📈 economics

Inattention to States and Characteristics

This paper introduces a rational inattention model where agents acquire costly information about both choice characteristics and payoff-relevant states, solving the resulting optimization problem by decomposing it into a strictly concave outer problem and an inner Schrödinger bridge problem to derive a unique, interior weighted multinomial logit choice probability.

Original authors: Chris Engh

Published 2026-01-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Chris Engh

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are walking into a massive grocery store to buy a jar of peanut butter. You have two main things to worry about:

The State (Your Needs): Are you hungry right now? Do you have a peanut allergy? (These are the "states" of the world that change and affect how good a choice is for you).
The Characteristics (The Options): Is the jar on the shelf "Creamy" or "Crunchy"? Is it "Organic" or "Regular"? (These are the "characteristics" of the choices available).

In the old way economists modeled how people make decisions (the "State-Action" model), they assumed you could instantly and perfectly see the label on every jar. You only had to pay a "mental cost" to figure out if you were hungry or not. If you picked the wrong jar, it was because you didn't know your own hunger level well enough.

The Problem with the Old Model
Christopher Engh argues this doesn't match real life. Sometimes, you don't just fail to know your hunger level; you also fail to read the label. Maybe the "Organic" label is faded, or there are so many jars that you just grab the first one you see without checking if it's actually the high-coverage insurance plan you wanted.

The old model couldn't separate these two failures. It treated the jar's label as if it were free and easy to read, which isn't true when you are overwhelmed.

The New Model: A Two-Part Mental Tax
Engh proposes a new model where your brain has to pay a "mental tax" for two different things:

Learning about the World (The State): Paying attention to whether you are actually hungry or allergic.
Learning about the Menu (The Characteristics): Paying attention to what the jars actually say.

He introduces a "knob" (called $\alpha$ ) that controls how much you care about one versus the other.

If the knob is turned all the way to one side, you are like a robot that instantly reads every label but is confused about your own hunger.
If it's turned to the other side, you know your hunger perfectly but just grab whatever jar is closest without reading it.
Most importantly: In the middle, you do a mix of both. You pay attention to your needs and you try to read the labels, but you have to balance the effort.

The "Weighted Logit" Formula: The Recipe for Choice
The paper's biggest discovery is a simple formula that predicts what you will buy. It says your choice is a tug-of-war between three forces:

Utility (The Taste): How much you want the peanut butter based on your needs (e.g., "I need crunchy because I'm making a sandwich").
The "Crowd" Effect ( $\phi$ ): How common that type of jar is in the store. If 90% of the jars are "Creamy," it's harder to avoid buying Creamy, even if you wanted Crunchy. The model says, "Lemons are harder to avoid when they are everywhere."
Inertia (The Habit): Because reading labels is expensive (mentally), you tend to stick with what you usually buy unless you really, really pay attention.

The result is a "Weighted Multinomial Logit." Think of it like a magnet. The magnet pulls you toward the best option, but the "crowd" (how common the option is) and your "laziness" (the cost of paying attention) pull you back toward the average.

Why This Matters: The "Unique" Answer
In the old models, sometimes the math couldn't decide exactly what you would do. It would say, "You might buy A, or you might buy B," without a clear reason why.

Engh's new model fixes this. Because it accounts for the cost of ignoring the labels, it forces a unique, specific prediction. It explains why people sometimes pick "bad" options (like a low-quality insurance plan) not because they are stupid, but because those bad options are so common and the mental effort to sort them out is too high.

The "Schrödinger Bridge" (The Math Magic)
To solve this, the author uses a fancy mathematical tool called a "Schrödinger bridge."

The Metaphor: Imagine you have a pile of sand (your current habits) and you want to move it to a new shape (your ideal choices) with the least amount of effort.
The math shows that there is only one perfect way to move that sand. This guarantees that the model gives a single, clear answer for how people will behave, rather than a confusing list of possibilities.

In Summary
This paper builds a better map of human decision-making. It admits that we are often too busy to read every label and too confused to know our own needs perfectly. By adding a cost for "not paying attention to the menu," the model creates a realistic, unique prediction of why we sometimes make the choices we do, especially when the options on the shelf are overwhelming.

Technical Summary: Inattention to States and Characteristics

Problem Statement
This paper addresses a limitation in the standard state–action rational inattention (RI) framework (Matějka and McKay, 2015) when applied to choice environments involving uncertainty over both payoff-relevant states ( $\theta$ ) and menu characteristics ( $\xi$ ). In settings such as insurance markets, decision-makers (DMs) face uncertainty regarding future states (e.g., risk of loss) and the characteristics of available options (e.g., coverage levels). The standard state–action model treats characteristics as either costlessly observed or part of the state vector. This approach fails to simultaneously discipline two distinct empirical phenomena:

The information gain regarding characteristics relative to a baseline prevalence (e.g., distinguishing high-coverage plans from low-coverage ones).
The informativeness of choices regarding the underlying state.

Specifically, the standard model's cost function, based on the mutual information between actions and the joint state-characteristic vector, does not account for the information gain between the state and the chosen characteristics. Consequently, it fails to generate unique predictions for conditional choice probabilities (CCPs) and cannot naturally explain the selection of dominated options when such options are prevalent.

Methodology
The author proposes a generalized RI model where the DM acquires costly information about both states ( $\theta \in \Theta$ ) and characteristics ( $\xi \in \mathcal{X}$ ). The DM chooses a joint distribution $P$ over characteristics and states to maximize expected utility minus an information cost, subject to:

Bayes Plausibility: The marginal over states must equal the prior $\mu$ ( $P_\theta = \mu$ ).
Marginal Feasibility: The marginal over characteristics must be absolutely continuous with respect to an exogenous prior $\phi$ ( $P_\xi \ll \phi$ ), representing the baseline prevalence of characteristics in the market.

The information cost $\kappa_{\alpha, \lambda}(P)$ is a weighted sum of two components:

Vertical Information: The average information gain of the conditional distribution $P(\xi|\theta)$ relative to the exogenous prior $\phi$ . This captures the cost of screening the menu.
Horizontal Information: The Shannon mutual information $I_P(\xi, \theta)$ (or equivalently, the information gain of $P(\theta|\xi)$ relative to $\mu$ ). This captures the cost of learning about states via choice.

The parameter $\alpha \in [0, 1]$ indexes the relative weight of these two margins. The model nests the standard state–action formulation at $\alpha = 0$ (costless characteristics) and a "characteristics-attention" limit at $\alpha = 1$ (costless states).

The mathematical approach decomposes the optimization into two steps:

Inner Problem: For a fixed marginal distribution $\nu$ over characteristics, the problem of finding the optimal joint distribution is formulated as an Entropic Optimal Transport (EOT) problem, specifically a Schrödinger bridge between $\nu$ and $\mu$ .
Outer Problem: The DM optimizes over the choice of the marginal $\nu$ . The author demonstrates that this outer problem is strictly concave.

Key Results

Weighted Multinomial Logit (WML): For intermediate $\alpha \in (0, 1)$ , the optimal conditional choice probability satisfies a unique, interior weighted logit rule:
$P(\xi|\theta) \propto \phi(\xi)^\alpha P_\xi(\xi)^{1-\alpha} e^{u(\xi, \theta)/\lambda}$
This formula integrates three forces: utility ( $u$ ), the pull of prevalent characteristics ( $\phi$ ), and the inertia induced by costly state learning (the endogenous marginal $P_\xi$ ).
Uniqueness and Interiority: Unlike the standard state–action model, which may yield multiple optimal marginals, this model guarantees a unique optimal marginal $P_\xi$ and, consequently, a unique CCP. Furthermore, the solution is interior relative to $\phi$ (i.e., $P_\xi \ll \phi$ and $\phi \ll P_\xi$ ). This interiority arises from an "Inada condition" on the cost function: as the probability of choosing a specific characteristic approaches zero, the marginal cost of further reducing that probability approaches infinity. This prevents the DM from completely ruling out "lemons" (low-quality options) if they are prevalent in the market.
Connection to Schrödinger Bridges: The paper establishes a novel link between rational inattention and entropic optimal transport. Fixing the marginal converts the inner problem into a Schrödinger bridge problem, which admits a unique solution characterized by Schrödinger potentials. The outer problem reduces to finding the optimal marginal that maximizes a strictly concave objective function derived via Jensen's inequality.
Testable Restrictions: The model provides falsifiable restrictions regarding how CCPs react to market entry (changes in $\phi$ ). Specifically, the ratio of choice probabilities for two characteristics across different states should be invariant to the state if the model holds, allowing for over-identification of the parameter $\alpha$ without requiring knowledge of the baseline $\phi$ .

Significance and Contributions
The paper claims three primary contributions:

Tractability and Testability: It provides a closed-form stochastic choice characterization (the WML) that separates the effects of utility, market prevalence, and information costs, offering a disciplined alternative to full-information models.
Point Predictions: By ensuring a unique, interior solution, the model resolves the non-uniqueness issues of the state–action framework and rationalizes the empirical observation that agents sometimes select dominated options when those options are common.
Theoretical Generalization: It extends rational inattention to general Polish spaces and non-discrete settings, bridging the gap between information economics and recent developments in entropic optimal transport (Schrödinger bridges).

The author concludes that this framework offers a more robust tool for analyzing welfare-relevant choice problems where uncertainty exists over both the environment and the menu of options, particularly in contexts like insurance where misunderstanding contract characteristics is a distinct failure mode from forecasting errors.

Technical Summary: Inattention to States and Characteristics

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