Abstract
We propose a robust solution concept for Bayesian persuasion that accounts for the Sender's concern that her Bayesian belief about the environment—which we call the conjecture—may be false. Specifically, the Sender is uncertain about the exogenous sources of information the Receivers may learn from, and about strategy selection. She first identifies all information policies that yield the largest payoff in the “worst-case scenario,” that is, when Nature provides information and coordinates the Receivers' play to minimize the Sender's payoff. Then she uses the conjecture to pick the optimal policy among the worst-case optimal ones. We characterize properties of robust solutions, identify conditions under which robustness requires separation of certain states, and qualify in what sense robustness calls for more information disclosure than standard Bayesian persuasion. Finally, we discuss how some of the results in the Bayesian persuasion literature change once robustness is accounted for, and develop a few new applications.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 2017-2051 |
| Number of pages | 35 |
| Journal | Econometrica |
| Volume | 90 |
| Issue number | 5 |
| DOIs | |
| State | Published - Sep 2022 |
Funding
“I am prepared for the worst but hope for the best,” Benjamin Disraeli, 1st Earl of Beaconsfield, UK Prime Minister. In the canonical Bayesian persuasion model, a Sender designs an information structure to influence the behavior of a Receiver. The Sender is Bayesian, and has beliefs over the Receiver's prior information as well as the additional information the Receiver might acquire after observing the realization of the Sender's signal. As a result, the Sender's optimal signal typically depends on the details of her belief about the Receiver's learning environment. In many applications, however, the Sender may be concerned that her belief—which we call a conjecture—is wrong. In such cases, the Sender may prefer to choose a policy that is not optimal under her conjecture but that protects her well in the event her conjecture turns out to be false. In this paper, we propose a solution concept for the persuasion problem that accounts for the uncertainty that the Sender may face over the Receiver's learning environment and that incorporates the Sender's concern for the validity of her conjecture. Specifically, we assume that the Sender discards all policies that do not provide her with the optimal payoff guarantee. The payoff guarantee is computed conservatively by considering all possible learning environments for the Receiver, without assuming that the Sender is last to speak or that the Receiver will break indifferences in the Sender's favor. We characterize properties of “robust solutions,” which we define as policies that maximize the Sender's payoff under her conjecture among those that provide the optimal payoff guarantee. The following example (inspired by the “judge example” from Kamenica and Gentzkow (2011)) illustrates our main ideas. 1 Example The Receiver is a judge, the Sender is a prosecutor, and there are three relevant states of the world, ω∈{i,m,f}, corresponding to a defendant being innocent, guilty of a misdemeanor, or guilty of a felony, respectively. The prior μ0 is given by μ0(i)=1/2 and μ0(m)=μ0(f)=1/4. The judge, who initially only knows the prior distribution, will convict if her posterior belief that the defendant is guilty (i.e., that ω∈{m,f}) is at least 2/3. In that case, she also chooses a sentence. Let x∈[x_,x¯], with x_>0, be the range of the number of years in prison that the judge can select from. The maximal sentence x¯ is chosen if the judge's posterior belief that a felony was committed conditional on the defendant being guilty is at least 1/2. Otherwise, the sentence is linearly increasing in the conditional probability of the state f. The prosecutor attempts to maximize the expected sentence (with acquitting modeled as a sentence of x=0). Formally, if μ is the induced posterior belief of the judge, with μ(ω) denoting the probability of state ω, the Sender's payoff is given by Vˆ(μ)=1{μ(m)+μ(f)≥23}min{x¯,x_+2μ(f)μ(f)+μ(m)(x¯−x_)}, where 1{a} is a function taking value 1 when the statement {a} is true and 0 otherwise. The Bayesian solution, as defined by Kamenica and Gentzkow (2011), is as follows: The prosecutor induces the posterior belief (μ(i),μ(m),μ(f))=(1,0,0) with probability 1/4 and the belief (1/3,1/3,1/3) with probability 3/4 (by saying “innocent” with probability 1/2 conditional on the state being i, and “guilty” in all other cases). The expected payoff for the prosecutor is (3/4)x¯. In the above situation, the prosecutor's conjecture is that she is the sole provider of information. However, this could turn out to be false. For example, after the prosecutor presents her arguments, the judge could call a witness. The prosecutor might not know the likelihood of this scenario, the amount of information that the witness has about the state, or the witness' motives.1 When confronted with such uncertainty, it is common to consider the worst case: Suppose that the witness knows the true state and strategically reveals information to minimize the sentence. Under this scenario, the prosecutor cannot do better than fully revealing the state. Indeed, if the prosecutor chose a disclosure policy yielding a strictly higher expected payoff, the adversarial witness could respond by fully revealing the state, lowering the prosecutor's expected payoff down to the full-disclosure payoff of (1/4)x_+(1/4)x¯. The key observation of our paper is that the prosecutor—even if she is primarily concerned about the worst-case scenario—should not fully disclose the state. Consider the following alternative partitional signal: reveal the state “innocent,” and pool together the remaining two states. Suppose that the witness is adversarial. When it is already revealed that the defendant is innocent, the witness has no information left to reveal. In the opposite case, because conditional on the state being m or f the prosecutor's payoff is concave in the induced posterior belief, the adversarial witness will choose to disclose the state. Thus, in the worst case, the prosecutor's expected payoff under this policy is the same as under full disclosure. At the same time, the policy is superior if the prosecutor's conjecture turns out to be true. Indeed, if the prosecutor is the sole provider of information, then the alternative policy yields (1/2)x¯, which is strictly greater than the full-disclosure payoff. It may be tempting to conclude from similar reasoning that the prosecutor might as well stick to the Bayesian solution, even if she is concerned about the worst-case scenario. After all, if the witness discloses the state in case she is adversarial, then it is irrelevant what signal the prosecutor selects, so should not she focus on maximizing her payoff under her conjecture? The problem with that argument is that the most adversarial scenario is not always that the witness fully discloses the state. In the Bayesian solution, when the prosecutor induces the posterior (1/3,1/3,1/3), the witness may instead reveal the state f with some small probability ϵ>0. With remaining probability, the judge's posterior belief that the defendant is guilty will then shift just below the threshold of 2/3. As a result, the judge acquits the defendant with probability arbitrarily close to one, not just when the latter is innocent but also when they are guilty. Thus, the payoff guarantee for the prosecutor from selecting the Bayesian solution is in fact 0, implying that the Bayesian solution need not be robust to misspecifications in the conjecture. When the prosecutor is unable to attach probability assessments to all relevant events (like the appearance of a witness), any policy she chooses results in a range of expected payoffs generated by the set of all possible scenarios. Thus, there are many ways in which any two information policies can be compared. Our solution concept is based on two pragmatic premises that are captured by a lexicographic approach. First, and foremost, the Sender would like to secure the best possible payoff guarantee. She does so by dismissing any policy that is not optimal in the “worst-case scenario.” Second, when there are multiple policies that are worst-case optimal, the Sender acts as in the standard Bayesian persuasion model. That is, she selects the policy that, among those that are worst-case optimal, maximizes her expected payoff under the conjecture. We refer to the case described by the conjecture as the base-case scenario. The base-case scenario may correspond to the specification that the Sender considers most plausible (e.g., after calibrating on some data), focal, or a good approximation (obtained, e.g., by ignoring events that appear unlikely). The combination of these two properties defines a robust solution: a policy that is base-case optimal among those that are worst-case optimal.2 The alternative policy described above is in fact a robust solution for the prosecutor. The Receiver is a judge, the Sender is a prosecutor, and there are three relevant states of the world, ω∈{i,m,f}, corresponding to a defendant being innocent, guilty of a misdemeanor, or guilty of a felony, respectively. The prior μ0 is given by μ0(i)=1/2 and μ0(m)=μ0(f)=1/4. The judge, who initially only knows the prior distribution, will convict if her posterior belief that the defendant is guilty (i.e., that ω∈{m,f}) is at least 2/3. In that case, she also chooses a sentence. Let x∈[x_,x¯], with x_>0, be the range of the number of years in prison that the judge can select from. The maximal sentence x¯ is chosen if the judge's posterior belief that a felony was committed conditional on the defendant being guilty is at least 1/2. Otherwise, the sentence is linearly increasing in the conditional probability of the state f. The prosecutor attempts to maximize the expected sentence (with acquitting modeled as a sentence of x=0). Formally, if μ is the induced posterior belief of the judge, with μ(ω) denoting the probability of state ω, the Sender's payoff is given by Vˆ(μ)=1{μ(m)+μ(f)≥23}min{x¯,x_+2μ(f)μ(f)+μ(m)(x¯−x_)}, where 1{a} is a function taking value 1 when the statement {a} is true and 0 otherwise. The Bayesian solution, as defined by Kamenica and Gentzkow (2011), is as follows: The prosecutor induces the posterior belief (μ(i),μ(m),μ(f))=(1,0,0) with probability 1/4 and the belief (1/3,1/3,1/3) with probability 3/4 (by saying “innocent” with probability 1/2 conditional on the state being i, and “guilty” in all other cases). The expected payoff for the prosecutor is (3/4)x¯. In the above situation, the prosecutor's conjecture is that she is the sole provider of information. However, this could turn out to be false. For example, after the prosecutor presents her arguments, the judge could call a witness. The prosecutor might not know the likelihood of this scenario, the amount of information that the witness has about the state, or the witness' motives.1 When confronted with such uncertainty, it is common to consider the worst case: Suppose that the witness knows the true state and strategically reveals information to minimize the sentence. Under this scenario, the prosecutor cannot do better than fully revealing the state. Indeed, if the prosecutor chose a disclosure policy yielding a strictly higher expected payoff, the adversarial witness could respond by fully revealing the state, lowering the prosecutor's expected payoff down to the full-disclosure payoff of (1/4)x_+(1/4)x¯. The key observation of our paper is that the prosecutor—even if she is primarily concerned about the worst-case scenario—should not fully disclose the state. Consider the following alternative partitional signal: reveal the state “innocent,” and pool together the remaining two states. Suppose that the witness is adversarial. When it is already revealed that the defendant is innocent, the witness has no information left to reveal. In the opposite case, because conditional on the state being m or f the prosecutor's payoff is concave in the induced posterior belief, the adversarial witness will choose to disclose the state. Thus, in the worst case, the prosecutor's expected payoff under this policy is the same as under full disclosure. At the same time, the policy is superior if the prosecutor's conjecture turns out to be true. Indeed, if the prosecutor is the sole provider of information, then the alternative policy yields (1/2)x¯, which is strictly greater than the full-disclosure payoff. It may be tempting to conclude from similar reasoning that the prosecutor might as well stick to the Bayesian solution, even if she is concerned about the worst-case scenario. After all, if the witness discloses the state in case she is adversarial, then it is irrelevant what signal the prosecutor selects, so should not she focus on maximizing her payoff under her conjecture? The problem with that argument is that the most adversarial scenario is not always that the witness fully discloses the state. In the Bayesian solution, when the prosecutor induces the posterior (1/3,1/3,1/3), the witness may instead reveal the state f with some small probability ϵ>0. With remaining probability, the judge's posterior belief that the defendant is guilty will then shift just below the threshold of 2/3. As a result, the judge acquits the defendant with probability arbitrarily close to one, not just when the latter is innocent but also when they are guilty. Thus, the payoff guarantee for the prosecutor from selecting the Bayesian solution is in fact 0, implying that the Bayesian solution need not be robust to misspecifications in the conjecture. When the prosecutor is unable to attach probability assessments to all relevant events (like the appearance of a witness), any policy she chooses results in a range of expected payoffs generated by the set of all possible scenarios. Thus, there are many ways in which any two information policies can be compared. Our solution concept is based on two pragmatic premises that are captured by a lexicographic approach. First, and foremost, the Sender would like to secure the best possible payoff guarantee. She does so by dismissing any policy that is not optimal in the “worst-case scenario.” Second, when there are multiple policies that are worst-case optimal, the Sender acts as in the standard Bayesian persuasion model. That is, she selects the policy that, among those that are worst-case optimal, maximizes her expected payoff under the conjecture. We refer to the case described by the conjecture as the base-case scenario. The base-case scenario may correspond to the specification that the Sender considers most plausible (e.g., after calibrating on some data), focal, or a good approximation (obtained, e.g., by ignoring events that appear unlikely). The combination of these two properties defines a robust solution: a policy that is base-case optimal among those that are worst-case optimal.2 The alternative policy described above is in fact a robust solution for the prosecutor. Our baseline model studies a generalization of the above example to arbitrary Sender-Receiver games with finite action and state spaces. To ease the exposition, we initially assume that the base-case scenario is that the Receiver does not have any exogenous information other than that contained in the common prior (the case considered in most of the literature).3 We capture the Sender's concern about the validity of her conjecture by introducing a third player, Nature, that may send an additional signal to the Receiver. We assume that Nature can condition on the Sender's signal realization, reflecting the Sender's uncertainty over the order in which signals are observed. Worst-case optimal policies maximize the Sender's expected payoff when Nature's objective is to minimize the Sender's payoff. Robust solutions maximize the Sender's base-case payoff among all worst-case optimal policies. Despite the fact that robust solutions involve worst-case optimality, they exist under standard conditions, and can be characterized by applying techniques similar to those used to identify Bayesian solutions (e.g., concavification of the value function). However, the economic properties of robust solutions can be quite different from those of Bayesian solutions. Our main result identifies states that cannot appear together in the support of any of the posterior beliefs induced by a robust solution. Separation of such states is both necessary and sufficient for worst-case optimality. Robust solutions thus maximize the same objective function as Bayesian solutions but subject to the additional constraint that the induced posteriors have admissible supports. The separation theorem permits us to qualify in what sense more information is disclosed under robust solutions than under standard Bayesian solutions: For any Bayesian solution, there exists a robust solution that is either Blackwell more informative or not comparable in the Blackwell order. A naive intuition for why robustness calls for more information disclosure is that, because Nature can always reveal the state, the Sender may opt for revealing the state herself. This intuition, however, is not correct, as we already indicated in the example above. While fully revealing the state is always worst-case optimal, it need not be a robust solution. In fact, if Nature's most adversarial response to any selection by the Sender is to fully disclose the state, then any signal chosen by the Sender yields the same payoff guarantee and hence is worst-case optimal—the Sender then optimally selects the same signal as in the standard Bayesian persuasion model. Instead, the reason why robustness calls for more information disclosure than standard Bayesian persuasion is that, if certain states are not separated, Nature could push the Sender's payoff strictly below what the Sender would obtain by fully disclosing these states herself. This is the reason why the Sender always reveals the state “innocent” in the robust solution in Example 1, whereas the Bayesian solution sometimes pools that state with the other two. When the Sender faces non-Bayesian uncertainty, it is natural for her to want to avoid dominated policies. A dominated policy performs weakly (and sometimes strictly) worse than some alternative policy that the Sender could adopt, no matter how Nature responds. We show that at least one robust solution is undominated, and that, provided that the conjecture satisfies a certain condition, all robust solutions are undominated. Thus, robust solutions can be of interest even if the Sender attaches no significance to any particular conjecture; they can be used to generate solutions that are worst-case optimal and undominated. Example 1 shows that focusing on worst-case optimal solutions is not enough for this purpose: Full disclosure is worst-case optimal but dominated. While we focus on a simple model to highlight the main ideas, we argue in Section 4 that our approach and results extend to more general persuasion problems, and can accommodate various assumptions about the Sender's conjecture and the worst case. With a single Receiver, we can allow the Sender to conjecture that the Receiver observes some exogenous signal; the non-Bayesian uncertainty is created by the possibility that the actual signal observed by the Receiver is different from the one conjectured by the Sender. Our results also generalize to the case of multiple Receivers under the assumption that the Sender uses a public signal. In the standard persuasion framework, it is typical to assume that the Sender not only controls the information that the Receivers observe but also coordinates their play on the strategy profile most favorable to her, in case there are multiple profiles consistent with the assumed solution concept and the induced information structure.4 In this case, a policy is worst-case optimal if it maximizes the Sender's payoff under the assumption that Nature responds to the information provided by the Sender by revealing additional information to the Receivers (possibly in a discriminatory fashion) and coordinating their play (in a way consistent with the assumed solution concept) to minimize the Sender's payoff. In contrast, if the Sender's conjecture turns out to be correct, the Receivers' exogenous information and the equilibrium selection are the ones consistent with the Sender's beliefs. As a result, robust solutions are a flexible tool that can accommodate various assumptions about the environment. For example, a Sender may conjecture that play will constitute a Bayes Nash equilibrium under the information structure induced by her signal. However, she may first impose a “robustness test” to rule out policies that deliver a suboptimal payoff in the worst Bayes correlated equilibrium. For any given specification of the worst-case and base-case Sender's payoffs, our separation theorem characterizes the resulting robust solutions. The rest of the paper is organized as follows. We review the related literature next. In Section 2, we present the baseline model, and then we derive the main properties of robust solutions in Section 3. Section 4 extends the model and the results to general persuasion problems, and Section 5 illustrates the results with applications. Finally, in Section 6, we discuss how our solution concept relates to alternative notions of robustness. Most proofs are collected in the Appendix. The Online Supplementary Material (Dworczak and Pavan (2022)) contains additional results, most notably a discussion of a version of our model in which Nature chooses her signal simultaneously with the Sender, rather than conditioning on the Sender's signal realization. Our paper contributes to the fast-growing literature on Bayesian persuasion and information design (see, among others, Bergemann and Morris (2019), and Kamenica (2019) for surveys). Several recent papers adopt a robust approach to the design of the optimal information structure. Inostroza and Pavan (2022), Morris, Oyama, and Takahashi (2020), Ziegler (2020), and Li, Song, and Zhao (2022) focus on the adversarial selection of the continuation strategy profile of the Receivers. Babichenko, Talgam-Cohen, Xu, and Zabarnyi (2021) characterize regret-minimizing signals for a Sender who does not know the Receiver's utility function. Most closely related are Hu and Weng (2021) and Kosterina (2021) who study signals that maximize the Sender's payoff in the worst-case scenario, when the Sender faces uncertainty over the Receivers' exogenous private information. Hu and Weng (2021) observe that full disclosure maximizes the Sender's payoff in the worst-case scenario, when the Sender faces full ambiguity over the Receivers' exogenous information (as in our solution concept). They also consider the opposite case of a Sender that faces small local ambiguity over the Receivers' exogenous information and show robustness of Bayesian solutions in this case. Kosterina (2021) considers a setting in which the Sender faces ambiguity over the Receiver's prior. This is similar to the version of our model (analyzed in the Online Appendix) in which the Sender and Nature move simultaneously; however, an important difference is that Nature in Kosterina's model chooses the Receiver's prior, while Nature in our model chooses a distribution of posteriors induced from a fixed (and known) prior.5 Our results differ from those in any of the above papers, and reflect a different approach to the design of the optimal signal. Once the Sender identifies all signals that are worst-case optimal, she considers their performance under the base-case scenario (as in the canonical Bayesian persuasion model). In particular, our solution concept reflects the idea that there is no reason for the Sender to fully disclose the state if she can benefit by withholding some information under the conjectured scenario while still guaranteeing the same worst-case payoff. Our lexicographic approach to the assessment of different information structures is in the same spirit as the one proposed by Börgers (2017) in the context of robust mechanism design. The literature on Bayesian persuasion with multiple Senders is also related, in that Nature is effectively a second Sender in the persuasion game that we study. Gentzkow and Kamenica (2016, 2017) consider persuasion games in which multiple Senders move simultaneously and identify conditions under which competition leads to more information being disclosed in equilibrium. Board and Lu (2018) consider a search model and provide conditions for the existence of a fully-revealing equilibrium. Au, Hung, and Kawai (2020) study multi-Sender simultaneous-move games where each Sender discloses information about the quality of her product (with the qualities drawn independently across Senders). They show that, as the number of Senders increases, each Sender discloses more information, with the information disclosed by each Sender converging to full disclosure as the number of Senders goes to infinity. Cui and Ravindran (2022) consider persuasion by competing Senders in zero-sum games and identify conditions under which full disclosure is the unique outcome.6 Li and Norman (2021), and Wu (2021), instead, analyze games in which Senders move sequentially and, among other things, identify conditions under which (1) full information revelation can be supported in equilibrium and (2) silent equilibria (that is, equilibria in which all Senders but one remain silent) sustain all equilibrium outcomes. These papers focus on equilibrium outcomes under competition, rather than robustness of the policy chosen by a single Sender. A key element in Li and Norman (2021)'s equilibrium analysis is the optimality for each Sender of inducing “stable beliefs” that are not further split by downstream Senders. Our paper shows that imposing a zero-sum payoff assumption generates a sharp implication on the structure of stable beliefs in terms of states that are separated under any of the induced posteriors. Kolotilin, Mylovanov, Zapechelnyuk, and Li (2017), Laclau and Renou (2017), and Guo and Shmaya (2019), among others, consider persuasion of privately informed Receivers. Additionally, Matysková and Montes (2021) and Ye (2021) study models in which the Receiver optimally acquires information in response to the signal selected by the Sender. Contrary to the present paper, in that literature, the distribution of the Receivers' private information is known to the Sender.
Keywords
- information design
- Persuasion
- robustness
- worst-case optimality
ASJC Scopus subject areas
- Economics and Econometrics