📈 economics

On Rational Inattention with Arbitrary Choice Sets

This paper demonstrates that rational inattention can be formulated as a nested regularized optimal transport problem, utilizing entropic optimal transport to reprove and extend key results from Matejka and McKay (2015) and Caplin, Dean, and Leahy (2019) to arbitrary choice sets.

Original authors: Chris Engh

Published 2026-03-17

📖 6 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

The Big Picture: The "Overwhelmed Brain" Problem

Imagine you are a chef in a busy kitchen. You have a menu of 50 dishes (your actions) and customers are ordering based on the weather, the time of day, and their mood (the states).

In the old days of economics, we assumed chefs had perfect brains. They knew exactly what every customer wanted and cooked the perfect dish instantly. But in reality, chefs get tired. They can't process 50 options perfectly every second. They have limited attention.

This is the problem of Rational Inattention. The chef wants to maximize tips (utility) but has to pay a "tax" for thinking too hard (information cost). The question is: How does a smart chef decide which dishes to focus on when they can't think about everything at once?

The Old Way vs. The New Way

The Old Way (Finite Sets):
Previously, economists (like Matějka and McKay) solved this for small menus. They said, "If you only have 3 dishes, here is the exact math for how you choose." But what if the menu is infinite? What if you can choose any temperature for your coffee, or any speed for your car? The old math got stuck.

The New Way (The Schrödinger Bridge):
This paper says: "Let's borrow a tool from a totally different field: Physics and Image Processing."

The author connects the chef's problem to something called Entropic Optimal Transport.

The Analogy: Imagine you have a pile of sand (the customers' moods) and a set of buckets (the dishes). You want to move the sand into the buckets to minimize the effort of carrying it, but you also want to keep the buckets looking "messy" (random) because you don't want to overthink the arrangement.
The "Bridge": In physics, a "Schrödinger Bridge" is the most likely path a particle takes to get from Point A to Point B while obeying the laws of thermodynamics. The author realizes that the chef's decision-making process is mathematically identical to this bridge.

The Two-Step Dance: The "Bridgehead" Strategy

The paper's main insight is that the chef doesn't solve the whole problem at once. They do it in two steps, like a dance:

Step 1: The Inner Dance (The Bridge):
First, the chef picks a guess about how often they will make each dish (e.g., "I'll make 20% coffee, 30% tea, 50% juice"). Let's call this the Bridgehead.
- Given this guess, the chef figures out the best way to match customers to dishes. This is the "Schrödinger Bridge." It's the most efficient way to serve customers if you stick to your guess.
Step 2: The Outer Dance (The Adjustment):
Now, the chef looks at the result. "Wait, I guessed I'd make 20% coffee, but actually, I'm making way more money on tea. I should change my guess."
- The chef adjusts the "Bridgehead" (the mix of dishes) to maximize profit.

The Magic: The paper proves that if you keep doing these two steps over and over, you eventually find the perfect strategy, even if the menu is infinite.

The Secret Sauce: "Potentials" (The Invisible Scorecards)

The paper introduces two invisible scorecards called Schrödinger Potentials. Think of them as the chef's internal "gut feelings" or "marginal benefits."

The State Potential (The "Weather" Score):
This tells the chef how valuable a specific customer mood is right now. If it's raining, the "rain score" goes up. This helps the chef decide which dish to recommend.
- Metaphor: It's like a weather app on your phone telling you, "It's a great day for soup."
The Action Potential (The "Menu" Score):
This tells the chef how much they should change their overall menu mix.
- The Big Insight: The paper shows that for the chef to be truly optimal, the "Menu Score" for every dish they actually make must be zero.
- Translation: If a dish has a positive score, you should make more of it. If it has a negative score, make less. If it's zero, you are in the perfect spot. You can't gain any more tips by shifting your focus.

The Algorithm: The "Blahut-Arimoto" Loop

How do you actually calculate this? The paper connects the chef's problem to a famous computer algorithm called Sinkhorn (used in AI to generate images) and Blahut-Arimoto (used in data compression).

Imagine the chef is playing a video game where they have to guess the right menu mix:

Guess: "I'll make 50% coffee."
Check: The computer (the algorithm) says, "Based on that, here is the best way to serve customers."
Update: The computer calculates the "Score" (Potential) for coffee. "Hey, coffee is actually under-valued! You should make 60%."
Repeat: The chef updates the guess to 60% and repeats.

The paper shows that this "guess-and-check" loop is exactly how the chef finds the perfect balance between thinking hard and making money.

Why Does This Matter?

It Unifies Everything: It shows that the math for a chef with 3 dishes and a chef with infinite options is actually the same. They are just different versions of the same "Bridge."
It Solves the "Infinite" Problem: Before this, economists struggled to model choices where you can pick any number (like speed or price). This paper says, "Treat it like a physics problem, and the math solves itself."
It Explains "Consideration Sets": Why do we only think about 3 or 4 options when buying a car, even though there are 500? The math shows that the "Score" for the other 497 cars is negative, so our brain naturally ignores them. We only focus on the ones where the score is zero (the "consideration set").

Summary in One Sentence

This paper tells us that when our brains are too busy to think about everything, we naturally act like a physics experiment trying to find the most efficient path (a "Bridge") between our limited attention and the world, and we can solve this complex puzzle using a simple "guess-and-adjust" loop borrowed from computer science.

1. Problem Statement

The paper addresses the Rational Inattention (RI) problem, originally pioneered by Sims (1998, 2003, 2005) and later formalized for discrete choice sets by Matějka and McKay (2015, MM) and Caplin, Dean, and Leahy (2019, CDL).

The Core Issue: An agent with limited information-processing capacity (Shannon information cost) must choose an information policy $P(\alpha|\omega)$ mapping states $\omega \in \Omega$ to actions $\alpha \in A$ to maximize expected utility minus the cost of information.
The Limitation: Previous tractable results were largely confined to finite action sets. Extending these results to arbitrary (including continuous) Polish spaces has been mathematically challenging due to the complexity of characterizing the optimal marginal distribution of actions ( $P_\alpha$ ), which typically lacks a closed form and is non-unique.
The Gap: While the optimal conditional choice probability $P(\alpha|\omega)$ is known to follow a multinomial logit structure given the marginal, solving for the optimal marginal itself remains difficult in general spaces.

2. Methodology

The author reframes the Rational Inattention problem as a nested Regularized Optimal Transport problem, specifically utilizing the Schrödinger Bridge (SB) framework.

Entropic Optimal Transport (EOT): The paper leverages the connection between RI and the Schrödinger Bridge problem, which seeks a joint coupling $P$ between two marginals ( $\nu$ on actions and $\mu$ on states) that minimizes the Kullback-Leibler (KL) divergence from a reference measure while satisfying marginal constraints, regularized by an entropy term.
The "Bridgehead" Decomposition: The author proposes a two-step decomposition of the RI problem:
1. Inner Problem (Schrödinger Bridge): For a fixed action marginal $\nu$ , find the optimal joint distribution $P$ that maximizes the objective. This is the standard SB problem.
2. Outer Problem (Marginal Optimization): Optimize over the choice of the marginal $\nu$ itself to maximize the value of the inner problem.
Mathematical Tools:
- Schrödinger Potentials: The dual variables ( $a_\nu$ for actions, $b_\nu$ for states) associated with the marginal constraints in the SB problem.
- Variational Analysis: Using Gateaux derivatives and the Envelope Theorem to characterize optimality conditions.
- Statistical Mechanics Analogy: Interpreting the problem through the lens of free energy minimization, where the information cost parameter $\lambda$ acts as temperature.

3. Key Contributions

The paper makes several theoretical and methodological contributions:

Generalization to Arbitrary Choice Sets: It extends the tractable RI framework from finite sets to general Polish spaces (continuous action spaces) by utilizing the machinery of entropic optimal transport.
The "Schrödinger Bridgehead" Approach: It introduces a novel decomposition that separates the matching problem (inner SB) from the marginal selection problem (outer optimization), clarifying the mathematical structure of the RI problem.
Economic Interpretation of Potentials:
- Action Potential ( $a_\nu$ ): Interpreted as the marginal benefit (directional derivative) of re-assigning probability mass to a specific action. The optimality condition implies $a_\nu(\alpha) = 0$ on the support of the optimal marginal and $a_\nu(\alpha) \leq 0$ elsewhere.
- State Potential ( $b_\nu$ ): Interpreted as the log of the partition function (normalizing factor) in the conditional choice probability. It encodes the state-dependent value and relates to the agent's willingness to pay for information.
Unification of Algorithms: It provides a new interpretation of the Blahut-Arimoto algorithm (used for RI) as an augmented Sinkhorn matrix-scaling algorithm. The update rule for the marginal $\nu$ is shown to be a "gradient ascent" step guided by the action potential $a_\nu$ .

4. Key Results and Theorems

Existence and Uniqueness: The paper establishes that for any fixed marginal $\nu$ , a unique Schrödinger Bridge exists (Theorem 1).
Additive Separability: The value function $V(\nu)$ of the inner problem is additively separable into the expected action potential and state potential:
$V(\nu) = \int a_\nu d\nu + \int b_\nu d\mu$
Equivalence of Objectives: The paper proves that maximizing the complex RI objective $V(\nu)$ is equivalent to maximizing a simpler function $f(\nu)$ (the Jensen envelope), defined as:
$f(\nu) = \int \log \left( \int e^{u(\alpha, \omega)/\lambda} d\nu(\alpha) \right) d\mu(\omega)$
(Theorem 9 and Corollary 9.1).
Optimality Conditions (Kuhn-Tucker): The necessary and sufficient conditions for the optimal marginal $\nu$ are derived as:
$\int \frac{e^{u(\alpha, \omega)/\lambda}}{\int e^{u(\alpha', \omega)/\lambda} d\nu(\alpha')} d\mu(\omega) = 1 \quad \text{for } \nu\text{-a.e. } \alpha$
This is re-interpreted via potentials as $a_\nu(\alpha) \leq 0$ everywhere, with equality on the support of $\nu$ (Theorem 15).
Invariant Likelihood Ratios (ILR): The paper recovers the ILR results of CDL, showing that the ratio of posteriors for different actions can be characterized by the difference in utilities adjusted by the state potential.
Thermodynamic Interpretation: The state potential $b_\nu(\omega)$ is shown to be the log-partition function. Its derivatives with respect to $1/\lambda$ yield the conditional moments of utility (Theorem 14), establishing a direct link between RI and free energy minimization.

5. Significance

Theoretical Unification: The paper bridges two distinct fields: Information Economics (Rational Inattention) and Optimal Transport (Schrödinger Bridges). It demonstrates that RI is a specific instance of a nested optimal transport problem.
Computational Insight: By interpreting the Blahut-Arimoto algorithm as a Sinkhorn-type scaling process guided by potentials, the paper offers a clearer intuition for why the algorithm converges and how it updates the agent's beliefs and action probabilities.
Modeling Flexibility: The extension to arbitrary (continuous) choice sets allows economists to model agents with finely discretized or truly continuous action spaces without losing the tractability of the logit structure, provided the optimal marginal is handled via the potential framework.
Conceptual Clarity: The "Bridgehead" perspective demystifies the "mysterious" optimal marginal in RI. It clarifies that the marginal is simply the solution to an outer optimization problem where the agent balances the gains from matching states to actions against the cost of information, with the potentials serving as the shadow prices for these constraints.

In summary, Engh's comment provides a rigorous mathematical foundation for Rational Inattention in general spaces, utilizing the powerful tools of entropic optimal transport to generalize, interpret, and compute solutions that were previously restricted to finite settings.