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Title: Persuading Risk-Conscious Agents: A Geometric Approach

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. more »« less

Su, ST.; Kempe, D.; Subramanian, V.G.(
, Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science())

Feldman, M.
(Ed.)

We study a Bayesian persuasion setting in which the receiver is trying to match the (binary) state of the world. The sender’s utility is partially aligned with the receiver’s, in that conditioned on the receiver’s action, the sender derives higher utility when the state of the world matches the action.
Our focus is on whether in such a setting, being constrained helps a receiver. Intuitively, if the receiver can only take the sender’s preferred action with smaller probability, the sender might have to reveal more information, so that the receiver can take the action more specifically when the sender prefers it. We show that with a binary state of the world, this intuition indeed carries through: under very mild non-degeneracy conditions, a more constrained receiver will always obtain (weakly) higher utility than a less constrained one. Unfortunately, without additional assumptions, the result does not hold when there are more than two states in the world, which we show with an explicit example.

Lingenbrink, David; Iyer, Krishnamurthy(
, 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.

Su, ST.; Subramanian, V.G.; Schoenebeck, G.(
, Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science())

Feldman, M.
(Ed.)

We consider a Bayesian persuasion problem where the sender tries to persuade the receiver to take a particular action via a sequence of signals. This we model by considering multi-phase trials with different experiments conducted based on the outcomes of prior experiments. In contrast to most of the literature, we consider the problem with constraints on signals imposed on the sender. This we achieve by fixing some of the experiments in an exogenous manner; these are called determined experiments. This modeling helps us understand real-world situations where this occurs: e.g., multi-phase drug trials where the FDA determines some of the experiments, start-up acquisition by big firms where late-stage assessments are determined by the potential acquirer, multi-round job interviews where the candidates signal initially by presenting their qualifications but the rest of the screening procedures are determined by the interviewer. The non-determined experiments (signals) in the multi-phase trial are to be chosen by the sender in order to persuade the receiver best. With a binary state of the world, we start by deriving the optimal signaling policy in the only non-trivial configuration of a two-phase trial with binary-outcome experiments. We then generalize to multi-phase trials with binary-outcome experiments where the determined experiments can be placed at arbitrary nodes in the trial tree. Here we present a dynamic programming algorithm to derive the optimal signaling policy that uses the two-phase trial solution’s structural insights. We also contrast the optimal signaling policy structure with classical Bayesian persuasion strategies to highlight the impact of the signaling constraints on the sender.

Yang, Pu; Iyer, Krishnamurthy; Frazier, Peter I.(
, 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.

Lingenbrink, David Alan(
, ProQuest Dissertations & Theses A&I)

In mechanism design, the firm has an advantage over its customers in its knowledge of the state of the system, which can affect the utilities of all players. This poses the question: how can the firm utilize that information (and not additional financial incentives) to persuade customers to take actions that lead to higher revenue (or other firm utility)?
When the firm is constrained to "cheap talk," and cannot credibly commit to a manner of signaling, the firm cannot change customer behavior in a meaningful way. Instead, we allow firm to commit to how they will signal in advance. Customers can then trust the signals they receive and act on their realization. This thesis contains the work of three papers, each of which applies information design to service systems and online markets.
We begin by examining how a firm could signal a queue's length to arriving, impatient customers in a service system. We show that the choice of an optimal signaling mechanism can be written as a infinite linear program and then show an intuitive form for its optimal solution. We show that with the optimal fixed price and optimal signaling, a firm can generate the same revenue as it could with an observable queue and length-dependent variable prices.
Next, we study demand and inventory signaling in online markets: customers make strategic purchasing decisions, knowing the price will decrease if an item does not sell out. The firm aims to convince customers to buy now at a higher price. We show that the optimal signaling mechanism is public, and sends all customers the same information.
Finally, we consider customers whose ex ante utility is not simply their expected ex post utility, but instead a function of its distribution. We bound the number of signals needed for the firm to generate their optimal utility and provide a convex program reduction of the firm's problem.

@article{osti_10128622,
place = {Country unknown/Code not available},
title = {Persuading Risk-Conscious Agents: A Geometric Approach},
url = {https://par.nsf.gov/biblio/10128622},
DOI = {10.1007/978-3-030-35389-6},
abstractNote = {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.},
journal = {Lecture notes in computer science},
author = {Anunrojwong, Jerry and Iyer, Krishnamurthy and Lingenbrink, David.},
}

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