Bayesian Persuasion Model

• Bayesian Persuasion Model is a strategic framework in which an informed sender commits to a signaling scheme to influence a receiver’s actions by selectively revealing information.
• It employs Bayes’ rule for belief updating and uses methods like linear programming and FPTAS to optimize sender utility across different prior distributions.
• The model has broad applications in economics, auction theory, and mechanism design, illustrating computational transitions from polynomial-time solvability to #P-hard cases.

The Bayesian-Persuasion Model is a foundational framework in information economics which formalizes how an informed agent (the sender) designs information disclosure policies (signaling schemes) to strategically influence the actions of a less informed agent (the receiver) and thereby maximize her own expected utility. The sender, holding private knowledge about the “state of nature” or payoff vector, publicly commits to a signaling scheme mapping states to observable signals before observing the realization; the receiver, observing the signal and knowing the disclosure policy, updates his prior belief to a posterior via Bayes’ rule and best-responds by choosing an action that optimizes his own expected payoff. The model captures the essence of informational advantage as leverage for directional influence, with broad relevance in economics, mechanism design, and computational social science.

1. Core Actors, Mechanism, and Influence

The canonical Bayesian persuasion model consists of two players: a sender (e.g., a prosecutor, adviser, or marketer) and a receiver (e.g., a judge, investor, or consumer). The receiver must select one action from a finite set; both sender and receiver utilities depend on an unknown state of nature (typically encoded as a vector of payoffs indexed by actions or states). The sender uniquely observes the realization of this state before play and commits in advance to a signaling policy—an information structure mapping each possible state to a probability distribution over signals.

Upon observing a signal, the receiver forms a posterior belief $p(\cdot | s)$ and maximizes expected utility, possibly taking into account the sender’s strategy and incentive structure. The sender, aware of this rational response, optimally “filters” her information revelation to sway the receiver’s action in her favor. Signals are often intentionally noisy or strategically partial, influencing the receiver’s posterior while exploiting the sender’s knowledge of the payoff dependencies.

The model captures a critical aspect: signals have value exclusively through their impact on induced posteriors and thus on the receiver’s subsequent best response; the substance of the message is otherwise inconsequential. This informational leverage underpins persuasion.

2. Computational Structure: Complexity by Input Model

A central contribution is the characterization of the computational complexity of signaling design under different information structures (Dughmi et al., 2015). The sender’s optimization problem—maximizing expected utility by selecting a (possibly randomized) mapping from states to signals—is polynomial-time solvable, intractable, or only approximately tractable depending on the prior’s structure:

Input Model Type	Complexity Result	Algorithmic Techniques
i.i.d. (actions identically, independently distributed; explicit marginal)	Polynomial time (exact)	Symmetry reduction, signature space, Border’s characterization, LP in reduced space
Independent, non-identical actions (explicit marginals)	#P-hard	Khintchine polytope reduction
Arbitrary joint prior (black-box sampling oracle only)	FPTAS with bi-criteria guarantee	Monte Carlo, relaxed LP, deferred decision principle

In the i.i.d. case, symmetry simplifies the model: one need only consider “symmetric signaling schemes.” The LP describing optimal persuasion collapses into a lower-dimensional space of “s-signatures,” leveraging Border’s theorem from auction theory. For independent, non-identical action priors, however, these symmetries break, and the problem becomes #P-hard—even strong analogs of auction-theoretic reduced forms do not yield tractable polytopes. In the most general case (black-box distributions, possibly correlated), the authors provide a fully polynomial-time approximation scheme (FPTAS) that simultaneously approximates sender utility and incentive compatibility constraints within additive error $\varepsilon$—a bi-criteria guarantee that is best possible due to information-theoretic sample complexity lower bounds.

3. Algorithmic Schemes and Structural Analogy with Auction Theory

For i.i.d. action priors, the optimal signaling scheme is characterized by a pair $(x, y)$: $x$ is the posterior for the recommended action, $y$ for non-recommended actions, with constraints $x + (n-1)y = q$ and $\|x\|_1 = \|y\|_1 = 1/n$, $q$ being the common marginal distribution. The reduced problem is a linear program solvable in polynomial time by separation oracles derived from Border’s theorem for single-item auction reduced forms. Each “action” in persuasion plays the algebraic role of a “bidder” in auction design, with feasible signatures corresponding to feasible reduced forms.

A “simple” $(1-1/e)$-approximation algorithm arises by relaxing the LP and independently signaling high/low for each action. If payoffs are nonnegative, signaling each action’s “high” outcome with probability $x^*$ (from a relaxed symmetric LP solution) and “low” otherwise assures the probability that at least one action receives a high signal is $1-(1-1/n)^n \geq 1-1/e$. This achieves a sender utility at least $(1-1/e)$ times optimal in polynomial time.

For the black-box model, the FPTAS samples $K = \mathrm{poly}(n, 1/\varepsilon)$ states, solves a relaxed LP with $\varepsilon$-deviation from incentive compatibility, and deploys empirical distribution anchoring. This sees bi-criteria loss as necessary due to unbounded sample complexity for joint exactness (sender’s utility and strict rationality) under only black-box access.

4. Main Technical Challenges and Hardness Results

The intractability of the independent, non-identical case is established via reduction from the #P-hard computation of the Khintchine constant. Accurately encoding the induced signatures and payoff space relies on intricate reductions, where small sender payoff boosts and careful parameterization force solution specificity, and marginalization structures in the “Khintchine polytope” surface directly as signature realizability conditions.

In the black-box setting, the information-theoretic lower bounds show that—even for near-optimal utility—exact preservation of incentive compatibility (even to constant factor) is generally impossible with bounded sample complexity; any FPTAS must tolerate a small additive loss in at least one axis (utility or rationality). These impossibility results demarcate the boundary between what is efficiently computable and what is only heuristically or approximately attainable in Bayesian persuasion.

5. Practical Applications and Broader Implications

The Bayesian persuasion model provides a unifying language for strategic information design and has penetrated multiple domains. In economics, classic examples include prosecutors selectively disclosing evidence to judges or financial advisers curating recommendations to influence investors, each manipulating beliefs to change downstream choices. The mapping to auction theory aids mechanism designers in analyzing and creating optimal information release policies, while applications in online advertising, recommendation systems, security signaling, and regulatory information disclosure all rely on variants of the basic model.

The computational insights inform practitioners: for instance, in environments where actions/alternatives are symmetric (i.i.d. case) or sampling is feasible, efficient or near-optimal policies can be constructed and deployed. When alternatives are heterogeneous, exhaustive or heuristic methods may be necessary due to intractability. Furthermore, the mathematical equivalence with reduced form auctions both enables modular algorithm design and clarifies modeling boundaries between strategic information and allocation mechanisms.

6. Significance in Algorithmic Game Theory

The analysis reveals that while persuasion and mechanism/auction design are conceptually parallel (information as instrument vs. allocation as instrument), their respective computational and geometric properties diverge sharply in the presence of action heterogeneity. The boundary between tractable and intractable design is demarcated not by the sender’s overall informational power, but by the underlying structure of state-action-inducible distributions. No generalization of Border’s theorem (feasible reduced forms) exists for nonidentical independent actions in persuasion, in sharp contrast to single-item auctions. This demarcates Bayesian persuasion as a benchmark for complexity transitions in information design.

7. Summary Table: Complexity and Algorithms in the Bayesian-Persuasion Model

Input Prior	Optimality Guarantee	Complexity	Main Technique
i.i.d. (explicit marginal)	Exact	P-time	Symmetric LP, reduced form, Border’s thm
Independent, non-identical	#P-hard	Intractable	Khintchine reduction
General, black-box	$(1-\varepsilon)$-optimal and $\varepsilon$-IC	FPTAS	Monte Carlo, relaxation

The model thus blends the core concepts of information economics (commitment to information structures, Bayes-plausible posteriors, and equilibrium best responses) with the algorithmic sophistication of modern discrete and convex optimization, offering a precise map of both theoretical tractability and application frontiers (Dughmi et al., 2015).