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1. Persuading Risk-Conscious Agents: A Geometric Approach

   https://doi.org/10.1007/978-3-030-35389-6  

   Anunrojwong, Jerry; Iyer, Krishnamurthy; Lingenbrink, David. (December 2019, Lecture notes in computer science)

   Motivated by practical concerns in applying information design to markets and service systems, we consider a persuasion problem between a sender and a receiver where the receiver may not be an expected utility maximizer. In particular, the receiver’s utility may be non-linear in her belief; we deem such receivers as risk-conscious. Such utility models arise, for example, when the receiver exhibits sensitivity to the variability and the risk in the payoff on choosing an action (e.g., waiting time for a service). In the presence of such non-linearity, the standard approach of using revelation-principle style arguments fails to characterize the set of signals needed in the optimal signaling scheme. Our main contribution is to provide a theoretical framework, using results from convex analysis, to overcome this technical challenge. In particular, in general persuasion settings with risk-conscious agents, we prove that the sender’s problem can be reduced to a convex optimization program. Furthermore, using this characterization, we obtain a bound on the number of signals needed in the optimal signaling scheme. We apply our methods to study a specific setting, namely binary per-suasion, where the receiver has two possible actions (0 and 1), and the sender always prefers the receiver taking action 1. Under a mild convexity assumption on the receiver’s utility and using a geometric approach,we show that the convex program can be further reduced to a linear program. Furthermore, this linear program yields a canonical construction of the set of signals needed in an optimal signaling mechanism. In particular, this canonical set of signals only involves signals that fully reveal the state and signals that induce uncertainty between two states.We illustrate our results in the setting of signaling wait time information in an unobservable queue with customers whose utilities depend on the variance of their waiting times. 

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2. Information Design in Spatial Resource Competition

   Yang, Pu; Iyer, Krishnamurthy; Frazier, Peter I. (December 2019, Lecture notes in computer science)

   We consider information design in spatial resource competition, motivated by ride sharing platforms sharing information with drivers about rider demand. Each of N co-located agents (drivers) decides whether to move to another location with an uncertain and possibly higher resource level (rider demand), where the utility for moving increases in the resource level and decreases in the number of other agents that move. A principal who can observe the resource level wishes to share this information in a way that ensures a welfare-maximizing number of agents move. Analyzing the principal’s information design problem using the Bayesian persuasion framework, we study both private signaling mechanisms, where the principal sends personalized signals to each agent, and public signaling mechanisms, where the principal sends the same information to all agents. We show: 1) For private signaling, computing the optimal mechanism using the standard approach leads to a linear program with 2 N variables, rendering the computation challenging. We instead describe a computationally efficient two-step approach to finding the optimal private signaling mechanism. First, we perform a change of variables to solve a linear program with O(N^2) variables that provides the marginal probabilities of recommending each agent move. Second, we describe an efficient sampling procedure over sets of agents consistent with these optimal marginal probabilities; the optimal private mechanism then asks the sampled set of agents to move and the rest to stay. 2) For public signaling, we first show the welfare-maximizing equilibrium given any common belief has a threshold structure. Using this, we show that the optimal public mechanism with respect to the sender-preferred equilibrium can be computed in polynomial time. 3) We support our analytical results with numerical computations that show the optimal private and public signaling mechanisms achieve substantially higher social welfare when compared with no-information and full-information benchmarks. 

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3. Leakage-Robust Bayesian Persuasion

   Haghtalab, Nika; Qiao, Mingda; Yang, Kunhe (July 2025, Twenty-Sixth ACM Conference on Economics and Computation (EC 2025))

   This paper introduces the concept of leakage-robust Bayesian persuasion. Situated between public Bayesian persuasion and private Bayesian persuasion, leakage-robust persuasion considers a setting where one or more signals privately communicated by a sender to the receivers may be leaked. We study the design of leakage-robust Bayesian persuasion schemes and quantify the price of robustness using two formalisms: - The first notion, k-worst-case persuasiveness, requires a signaling scheme to remain persuasive as long as each receiver observes no more than k leaked signals from other receivers. We quantify the Price of Robust Persuasiveness (PoRPk)— i.e., the gap in sender's utility as compared to the optimal private persuasion scheme—as Θ(min{2k,n}) for supermodular sender utilities and Θ(k) for submodular or XOS sender utilities, where n is the number of receivers. This result also establishes that in some instances, Θ(log k) leakages are sufficient for the utility of the optimal leakage-robust persuasion to degenerate to that of public persuasion. - The second notion, expected downstream utility robustness, relaxes the persuasiveness requirement and instead considers the impact on sender's utility resulting from receivers best responding to their observations. By quantifying the Price of Robust Downstream Utility (PoRU) as the gap between the sender's expected utility over the randomness in the leakage pattern as compared to private persuasion, our results show that, over several natural and structured distributions of leakage patterns, PoRU improves PoRP to Θ(k) or even Θ(1), where k is the maximum number of leaked signals observable to each receiver across leakage patterns in the distribution. En route to these results, we show that subsampling and masking serve as general-purpose algorithmic paradigms for transforming any private persuasion signaling scheme to one that is leakage-robust, with minmax optimal loss in sender's utility. A full version of this paper can be found at https://arxiv.org/abs/2411.16624. 

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4. Optimal Signaling Mechanisms in Unobservable Queues with Strategic Customers

   https://doi.org/10.1145/3033274.3085135  

   Lingenbrink, David; Iyer, Krishnamurthy (June 2017, Proceedings of the 2017 ACM Conference on Economics and Computation (EC))

   We study the problem of optimal information sharing in the context of a service system. In particular, we consider an unobservable single server queue offering a service at a fixed price to a Poisson arrival of delay-sensitive customers. The service provider can observe the queue, and may share information about the state of the queue with each arriving customer. The customers are Bayesian and strategic, and incorporate any information provided by the service provider into their prior beliefs about the queue length before making the decision whether to join the queue or leave without obtaining service. We pose the following question: which signaling mechanism and what price should the service provider select to maximize her revenue? We formulate this problem as an instance of Bayesian persuasion in dynamic settings. The underlying dynamics make the problem more difficult because, in contrast to static settings, the signaling mechanism adopted by the service provider affects the customers' prior beliefs about the queue (given by the steady state distribution of the queue length in equilibrium). The core contribution of this work is in characterizing the structure of the optimal signaling mechanism. We summarize our main results as follows. (1) Structural characterization: Using a revelation-principle style argument, we find that it suffices to consider signaling mechanisms where the service provider sends a binary signal of "join" or "leave", and under which the equilibrium strategy of a customer is to follow the service provider's recommended action. (2) Optimality of threshold policies: For a given fixed price for service, we use the structural characterization to show that the optimal signaling mechanism can be obtained as a solution to a linear program with a countable number of variables and constraints. Under some mild technical conditions on the waiting costs, we establish that there exists an optimal signaling mechanism with a threshold structure, where service provider sends the "join" signal if the queue length is below a threshold, and "leave" otherwise. (In addition, at the threshold, the service provider randomizes.) For the special case of linear waiting costs, we derive an analytical expression for the optimal threshold i terms of the two branches of the Lambert-W function. (3) Revenue comparison: Finally, we show that with the optimal choice of the fixed price and using the corresponding optimal signaling mechanism, the service provider can achieve the same revenue as with the optimal state-dependent pricing mechanism in a fully-observable queue. This implies that in settings where state-dependent pricing is not feasible, the service provider can effectively use optimal signaling (with the optimal fixed price) to achieve the same revenue. 

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5. Algorithmic Persuasion with Evidence

   https://doi.org/10.1145/3696470  

   Hoefer, Martin; Manurangsi, Pasin; Psomas, Alexandros (December 2024, ACM Transactions on Economics and Computation)

   In a game of persuasion with evidence, a sender has private information. By presenting evidence on the information, the sender wishes to persuade a receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether the receiver takes the action. The receiver’s utility depends on both the action and the sender’s private information. We study three natural variations. First, we consider the problem of computing an equilibrium of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, and polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm in algorithmic economics. 

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