Costly miscalibration

Guo, Yingni; Shmaya, Eran Article Costly miscalibration Theoretical Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Guo, Yingni; Shmaya, Eran (2021) : Costly miscalibration, Theoretical Economics, ISSN 1555-7561, The Econometric Society, New Haven, CT, Vol. 16, Iss. 2, pp. 477-506, https://doi.org/10.3982/TE3991 This Version is available at: https://hdl.handle.net/10419/253509 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), 477–506 1555-7561/20210477 Costly miscalibration Yingni Guo Department of Economics, Northwestern University Eran Shmaya Department of Economics, Stony Brook University We consider a platform that provides probabilistic forecasts to a customer using some algorithm. We introduce a concept of miscalibration, which measures the discrepancy between the forecast and the truth. We characterize the platform’s optimal equilibrium when it incurs some cost for miscalibration, and show how this equilibrium depends on the miscalibration cost: when the miscalibration cost is low, the platform uses more distant forecasts and the customer is less responsive to the platform’s forecast; when the miscalibration cost is high, the platform can achieve its commitment payoff in an equilibrium and the only extensive-form rationalizable strategy of the platform is its strategy in the commitment solution. Our results show that miscalibration cost is a proxy for the degree of the platform’s commitment power and, thus, provide a microfoundation for the commitment solution. Keywords. Calibration, miscalibration, cheap talk, commitment, Bayesian persuasion, e-commerce platform. JEL classification. D81, D82, D83. 1. Introduction E-commerce platforms often provide information to customers about their products. For example, the fare aggregator Kayak.com provides forecasts of future prices. The real estate aggregator Redfin.com identifies “hot homes” that are likely to sell quickly. The platform generates information using some algorithm, which it applies to many products. This algorithm is usually a trade secret and is, therefore, not observed by outsiders. For example, Kayak states only that “our scientists develop these flight price trend forecasts using algorithms and mathematical models.” Redfin states only that “the hot homes algorithm automatically calculates the likelihood by analyzing more than 500 attributes of each home.” In this paper, we develop a general model to analyze a platform’s communication with its customers, in which customers trust the platform’s forecasts in an equilibrium even though they do not observe the algorithm. Yingni Guo: [email protected] Eran Shmaya: [email protected] Financial support from National Science Foundation Grant SES-1530608 is gratefully acknowledged. ©2021 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at https://econtheory.org.https://doi.org/10.3982/TE3991 478 Guo and Shmaya Theoretical Economics 16 (2021) Environment We model the interaction between the platform and a customer as a sender–receiver game: the forecast is the sender’s message and the algorithm induces a strategy for the sender. The sender provides information in the form of probabilistic distributions. For example, Redfin defines a “hot home” as one that has a 70 percent chance or higher of having an accepted offer within 2weeks of its debut. Given the sender’s strategy, for each claim that the sender makes, we can calculate the true conditional distribution over the states. We say that the sender’s strategy is miscalibrated if there is a discrepancy between what he claims and the true conditional distribution, for example, if only 50 percent of the homes that Redfin identifies as hot homes have an accepted offer within 2weeks of their debut. Likewise, the sender’s strategy is calibrated if what he claims is always the same as the true conditional distribution. The sender incurs a cost that is a function of the miscalibration measure, which we define in the paper. The sender’s cost also depends on a parameter, which we call the cost intensity. The cost intensity indicates how severely the sender is punished for each unit of miscalibration or how easy it is to collect data. The sender’s cost is a proxy for the reputation damage from making incorrect assertions. The fact that platforms have various degrees of reputation concern is well documented in the empirical literature. Mayzlin et al. (2014) show that small or independent hotels are more likely to engage in review manipulation than multi-unit or branded chain hotels, and Luca and Zervas (2016) show that chain restaurants are less likely to commit review fraud. There is also a large experimental literature showing that people have a preference for being seen as honest (e.g., Abeler et al. 2019). As in the cheap-talk literature, the receiver observes only the message, but not the sender’s strategy. When the cost intensity is zero, our game is a cheap-talk game (e.g., Crawford and Sobel 1982,Green and Stokey 2007). When the cost intensity is positive, the sender’s talk is not cheap, because of the miscalibration cost. As the cost intensity increases, the sender becomes more concerned about the validity of his assertions. We examine how this miscalibration cost affects the sender’s optimal equilibrium and the receiver’s response to the messages. Main results In the sender–receiver game between the platform and the customer, the sender promotes a product and the receiver decides whether to buy it. We call such a game a promotion game. In addition to our platform–customer application, promotion games include many applications studied in recent papers, such as the interaction between a prosecutor and a judge as in Kamenica and Gentzkow (2011), the certification game in Henry and Ottaviani (2019)andPerez-Richet and Skreta (2021), the informational lobbying game in Bardhi and Guo (2018), Guo and Shmaya (2019), and Minaudier (2020), and the media censorship game in Gehlbach and Sonin (2014)andKolotilin et al. (2017). Our first result is that the sender’s has an optimal equilibrium in which his strategy is calibrated and has the support of two messages, one of which induces the receiver to purchase with positive probability. Using this characterization, we study how cost intensities affect the sender’s optimal equilibrium. For high cost intensities, the commitment solution is an equilibrium, since Theoretical Economics 16 (2021) Costly miscalibration 479 even a small deviation incurs a big miscalibration cost. When the cost intensity is low, the equilibrium exhibits two features that are distinct from the commitment solution. First, the receiver purchases with only some probability after the purchase message. Second, the sender uses messages that are more distant from each other than the messages that are used in the commitment solution. Both these features reduce the sender’s incentive to deviate. These features show how the sender leverages his low commitment power to gain credibility. Our result suggests that the sender makes more “extreme” statements than in the commitment solution, so as to lend himself credibility. For instance, Airbnb labels exceptional hosts as “superhosts.” One criterion is that superhosts cancel less than 1percent of the time (i.e., a maximum of 1cancellation in 100 bookings). Given this extremely high standard, a customer who experienced a cancellation can confidently suspect miscalibration if another customer recently experienced the same.1Hence, Airbnb retains the credibility of its superhost label. We also show that the sender’s optimal equilibrium payoff is monotone-increasing in the cost intensity. This is because the set of calibrated equilibria expands as the cost intensity increases. Given that the sender’s optimal equilibrium is calibrated, a higher cost intensity thus leads to a higher optimal equilibrium payoff for the sender. Recall that the cost intensity models the degree of reputation damage to the sender from making incorrect forecasts. The monotonicity result shows that the cost intensity can also be interpreted as a proxy for the degree of the sender’s commitment power. The more the sender is likely to suffer from making incorrect assertions, the more credible he becomes in his communication with the receiver. A higher payoff to the sender thus results. Our second result is that our model bridges the cheap-talk and the persuasion models. For any sender–receiver game, when the cost intensity is zero, our game is (by definition) a cheap-talk game. We show that when the cost intensity is high, the sender’s optimal equilibrium payoff is the payoff he could get if he could commit to a strategy. This result asserts some lower hemicontinuity of the equilibrium correspondence. As usual, lower hemicontinuity is not straightforward. In our setup, it requires a generic assumption on the sender–receiver game. We also show that the only extensive-form rationalizable strategy of the sender is his strategy in the commitment solution. To summarize, our contribution is threefold. We develop a general framework to analyze an e-commerce platform’s communication with its customer. We characterize how miscalibration cost affects the platform’s optimal equilibrium and the customer’s response to the messages. We provide a microfoundation for the commitment solution; our model bridges the cheap-talk and the persuasion models, and can be used to analyze the middle ground where neither the talk is completely cheap nor the commitment absolute. Related literature Our paper contributes to the literature on strategic communication with a lying cost (e.g., Kartik et al. 2007,Kartik 2009). The key difference is that in the 1For a case in which miscalibration occurred, see Patrick Collinson, “My Airbnb Superhost Stay Turned into a Super Disaster,” The Guardian (https://www.theguardian.com/money/2019/dec/21/my-airbnb -superhost-stay-turned-into-a-super-disaster), December 21, 2019. 480 Guo and Shmaya Theoretical Economics 16 (2021) costly-lying model, the message space is typically the state space, so the sender’s only option is to announce a state. Consequently, the only way for the sender to avoid a lying cost is to reveal everything he knows. In contrast, our sender avoids a lying cost as long as what he claims is true (i.e., his asserted distribution is the same as the true distribution). The implication of this modeling difference is best seen when the lying cost is high. High lying costs translate into commitment power in our model, but into full disclosure in the costly-lying model. These different high cost results capture different intuitions about what happens when lying becomes very costly. In Remark 1 in Section 2,weformalize the notion that our model nests the costly-lying model. Our notion of miscalibration differs from the notion of lying in Sobel (2020), where the message space includes some subsets of the state space. The sender has lied if the state does not belong to the subset of the state space that he has announced. Our paper is also related to Perez-Richet and Skreta (2021), which characterizes the receiver’s optimal test when the sender can falsify the state. In their baseline model, the sender’s falsification is observable. Falsification improves the test results, but devalues their meanings. They show that a receiver’s optimal test in which the sender does not falsify always exists. In our paper, the sender can deviate to any communication strategy. His strategy is not observable, but the potential miscalibration cost allows for meaningful communication. We characterize an environment in which the sender does not miscalibrate in his optimal equilibrium, and also show that this is not always the case. Our paper is also related to Nguyen and Tan (2021). Their sender first publicly announces a test. Then he privately observes the message generated and can manipulate the message at a cost. The receiver’s action depends on both the announced test and the final message. They characterize how the chance of manipulation affects the test design. Our model differs in that our sender chooses a test that is not observable, so the receiver observes only the message (i.e., the customer sees only the platform’s message, not its algorithm). Our results show that our model bridges the cheap-talk and persuasion models (e.g., Rayo and Segal 2010,Kamenica and Gentzkow 2011). In this aspect, our paper is related to Fréchette et al. (2020), Lipnowski et al. (2019), and Min (2020). In these papers, the sender first publicly announces a test. With an exogenously given probability, the message is given by this test. With complementary probability, the sender can secretly choose a different message. The receiver in all their models observes both the announced test and the final message. Our model and results propose a different measure of commitment and generate qualitatively different predictions than their probabilistic commitment models. Lipnowski and Ravid (2020) is another recent paper that relates to the relationship between the commitment world and the cheap-talk world, and models cheap talk from a belief-based perspective. The concept of calibration is central in the forecasting literature (e.g., Dawid 1982, Murphy and Winkler 1987,Foster and Vohra 1998,Ranjan and Gneiting 2010). It is used for two closely related ideas. The first is in the purely probabilistic sense, namely, that a forecast (a message, in our setup) matches the conditional distribution over states given the forecast. The second is the idea of calibration with the data, namely, that a Theoretical Economics 16 (2021) Costly miscalibration 481 forecast matches the realized distribution of states at those times in which the forecast was given. Building on these ideas, we introduce the concept of calibration to sender– receiver games and develop a miscalibration measure to study these games.2 This paper is also related to the empirical literature on firms’ online communication with customers. Chevalier and Mayzlin (2006) show that information on platforms has a significant impact on sales. Mayzlin et al. (2014) explore the difference between a website on which faking is difficult and a website on which faking is relatively easy. They show that the cost of review manipulation determines the amount of manipulation in equilibrium and that different firms have different incentives to manipulate. We contribute to this literature by developing a general framework to analyze platforms’ communication with customers. The results shed light on how firms’ reputation concerns affect the effectiveness of their communication. Structure of the paper Section 2 presents the model. In Section 3, we characterize the sender’s optimal equilibrium for promotion games. Section 4 presents our results when the cost intensity is high. Section 5 extends our analysis to setups in which the sender has some information on the state, but does not necessarily know it. Section 6 contains the proofs. 2. Environment We consider a game with incomplete information between two players, Sender and Receiver. Sender sends Receiver a message that depends on a state of nature that is unobserved by Receiver. Receiver then chooses an action. Let Sbe a finite set of states equipped with a prior distribution pwith full support and let Mbe a Borel space of messages. A Sender’s strategy with message space Mis givenbyaMarkovkernelσfrom Sto M: when the state is s, Sender randomizes a message from σ(·|s).ForSender’sstrategyσ,letτσ:M→(S) be such that τσ(m) is the conditional distribution over states given m. We sometimes use τσ(s|m) for τσ(m)(s), both of which denote the conditional probability of sgiven m. We say that a strategy σhas finite support if σ(·|s) has finite support for every s, in which case we let support(σ) =ssupport(σ(·|s)).Whenσhas finite support, the conditional probability τσ(s|m) is given by τσ(s|m) =p(s)σ(m|s)  s psσm|s for every m∈support(σ), and is defined arbitrarily for m/∈support(σ). For the rest of the paper, we assume that M=(S).InSection 2.1, we introduce the concepts of calibrated strategies and miscalibration. These concepts are independent of Sender’s interaction with Receiver. In Section 2.2, we review the model of sender– receiver games and the definition of a cheap-talk equilibrium. Section 2.3 presents our definition of an equilibrium with costly miscalibration. 2There is also literature about the strategic manipulation of calibration tests. See, for instance, Foster and Vohra (1998), Lehrer (2001), and Olszewski (2015). 482 Guo and Shmaya Theoretical Economics 16 (2021) 2.1 Sender’s calibrated strategies and miscalibration Sender’s message mis an asserted distribution over states. We make the assumption that M=(S) for two reasons. First, it is hard to define a notion of costly miscalibration when the message space is not specified. Second, the space of beliefs is exactly the space that captures the finest grain of information needed by Receiver in a sender–receiver game. We thus refer to τσ(m) as the true conditional distribution over states given a message m. We say that a strategy σis calibrated if τσ(m) =ma.s.3Under a calibrated strategy, messages mean what they say, i.e., they can reliably be taken at face value. We now introduce a key component of our model: a measure of miscalibration κ(σ) for Sender’s strategy σ. To define this measure, let d:(S) ×(S) →R+be a continuous function, where d(qm) measures the distance between a message m∈(S) and a truth q∈(S). We assume that d(qm) =0if and only if m=q.Let κ(σ) = s p(s) dτσ(m)mσ(dm|s) be the expected distance between the distribution asserted by Sender’s message and the true conditional distribution given that message, when Sender’s strategy is σ.Astrategy σis calibrated if and only if κ(σ) =0. The following example illustrates the concept of miscalibration in the case of Redfin. Redfin’s algorithm is not observed by Receiver or any third party. When the algorithm is applied to many products, however, the true conditional distribution given a message can be estimated. Example 1. The Redfin example. Redfin defines a hot home as one that has a 70 percent chance or higher of having an accepted offer within 2weeks of its debut. Its hot home algorithm “identifies hot homes based on real estate conditions in each market.” This allows us to focus on a local market. We collected 2,150 hot homes and tracked their status.4When a home’s status becomes contingent, pending, or sold, it is considered to have an accepted offer. The percentage of hot homes having an accepted offer within 2 weeks of its debut was 531. This is different from 70 percent. Let τbe the true probability that a hot home has an accepted offer in 2weeks. The p-value for the null hypothesis τ≥70% against the alternative hypothesis τ<70% is smaller than 00001. Redfin’s forecast was not calibrated.5♦ 2.2 Sender–receiver games Let Abe a finite set of actions by Receiver. Let vu :S×A→Rbe, respectively, Sender’s and Receiver’s payoff functions. Receiver’s strategy is given by a Markov kernel ρfrom 3The term a.s. in our paper means “almost surely with respect to the probability distribution over messages induced by σ.” Recall that τσis defined up to a set of messages with probability 0. 4We examined Cook County in Illinois, which includes 165 zip codes, and collected homes that Redfin had identified as hot homes over 2weeks (03/14/2018–03/27/2018). 5We thank Cassiano Alves and Samuel Goldberg for excellent research assistance. Theoretical Economics 16 (2021) Costly miscalibration 483 Mto A, with the interpretation that Receiver randomizes an action from ρ(·|m) after message m. For a profile (σρ) of Sender’s and Receiver’s strategies, we let πσρ ∈(S ×A) be the induced distribution over states and actions when the players follow this profile. The payoffs to Sender and Receiver under (σρ) are given by V0(σρ) =vdπσρ and U(σρ)=udπσρ respectively. The reason we add the subscript 0in Sender’s payoff function V0becomes clear later when we define Vλfor every λ≥0. ABayesian Nash equilibrium (BNE) for a cheap-talk game is a Nash equilibrium in the normal-form game defined by the payoff functions V0and U. 2.3 Equilibrium with costly miscalibration We now define a BNE for the game with costly miscalibration. The definition is the same as that for a cheap-talk game except that Sender’s payoff is given by Vλ(σρ) =V0(σ ρ) −λκ(σ) where κ(σ) is the miscalibration measure and λ≥0is a parameter that indicates the intensity of Sender’s miscalibration cost. A BNE is a Nash equilibrium in the normal-form game defined by the payoff functions Vλand U. We say that an equilibrium is calibrated (or miscalibrated) if Sender’s strategy is calibrated (or miscalibrated). Before proceeding, we show that any calibrated equilibrium for λis also an equilibrium for a higher λ>λ. Intuitively, if Sender has no incentive to miscalibrate for a low cost intensity, he surely has no incentive to do so when the cost is higher. However, a miscalibrated equilibrium for λis not necessarily an equilibrium for a higher λ>λ. This is because moving to a higher intensity means a higher miscalibration cost, which might prompt Sender to deviate. Claim 1. For any sender–receiver game and any distance function, let (σρ) be an equilibrium for λ≥0.Ifσis calibrated so Sender pays no miscalibration cost on path, then (σρ) is an equilibrium for a higher λ>λ. Proof. Since (σ ρ) is an equilibrium for λ≥0,ρis a best response by Receiver to σ. To show that (σρ) is an equilibrium for λ, we need to show that Sender’s payoff from deviating to any σis smaller than his equilibrium payoff: Vλ(σρ)≤Vλ(σρ).Thisis the case, since Vλσρ=V0σρ−λκσ≤V0σρ−λκσ=Vλσρ≤Vλ(σρ) =Vλ(σ ρ) The first inequality follows from the fact that λ>λand κ(σ)≥0. The second inequality follows from the fact that (σρ) is an equilibrium for λ. The last equality follows from the fact that σis calibrated so κ(σ) =0. 484 Guo and Shmaya Theoretical Economics 16 (2021) Remark 1. We now formalize the relationship between our model and the costly-lying model (Kartik et al. 2007,Kartik 2009). In the costly-lying model, Sender’s lying measure is given by some ˆ d(ss)if the state is sand Sender declares some s∈S. For example, ˆ d(ss)=(s −s)2. In our model, if Sender is restricted to announce deterministic messages (i.e., m∈{δs:s∈S}) and for any such mthe distance function d(qm) is linear in q, d(qδs)= s q(s) ˆ dss then our model reduces to the costly-lying model. 3. Promotion games Given our interest in platforms, we study a class of sender–receiver games that we call promotion games: Receiver decides whether to buy, and Sender’s payoff is 1if Receiver buys and 0otherwise. Formally, we assume that A={BNB},andthatv(sB)=1and v(sNB)=0for each s. It is without loss to set u(s NB)=0for each s. It is also without loss to identify each state with Receiver’s payoff from action B in that state. Thus, we set s=u(s B)for each s. Promotion games are rich enough to capture games with binary actions for Receiver and state-independent payoff for Sender. To avoid triviality, we assume that sp(s)s < 0 and that s>0for some s. Hence, Receiver strictly prefers not to buy given the prior belief and strictly prefers to buy for some state. It is well known that in the cheap-talk case (i.e., λ=0), no promotion is possible in the sense that Receiver never buys. Indeed, suppose on the contrary that Receiver buys with positive probability after some message. Then Sender will announce only those messages that induce the highest buying probability. This means that Receiver is willing to buy after every message on path, contradicting the assumption that Receiver strictly prefers not to buy given the prior belief. We now characterize Sender’s optimal equilibrium for any λ>0, showing that (i) even for a low λ, Sender gets a positive payoff, and (ii) for a high λ, Sender gets his commitment payoff. Before presenting our result, we introduce a promotion-game example to illustrate the main concepts introduced in the previous section. We also use this example later to illustrate our result. Example 2. Consider a promotion game with S={−2−11}and the prior p= (1/81/23/8). Consider Sender’s strategy in Table 1. He sends either message m0or m1. Message m0says that the state is −1. Message m1says that the state distribution is (1/51/53/5). Each column shows the probabilities with which Sender sends m0or m1in each state. Under this strategy, the true conditional distribution given m0is τ(m0)=(010). The true conditional distribution given m1is τ(m1)=(1/51/53/5). Each message coincides with the true conditional distribution given that message, so this strategy is calibrated. Theoretical Economics 16 (2021) Costly miscalibration 491 Lemma 1. Let dbe a Wasserstein distance over (S) and let qq∈(S).Letq=t1q1+ ···+tnqnbe a splitting of q,whereti≥0,qi∈(S) for all i,andn i=1ti=1. Then there exists a splitting q=t1q 1+···+tnq nof qwith the same weights such that dqq=t1dq1q 1+···+tndqnq n(3) To prove Theorem 1, we begin with an arbitrary equilibrium and construct a new equilibrium that has the properties in Theorem 1 and generates the same payoff for Sender. Roughly speaking, the proof can be divided into two steps. In the first, we make the equilibrium calibrated. In the second, we combine into a single message all the messages under which Receiver buys. Both steps use the same logic. We first construct a new strategy profile that has the desired property and generates the same payoff for Sender. We then show that for every deviation of Sender from the new profile he has a more profitable deviation from the original profile. Since the original profile was an equilibrium, the new profile must also be an equilibrium. The first step uses the fact that Wasserstein distances satisfy the triangular inequality. Consider, for example, an equilibrium under which (i) with some probability, Sender announces a message m, after which Receiver buys with probability 1, and (ii) the true conditional distribution given mis q=τσ(m). So Sender’s payoff when he announces mis 1−λd(qm). Now we change Receiver’s strategy so that he buys with probability 1−λd(qm) after message q, and we change Sender’s strategy so that he announces the true distribution qinstead of m. This new strategy profile gives Sender the same payoff and his message is now calibrated. We now argue that Sender has no profitable deviation. Suppose that Sender considers deviating to some miscalibrated strategy, so that the true conditional distribution when he announces qis in fact q. Sender then suffers a miscalibration cost of λd(qq), so his payoff when he sends qis 1−λd(qm) −λdqq≤1−λdqm where the inequality follows from the triangular inequality. The right-hand side is the payoff to Sender under the original profile if Sender deviates by announcing message m when the truth is q, and he suffers a miscalibration cost of λd(qm). Since the original profile was an equilibrium, this implies that the new profile must also be an equilibrium. We note that this argument is not completely accurate, because if Receiver strictly prefers to buy when his belief about the states is q, then he must buy with probability 1 after message qin a calibrated equilibrium. Because of this nuisance, the proof has an additional modification to the equilibrium profile: it replaces qwith a belief that makes Receiver indifferent. The second step of the proof combines all the messages under which Receiver buys into a single message. Assume that, for example, we had a calibrated equilibrium in which there are nmessages q1qn, after which Receiver buys with probability 1. Sender announces message qiwith probability ri. We replace this strategy profile with a new profile. In the new profile, Sender announces the message q=t1q1+···+tnqnwith probability r=r1+···+rn,whereti=ri/r, and Receiver buys with probability 1after message q. We need to show that the new profile is an equilibrium. We again construct, 492 Guo and Shmaya Theoretical Economics 16 (2021) for every deviation that Sender has in the new profile, a deviation strategy in the original profile that is more profitable. Suppose Sender deviates to a miscalibrated strategy, under which he sends the message qwith some probability r, but the true distribution when he sends qis q. Under this deviation, Sender will suffer a miscalibration cost of λd(qq) when he sends q.Lemma 1 states that Sender has a deviation strategy in the original profile such that he sends message qiwith probability rtiand the true distribution when he sends qiis q i. Moreover, the overall miscalibration cost Sender suffers, conditional on sending one of q1qn,isalsoλd(qq). Hence, the aforementioned deviation to the new profile cannot be profitable. 4. High cost intensity We now turn to general sender–receiver games, in which the set Aof Receiver’s actions is finite and Sender’s payoff v(sa) can depend on s. The next two propositions formalize the intuition that when the cost intensity is high, Sender gains the commitment power not to make false assertions. Proposition 1 shows that Sender can achieve his commitment payoff in an equilibrium. Proposition 2 shows that if the distance function has a kink, the only extensive-form rationalizable strategy of Sender is his strategy in the commitment solution. Formally, the commitment payoff is given by CP =maxV0(σ ρ),wherethemaximum ranges over all profiles (σρ) such that ρis a best response to σ.Wecallaprofile(σρ) that achieves the maximum a commitment solution. 4.1 Equilibrium result On the surface, Proposition 1 can be understood from the following two observations. First, if Sender were exogenously restricted to using only calibrated strategies, then Receiver could take messages at face value. Hence, Sender’s optimal commitment strategy, along with Receiver’s best response to the face value of each message, would be an equilibrium. Second, as the cost intensity increases, in any equilibrium Sender uses only strategies that are close to being calibrated, in the sense that each message is close to the true posterior over states given that message. Thus, our result asserts some lower hemicontinuity of the equilibrium correspondence. As usual, lower hemicontinuity is not straightforward. The difficulty is that Sender’s payoff is not a continuous function of his strategy, because Receiver’s strategy is typically not a continuous function of the message. Therefore, a small deviation from the optimal commitment strategy might have a big impact on Sender’s payoff. We make the assumption that the sender–receiver game is generic. By “a generic set of games,” we mean that the set of payoff functions u:S×A→Rfor which the assertion does not hold (viewed as a subset of RS×A) is a closed set with an empty interior and a Lebesgue measure of 0. Our generic assumption requires that every Receiver’s action be a unique best response to some beliefs over the states. The following proposition requires only that dbe convex in q. Wasserstein, Kullback–Leibler, and Euclidean distances all satisfy this condition. Theoretical Economics 16 (2021) Costly miscalibration 493 Proposition 1. Assume that dis convex in q. In a generic set of games, for every ε> 0,thereexists¯ λsuch that for every λ>¯ λ, there exists an equilibrium in the game with intensity λsuch that Sender’s payoff is at least CP −ε. To illustrate the proof idea, consider the simplifying assumption that Receiver has some “punishment” action that yields a bad payoff for Sender in every state. For a generic game, any approximation for the commitment payoff can be achieved by a calibrated strategy for which Receiver’s best response is unique. Then we construct an equilibrium in which Receiver best responds to the face value of each message in the support of this strategy and uses the punishment action for any other message. Sender’s equilibrium strategy may well be miscalibrated, but for a high cost intensity, the amount of miscalibration will be small enough so that Receiver’s response is still uniquely optimal. Sender thus gets an equilibrium payoff close to the commitment payoff. The proof for the case in which the simplifying assumption does not hold is more involved, but relies on a similar idea. In the following example, Sender’s optimal equilibrium must be miscalibrated, yet his optimal equilibrium payoff converges to his commitment payoff. Example 4(Withadifferent distance function). Consider the promotion game in Example 3, but with this difference: the distance between a message and a truth is the Kullback–Leibler distance dτ(m)m= s∈S τ(m)(s) log τ(m)(s) m(s)  As in Example 3,weusetheprobabilityofstate1to represent the state distribution. In the commitment solution, Sender splits the prior 1/4into 0and 1/2. Receiver buys for sure after the message 1/2. However, for any intensity λ≥0, this strategy profile cannot be an equilibrium: Sender will deviate by announcing 1/2more often. This is because, in the case of the smooth Kullback–Leibler distance, for any message with full support, a small amount of miscalibration has no first-order impact on the miscalibration cost. Sender has to “overshoot” to gain credibility. Consider the following equilibrium. Sender splits the prior into 0and 1/2.When1/2is realized, Sender overshoots by saying (1−e−1/λ/2)>1/2. Receiver buys for sure after (1−e−1/λ/2)and does not buy after other messages. Sender’s equilibrium payoff is λlog(2e1/λ −1)/4,whichgoesfrom0to the commitment payoff 1/2as λgoes from 0to ∞.OurExample 1 shows that Redfin indeed overshoots. ♦ We have shown that Sender achieves his commitment payoff (up to ε)inanequilibrium with some generic assumption about the game. Example 5 is a nongeneric game in which Sender’s optimal equilibrium payoff for any λis bounded away from his commitment payoff. 494 Guo and Shmaya Theoretical Economics 16 (2021) Example 5. Consider a promotion game with S={0−1}and the prior p=(p01− p0). Sender’s commitment payoff is p0: Sender fully reveals the state and Receiver takes action B if the state is 0and action NB if the state is −1. Unlike Example 4, there exists no belief over the states such that B is uniquely optimal for Receiver, so there is no room for Sender to overshoot. Hence, for a smooth distance function d, in every equilibrium, Receiver chooses only the safe action NB, so Sender’s payoff is 0.♦ 4.2 A rationalizablility result Up to now, following the cheap-talk literature, we used equilibrium as our solution concept. We argued that Sender’s optimal equilibrium achieves the commitment solution for a high λ, but the game still admits other equilibria. For example, even for a high λ, there exists an uninformative equilibrium. Under this equilibrium, Sender always announces the prior, and after every possible Sender’s message, Receiver believes that the state is distributed according to the prior and best responds to the prior. There is, however, an unsatisfactory aspect to the uninformative equilibrium in our context. Consider Example 2 with a high λ. Assume that Receiver plays according to the uninformative equilibrium and that the message (001), which says that the state is 1with probability 1, arrives. What should Receiver do? The message is a surprise, but given the high λ, Sender suffers a very high cost if the truth is far away from this message. It seems reasonable that Receiver will deduce that the truth is close enough to this message, in which case Receiver will buy the product. BNE and refinements such as the perfect Bayesian equilibrium do not capture this intuition because they allow arbitrary behaviors or beliefs off-path. These behaviors do not have to be rationalizable. To incorporate this earlier intuition, we turn to extensiveform rationalizability (hereafter, EFR; see Pearce 1984,Battigalli 1997,andBattigalli and Siniscalchi 2002). EFR dispenses with the assumption that players’ beliefs are correct, but requires that, at every point in the game, each player form a belief that is, as much as possible, consistent with the opponent being rational. EFR is usually defined in an environment with only countable information sets. How to extend such a definition to sender–receiver games with a continuum message space is not obvious. (See Remark 2 in the Appendix and Friedenberg 2019, for example, where similar issues arise.) However, the issue is somewhat simpler in our setup, because each player takes action only once. The important assumption behind EFR is that Receiver strongly believes that Sender does not use strictly dominated strategies. This means that after observing a message, Receiver has a belief about Sender’s strategies that is concentrated on Sender’s strategies that are not strictly dominated. Let BR(q) be the set of all of Receiver’s best actions when his belief about the state is q: BR(q) =arg max a s q(s)u(s a) (4) Theoretical Economics 16 (2021) Costly miscalibration 495 Proposition 2. Assume that there exists some γ>0such that d(qm) ≥γ|q−m|1.Then, in a generic set of games, there exists ¯ λsuch that for every λ>¯ λ, the following statements hold: (i) Receiver’s extensive-form rationalizable strategies are those that satisfy supportρ(·|m)⊆BR(m) (ii) Sender’s extensive-form rationalizable strategies are his calibrated strategies σsuch that Vλ(σρ) =CP for some best response ρof Receiver to σ. To prove this result, we use the result (Lemma 5 in the Appendix) that if the distance function dhas a kink, then for sufficiently high λ,aSender’sstrategyσis not strictly dominated if and only if it is calibrated.8Therefore, only Sender’s calibrated strategies survive the first round of elimination. Given that Sender uses only calibrated strategies, Receiver takes any message at face value and best responds to the face value. In a generic game, any approximation for the commitment payoff can be achieved by a calibrated strategy for which Receiver’s best response is unique. Therefore, the only Sender’s strategy that will survive is one that gives him the commitment payoff against some Receiver’s best response. Note that Proposition 2 does not hold in the nongeneric game in Example 5 even when dhas a kink. In that game, Receiver’s strategy to always take NB is rationalizable and, therefore, every calibrated strategy is rationalizable for Sender. 5. Discussion:Partial information on the state By allowing Sender to choose any strategy σ:S→(S), we implicitly assume that Sender knows the state. This assumption is natural since our main focus is on Sender’s incentives. However, because we use terminology from the forecasting literature and because of our interest in e-commerce platforms like Redfin, it would be more realistic to assume that Sender has some information on the state, but does not necessarily know it. We argue that our analysis and results extend to this environment as well. To model such an environment, we can add an exogenous set Tof Sender’s types and an exogenous information structure σE:S→(T),suchthatσE(s) is the distribution over Sender’s types. When Sender’s type is t, his belief about the state is τσE(t).Our definitions and results carry through mutatis mutandis if we assume that the message space Mis the convex hull of {τσE(t) :t∈T}and that Sender is restricted to strategies that are less informative than σE(in Blackwell’s sense). 6. Proofs 6.1 Preliminaries We extend the distance function dto a function d:RS +×(S) →R+given by d(γqm) = γd(qm) for every q∈(S) and γ≥0. We extend the best-response correspondence BR(·)to a correspondence from RS +to Agiven by the same formula (4). 8The assumption that d(qm) ≥γ|q−m|1for some γ>0holds for Wasserstein distances and, in particular, the total-variation distance. It also holds for the Euclidean distance d(qm) =|q−m|2and any other distance that is derived from a norm. 496 Guo and Shmaya Theoretical Economics 16 (2021) For a Sender’s strategy σ,welet χσ= s p(s)σ(·|s) (5) be the distribution over messages induced by σ.Lemma 2 below is essentially Aumann and Maschler’s splitting lemma (see, for example, Zamir 1992, Proposition 3.2). It says that a distribution over messages in (S) is induced by some calibrated strategy if and only if its barycenter is the prior p. Lemma 2. (i) For every strategy σ, we have τσ(m)χσ(dm) =p (6) (ii) If χis a distribution over messages in (S) such that mχ(dm) =p, then a calibrated strategy σsuch that χσ=χexists. We also say that a finite splitting of a probability measure p∈(S) is a representation p=itiqisuch that qi∈(S),ti≥0and iti=1. We sometimes call tithe weights. The splitting lemma implies that for every such finite splitting, there exists a calibrated strategy that announces the message qiwith probability ti. 6.2 Proof of Theorem 1 6.2.1 Preliminaries We can assume without loss that λ=1, since all properties of the distance function dthat are used in the proof still hold if we replace dby λd. We denote a Receiver’s strategy by a function ρ:(S) →[01],sothatρ(m) is the probability that Receiver buys after message m. For a bounded function f:(S) →R,letLip f:(S) →Rbe given by Lipf(q)=sup m∈(S)f(m)−d(qm) Note that if dsatisfies the triangular inequality, then Lipfis the least 1-Lipschitz (w.r.t. d) majorant of f. For a bounded function f:(S) →R,wedenotebyCav fthe least concave majorant of f, sometimes called the concave envelope of f. The following proposition summarizes several properties of the operators Lip and Cav. For functions f g :(S) →R,wedenotef≤gwhen f(q) ≤g(q) for every q∈(S). Proposition 3. Let f g :(S) →Rbe bounded. Then the following statements hold: (i) If dsatisfies the triangular inequality, then Lip Lipf=Lipf. (ii) We have CavCav f=Cavf. (iii) If f≤g,thenLip f≤Lip gand Cavf≤Cavg. Theoretical Economics 16 (2021) Costly miscalibration 497 Claim 2. If d(qm) is convex in q, then Sender’s optimal payoff against a Receiver’s strategy ρ:(S) →[01]when the prior distribution over states is pis CavLipρ(p). Proof. Consider some Sender’s strategy σ. Then Sender’s payoff under (σ ρ) is V1(σρ) =ρ(m) −dτσ(m)mχσ(dm) ≤Lip ρτσ(m)χσ(dm) ≤CavLip ρ(p) where the first inequality follows from ρ(m) −d(τσ(m) m) ≤Lipρ(τσ(m)) by the definition of Lip ρ, and the second follows from (6) and the definition of the concave envelope Cav. For the converse, fix ε>0. From the definition of Cav and Lip, it follows that there exist elements qi∈(S),weightsti≥0, and messages mi,suchthatp=itiqi,iti=1, and  i tiρ(mi)−d(qimi)≥Cav Lipρ(p) −ε (7) We can assume that mi= mjif i= j;otherwise,ifmi=mj=m,welett=ti+tjand q=(tiqi+tjqj)/t. Since dis convex in q, we can replace the elements qiand qjwith a single element q, whose weight and corresponding message are, respectively, tand m, without violating (7). We now consider Sender’s strategy σsuch that χσ=itiδmiand τσ:(S) →(S) is a function such that τσ(mi)=qi. Such a strategy exists by Lemma 2.Then(7)implies that Sender’s payoff when using σis at least Cav Lipρ(p) −ε. Lemma 1 shows that Wasserstein distances are well behaved under the splitting of probability distributions. See Laraki (2004) for related results on other metrics and generalizations to infinite-dimensional spaces, including implications for Lipschitz continuity as in our Corollary 2. Proof of Lemma 1.Thedirectiond(qq)≤t1d(q1q 1)+···+tnd(qnq n)follows from convexity of dfor every splitting q=t1q 1+···+tnq nof qwith the weights t1tn. For the other direction, let Qand Qbe S-valued random variables with marginal distributions qand q, respectively, such that d(qq)=Ed(QQ).LetXbe a random variable that assumes values in {1n}such that ti=P(X =i) and such that qi(s) = P(Q =s|X=i) for every i∈{1n}and s∈S. The existence of such a variable follows from the splitting lemma. Finally, let q i(s) =P(Q=s|X=i).Then q(s) =PQ=s= i P(X =i)PQ=s|X=i= i tiq i(s)  i tidqiq i≤ i P(X =i)EdQQ|X=i=EdQQ=dqq where the inequality follows from the definition of d(qiq i), since the marginal distributions of Qand Qconditioned on the event X=iare qiand q i, respectively. 498 Guo and Shmaya Theoretical Economics 16 (2021) Corollary 2. Let dbe a Wasserstein distance. Then, for every function f:(S) →[01] that is 1-Lipschitz w.r.t. d, the concave envelope Cav fis also 1-Lipschitz w.r.t. d. Proof.Letqq∈(S) and let q=t1q1+···+tnqnbe a splitting of qsuch that Cavf(q)= t1f(q1)+···+tnf(qn).ByLemma 1, there exists a splitting q=t1q 1+···+tnq nof qsuch that (3) holds. Therefore, Cavfq≥ i tifq i≥ i tif(qi)−dqiq i=Cavf(q)−dqq where the first inequality follows from the definition of Cav fand the second from the fact that fis 1-Lipschitz. 6.2.2 Proof of Theorem 1 Let (σ ρ) be an equilibrium. From the equilibrium condition for Sender and Claim 2, it follows that CavLipρ(p) =ρ(m) −dτσ(m) mχσ(dm) (8) since the right-hand side is Sender’s payoff under the profile (σρ). Let A={q∈(S) :sq(s)s ≥0}be the set of beliefs over states under which Receiver is willing to buy. Let t=χσ(τ−1 σ(A)),andletχσ=χσ(·|τ−1 σ(A)) and χσ=χσ(·|τ−1 σ(Ac)).Letq= τσdχσand q=τσdχσ.Then χσ=(1−t)χσ+tχσ(9) p=(1−t)q +tq (10) From the equilibrium condition for Receiver, it follows that ρ=0,χσ-almost surely. Therefore, from (8)and(9), it follows that CavLipρ(p) ≤tρ(m) −dτσ(m) mχσ(dm) (11) It follows from the definition of qand the convexity of Athat q∈A. Since p/∈A, there exists a belief q∗on the interval [p q]that is on the boundary of A, i.e., sq∗(s)s = 0.From(10), q∗=1−t∗q+t∗q=1−t∗q+t∗τσdχσ(12) for some t∗≥t.Also, p=1−t t∗q+t t∗q∗ We now define a strategy profile as follows: (i) Receiver’s strategy ρ∗is given by ρ∗(q∗)=Cav Lipρ(q∗)and ρ∗(m) =0for m= q∗. Theoretical Economics 16 (2021) Costly miscalibration 499 (ii) Sender’s strategy σ∗is the calibrated strategy induced by the distribution over messages given by χ∗=(1−t/t∗)δq+t/t∗δq∗. WeclaimthatthisisanequilibriumthatgivesSenderthesamepayoffCav Lipρ(p) as the original equilibrium. First, note that the equilibrium condition on Receiver’s side is satisfied since Sender’s strategy is calibrated, q/∈A, and Receiver is indifferent under q∗. Second, Sender’s payoff under the strategy profile (σ∗ρ∗)satisfies V1σ∗ρ∗=t t∗ρ∗q∗=t t∗CavLipρq∗ ≥tLipρτσ(m)χσ(dm) ≥tρ(m) −dτσ(m)mχσ(dm) ≥CavLipρ(p) where the first inequality follows from (12), the second inequality follows from the definition of Lip, and the third inequality follows from (11). Last, since ρ∗(q) ≤Cav Lipρ(q) for every q∈(S), then CavLipρ∗(p) ≤Cav LipCavLip ρ(p) =CavCavLip ρ(p) =CavLipρ(p) The first inequality and the last equality follow from Proposition 3(iii) and (ii), respectively. For the second equality, according to Corollary 2,Cav Lipρ(p) is 1-Lipschitz w.r.t. d. Therefore, LipCavLip ρ(p) =CavLipρ(p).ByClaim 2 this implies that Sender’s optimal payoff against ρ∗is at most Cav Lipρ(p), as desired. 6.3 Proving Proposition 1 6.3.1 Lemmas Lemma 3. In a generic set of games, for every ε>0,thereexistsxa∈(S) for each a∈Aand a Sender’s calibrated strategy σ,suchthat(i)BR(xa)={a}for each a∈A, (ii) support(σ) ={xa}a∈A, and (iii) if ρis Receiver’s best response to σ,thenVλ(σ ρ) > CP −ε. We prove Lemma 3 with the help of Lemma 4 below. The assertion in Lemma 4 is standard. See Balkenborg et al. (2015)andBrandenburger et al. (2021) for similar arguments. Lemma 4. In a generic set of games, for every action athat is not strictly dominated for Receiver, there exists a belief q∈(S) such that BR(q) ={a}. Returning to the proof of Lemma 3, since strictly dominated actions are not played in any equilibrium or commitment solution, we can discard them and consider a generic game with no strictly dominated strategies. We divide the proof into two claims. Claim 3. There exists ya∈RS +\{0}such that p=a∈Ayaand BR(ya)={a}. 500 Guo and Shmaya Theoretical Economics 16 (2021) Proof.ByLemma 4, for every action a, there exists some belief qa∈(S) such that a is the unique best response to the belief qa, i.e., BR(qa)={a}. Since phas full support, there exists a small t>0such that pa∈Atqa.Let¯ abe such that ¯ a∈BR(p −atqa). Let y¯ a=p−atqa+tq¯ aand ya=tqafor every a= ¯ a. Claim 4. There exists xa∈(S),ta>0,foreacha∈Asuch that p=a∈Ataxa, a∈Ata=1,BR(xa)={a},andas taxa(s)v(s a) > CP −ε. Proof.Let(σρ) be a commitment solution such that CP =sa v(sa)πσρ(s a), where πσρ is the distribution induced by the profile (σ ρ) over S×A.Letza(s) = ε/(2B)ya(s) +(1−ε/(2B))πσρ(s a),whereyais given by Claim 3 and Bis the bound on Sender’s payoff function v.Thenza∈RS +\{0},BR(za)={a},andsa v(sa)za(s) > CP −ε.Letxa∈(S) and ta>0be such that za=taxa. Since p∈(S),xa∈(S) for a∈A,andp=a∈Ataxa, it follows that a∈Ata=1. By the splitting lemma, exists a calibrated strategy σsuch that χσ=ataδxaexists. From Claim 4,thestrategyσsatisfies the requirements in Lemma 3. 6.3.2 Proof of Proposition 1 We consider the following auxiliary game, in which Sender’s set of messages is A∪{},whereis a message that says “silent.” In the auxiliary game, if Sender sends message a∈A, then Receiver must play action aand Sender pays a miscalibration cost relative to xagiven in Claim 4; alternatively, if Sender sends the silent message , then Receiver can choose any action from Aand Sender pays no miscalibration cost. Sender’s strategies can be represented by w={wm∈RS +}m∈A∪{} such that m∈A∪{} wm=p, with the interpretation that wm(s) is the probability that (i) the state is sand (ii) Sender sends m. Receiver’s strategies are given by elements ρ(·|)∈(A) (mixed actions, to be played after the silent message). The payoff to Sender in the auxiliary game under the profile (w ρ(·|)) is given by ˜ Vλwρ(·|)= a∈As∈Swa(s) +w(s)ρ(a|)v(sa) −λ a∈A d(waxa) The payoff to Receiver under this profile is given by ˜ Uwρ(·|)= a∈As∈Swa(s) +w(s)ρ(a|)u(s a) In the auxiliary game, Sender has a convex, compact set of strategies and a concave payoff function (which follows from the convexity of the distance function d), and Receiver has a finite set of pure actions. Therefore, the auxiliary game admits a Nash equilibrium. Let w∗={w∗ m∈RS +}m∈A∪{} be Sender’s strategy under the Nash equilibrium and let ρ∗(·|)∈(A) be Receiver’s strategy. For Sender’s equilibrium strategy w∗,welet|w∗ m|=sw∗ m(s) be the probability that Sender sends m.If|w∗ m|>0, then the posterior distribution over states conditioned on Sender announcing mis w∗ m/|w∗ m|. The following claim says that in the auxiliary game, Sender does not miscalibrate too much.