Bayesian Persuasion Research Articles

Online learning in sequential Bayesian persuasion: Handling unknown priors

We study a repeated information design problem faced by an informed sender who tries to influence the behavior of a self-interested receiver, through the provision of payoff-relevant information. We consider settings where the receiver repeatedly faces a sequential decision making (SDM) problem. At each round, the sender observes the realizations of random events in the SDM problem, which are only partially observable by the receiver. This begets the challenge of how to incrementally disclose such information to the receiver to persuade them to follow (desirable) action recommendations. We study the case in which the sender does not know random events probabilities, and, thus, they have to gradually learn them while persuading the receiver. We start by providing a non-trivial polytopal approximation of the set of the sender's persuasive information-revelation structures. This is crucial to design efficient learning algorithms. Next, we prove a negative result which also applies to the non-sequential case: no learning algorithm can be persuasive in high probability. Thus, we relax the persuasiveness requirement, studying algorithms that guarantee that the receiver's regret in following recommendations grows sub-linearly. In the full-feedback setting—where the sender observes the realizations of all the possible random events—, we provide an algorithm with O˜(T) regret for both the sender and the receiver. Instead, in the bandit-feedback setting—where the sender only observes the realizations of random events actually occurring in the SDM problem—, we design an algorithm that, given an α∈[1/2,1] as input, guarantees O˜(Tα) and O˜(Tmax⁡{α,1−α2}) regrets, for the sender and the receiver respectively. This result is complemented by a lower bound showing that such a regret trade-off is tight for α∈[1/2,2/3].

Signaling through Bayesian persuasion

This paper examines the conditions under which separating outcomes occur in informed persuasion, that is, in Bayesian persuasion settings in which the sender is privately informed about the payoff-relevant state prior to committing to an experiment. We consider a setting with finite payoff-relevant states and sender payoffs that are continuous and monotonic in the receiver’s posterior beliefs. The paper finds that if full disclosure of the payoff-relevant state reduces the sender’s expected payoff under any common prior (i.e., if the sender’s payoff function is outer concave), then single-crossing properties arise such that the high sender type can separate from the low type by choosing more informative experiments. This single-crossing condition leads to the selection of “least costly” separating equilibria by the D1 criterion, i.e., the sender’s choice of experiment signals his type. Further, separating equilibria are characterized by simple constrained maximization problems.

Monotone Function Intervals: Theory and Applications

A monotone function interval is the set of monotone functions that lie pointwise between two fixed, -monotone functions. We characterize the set of extreme points of monotone function intervals and apply this to a number of economic settings. First, we leverage the main result to characterize the set of distributions of posterior quantiles that can be induced by a signal, with applications to political economy, Bayesian persuasion, and the psychology of judgment. Second, we combine our characterization with properties of convex optimization problems to unify and generalize seminal results in the literature on security design under adverse selection and moral hazard. (JEL C61, C65, D72, D82, D91, G12)

Reputation-based persuasion platforms

In this paper, we introduce a two-stage Bayesian persuasion model in which a third-party platform controls the information available to the sender about users' preferences. We aim to characterize the optimal information disclosure policy of the platform, which maximizes average user utility, under the assumption that the sender also follows its own optimal policy. We show that this problem can be reduced to a model of market segmentation, in which probabilities are mapped into valuations. We then introduce a repeated variation of the persuasion platform problem in which myopic users arrive sequentially. In this setting, the platform controls the sender's information about users and maintains a reputation for the sender, punishing it if it fails to act truthfully on a certain subset of signals. We provide a characterization of the optimal platform policy in the reputation-based setting, which is then used to simplify the optimization problem of the platform.

Designing contracts and information jointly

We consider a Bayesian persuasion problem with hidden action. When the principal can decide on both information structure and monetary transfer, which relies on the stochastic output, then full separation or full pooling contract is optimal.

Procuring Unverifiable Information

We study settings where information in the form of Bayesian signals is acquired by an expert on behalf of a principal. Information acquisition is costly for the expert and crucially not verifiable by the principal. The expert is compensated by the principal with a menu of state-contingent payments. We provide a full characterization of the set of all menus that implement (respectively, strictly implement) each signal. Moreover, we provide a closed-form characterization for the expected cost for the cheapest such menu, which we call proxy cost of the signal. Surprisingly, in general, the proxy cost is neither increasing in the Blackwell order nor posterior separable, even when the expert’s cost function is posterior separable itself. Subsequently, we study the full-agency problem (by introducing a downstream decision), thus endogenizing the signal. We show that there is always an optimal signal that can be strictly implemented, meaning that it is without loss of generality to exogenously restrict attention to strict implementation. As a result, similarly to Bayesian persuasion, the complexity of the principal’s optimal signal is bounded by the cardinality of the state space. Finally, we present some applications of interest. Funding: Financial support by the Wallander-Hedelius Foundation [Grants P2016-0072:1 and P20-0035 (to S. Sharma and M. Voorneveld)] is gratefully acknowledged.

Inside and Outside Information

ABSTRACTWe study an economy with financial frictions in which a regulator designs a test that reveals outside information about a firm's quality to investors. The firm can also disclose verifiable inside information about its quality. We show that the regulator optimally aims for “public speech and private silence,” which is achieved with tests that give insiders an incentive to stay quiet. We fully characterize optimal tests by developing tools for Bayesian persuasion with incentive constraints, and use these results to derive novel guidance for the design of bank stress tests, as well as benchmarks for socially optimal corporate credit ratings.

Starting rough, Dynamic persuasion with partial information

This study builds upon a Bayesian persuasion model (Kamenica and Gentzkow, 2011), extending it to two stages where the sender observes signals correlated with the persistent state instead of directly observing it. We demonstrate that the uncertainty about the state yields an optimal persuasion policy involving starting rough: the sender maximizes the second stage’s payoff by forgoing persuasion in the first stage.

Bayesian persuasion in lawyer–client communication

When considering whether to bring a lawsuit, a plaintiff may delegate the decision to her lawyer- better informed about the case’s merits- or consult with her lawyer and decide for herself. Focusing on the latter, we assess how a lawyer communicates with his client about the case’s merits if the client relies on the lawyer’s information to decide whether to bring suit. In some circumstances, the lawyer only partially reveals the case’s merits to persuade the client to bring a suit. In addition, in anticipation of the lawyer’s strategic communication, the plaintiff sometimes adjusts her questions to the lawyer about the case’s merits. Focusing on the communication between lawyers and clients, our paper explores a new consequence of misaligned incentives between the two parties.

Persuading Skeptics and Fans in the Presence of Additional Information

We consider the information design problem of a demand-maximizing firm launching a product of unknown quality to a market consisting of customers who have heterogeneous prior beliefs about quality. The firm publicly discloses information about quality to all customers. These customers can subsequently opt to acquire additional information about the product at a cost from sources beyond the firm’s control. Our study is motivated by the common practice of firms conducting public pilot tests or soliciting reviews from opinion leaders before launching a new product to inform potential customers about its quality. To analyze this problem, we construct a game-theoretic model of Bayesian persuasion between the firm and its customers. We characterize the firm’s optimal information policy and show that it can range from fully disclosing quality to exaggerating or downplaying quality to not disclosing quality at all depending on market characteristics. We delineate the impact of market heterogeneity and access to additional information on the optimal information disclosure policy of the firm. Our analysis provides managerial guidance for firms in designing information provision strategies and operationalizing them for different market characteristics.

Delegation to incentivize information production

This paper examines the impact of delegating decision-making authority to a biased and uninformed agent in a Bayesian persuasion model. We find that delegation can incentivize the sender to produce more information about the merits of the sender’s proposed project. This is because an agent with a small negative bias requires more compelling evidence than the principal to implement the project. The improvement in evidence quality does not occur if the agent’s bias is positive. When the informational benefits exist, delegation lead to higher organizational welfare than centralization. The informational benefits of delegation extend to cases where the principal delegates authority through a general delegation mechanism and when the agent’s bias is privately known.

Optimal auction design with aftermarket Cournot competition

We study an optimal information/mechanism design problem for selling an object to a number of asymmetric, privately informed bidders in which the winning bidder competes with a third party under differentiated Cournot competition afterwards. We show how to decompose the problem into two sub-problems: Bayesian persuasion and standard mechanism design. Full disclosure of the winner's marginal cost emerges as the unique optimal information policy, combined with allocating the auctioned object to the bidder with the highest “virtual surplus.” Bidders' signaling incentives increase the seller's expected revenue and lower discriminatory reserve prices in the optimal mechanism, improving ex post efficiency, measured by the probability of no-sale.

The Role of Messenger in Advertising Content: Bayesian Persuasion Perspective

This paper frames the choice of messenger in advertising content as a Bayesian persuasion problem and characterizes the optimal messenger choice.

Maximal Disclosure in Bayesian Persuasion

Maximal Disclosure in Bayesian Persuasion

Optimal Bayesian Persuasion for Containing SIS Epidemics

Optimal Bayesian Persuasion for Containing SIS Epidemics

The persuasion duality

We present a unified duality approach to Bayesian persuasion. The optimal dual variable, interpreted as a price function on the state space, is shown to be a supergradient of the concave closure of the objective function at the prior belief. Strong duality holds when the objective function is Lipschitz continuous. When the objective depends on the posterior belief through a set of moments, the price function induces prices for posterior moments that solve the corresponding dual problem. Thus, our general approach unifies known results for one‐dimensional moment persuasion, while yielding new results for the multidimensional case. In particular, we provide a condition for the optimality of convex‐partitional signals, derive structural properties of solutions, and characterize the optimal persuasion scheme when the state is two‐dimensional and the objective is quadratic.

Herd Design

The classic herding model examines the asymptotic behavior of agents who observe their predecessors’ actions as well as a private signal from an exogenous information structure. In this paper, we introduce a self-interested sender into the model and study her problem of designing this information structure. If agents cannot observe each other, the model reduces to Bayesian persuasion. However, when agents observe predecessors’ actions, they may learn from them, potentially harming the sender. We identify necessary and sufficient conditions under which the sender can nevertheless obtain the same utility as when the agents are unable to observe each other. (JEL D82, D83, D91)

Optimal disclosure in all-pay auctions with interdependent valuations

We study all-pay auctions with one-sided private information and interdependent valuations. To sharpen the competition and maximize revenue, the auction organizer can design an information disclosure policy through Bayesian persuasion about the bidder with private information. We characterize optimal disclosure and find that optimal disclosure exhibits almost full disclosure, where the uninformed bidder can always narrow the informed bidder's private information down to at most two types. We also illustrate our characterization in a simple binary-type setting and investigate issues such as comparative statics, welfare, and efficiency.

A population's feasible posterior beliefs

We consider a population of Bayesian agents who share a common prior over some finite state space and each agent is exposed to some information about the state. We characterize which distributions over the empirical distribution of posterior beliefs in the population are feasible. We apply this result in several domains. First, we study the problem of maximizing the polarization of beliefs in a population. Second, we provide a characterization of the feasible agent-symmetric product distributions of posteriors. Finally, we study an instance of a private Bayesian persuasion problem and provide a clean formula for the sender's optimal value.

Sequential Bayesian persuasion

I study a Bayesian persuasion model in which multiple senders sequentially persuade one receiver, after observing signal structures of prior senders and their realizations. I develop a geometric method, recursive concavification, to characterize the Perfect Bayesian Equilibrium paths. I prove the existence of the silent equilibrium, where at most one sender provides nontrivial information. I also show that when there are only two senders and the receiver has a finite action space, it is generically without loss to focus on silent equilibrium. Finally, I show that if there are two senders who have zero-sum payoffs, the truth-telling signal structure is always supported in equilibrium.