The tension between persuasion and privacy preservation is common in real-world settings. Online platforms should protect the privacy of web users whose data they collect, even as they seek to disclose information about these data (e.g., to advertisers). Similarly, hospitals may share patient data to attract research investments with the obligation to preserve patients' privacy. To address these issues, we study Bayesian persuasion under differential privacy constraints, where the sender must design an optimal signaling scheme for persuasion while guaranteeing the privacy of each agent's private information in the database. To understand how privacy constraints affect information disclosure, we explore two perspectives within Bayesian persuasion: one views the mechanism as releasing a posterior about the private data, while the other views it as sending an action recommendation.

Information disclosure can compromise privacy when revealed information is correlated with private information. We consider the notion of inferential privacy, which measures privacy leakage by bounding the inferential power a Bayesian adversary can gain by observing a released signal. Our goal is to devise an inferentially-private private information structure that maximizes the informativeness of the released signal, following the Blackwell ordering principle, while adhering to inferential privacy constraints. To achieve this, we devise an efficient release mechanism that achieves the inferentially-private Blackwell optimal private information structure for the setting where the private information is binary. Additionally, we propose a programming approach to compute the optimal structure for general cases given the utility function. The design of our mechanisms builds on our geometric characterization of the Blackwell-optimal disclosure mechanisms under privacy constraints, which may be of independent interest.

We consider the single-item interdependent value setting, where there is a single item sold by a monopolist, n buyers, and each buyer has a private signal si describing a piece of information about the item. Additionally, each bidder i has a valuation function vi(s1, …, sn) mapping the (private) signals of all buyers into a positive real number representing their value for the item. This setting captures scenarios where the item's information is asymmetric or dispersed among agents, such as in competitions for oil drilling rights, or in auctions for art pieces. Due to the increased complexity of this model compared to the standard private values model, it is generally assumed that each bidder's valuation function vi is public knowledge to the seller or all other buyers. But in many situations, the seller may not know the bidders' valuation functions—how a bidder aggregates signals into a valuation is often their private information. In this paper, we design mechanisms that guarantee approximately-optimal social welfare while satisfying ex-post incentive compatibility and individually rationality for the case where the valuation functions are private to the bidders, and thus may be strategically misreported to the seller. When the valuations are public, it is possible for optimal social welfare to be attained by a deterministic mechanism when the valuations satisfy a single-crossing condition. In contrast, when the valuations are the bidders' private information, we show that no finite bound on the social welfare can be achieved by any deterministic mechanism even under single-crossing. Moreover, no randomized mechanism can guarantee better than n-approximation. We thus consider valuation functions that are submodular over signals (SOS), introduced in the context of combinatorial auctions in a recent breakthrough paper by Eden et al. [EC'19]. Our main result is an O(log2 n)-approximation randomized mechanism for buyers with private signals and valuations under the SOS condition. We also give a tight Θ(k)-approximation mechanism for the case each agent's valuation depends on at most k other signals even for unknown k.

When selling information products, sometimes the seller can provide some free partial information to change people's valuations so that the overall revenue can possibly be increased. In this work, we study the general problem of advertising information products by revealing partial information. We consider buyers who are decision-makers. The outcomes of the decision problems depend on the state of the world that is unknown to the buyers. The buyers can make their own observations and thus can hold different personal beliefs about the state of the world. There is an information seller who has access to the state of the world. The seller can promote the information by revealing some partial information. We assume that the seller chooses a long-term advertising strategy and then commits to it. The buyers decide whether to purchase the full information product after seeing the partial information. The seller's goal is to maximize the expected revenue. We study the problem in two settings. (1) The seller targets buyers of a certain type. In this case, we prove that finding the optimal advertising strategy is equivalent to finding the concave closure of a simple function. The function is a product of two quantities. The first one is what we call the likelihood ratio, which depends on the buyer's personal belief about the state of the world. The second one is the cost of uncertainty, which represents the value of the information to the buyer. Based on this observation, we prove some properties of the optimal mechanism, which allow us to solve for the optimal mechanism by a finite-size convex program. The convex program will have a polynomial size if the state of the world has a constant number of possible realizations or the buyers face a decision problem with a constant number of options. For the general problem, we prove that it is NP-hard to find the optimal mechanism. (2) For the general problem when the seller faces buyers of different types and only knows the distribution of their types, we provide an approximation algorithm when it is not too hard to predict the possible type of buyers who will make the purchase. For the general problem, we prove that it is NP-hard to find a constant-factor approximation.

Recent advances in multi-task peer prediction have greatly expanded our knowledge about the power of multi-task peer prediction mechanisms. Various mechanisms have been proposed in different settings to elicit different types of information. But we still lack understanding about when desirable mechanisms will exist for a multi-task peer prediction problem. In this work, we study the elicitability of multi-task peer prediction problems. We consider a designer who has certain knowledge about the underlying information structure and wants to elicit certain information from a group of participants. Our goal is to infer the possibility of having a desirable mechanism based on the primitives of the problem. Our contribution is twofold. First, we provide a characterization of the elicitable multi-task peer prediction problems, assuming that the designer only uses scoring mechanisms. Scoring mechanisms are the mechanisms that reward participants' reports for different tasks separately. The characterization uses a geometric approach based on the power diagram characterization in the single-task setting. For general mechanisms, we also give a necessary condition for a multi-task problem to be elicitable. Second, we consider the case when the designer aims to elicit some properties that are linear in the participant's posterior about the state of the world. We first show that in some cases, the designer basically can only elicit the posterior itself. We then look into the case when the designer aims to elicit the participants' posteriors. We give a necessary condition for the posterior to be elicitable. This condition implies that the mechanisms proposed by Kong and Schoenebeck are already the best we can hope for in their setting, in the sense that their mechanisms can solve any problem instance that can possibly be elicitable.

We consider the problem of purchasing data for machine learning or statistical estimation. The data analyst has a budget to purchase datasets from multiple data providers. She does not have any test data that can be used to evaluate the collected data and can assign payments to data providers solely based on the collected datasets. We consider the problem in the standard Bayesian paradigm and in two settings: (1) data are only collected once; (2) data are collected repeatedly and each day's data are drawn independently from the same distribution. For both settings, our mechanisms guarantee that truthfully reporting one's dataset is always an equilibrium by adopting techniques from peer prediction: pay each provider the mutual information between his reported data and other providers' reported data. Depending on the data distribution, the mechanisms can also discourage misreports that would lead to inaccurate predictions. Our mechanisms also guarantee individual rationality and budget feasibility for certain underlying distributions in the first setting and for all distributions in the second setting.

We consider a monopoly information holder selling information to a budget-constrained decision maker, who may benefit from the seller's information. The decision maker has a utility function that depends on his action and an uncertain state of the world. The seller and the buyer each observe a private signal regarding the state of the world, which may be correlated with each other. The seller's goal is to sell her private information to the buyer and extract maximum possible revenue, subject to the buyer's budget constraints. We consider three different settings with increasing generality, i.e., the seller's signal and the buyer's signal can be independent, correlated, or follow a general distribution accessed through a black-box sampling oracle. For each setting, we design information selling mechanisms which are both optimal and simple in the sense that they can be naturally interpreted, have succinct representations, and can be efficiently computed. Notably, though the optimal mechanism exhibits slightly increasing complexity as the setting becomes more general, all our mechanisms share the same format of acting as a consultant who recommends the best action to the buyer but uses different and carefully designed payment rules for different settings. Each of our optimal mechanisms can be easily computed by solving a single polynomial-size linear program. This significantly simplifies exponential-size LPs solved by the Ellipsoid method in the previous work, which computes the optimal mechanisms in the same setting but without budget limit. Such simplification is enabled by our new characterizations of the optimal mechanism in the (more realistic) budget-constrained setting.