Rationality-Robust Information Design: Bayesian Persuasion under Quantal Response

Abstract

Bayesian persuasion in information design studies how an information-advantaged sender can influence a receiver to take the sender's preferred action by designing an information disclosure policy, a.k.a., a signaling scheme. Most works in Bayesian persuasion assume that the receiver is fully rational, i.e., she always selects the action that maximizes her expected utility. Yet empirical evidence suggests that human decisions may deviate from this expected utility theory. In this work, we relax the full rationality assumption and consider a boundedly rational receiver. In particular, we formulate the boundedly rationality of the receiver by adopting the classical model of quantal response, which captures the receiver's possible deviation from full rationality through incorporating noises in the receiver's process of utility estimation during decision making. Focusing on a fundamental persuasion model with binary receiver action, we seek to understand how the receiver's boundedly rational behavior impacts the design of optimal signaling schemes, and whether there exists robust signaling scheme when the receiver's boundedly rational behavior remains unknown.For both questions, we provide both positive and negative answers. At a high-level, a crucial condition which governs our answers is whether the sender's utility depends on the realized state. For the first question, in contrast to the setting with a fully rational receiver where censorship signaling scheme is always optimal (Renault et al., 2017), we show that for a boundedly rational receiver, the censorship signaling scheme remains optimal in state independent sender utility (SISU) environments, but is no longer optimal in state dependent sender utility (SDSU) environments. Nonetheless, we show that censorship signaling scheme is Θ(m)-approximation in SDSU environments, where m denotes the number of states. En route, we characterize the optimal signaling scheme in SDSU environments, in which the sender's signal either reveals the true state, or randomizes the receiver's uncertainty on only two states. For the second question, we introduce rationality-robust information design -a framework in which a signaling scheme is designed for a receiver with unknown boundedly rational behavior. We show that in SISU environments, up to a 2-approximation factor, implementing optimal censorship signaling scheme of a fully rational receiver is rationality-robust for any boundedly rational receiver. In marked contrast to this positive result, in SDSU environments, there exists no robust signaling scheme that has any bounded approximation. Nonetheless, for any problem instance with binary state, under a reasonable multiplicative boundedness condition on the receiver's behavior, the sender is able to design a signaling scheme whose approximation ratio depends linearly on the multiplicative error.

Persuading a Behavioral Agent: Approximately Best Responding and Learning

“…Closest to our work is the line of works on Robust Bayesian persuasion, which study the Bayesian persuasion problem with various assumptions in the classic model being relaxed. For example, the sender does not know the receiver's prior belief (Kosterina, 2022;Zu et al, 2021); the sender does not know the utility function of the receiver (Babichenko et al, 2021); the receiver may receive additional signals besides the signal sent by the sender (Dworczak and Pavan, 2020;Ziegler, 2020); the receiver can make mistakes in Bayesian update (de Clippel and Zhang, 2022); the receiver is boundedly rational and responds to the sender according to the quantal response model (Feng et al, 2022a). Many of these works take the maximin approach of maximizing the sender's worst-case utility when the classic assumptions are violated, aiming to find the exact solution to the maximin problem.…”

“…We also take a maximin approach, but unlike previous works, instead of finding the exact maximin solution, we are interested in bounding the maximin solution and finding conditions under which the maximin solution is close to the original solution. This allows us to study a very general model that captures several previous models as special cases (e.g., de Clippel and Zhang (2022) and Feng et al (2022a)). We also study a maximax problem, which is typically not studied in the robust Bayesian persuasion literature.…”

“…Our definition of (γ, δ)-best-responding strategies includes, for example, two other models in the literature that also relax the receiver's rationality in the Bayesian persuasion problem: the quantal response model (proposed by McKelvey and Palfrey (1995) in normal-form games, recently studied by Feng et al (2022a) in Bayesian persuasion) and a model where the receiver makes mistakes in Bayesian update (de Clippel and Zhang, 2022).…”