Posterior-Separable Costs and Menu Preferences

TL;DR

The paper axiomatizes menu-preference representations that arise from posterior-separable information costs in a rational-inattention setting, showing that Independence of Irrelevant Alternatives (IIA) and Ignorance Equivalence (IE) are necessary and sufficient for such a representation to exist with a canonical measure of uncertainty $\psi$ that satisfies Joint-Directional Differentiability (JDD) and, equivalently, the Unique Hyperplane Property (UHP). It demonstrates an equivalence among IIA+IE, posterior-separable representations with JDD, and posterior-separable representations with UHP, with the representing $\psi$ being unique once the utility is fixed. The analysis connects to Bayesian persuasion through the concavification of $N_F(p)=\phi_F(p)-\psi(p)$ and clarifies when costs can be uniformly posterior separable across priors, yielding differentiability of $\psi$ on the effective-domain interior. These results provide a structural foundation for information design with rationally inattentive agents and offer precise conditions under which the optimal information structure is characterized by a single tangent hyperplane, improving tractability and robustness of applications.

Abstract

We consider an agent with a rationally inattentive preference over menus of acts, as in de Oliveira et al (2017). We show that two axioms, Independence of Irrelevant Alternatives and Ignorance Equivalence, are necessary and sufficient for this agent to have a posterior-separable cost satisfying a mild smoothness condition, called joint-directional differentiability. Viewing the decision-maker's problem as a Bayesian persuasion problem, we also show that these axioms are necessary and sufficient for solvability by a unique hyperplane. When the cost function remains invariant for different priors, we show that these axioms imply uniformly posterior separable costs that are differentiable.

Figures (5)

• Figure 1: $p_0$ not in relative interior
• Figure 2: Linear transformation $T$ for dimension matching
• Figure 3: Non-differentiability in direction $\delta$
• Figure 4: Posterior-separable cost function violating IIA
• Figure 5: Constructing the ignorance equivalent given unique hyperplane

Theorems & Definitions (11)

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