Posterior-Separable Costs and Menu Preferences

1. Problem Statement

The paper addresses the problem of axiomatically characterizing the preferences of a Rationally Inattentive (RI) Decision Maker (DM) over menus of acts. While the literature (e.g., De Oliveira et al., 2017) has established that RI preferences can be represented by a general cost function, a specific and widely used subclass is posterior-separable costs.

In posterior-separable models, the cost of an information structure (experiment) $\pi$ is given by:
$$c(\pi) = \int \psi(p) \pi(dp)$$
where $\psi$ is a "measure of uncertainty" and $p$ represents posterior beliefs. This structure is crucial because it allows the RI problem to be solved using tools from Bayesian persuasion (specifically, the concavification of the net utility function).

However, a gap exists in the literature:

1. Not all posterior-separable costs yield a unique solution to the DM's optimization problem.
2. The standard axioms for RI preferences do not fully pin down the specific properties of $\psi$ required for the "unique hyperplane" property, which is essential for the Bayesian persuasion formulation to be well-behaved (i.e., having a unique optimal information structure).

The authors seek to identify the specific behavioral axioms that are necessary and sufficient for a DM to have a posterior-separable cost function that satisfies a mild smoothness condition, ensuring a unique solution to the information acquisition problem.

2. Methodology

The authors employ an axiomatic approach within the framework of menu preferences.

• Framework: The DM chooses a menu $F$ (a set of acts) knowing they will acquire information before selecting a specific act. The utility is represented as:
  $$V(F) = \max_{\pi \in \Pi(p_0)} \left[ \int \phi_F(p) \pi(dp) - c(\pi) \right]$$
  where $\phi_F(p) = \max_{f \in F} \sum u(f(\omega))p(\omega)$ is the indirect utility given belief $p$.
• Geometric Interpretation: The problem is mapped to finding the concavification of the net utility function $N_F(p) = \phi_F(p) - \psi(p)$. Geometrically, this corresponds to finding the lowest hyperplane (supporting hyperplane) lying above the graph of $N_F$.
• Key Concepts:
  • Unique Hyperplane Property (UHP): The condition that for any menu, there is a unique optimal hyperplane (and thus a unique optimal information structure).
  • Joint-Directional Differentiability (JDD): A novel smoothness condition on the cost function $\psi$. It requires that the subdifferential of $\psi$ does not allow for "wobbling" hyperplanes in a specific direction across a set of points whose convex hull contains the prior.

3. Key Axioms

The paper introduces two specific axioms to characterize the desired class of preferences:

1. Independence of Irrelevant Alternatives (IIA):

   • Statement: If a DM is indifferent between two menus $F$ and $G$, and also indifferent between their intersection $F \cap G$, then they must be indifferent to their union $F \cup G$.
   • Intuition: If the options outside the intersection are irrelevant to the choice between $F$ and $G$, adding them together should not create new value. This prevents the "synergy" of information costs that arises when multiple non-optimal options interact to change the optimal information structure.

2. Ignorance Equivalence (IE):

   • Statement: For every menu $F$, there exists an act $h$ (the ignorance equivalent) such that $h \sim F \sim F \cup \{h\}$.
   • Intuition: There is always a "certainty equivalent" for a menu that requires no information acquisition. Adding this act to the menu does not increase the value of the menu because the agent would simply choose $h$ without learning. This axiom rules out cost functions where the prior belief is not "grounded" (i.e., where the cost of learning is negative relative to the prior).

4. Main Results

Theorem 1: Characterization of Menu Preferences

The authors prove the equivalence of the following three statements:

1. The preference relation $\succsim$ satisfies IIA and IE.
2. The preference has a posterior-separable representation with a canonical measure of uncertainty $\psi$ satisfying Joint-Directional Differentiability (JDD).
3. The preference has a posterior-separable representation satisfying the Unique Hyperplane Property (UHP).

Significance of JDD:

• JDD is a weak smoothness condition. It is satisfied by all differentiable functions but also includes many non-differentiable functions commonly used in economics (e.g., entropy, log-likelihood, residual variance).
• The theorem establishes that UHP is equivalent to JDD. Without JDD, multiple optimal hyperplanes can exist, leading to non-unique solutions for the information structure.
• The set of functions satisfying JDD is dense and includes all standard cost functions used in the literature.

Theorem 2: Uniform Posterior-Separability

When the cost function is prior-invariant (the cost structure remains the same regardless of the prior belief), the authors show that IIA and IE imply that the cost is uniformly posterior separable and represented by a differentiable function $\psi$.

• This result strengthens the characterization for models where the information cost is independent of the specific prior (e.g., Mutual Information).

5. Technical Contributions

1. Novel Smoothness Condition (JDD): The paper defines "Joint-Directional Differentiability," a condition weaker than full differentiability but strong enough to guarantee the uniqueness of the optimal hyperplane. This bridges the gap between the behavioral axioms and the geometric requirements of the concavification approach.
2. Uniqueness of Representation: The paper demonstrates that without JDD (or UHP), the measure of uncertainty $\psi$ is not uniquely identified by the preference relation. The axioms ensure that the canonical representation is unique.
3. Connection to Bayesian Persuasion: The authors rigorously link the behavioral axioms of RI to the solvability of the problem via the "lowest hyperplane" formulation used in Bayesian persuasion (Kamenica and Gentzkow, 2011). They show that IIA and IE are the behavioral counterparts to the mathematical requirement of a unique supporting hyperplane.
4. Canonical Measure of Uncertainty: The paper refines the definition of a "canonical" cost function, adding the condition that the prior must lie in the relative interior of the effective domain of $\psi$. This is shown to be without loss of generality but necessary for the uniqueness proofs.

6. Significance and Implications

• Validation of Standard Models: The results provide a behavioral foundation for the vast literature using posterior-separable costs (e.g., Sims, 2003; Matějka and McKay, 2015). It confirms that these models implicitly assume IIA and IE, which in turn guarantees the existence of a unique solution.
• Identification: The paper clarifies identification issues in rational inattention. If a DM's preferences violate IIA or IE, one cannot simply assume a posterior-separable cost with a unique solution; the cost function might be non-smooth in a way that creates multiple equilibria for information acquisition.
• Mechanism Design: For mechanism designers (e.g., in pricing or security design) who assume agents are rationally inattentive, this paper suggests that assuming IIA and IE is necessary to ensure the agent's information acquisition problem has a unique, well-defined solution that can be analyzed using standard concavification techniques.
• Robustness: By showing that JDD is satisfied by a dense set of functions, the authors argue that the "unique hyperplane" assumption is robust and holds for almost all practically relevant cost functions, even those that are not fully differentiable.

In summary, the paper provides the missing behavioral axioms that justify the mathematical convenience of posterior-separable costs with unique solutions, bridging the gap between abstract preference theory and the practical application of information design tools.