Persuasion with Ambiguous Communication

We explore whether ambiguous communication can be beneficial to the sender in a persuasion problem, when the receiver (and possibly the sender) is ambiguity averse. Our analysis highlights the necessity of using a collection of experiments that form a splitting of an obedient experiment. Some experiments in the collection must be Pareto-ranked in that both players agree on their payoff ranking. If an optimal Bayesian persuasion experiment can be split in this way, then any not-too-ambiguity-averse sender as well as the receiver benefit. There are no benefits when the receiver has only two actions.

💡 Research Summary

The paper investigates whether a sender can profit from deliberately using ambiguous communication in a persuasion setting when the receiver (and possibly the sender) is ambiguity‑averse. Building on the Bayesian persuasion framework of Kamenica and Gentzkow (2011), the authors extend the sender’s strategy set to include “ambiguous experiments,” i.e., collections of statistical experiments together with a probability distribution over which experiment will actually be used. The receiver treats the uncertainty about which experiment will be realized as ambiguity and evaluates the resulting lottery of utilities using the smooth‑ambiguity model of Klibanoff, Marinacci and Mukerji (2005). In this model the receiver’s utility from an ambiguous experiment (σ, μ) is ϕ⁻¹(∑_θ μ_θ ϕ(u_r(σ_θ, τ*))), where ϕ is increasing and concave; the degree of concavity captures the intensity of ambiguity aversion.

The central contribution is a set of necessary and sufficient conditions under which ambiguous communication strictly improves on the optimal Bayesian persuasion outcome. The key objects are Pareto‑ranked experiments—pairs of experiments σ¹ and σ² such that both the sender and the receiver rank σ¹ above σ² in terms of expected utility—and splitting of an obedient experiment. An obedient experiment is one that induces the receiver to follow the sender’s recommended actions. The authors show that if the optimal Bayesian persuasion experiment σ* can be expressed as a convex combination σ* = λσ¹ + (1‑λ)σ² with 0 < λ < 1 and the two component experiments are Pareto‑ranked, then the sender can construct an ambiguous experiment that places probability λ on σ¹ and (1‑λ) on σ². Because the receiver is ambiguity‑averse, they hedge against the possibility that the inferior experiment σ² will be realized; this hedging makes the receiver more tolerant of the superior experiment σ¹ being used more frequently than in the unambiguous case. Consequently, the sender’s expected payoff exceeds the Bayesian persuasion benchmark, while the receiver’s payoff also improves (Theorem 4, Corollary 2).

A particularly rich class of games where the splitting condition holds is threshold‑preference games. In these games each state has a unique “risky” action that is optimal for the receiver only if the posterior belief exceeds a state‑specific threshold; otherwise a default action is chosen. The sender prefers the risky actions in all states. The authors show (Proposition 3) that in generic threshold‑preference environments with at least three states and three actions, the optimal Bayesian persuasion experiment can indeed be written as a convex combination of two Pareto‑ranked experiments, so ambiguous communication yields a strict gain for any sender who is not infinitely ambiguity‑averse.

Conversely, the paper proves a negative result: when the receiver has only two available actions, no pair of Pareto‑ranked experiments can exist, and therefore any ambiguous experiment collapses to the Bayesian persuasion optimum (Corollary 1). This establishes a sharp boundary on the applicability of ambiguous persuasion.

The analysis also clarifies the role of the sender’s own ambiguity aversion. If the sender is infinitely ambiguity‑averse (max‑min preferences), the ambiguous experiment offers no advantage, reproducing the findings of Beauchene, Liao, and Li (2019) (BLL). The present work therefore highlights that benefits from ambiguity require the sender to be at most moderately ambiguity‑averse, while the receiver must exhibit some degree of ambiguity aversion.

Methodologically, the paper contributes by (i) formalizing ambiguous experiments as a pair (σ, μ) and embedding them in the smooth‑ambiguity framework; (ii) proving that canonical (message‑as‑action) and obedient experiments remain without loss of generality even with ambiguity (Proposition 1); (iii) establishing that any optimal ambiguous persuasion strategy must contain a Pareto‑ranked pair of experiments (Theorem 1, Theorem 2); and (iv) showing that the existence of a splitting of an obedient experiment into Pareto‑ranked components is both necessary and sufficient for a sender to gain (Theorem 3).

The paper situates its contributions relative to prior literature. BLL (2019) also examined ambiguous communication but assumed the receiver’s updating rule leads to dynamic inconsistency, effectively allowing the sender to exploit sub‑optimal receiver behavior. By contrast, the current model assumes a sequentially optimal, dynamically consistent receiver, thereby isolating the genuine value of ambiguity rather than a by‑product of behavioral inconsistency. Cheng (2025) demonstrated that BLL’s gains disappear under a dynamically consistent receiver, reinforcing the importance of the present paper’s assumptions.

Finally, the authors discuss practical implications. In financial regulation, for instance, a regulator could introduce ambiguity by allowing banks to run stress‑test models contingent on an exogenous, payoff‑irrelevant parameter (e.g., a random draw from an Ellsberg urn). The regulator’s ambiguous specification forces the market (investors) to hedge against the worst‑case model, making them more receptive to favorable messages generated under the better model. Similar mechanisms could be employed in political campaigning, marketing, or any setting where a sender wishes to persuade a receiver while the receiver is naturally cautious about unknown sources of information.

In sum, the paper provides a rigorous characterization of when and how ambiguous communication can be a strategic advantage in persuasion, delineates its limits (no benefit with only two actions, or with infinitely ambiguity‑averse senders), and offers a clear roadmap for applying these insights in real‑world information design problems.