talk communication environments in which, after a lie, the informed party has no memory of the content of the lie. I characterize the equilibria with forgetful liars in such settings assuming that a liar’s expectation about his past lie coincides with the equilibrium distribution of lies aggregated over all possible realizations of the states. The approach is used to shed light on when the full truth is almost surely elicited, and when multiple lies can arise in equilibrium. Elaborations are proposed to shed light on why nontrivial communication protocols are used in criminal investigations. Keywords. Forgetful liars, lie detection, analogy-based expectations, cheap talk. JEL classification. C72, D82. 1. Introduction In criminal investigations, it is of primary importance to detect when a suspect is lying. Quite commonly, suspects are requested to tell an event several times, possibly in different frames, and inconsistencies across the reports are typically used to detect lies, and obtain admission of guilt. As formulated in Vrij et al. (2011), the benefit of repeating the request is that a liar’s memory of a fabricated answer may be more unstable than a truth-teller’s memory of the actual event. As a result, it may be harder for a lying suspect than for a truth-teller to remain consistent throughout, which can then be exploited by investigators. Such a view about the potential instability in liars’ memory has been investigated experimentally by a number of scholars typically outside economics (see the discussion and literature review in Vrij et al. 2011). The objective of this paper is to develop a game theoretic framework and corresponding solution concept that formalize it. Specifically, I am interested in understanding how the asymmetry in memory between liars and truthtellers can affect the strategy of communication of informed parties. To this end, I consider standard communication settings in which there is a conflict of interest between an informed party (denoted I) who knows an event sand an uninformed party (denoted U) who does not know sbut would like to learn about it. Communication about stakes Philippe Jehiel: [email protected] I wish to thank the three anonymous referees for constructive comments. I also that Johannes Hörner, Navin Kartik, Frédéric Koessler, Joel Sobel, Rani Spiegler as well as seminar participants at PSE, Warwick theory workshop, Barcelona workshop, Glasgow University, Lancaster game theory workshop, D-Tea 2018, University of Bonn, ESSET 2018, and Stockholm University for useful comments. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement 742816). ©2021 The Author. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE4154 606 Philippe Jehiel Theoretical Economics 16 (2021) place in more than one round so that there is room for a liar to forget some of what he previously said. Key questions of interest are: Does the informed party engage into lying, and if so in what kind of events sand with what kind of lies? Do inconsistencies trigger harmful consequences? Are there circumstances in which the full truth about the event is almost surely elicited? Addressing such questions is of clear interest to the understanding of any strategic communication setting to the extent that the memory asymmetry between liars and truth-tellers seems widespread.1An important game theoretic insight obtained for such interactions in the absence of memory imperfections has been that full information transmission should not be expected, as soon as there are conflicts of interest (Crawford and Sobel 1982). But, how is this insight affected when liars are forgetful? A key modeling choice concerns the expectations of liars with respect to the content of their past lies. I will have in mind environments in which a given individual in the role of party Iwould not engage himself very often in the communication game. Thus, he would not know how he (routinely) communicates as a function of s. However, he would know, through learning from others’ experiences, the empirical distribution of lies (as aggregated over different realizations of s). I will be assuming that when party Ilies, he later believes he used a communication strategy that matches this aggregate empirical distribution.2 To state the main insights, let me complete the description of the communication setting. The events referred to as sates scan take discrete values in S⊆[01], and each realization of scan occur with a probability known to party U. In the criminal investigation application, the various scorrespond to different levels of guilt where s=1can be interpreted as complete innocence and s=0as full guilt. After hearing party I,partyU chooses the action that matches her expectation of the mean value of s,anactionthat affects party I’s well-being. Communication does not take place at just one time. Specifically, two messages m1 and m2are being sent by party Iat two different times t=12.IfpartyIin state stells the truth by communicating m1=sat time t=1, he remembers it at time t=2,butif he lies by communicating m1= s, he does not remember at time t=2what message was sent at time t=1.3He is always assumed to know the state sthough. That is, the imperfect memory is only about the message sent at time t=1, not about the state. Party Uis assumed to make the optimal choice of action given the messages (m1m2) she receives. 1To substantiate this, it may be relevant to mention a popular quote attributed to Mark Twain “When you tell the truth you do not have to remember anything,” which subtly suggests a memory asymmetry whether you tell the truth or you lie. 2Such an assumption implicitly requires that previous messages not corresponding to the truth are disclosed and tagged as being lies before the details of the state are disclosed, thereby ensuring that there is no access to the joint distribution of previous messages and states. See below for elaborations on this where I also discuss alternative specifications. 3My approach thus assumes that messages have an accepted meaning so that lying can be identified with sending a message that differs from the truth (see Sobel 2020 for a recent contribution that provides a definition of lying in communication games that agrees with this view). Theoretical Economics 16 (2021) Communication with forgetful liars 607 As highlighted above, I assume that when party Ilies at t=1,hebelievesatt=2that he sent a message at t=1that matches the aggregate distribution of lies as occurring in equilibrium across the various states. All other expectations of party Iare assumed to be correct, and strategies are required to be best-responses to expectations, as usual. The corresponding equilibria are referred to as equilibria with forgetful liars. I characterize such equilibria in the communication setting just described adding the (small) perturbations that, party Iincurs a tiny extra cost when lying, and with a tiny probability, party Ialways communicates the truth.4The main findings are as follows. I first consider pure persuasion situations in which party I’s objective is the same for all states and consists in inducing a belief about sas high as possible in party U’s mind. For such specifications, the equilibria employing pure strategies have the following form. Either party Ialways tells the truth or there is exactly one lie made in equilibrium. In the latter case, the unique lie shbelongs to the state space S,andpartyIchooses to lie when the state sis below a threshold sldefined so that E(s ∈Ss ≤slor s=sh)is in between sland the state in S\{sh}just above sl. Moreover, when considering the fine grid case in which two consecutive states are close to each other and all possible states can arise with a probability of similar magnitude, I show that all pure strategy equilibria with forgetful liars lead approximately to the first-best in which party Uperfectly infers the state whatever s, and chooses the action a=saccordingly. The reason why some one-lie communication strategies can be sustained as equilibria is that then the aggregate distribution of lies is concentrated on just one realization so that a liar by making this common lie can ensure he will not be caught being inconsistent. Arguments similar to the unraveling argument are next used to complete the characterization of pure strategy equilibria. I also briefly discuss a class of mixed strategy equilibria,5and observe for those that multiple lies can occur, inconsistent messages can happen leading to less good outcomes for party I, and, as for the pure strategy equilibria, the first-best is asymptotically approached in the fine grid case. Thus, in pure persuasion situations, when liars are forgetful, simple multiround communication protocols ensure that party Uobtains much more information transmission from party Ias compared with one-shot communication protocols in which party Iwould not disclose any information. Moreover, when there is some significant lying activity (i.e., moving away from the fine grid case), there is only one lie occurring in pure strategy equilibria, this unique lie is made only for low levels of s,andpartyIis never caught making inconsistent lies. I next explore the effect of letting the objective of the informed party Idepend also on the state s. The main observation in this case is that multiple lies can sometimes arise 4These perturbations ensure that if party Iis indifferent between lying and truth-telling, he chooses truth-telling, and if off-the-path messages m1=m2=s∈Swere received, party Uwould believe the state is s. I will also assume that when off-the-path message profiles other than (ss) are received, the action chosen by party Uis 0(or small enough), which is required to support pure strategy equilibria (see discussion below). 5These can be shown to be the only ones under additional perturbations (see the working paper version Jehiel 2019). 608 Philippe Jehiel Theoretical Economics 16 (2021) in equilibria employing pure strategies. The reason is as follows. After a lie at t=1,even though, irrespective of s,partyIat t=2holds the same belief about the message sent at t=1, he may now opt for different m2depending on the state sbecause party Irightly understands how party U’s action varies with the messages and party I’s payoff depends on s, unlike in the pure persuasion case. This observation can be used to construct equilibria in which depending on the state, liars find it strictly beneficial to sort into different lies without ever being inconsistent. In the final part of the paper, I briefly consider an extension (with the criminal investigation application in mind) in which the state takes a more complex form with two attributes sAand sBwhose sum s=sA+sBdetermines the level of guilt, and the imperfect memory of a liar concerns the details describing the lie (the exact profile of reported attributes) but not the targeted level of guilt (as represented by the sum of the reported attributes). When the communication protocol takes a sufficiently nontrivial form (with randomization on the order in which the details are requested at t=1and randomization on which attribute is requested at t=2), the equilibrium outcomes of the communication game with forgetful liars (to be extended appropriately) are very similar to the ones arising in the basic model (with only one lie being made in the pure strategy equilibria in the pure persuasion scenario and almost perfect information elicitation in the fine grid case). Interestingly, more equilibrium outcomes (including ones which are bounded away from the first-best in the fine grid case) can be supported if the communication protocol is too simple (e.g., as resulting from protocols in which at t=2,party Iis always asked to report the realization of the same prespecified attribute). Such additional insights while obtained in a stylized model can be viewed as shedding light on why nontrivial communication protocols are generally used in criminal investigations (see also Vrij et al. 2008 for experimental finding showing the benefit of increasing the cognitive load for information transmission purposes). Related literature The paper can be related to different strands of literature. First, there is a large literature on cheap talk initiated by Crawford and Sobel (1982)(seealsoGreen and Stokey 2007), which has emphasized that in the presence of conflicts of interest, some information would be withheld by the informed party. The analysis of strategic communication is deeply affected by the forgetfulness of liars, as exemplified by the almost perfect information transmission obtained in a two-round communication protocol when the grid of states is fine. In relation to the cheap talk literature, it should be mentioned that while most of this literature has considered one-round communication protocols, it has also observed that with multiple rounds, more equilibrium outcomes can be supported. The logic of such results is however unrelated to the memory imperfections considered in this paper and, for example, the almost perfect elicitation of the state would never arise even with multiple rounds when there is no memory imperfection.6 6With perfect memory, multiround communication protocols allow to implement a larger spectrum of the communication equilibria that could be obtained through the use of a mediator as compared with the Theoretical Economics 16 (2021) Communication with forgetful liars 609 Second, the equilibria with forgetful liars turn out to be similar to the perfect Bayesian Nash equilibria that would arise in certification games in which all types but those corresponding to the lies could certify all what they know (see Grossman and Hart 1980,Grossman 1981,Milgrom 1981,Dye 1985 or Okuno-Fujiwara et al. 1990 for some key references in the certification literature).7In particular, when there is only one lie shas in the pure strategy equilibria of the pure persuasion games, the equilibrium outcome is similar to that in Dye’s (1985) model identifying type shin my model with the type that cannot be certified (the uninformed type) in his. Of course, a key difference is that, in this analogy, the set of types that cannot be certified is not exogenously given in the present context, as it is determined by the set of lies made in equilibrium, which is endogenously determined. Third, the proposed modeling of the expectation of a forgetful liar is in the spirit of the analogy-based expectation equilibrium (Jehiel 2005 and Jehiel and Koessler 2008) to the extent that the considered distribution of messages is the overall distribution of lies aggregated over all states, and not the corresponding distribution conditioned by the state. I briefly discuss below the case in which a forgetful liar would use the conditional distribution instead (this alternative modeling would be in the spirit of either of the multi-selves approaches considered by Piccione and Rubinstein 1997 and fits applications in which party Iwould know how his lying strategy varies with s, e.g., because he would have played the game himself many times). I note that with such a modeling, many more equilibrium outcomes can be supported. In particular, in the pure persuasion case, I construct such pure strategy equilibria the outcome of which is far away from the first-best, even in the fine grid case. Fourth, putting the present paper in the perspective of other behavioral models of strategic communication, I note that a number of these consider the modeling of deception, which is concerned with how the informed party can manipulate the belief of the uninformed party (a behavioral dimension not present here). These include Crawford (2003) who adds in a game in which players communicate about their intended action the possibility that players interpret the declared intention naively as in the level-k approach, Kartik et al. (2007) who consider one-shot cheap talk games with unbounded state space again when some share of receivers interpret naively what they are told (with the observation that the cheap talk game is then transformed into a signaling game admitting a separating equilibrium when the state space is unbounded) or Ettinger and Jehiel (2010) who consider a different application of the analogy-based expectation equilibrium to communication games this time focused on the coarse understanding of the smaller set of Nash equilibria that can be implemented with one round of direct communication between the two parties (see Forges 1990,Aumann and Hart 2003,orKrishna and Morgan 2004 for discussion of this). 7Interestingly, Mark Twain’s quote as reported in Footnote 1 has sometimes been used to motivate that explicit lies (as opposed to lies by omission) may be costly or simply impossible as in certification games (see, e.g., Hart et al. 2017). By contrast, my approach can be viewed as offering an explicit formalization of memory asymmetry between liars and truth-tellers as suggested in that quote. It may be mentioned here that the same Twain quote appears also in a recent paper by Hörner et al. (2017) on dynamic communication with Markovian transitions between states, but the link to the present study in which there is no evolution of states is even less immediate. 610 Philippe Jehiel Theoretical Economics 16 (2021) uninformed party rather than the memory limitations of the informed party. Concerning behavioral twists on the informed party’ side, one may mention the work of Kartik (2009) who adds explicit (and not vanishingly small, as considered here) lying costs to the standard cheap talk game, and observe in a setup with bounded state space that every type has an incentive to inflate his type with some pooling at the highest messages (which sharply contrasts with the shape of the equilibria with forgetful liars in pure persuasion situations as described above in which pooling occurs for low types) or the work of Deneckere and Severinov (2018) with similar lying costs this time considered in more flexible mechanism design settings.8 Finally, it may be worth mentioning the work of Dziuda and Salas (2018)whoconsider one-round communication settings similar to those in Crawford and Sobel in the pure persuasion game scenario (see also Balbuzanov 2019 for the case of statedependent preferences) in which a lie made by the Sender may sometimes be detected by the receiver. Thinking of the observation of inconsistencies by the uninformed party as a lie detection technology, it would seem the present paper proposes an endogenous channel through which lies are detected. Yet, this is not the driving force behind the analysis here as in many equilibria with forgetful liars (in particular those employing pure strategies), there is no inconsistency in equilibrium, and thus no lie detection as in Dziuda and Salas (it is rather the fear of being inconsistent if lying that drives the equilibrium choice of strategy of the informed party).9 The rest of the paper is organized as follows. Section 2 describes the model and solution concept. Section 3 analyzes pure persuasion situations. Section 4 analyzes a simple class of state-dependent preferences. Section 5 offers a discussion. Section 6 concludes. 2. The model Events s—referred to as states—can take npossible values s1<s 2<···<s nwith s1=0 and sn=1where S={sk}n k=1denotes the state space. The ex ante probability that state skarises is p(sk), which is commonly known. There are two parties, an informed party I and an uninformed party U. The informed party knows the realization of the state s∈S, the uninformed party does not. 8Deneckere and Severinov (2018) assumed that each time the informed party misreports his type, he incurs an extra cost. They observe that using multiround mechanisms (in which if consistently lying the informed party would have to incur prohibitive cost) may help extract the private information at no cost. While the benefit of multiround communication is common to my approach and theirs, the main contribution of the present study concerns the endogenous derivation of lying costs as arising from memory limitations in given communication games. This is clearly complementary to the mechanism design perspective of their approach in which lying costs are exogenously given. 9Clearly, the informational settings are very different in the two papers: there is no memory issue on the informed party side in Dziuda and Salas and there is no technology for lie detection in my setting. Yet, a common feature of the analysis is that Senders in favorable states prefer telling the truth. But, note that the shape of the lying strategy of those senders in unfavorable states is different as these randomize over a full range of messages above a threshold in Dziuda and Salas, which is not so in my setting. Theoretical Economics 16 (2021) Communication with forgetful liars 611 Party Ifirst communicates about saccording to a protocol to be described shortly. At the end of the communication phase, party Uhas to choose an action a∈[01].Theobjective of party Utakes the quadratic form −(a−s)2so that she chooses the action athat corresponds to the expected value of sgiven what she believes about its distribution. Party Icares about the action achosen by Uand possibly (but not necessarily) about the state s. Ignoring for now the messages sent during the communication phase, party I’s payoff can be written as u(as). I will start the analysis with pure persuasion situations in which party Iwould like the action ato be as large as possible independently of s.Iwillnextdiscusshowthe analysis should be modified when party I’s objective may depend on the state sas well as a, focusing on the specification u(as) =−(a −b(s))2where b(s)—assumed to be strictly increasing—represents the action amost preferred by party Iin state s. Communication game In standard communication games, à la Crawford and Sobel (1982), party Isends a message monce to party Uwho then chooses an action a. Message mneed not have any accepted meaning in that approach. That is, the message space Mneed not be related to the state space S. I consider the following modifications. First, in order to identify messages as lies or truths, I explicitly let all the states s∈Sbe possible messages, that is, S⊆M.When message m=sis sent, it can be thought of as party Isaying “The state is s.” I also allow party Ito send messages outside Ssuch as “I do not know the state” when everybody knows that Iknows s,thatis,M\S= ∅. While the set Mwill be assumed to be finite, in applications the set Mis likely to be much larger than S. Second, in order to let memory play a role, I assume that party Isends two messages m1,m2∈Mone after the other, at times t=1and 2.PartyUobserves the messages m1, m2, and she chooses her action a(m1m2)as a function of these. Letting party Isend two messages instead of one would make no difference if after sending message m1,partyIalways remembered what message m1he previously sent, and if both parties Iand Uwere fully rational, as usually assumed. While party Uwill be assumed to be rational, I consider environments in which party Iat time t=2has imperfect memory about the message m1sent at time t=1. More precisely, I assume that when party Iin state stells the (whole) truth at time t=1,thatis,sendsm1=s, he remembers that m1=sat t=2, but when he lies (identified here with not telling the whole truth) and sends m1= s, he does not remember what message m1he previously sent (he may still think that he sent m1=s, as I do not impose in the basic approach that he is aware that he lied; see below for further discussion). A key modeling choice concerns how party Iat time t=2forms his expectation about the message sent at t=1when he lied lie at t=1. I adopt the following approach. Solution concept A multiself approach is considered, which is standard in situations with imperfect recall (see Piccione and Rubinstein 1997). That is, think of the state sas a type for party I,and 612 Philippe Jehiel Theoretical Economics 16 (2021) envision party Iwith type sat times t=1and 2as two different players I1(s) and I2(s) having the same preference as party I. To model the belief of a forgetful liar, let σ1(m |s) denote the (equilibrium) probability with which message m1=mis sent at t=1by party Iwith type s. Assuming that at least one type slies with positive probability at t=1,that is, σ1(m |s) > 0foratleastone(m s) with m= s, one can define the distribution of lies at t=1aggregating lies over all possible realizations of s. The probability of message m in this aggregate distribution is  s∈Ss=m σ1(m |s)p(s) (ms)∈M×Sm=s σ1m|sps(1) In an equilibrium with forgetful liars σ,whenI1(s) lies at t=1(i.e., sends m1= s), player I2(s) at time t=2believes that player I1(s) sent mwith probability as expressed in (1). Ifnolieiseversentattimet=1in equilibrium, the belief after a lie can be arbitrary. By contrast, when I1(s) tells the truth (i.e., sends m1=s), player I2(s) knows that m1=s. The other features of the equilibrium with forgetful liars are standard. All expectations other than that of I2(s) about m1after a lie at t=1are correct, and all players are requested to choose best-responses to their beliefs given their preferences (deviations of Iare local and not joint between t=1and 2due to the multiself specification). To give a concrete illustration of how the beliefs about m1are formed by party Iat t=2, assume there are four states s1,s2,s3,ands4, with states s2and s3being equally likely (the other two states may have different ex ante probabilities). Assume that party Iat time t=1sends message m1=s1in states s1and s2, and message m1=s4in states s=s3and s4.Att=2, in states s=s1and s4,partyIremembers that he sent m1=s as there is no lie in these states. By contrast, in states s=s2and s3, there is a lie so that party Idoes not remember at t=2what message he previously sent. In these two states, party Ibelieves that he sent as first message s1and s4with equal probability where the weighting of the two lies is imposed by the assumed time t=1communication strategy and the assumption that s2and s3are equally likely. As is common in many studies of communication games (see, e.g., Chen 2011 or Hart et al. 2017), I consider refinements/perturbations which I view as natural and serve the purpose of ruling out implausible equilibria and/or ensuring the existence of pure strategy equilibria. Refinements/perturbations First, I assume that in case of indifference between lying and truth-telling, party Iopts for truth-telling. Formally, for some positive εassumed to be sufficiently small, party I’s payoff as a function of (sam1m2)is UI(sam1m2)=u(a s) −ε(1m1=s+1m2=s) That is, a lie whether at t=1or 2is assumed to inflict an extra εcost. This will be referred to as the TP-perturbation. Second, I assume that were party Utoreceivetwicethesamemessagem1=m2=s corresponding to a state s∈Sthat would never been sent in equilibrium, party Uwould Theoretical Economics 16 (2021) Communication with forgetful liars 619 Finally, while I regard the perturbations/refinements introduced at the end of Section 2 as fairly reasonable ones in the context of communication games, it is instructive to know how the analysis would be affected when alternative specifications for off-path actions are considered, when perturbations TP or TB are removed, or when the genericity assumption GE is dropped. I now provide a sketchy description of this. (1) The above analysis of pure strategy equilibria remains unchanged for alternative specifications of a(m1m2)for off-the-path message profiles (m1m2),aslongasthese are set to be small enough. If by contrast such a(m1m2)are set too big, then there is no pure strategy equilibrium, and one has to look for mixed strategy equilibria. Thus, the assumption that a(m1m2)=0for off-the-path message profiles is without loss of generality for the analysis of pure strategy equilibria.23 (2) If the genericity assumption GE is dropped, more equilibrium outcomes possibly with multiple lies and different welfare consequences can sometimes be supported. For example, suppose one considers a situation with an even number nstates such that sn−k+1+sk=1for all kand an equal probability that each state arises. One can support an equilibrium with forgetful liars in which for each s∈{sksn+1−k}party Iconsistently reports that the state is max{sksn+1−k}at t=12. In this case, the aggregate distribution of lie is the uniform distribution over max{sksn+1−k}for the various k. After any such (consistent) lie, party Uwould choose a=1 2,andpartyIafter a lie at t=1would be indifferent as to which lie in the set {max{sksn+1−k}}kto choose. By requiring that in state min{sksn+1−k}the lying party Ichooses max{sksn+1−k}at t=2, one can support this as an equilibrium. Note though the fragility of such a construction, as it would require that, in state s=min{sksn+1−k},partyIat t=2chooses the specific best-response max{sksn+1−k}when in fact he is indifferent between all max{sksn+1−k}obtained when kvaries.24,25 (3) If perturbation TB concerning the interpretation of m1=m2=skoff-the path is removed, then one can support equilibria in which party Ilies in more states. In particular, party Iconsistently lying and sending m1=m2=1in all states s= 1can be part of an equilibrium with forgetful liars if the expectation is that when m1=m2=skwith sk= 1 are received a sufficiently low action (e.g., a=0) would be chosen by party U.Tosee this, observe that with the proposed strategy, the action after m1=m2=1would just be E(sk)the expected value of the state, there would only be one lie m∗=1in equilibrium, 23To ensure that there are no other pure strategy equilibria for generic values of the states, one should assume that the perturbations giving rise to the choices of such a(m1m2)are not fine-tuned to the values of sk, as would result from trembling behaviors assumed to be solely determined by the order of the states (and not their exact values). To ensure that the truth-telling trembling dominates the alternative trembling possibly resulting in inconsistencies (so that a(s s) =swhenever m1=m2=s∈Sis off-the-path), one should have in mind that the message space is much larger than the state space so that being consistently truthful by chance (i.e., without being a truth-teller) would be very unlikely. 24If party Iwere choosing another best-response at t=2, he would be caught sending inconsistent messages, and party Iwould rather avoid sending message max{sksn+1−k}at t=1. 25This argument illustrates why an alternative to the GE assumption to get the same result as in Proposition 1 is to assume that in case of indifferences, (the lying) party Ialways picks the same best-response irrespective of the state. 620 Philippe Jehiel Theoretical Economics 16 (2021) and telling the truth consistently at t=1and 2would not be attractive to party Iwhen the state is sk= 1. More technically, when perturbation TB is removed, the unraveling argument breaks down, and party Iin states sthat are different from the common lie but above the action resulting from the common lie may prefer lying to telling the truth. The breakdown of the unraveling argument in turn allows to support more equilibrium outcomes with different welfare consequences, thereby revealing the essential role of perturbation TBin the derivation of Proposition 1. (4) If perturbation TP concerning the slight preference for truth-telling is removed, the insight that inconsistent messages cannot arise in equilibria employing pure strategies (Lemma 4) no longer holds. That is, new equilibria in which party Iin states s>a(m ∗m∗)with s= m∗wouldbesendingamessagem2= safter m1=swas sent can now be supported (this is so because the inconsistency (s m2)would safely be attributed to state s, thereby leading to action a=sin this case). I note that such equilibria (which are outcome equivalent to those considered in Proposition 1) would not be robust to other (natural) perturbations in which party Iin sufficiently low states would be viewed as randomly sending inconsistent messages with positive probability (since with such additional perturbations, (sm2)would now be followed by a lower action than when (s s) is chosen).26 3.2 Approximate first-best with fine grid So far, states skcould be distributed arbitrarily on [01]. What about the case when consecutive states are close to each other and all states have a comparable ex ante probability? I show that in such a case, all equilibria in pure strategies are close to the truthtelling equilibrium, resulting in the approximate first-best outcome for party U.More precisely, Definition 1. A state space Sn={s1sn}satisfies the n-fine grid property if sk+1− sk<2 nfor all k,andforsome(αα),0<α< α, set independently of n,α<p(s k)/p(sk)< α, for all k,k. I will be considering sequences of state spaces Snsatisfying the n-fine grid assumption and of lying costs εnwhere for each n(the above genericity assumption (GE) is satisfied and) εnis smaller than half the minimum value of kp(T kn a)(e(Tkn b)−e(Tkn c)) when allowing Tn a=(Tkn a)kand Tn b=(Tkn b)k= (Tkn c)k=Tn cto be any families of disjoint subsets of Sn. Proposition 2. Consider a sequence (Snεn)∞ n=nsatisfying the above conditions, and a sequence (σn)∞ n=nof pure strategy equilibria with forgetful liars associated with (Snεn). For any  a>0,thereexistsnsuch that for all n>n, the equilibrium action of party Uafter a lie prescribed by σnis smaller than a.Asnapproaches ∞, the expected utility of party U approaches the first-best (i.e., converges to 0). 26Based on this, I would argue that perturbation TP may not be needed for the derivation of Proposition 1 if other (plausible) perturbations are considered instead. Theoretical Economics 16 (2021) Communication with forgetful liars 621 To prove Proposition 2, I make use of the characterization result of Proposition 1.Let a∗be the expected payoff obtained by party Iwhen lying at t=1in an equilibrium in pure strategy. If a∗is significantly away from 0, say bigger than  aassumed to be strictly positive, then under the fine grid property the expectation of sover the set of states that are either below a∗or else equal to 1must be significantly below a∗.ButthenI(s) for some s=skstrictly below a∗would strictly prefer telling the truth rather than lying undermining the construction of the equilibrium (that requires party I(s) with s<a ∗to be lying). This argument shows that a∗must get close to 0as napproaches ∞,thereby paving the way to prove Proposition 2. The intuition for Proposition 2 can be understood as follows. For a given lie m∗to possibly emerge in equilibrium, it should be that the probability that the state s=m∗ arises is not too small relative to the probability that the lie m∗is used. In the fine grid case, this implies that there can be little lying in such an equilibrium, since the probability of each state becomes increasingly small in this case. Another way to think of this result is to build on the observation made after Proposition 1. An equilibrium with forgetful liars in pure strategy with lie m∗can be viewed as a perfect Bayesian Nash equilibrium of a certification game in which party Ican certify his type when s= m∗,but not when s=m∗. In such a certification framework, if the ex ante probability of m∗gets small —which must be so in the fine grid case—one gets an equilibrium outcome close to that in the classic persuasion game in which the unraveling argument leads to full disclosure (and no lying). 3.3 Mixed strategy equilibria I now consider mixed strategy equilibria. I will not aim at characterizing all such equilibria, but instead I will consider a subclass of those having the property that, irrespective of s,whenpartyIlies, he randomizes, and, for some (m∗ k)kand some (μk)k, chooses message m∗ kwith probability μkin the same way and independently at t=1and 2.It should be noted that such a restriction would arise in all mixed strategy equilibria, if I were to assume that in case of indifference, the chosen randomization over messages is not allowed to depend on the state snor on the calendar time t(see the discussion paper version Jehiel 2019 in which such a feature is imposed as a refinement). Consider such a mixed strategy equilibrium that necessarily involves multiple lies. I note that some inconsistencies must arise with positive probability on-the-path, and any inconsistent messages (m1m2)with m1= m2arising on-the-path must result in the same action of party Udenoted hereafter ainc, since any such message profile would be equally informative about the state s. Moreover, the optimality condition for liars would impose, letting ak=a(m∗ km∗ k),thatμkak+(1−μk)ainc is independent of k.This common value will be denoted a∗hereafter. I next observe that all m∗ kmust belong to S(as results from an unraveling argument) and that party Iin state m∗ kshould be telling the truth twice (given that some types must find the lie m∗ kweakly optimal, it must be that in state m∗ k,partyIstrictly prefers telling the truth rather than lying that would impose an extra lying cost). Moreover, take any s∈Sother than m∗ kfor k=1K.Ifs<a ∗−2ε,I1(s) would strictly prefer sending any m∗ kexpecting to get a∗−2εrather than telling the truth that 622 Philippe Jehiel Theoretical Economics 16 (2021) would only yield s.Ifs>a ∗−2ε,I1(s) would strictly prefer telling the truth (anticipating that I2(s) would also do so) rather than lying. These observations yield. Proposition 3. The following define a class of mixed strategy equilibria. For some a∗, m∗ k,k=1K,withm∗ k>a ∗,andμk>0with kμk=1, satisfying μkak+(1−μk)ainc =a∗ ainc =Es∈Ss < a∗−2ε am∗ km∗ k=akwith ak=μkPrs∈Ss < a∗−2εainc +pm∗ km∗ k/μkPrs∈Ss < a∗−2εainc +pm∗ km∗ k and a(s s) =sfor s∈Ss = m∗ kk=1K It(s) with s<a ∗−2εsends m∗ kwith probability μkindependently at t=12 It(s) with s>a ∗−2εtells the truth at t=12 Observe that ak>a inc for all k, and thus being inconsistent is never profitable relative to what happens when the same message is being sent at t=1and 2on-the-path. Such a finding can be viewed as formalizing that being inconsistent in a strategic communication setting with forgetful liars must be harmful. Observe also that as for the pure strategy equilibria, in the fine grid case, the proposed mixed strategy equilibria approach the first-best for party U, as can be inferred from the observation that a∗must be converging to 0in such a limit (see the working paper version for details on this). 3.4 On alternative modeling of forgetful liars 3.4.1 When the informed party knows his lying strategy How is the analysis affected when considering the scenario in which a forgetful liar would know the distribution of lies conditional on the state (and not just in aggregate over the various states as assumed above; see expression (2)). While the equilibria arising with the main proposed approach would continue to be equilibria with this alternative approach, the main observation is that many additional equilibrium outcomes can also arise. In particular, even in the fine grid case, equilibrium outcomes significantly away from the first-best can now be supported. To illustrate this, I focus on equilibria employing pure strategies. Consider a setup with an even number nof states and a pairing of states according to Sk={sk sk}with (Sk)kbeing a partition of the state space and sk< skfor all k. I claim that with this alternative approach, one can support an equilibrium in which for every k,I(sk)lies consistently by sending mt=skat t=12while I(sk)tells the truth. To complete the description of the equilibrium, party U’s action when hearing twice skshould be a(sk sk)=E(s ∈Sk),and I let the belief of I2(sk)if I1(sk)weretolietobethatmessage0was sent at t=1.27 27As in the main model, one also requires that when hearing hearing off-the-path message profiles, party Uchooses a=0. Theoretical Economics 16 (2021) Communication with forgetful liars 623 The reason why such an equilibrium can arise now is that with the new expectation formulation, when I1(sk)lies at t=1,playerI2(sk)(rightly) believes that player I1(sk) sent m1=skgiven that this is the only lie made by I1(sk)in equilibrium. As a result, player I2(sk)after a lie at t=1finds it optimal to send m2=skas any other message is perceived to trigger action a=0, which is less than E(s ∈Sk). Given that I1(sk)has the correct expectation about I2(sk)’ strategy, I1(sk)either lies and sends m1=skor else he tells the truth. Given that E(s ∈Sk)>s k, he strictly prefers lying (whenever εis small enough), thereby showing the optimality of It(sk)’ strategy for t=12. Showing the optimality of It(sk)’ strategy is easily derived using the off-path beliefs proposed above.28 The key reason why multiple lies can be sustained now and not previously is that the belief of I2(sk)after a lie at t=1now depends on skgiven that the mere memory of the state sktogether with the knowledge of the equilibrium strategy of I1(sk)allows player I2(sk)to recover the lie made by I1(sk), even if he does not directly remember m1. It is also readily verified that such equilibrium outcomes can lead party Uto get payoffs bounded away from the first-best, even in the fine grid case as the number of states gets large, in contrast to the insight derived in Proposition 2 (think, e.g., of the limit pairing of sand 1−sin the approximately uniform distribution case that would result in party Uchoosing approximately action a=1 2in all states, which corresponds to what happens in the absence of any communication). Thus, when party Iknows his lying strategy (possibly as a consequence of playing the game many times), party Imay still withhold a lot of information, even when physically forgetting his past lies. This was not so (in particular in the fine grid case) when subjects in the role of party Iwere viewed as occasional players and access to past interactions was focused on the distribution of lies (and not the joint distribution of lies and states). 3.4.2 When others’ lies are not tagged as such Having again in mind that subjects in the role of party Iare occasional players and learning environments in which there is no access to the joint distribution of messages and states, one may in contrast to the main modeling approach consider situations in which the time t=1messages m1would not, for learning purposes, be tagged as lies before the state is disclosed. In this case, there would be no easy access for newcomers to the aggregate distribution of lies, and it is then more natural to assume that when party Ilies at t=1,hebelievesatt=2that he sent a message according to the aggregate equilibrium distribution of messages used at t=1(aggregating this time not only over the states but also whether or not messages correspond to the truth). I will not develop a full analysis with this alternative formulation, but it may be interesting to note that whenever the probability p(sn)of state sn=1is no smaller than p(sk) 28One may be willing to refine the off-path beliefs of I2(sk)in the above construction, for example, by requiring that a lie m1=1(instead of m1=0) is more likely to occur when I1(sk)lied (and sk= 1). Note that the above proposed strategies would remain part of an equilibrium with this extra refinement, assuming that {01}is one of the pairs Skand E(s =0or 1)takes the smallest value among all E(s ∈Sk)(think of assigning sufficient weight on the state being s=0). Indeed, in such a scenario, if I1(sk)were to lie, he would send m1=1anticipating that I2(sk)would send m2=1next, and this would be worse than truthtelling. 624 Philippe Jehiel Theoretical Economics 16 (2021) for every k<n, then the (slsh)-communication strategy with sh=sn=1and sldefined appropriately so that (slsh)satisfies the conditions shown in Proposition 1 would continue to be a pure strategy equilibrium in this alternative approach. Roughly, the reason why this holds true is that with such a communication strategy, there would be enough probability on the message shin the aggregate distribution of time t=1messagessothataliarattimet=2would always find it optimal to send message sh.29 It is also not difficult to see using arguments similar to the ones developed above that with this alternative approach, pure strategy equilibria will only have one lie, there would be no inconsistent messages, and party Iwould have to be using (slsh)-communication strategies with the restrictions imposed in Proposition 1. Possibly, not all of the communication strategies shown in Proposition 1 could arise as equilibria, as for some low enough sh, a liar at t=2would end up preferring message m2= sh, undermining the equilibrium construction. In particular, the truth-telling strategy would not longer be part of an equilibrium. Overall, the implications of this alternative approach are very similar to the ones obtained with the main model, as far as pure strategy equilibria are concerned. 3.4.3 When liars remember that they lied In the main approach, I assumed that a forgetful liar in state scould consider that he previously sent swith some positive probability if shappened to be a lie made in another state s. If instead player I2(s) after a lie at t=1were to be aware that party Ilied at t=1, it would be more natural to assume that player I2(s) would rule out that player I1(s) sent m1=s. With such an alternative approach, a liar would consider the aggregate distribution of lie and (possibly) update it by conditioning on the information that m1= s. Clearly, the pure strategy equilibria shown in Proposition 1 would be unaffected by this alternative modeling to the extent that in such equilibria there is (at most) one lie sh(and thus the extra conditioning has no bite for the lying party I). In the Appendix, I show that there cannot be pure strategy equilibria with multiple lies under this alternative modeling, thereby establishing the robustness of the analysis to such a variant. 4. Communicating with state-dependent objectives I consider now situations in which party I’s blisspoint action may depend on the state. Specifically, I let u(as) =−(a −b(s))2where b(s) is assumed to be increasing with s. I wish to characterize the equilibria with forgetful liars as defined in Section 2 restricting attention to pure strategy equilibria. The main observation is that multiple lies may arise in pure strategy equilibria when party I’s objective is state-dependent. The key reason for this is that party Iat t=2, after a lie at t=1, may end up choosing different messages as a function of the state 29This follows because letting a∗=E(s such that s≤slor s=sh), one would have: a∗=Prs≤slEs≤sl+p(sn)sn Prs≤sl+p(sn) and thus (Pr(s ≤sl)+p(sn))a∗>p(s n)sn>p(s k)skfor any k<n k. Theoretical Economics 16 (2021) Communication with forgetful liars 625 despite having the same belief about what the first message was. This is so because the objective of party Iis state-dependent and party Irightly anticipates which action is chosen by party Uas a function of the messages. This, in turn, allows party Iat t=1to safely engage in different lies as a function of the state, while still ensuring that he will remain consistent throughout. Another observation concerns the structure of lies in equilibrium. I show that in all pure strategy equilibria, lies inducing larger actions aare associated with higher states, which eventually leads to a characterization of equilibria that borrow features both from cheap talk games (the interval/monotonicity aspect) and certification games (as seen in pure persuasion situations). An example with multiple lies. Assume that Sconsists of four equally likely states s=0,s∗ 1,s∗ 2,and1. Let the blisspoint function be b(s) =s+for some satisfying 1 2>>0. I will look for conditions on s∗ 1,s∗ 2so that, in equilibrium, party Isends messages m1=m2=s∗ 1in states s=0and s∗ 1,andpartyIsends messages m1=m2=1in states s=s∗ 2and 1. In such a proposed equilibrium, party Umust choose a(s∗ 1s∗ 1)=s∗ 1 2,a(11)=s∗ 2+1 2 and a(00)=0,a(s∗ 2s∗ 2)=s∗ 2as well as a(m1m2)=0for all other message profiles. With such strategies, two lies m∗ 1=s∗ 1and m∗ 2=1are made in equilibrium, and these two lies occur with the same probability. Thus, party I2(s) after a lie m1= sat t=1believes at t=2that at t=1player I1(s) either sent m1=s∗ 1or 1, each with probability half. To be an equilibrium, it should be that player I1(s∗ 1)weakly prefers a(s∗ 1s∗ 1)to a(11), as otherwise, player I1(s∗ 1)would strictly prefer lying by sending m1=1anticipating that player I2(s∗ 1)would also send m2=1(given that I2(s∗ 1)would perceive that m1=s∗ 1or 1are equally likely and inconsistent messages result in a=0). Thus, s∗ 1+−a(s∗ 1)≤ a(1)−s∗ 1−or 1+s∗ 1+s∗ 2/2−2≥2s∗ 1.(3) More generally, it turns out that the incentives of I1(s) and I2(s) are aligned for all s. Thus, the remaining equilibrium conditions require that party Iin state s∗ 2weakly prefers a(1)to a(s∗ 1)as otherwise, party Iwould strictly prefer the lie s∗ 1to the lie 1(both at t=1and 2). That is, a(1)−s∗ 2−≤s∗ 2+−a(s∗ 1)or 2s∗ 2≥1+s∗ 1+s∗ 2/2−2 (4) Moreover, it should be that party Iin state s=0strictly prefers a(s∗ 1)to a(0)=0(what he can get by telling the truth). That is, a(s∗ 1)<2or 4>s ∗ 1(5) Finally, it should be that party Iin state s=s∗ 2strictly prefers a(1)to a(s∗ 2)=s∗ 2(what he can get by telling the truth). That is, a(1)<s ∗ 2+2or 4>1−s∗ 2(6) 626 Philippe Jehiel Theoretical Economics 16 (2021) Whenever conditions (3)–(4)–(5)–(6) are satisfied (which is so whenever s∗ 1is small enough and s∗ 2is large enough, as soon as <1 2), the above two-lie communication strategy can be sustained as an equilibrium with forgetful liars. ♦ Characterization of equilibria employing pure strategies Pure strategy equilibria are characterized as follows where for any subsets Aand Bof S, IletA<Bwhenever for all sA∈Aand sB∈B,wehavethatsA<s B. No inconsistent messages are sent in equilibrium, as follows from an unraveling argument. Let m∗ kdenote a consistent lie made by at least one type s= m∗ kin equilibrium, and let Lkdenote the set of types ssuch that party Iwith type ssends twice m∗ k,thatis, m1=m2=m∗ k.LetL− k=Lk\{max(s ∈Lk)}and L=(Lk)k.Let  ak(L) =E(s ∈Lk)and ( pk(L))kbe such that  pk(L)/ pk(L) =p(L− k)/p(L− k)(with k pk(L) =1). The following proposition (proven in the Appendix) summarizes the main properties of the pure strategy equilibria with forgetful liars. Proposition 4. There always exists an equilibrium with forgetful liars in pure strategies and any nonuniformly truthful such equilibrium satisfies the following properties. There is a disjoint family of lie sets L=(Lk)K k=1,withL− 1<···<L − K,m∗ k=max(s ∈Lk)such that (1) Party Iwith type s∈L− klies twice by sending m1=m2=m∗ k;(2)PartyIwith type s∈S\∪ kL− ktells twice the truth; (3) A liar’s belief assigns probability  pk(L) to m1=m∗ k; (4) Party Uwhen hearing m1=m2=m∗ kchooses a= ak(L);whenhearingm1=m2=s∈ S\{m∗ 1m∗ K}chooses a=s; and when hearing any other message profile chooses a=0. In other words, lie sets L− kare ordered and the common lie in L− kis m∗ k=max(s ∈ Lk).PartyIin state santicipates that if he lies at t=1he will lie next by sending m∗ k(s) where k(s) =arg maxkv(ks) and v(ks) =−  pk(L)( ak(L) −b(s))2−(1− pk(L))b(s)2is party I’s time t=2perceived expected utility of sending m2=m∗ kafter he lied at t=1 (the probability attached to m1=m∗ kis  pk(L) as follows from the consistency requirement (1)). To avoid being inconsistent, party Iin state swill either send m∗ k(s) both at t=1and t=2or he will be truthful (both at t=1and t=2) depending on what he likes best. Comment.Whenmultipleliesm∗ kcan be sustained in equilibrium, it is worth noting some similarity with the perfect Bayes Nash equilibria that would arise in the one shot communication game in which all types except those corresponding to lies m∗ kcould be certified (the similarity comes from the observation that types other than m∗ keither tell the truth (and get a payoff corresponding to the one they would get if they could fully disclose their type) or they consistently send message m∗ k).30 Yet, a notable difference concerns the belief of a liar regarding which m∗ khe previously sent, which in turn induces incentive constraints typically more stringent than in the usual certification setup. Another difference already mentioned in the context of pure persuasion is that which type can be certified is endogenously determined by the equilibrium set of lies in the present context. 30Such a richer certification setup falls in the general framework defined in Green and Laffont (1986)or Okuno-Fujiwara et al. (1990). Theoretical Economics 16 (2021) Communication with forgetful liars 627 First-best with fine grid While multiple lies can arise in equilibrium when party I’s objective may depend on the state, in the fine grid case (as defined in pure persuasion situations), it is not possible to sustain equilibria with multiple lies. Considering the general characterization shown in Proposition 4, in the fine grid case, all ak(L) must be approaching 0as otherwise party I in too many states s∈Ssmaller than  ak(L) would be willing to make the lie m∗ k, making it in turn impossible to have that  ak(L) =E(s ∈Lk)(it is readily verified that there is only one state in Lkthat lies above  ak(L) and this is s=m∗ k). As a result, in the fine grid case, assuming that b(s) ≥s+for some >0, there can only be one lie in a pure strategy equilibrium, and the first-best for party Uis being approached in the limit. This is similar to what was obtained in the pure persuasion case. 5. Discussion 5.1 Back to criminal investigations A key assumption driving the main insights is the memory asymmetry whether the informed party Ilies or tells the truth at t=1. With the criminal investigation application in mind, one may legitimately raise the concern that if a lying suspect pretends he is not guilty (i.e., by sending m1=1at t=1) this may not be so hard to remember at t=2,making the memory asymmetry assumption as considered in the main model not so clearly compelling in this case. In most applications (criminal or otherwise), the full description of the state (or event) typically consists of many details, and not just a summary statistic, such as the level of guilt, that pins down parties’ payoffs. When party Ilies, he has to report details about the fabricated state, and it may be difficult later for party Ito remember all these details when asked extra questions. In this case, it is not the overall level of guilt (as determined by the full description of the state) that is not remembered by the lying suspect, but the details reported in the lie. I would like to think of the main model as a simplified representation of such a richer specification. But, a question arises as to whether this view is legitimate and for what kind of communication protocols. Making progress on this may also be of interest to shed light on some experimental studies reported in the context of criminal investigation. In particular, Vrij et al. (2008) has experimentally observed that when the communication protocol is too simple (e.g., always asking subjects to report the details in the same chronological order), subjects in the lab who are instructed to make lies tend to consistently report these details more correctly than when the communication protocol is less straightforward (e.g., asking subjects to report the details in reverse order). This may suggest that the type of communication protocol whether simple or nonstraightforward may have implications on whether one may remain consistent when lying, which in turn may affect the incentive to lie in the first place. While a full understanding of this would require further work, I would like now to propose a stylized modification of the main model that is suggestive of the directions such future research could take. Specifically, let me enrich the model as follows. Every 628 Philippe Jehiel Theoretical Economics 16 (2021) state now denoted θconsists of (sAsB)where sAand sBassumed to be nonnegative numbers correspond to the Aand Battributes (or details) of the state θ,ands=sA+sB summarizes the characteristics of the state (guilt level) parties Iand Ucare about. As in Section 2, I assume that party Uforms the best guess aabout the expected value of safter the hearing of party I(she chooses action aand her objective is −(a −s)2), and as in Section 3,partyIwho is informed of the state θseeks to maximize a.Thereare finitely many states θin and the possible values of sare s1=0sn=1where skhas probability p(sk). There is a small lying cost εand the same genericity assumption as in the main model holds. I will focus the analysis on a communication protocol that is clearly nonstraightforward and I will then discuss how the analysis is modified when a simpler communication protocol is considered instead. Communication takes place at two times t=12.At t=1,partyIis asked to send a message m1describing the state either in normal order −→ m1=(  sA sB)or in reverse order ←− m1=(  sB sA)each with probability half. At t=2,party Iis asked to send a message about attribute Xwith X=A(i.e., mA 2= sA)orX=B(i.e., m2= sB), each with probability half.31 If the two messages are consistent (in the sense that sX= sX)thenpartyUis informed of s= sA+ sBand makes the best guess of sbased on s, denoted a(  s). To simplify the exposition of the arguments, I will be assuming that if the two messages are inconsistent, then party Uis only informed of the inconsistency and chooses ainc =0in such a case.32 Concerning party I’s memory of m1at t=2, I consider the following modification of the main model. As before, I distinguish the memory of party Iat t=2about m1 according to whether party Itold the truth or lied at t=1.IfpartyItold the truth at t=1, party Ihas perfect memory of m1at t=2. If however party Iat t=1lied, then party I at t=2has no memory of which sXfor X=AB was reported. Party I’s belief about sX is then the equilibrium aggregate distribution of first attribute (Aor B)reportedinm1 when there was a lie at t=1.33 In all cases, party Iremembers the state θ=(sAsB). The novelty compared to the main model is that after a lie at t=1,partyIis now only supposed to be confused (not remembering) the exact description of attribute X (Aor B) in his message m1. That is, unlike what was assumed in the main model, party Iafter a lie may remember the targeted level of guilt (as represented by sA+ sBin m1).34 31The possible request of describing the state in reverse order is in reference to some of the experimental settings considered in Vrij et al. (2008). The randomization on the requested attribute is an extra level of complication with no clear link to Vrij et al.’s work. 32Letting party Uchoose freely the action in case of inconsistency as well letting party Uobserve the exact choices of inconsistent messages would not affect the conclusions, but it would require extra analytical steps. 33That is, aggregating for every state θ=(sAsB)(with a weight proportional to the probability of θ), for every normal order request,  sAwhenever −→ m1=(  sA sB)= (sAsB), and for every reverse order request,  sB whenever ←− m1=(  sB sA)= (sBsA). 34When asked about  sX, I am assuming that party Ibelieves that  sXis distributed according to the marginal aggregate distribution of  sX,asexplainedabove. PartyIcould possibly refine this belief when remembering  sA+ sB, and realizing that the marginals of both  sXand  s−X(where −Xis attribute other than X) are the same. Such extra inferences are not trivial and they would become increasingly complex and weak if I were to consider a scenario with sufficiently many attributes. This leads me not to consider such inferences here. Theoretical Economics 16 (2021) Communication with forgetful liars 635 Proof of Proposition 4.Letm∗ kdenote a consistent lie made by at least one type s= m∗ k,thatis,partyIwith type ssends twice the message m∗ k, and assume there are K different such lies in equilibrium. Define then Lkas the set of types ssuch that party Iwith type ssends twice m∗ k,thatis,m1=m2=m∗ k(this includes those types who lie and say consistently m∗ kand possibly type s=m∗ kif this type tells the truth), and let L=(Lk)k. Clearly, in such an equilibrium, after the message m∗ khas been sent twice, party Uwould choose ak=E(s ∈Lk).Iletskdenote maxLkand observe that skshould be one of the consistent lies m∗ rfor r=1K. Lemma 5. For all k,sk=max Lkshould be a consistent lie. Proof. Suppose this is not the case. Then party Iwith type skwould induce action a= skby telling twice the truth. This would be strictly better for him than what he obtains by sending twice m∗ k, which gives action ak=E(s ∈Lk)≤sk=maxLk(and inflicts an extra 2εpenalty for not telling the truth—this is needed to take care of the case in which Lkwould consist of skonly).  A simple implication of Lemma 5 is the following. Corollary 1. There is a bijection between {L1LK}and {s1sK}. Another observation similar to that obtained in pure persuasion games is the following. Lemma 6. There can be no (voluntary) inconsistent messages sent by any type s= 0in equilibrium. Proof.LetSinc(m1m2)={ssuch that σ1(s) =m1and σ2(s) =m2}with m1= m2and assume by contradiction that Sinc(m1m2)= ∅.ByCorollary 1, on can infer that m∗ k/∈Sinc(m1m2).Lets∗ inc(m1m2)=max Sinc(m1m2). It is readily verified that I1(s∗ inc(m1m2)) and I2(s∗ inc(m1m2)) are strictly better off telling the truth, thereby leading to a contradiction.  Let μkdenote the overall probability (aggregating over all s)withwhichm∗ kis sent at t=1conditional on a lie being sent then (i.e., conditional on m1= s). Without loss of generality, reorder the kso that μkakincreases with k. The single crossing property of u(as) implies the following. Lemma 7. For any k1<k 2, if in equilibrium I(s) lies by sending twice m∗ k1and I(s)lies by sending twice m∗ k2,itmustbethats<s . Moreover, for every k,itmustbethatthe consistent lie m∗ kin Lkcoincides with max Lk,thatis,sk=m∗ k. Proof. For the first part, note that after a lie, player I2(s) would send m2=m∗ k(s) where k(s) =arg max k v(ks) and v(ks) =−μkak−b(s)2−(1−μk)ainc −b(s)2 636 Philippe Jehiel Theoretical Economics 16 (2021) Given that ainc =0,andμ1a1<μ 2a2<··· <μ KaK(they cannot be equal by the genericity assumption), it is readily verified that for any s1<s 2,andk1<k 2,ifv(k2s1)> v(k1s1)then v(k2s2)>v(k 1s2).39 Thus if party Iwith type s2finds lie m∗ k2optimal, he must find it better than m∗ k1,and thus by the property just noted, party Iwith any type s>s 2must also find m∗ k2better than m∗ k1, making it impossible that he finds m∗ k1optimal. To show the second part (sk=m∗ k), I make use of Corollary 1 to establish that if it were not the case there would exist an increasing sequence k1<k 2<···<k Jsuch that type skjwould lie by sending skj+1for j<Jand skJwould lie by sending sk1,whichwould violate the property just established.  To complete the description of equilibria, let L− k=Lk\{m∗ k}where m∗ k=sk= maxLk;p(L− k)denote the probability that s∈L− k;μk(L) =p(L− k)/(rp(L− r)) the probability that the lie m∗ kis made at t=1in the aggregate distribution of lies at t=1; k(s) =arg maxkv(ks) where v(ks) =−μk(L)(ak−b(s))2−(1−μk(L))(b(s))2and ak(L) =E(s ∈Lk). Realizing that party Iwith a type sthat lies outside {m∗ 1m∗ K}will either tell the truth or lie by sending m∗ k(s) depending on what he likes best, and that by Lemma 7 party Iwith type sk=m∗ kshould prefer telling the truth to lying by sending m∗ k(sk), the conditions shown in Proposition 4 follow. 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Manuscript received 12 February, 2020; final version accepted 21 August, 2020; available online 4 September, 2020.