📈 economics

Making Information More Valuable

This paper establishes that an agent's value for information increases if and only if the convexity of her reduced-form payoff in beliefs increases, a condition that is geometrically characterized when new actions are added and applied to monopolistic screening and delegation scenarios.

Original authors: Mark Whitmeyer

Published 2026-04-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Mark Whitmeyer

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 a chef trying to cook the perfect meal. You have a set of ingredients (your actions) and you don't know exactly what kind of weather you'll have tomorrow (the state of the world). If you knew the weather, you could pick the perfect dish. Since you don't, you have to guess.

Information is like a weather forecast. It helps you make a better choice.

This paper asks a simple but deep question: When does a chef value a weather forecast more?

Does a chef with a tiny kitchen (few ingredients) value the forecast more than a chef with a massive, fully stocked pantry? Or does it depend on what new ingredients you add to the pantry?

Here is the breakdown of the paper's findings, translated into everyday language.

1. The Core Rule: "The More Flexible, The More You Need to Know"

The paper's main discovery is that information becomes more valuable when your ability to react to that information gets sharper.

Think of your decision-making as a bumpy landscape (a graph).

Flat spots mean: "No matter what the weather is, I'm going to cook pasta." (You don't need a forecast; you're stuck on pasta).
Bumpy, jagged spots mean: "If it's sunny, I'll grill. If it's rainy, I'll stew. If it's windy, I'll bake." (You are highly sensitive to the forecast).

The paper proves that information is more valuable if and only if your decision landscape becomes "bumpier" (more convex). If adding a new option makes you switch your plan more drastically based on small changes in the forecast, you will pay more for that forecast.

2. The "Refining" Action: Adding a New Tool

Imagine you have a basic toolkit: a hammer and a screwdriver.

Scenario A: You add a sledgehammer.
- Result: Now, if the job is huge, you use the sledgehammer. If it's medium, the hammer. If it's tiny, the screwdriver.
- The Catch: The sledgehammer is so big that it only works for huge jobs. It doesn't mess with your choices for small or medium jobs. You are now more sensitive to the size of the job. Information is more valuable.
Scenario B: You add a tiny, weak screwdriver that is slightly better than your current one for some jobs, but worse for others.
- Result: This new tool creates a "middle ground" where you might just stick with your old tools because the new one isn't quite right. It blurs the lines. You become less sensitive to the details of the job. Information becomes less valuable.

The paper calls the first type of addition a "Refining Action." It's a new tool that is so specific it only takes over a tiny, specific slice of your decision-making, leaving the rest of your choices sharp and distinct. If you add a refining action, you value information more.

3. The "Subtracting" Trap: Taking Things Away

What happens if you take a tool away?

The Bad News: If you take away a tool that was actually useful (one you would have used in some situation), you become less flexible. You have fewer ways to react to the world. Consequently, information becomes less valuable.
The Only Exception: The only time taking a tool away helps is if the tool you removed was useless trash (dominated) to begin with. Removing trash doesn't change your flexibility, so your value for information stays the same.
The "New World" Exception: If you remove so many tools that you are left with a completely different set of options (a totally new decision problem), the rules change, but generally, the paper argues that simply "pruning" your options usually hurts your desire for information.

4. The "Risk" Analogy: Making the Stakes Bigger

Imagine you are playing a game for money.

Scaling Up: If you double the prize money for every outcome, the "bumps" in your decision landscape get steeper. The difference between winning big and winning small becomes huge. You become more sensitive to the forecast. Information is more valuable.
Risk Aversion: If you become more scared of losing (more risk-averse), the paper says the effect is messy. It depends entirely on how your fear changes your choices. Sometimes it makes you more sensitive; sometimes less. There is no simple rule here, unlike the "adding a tool" or "doubling the prize" scenarios.

5. Real-World Examples from the Paper

The author uses a doctor to illustrate this:

The Doctor's Toolkit: A doctor can do nothing (n) or cast a broken bone (c).
Adding Surgery (s): Surgery is high-risk and only for severe breaks. It creates a new, specific zone where the doctor must know the severity to act. This is a refining action. The doctor now cares more about getting a precise diagnosis.
Adding Rehab (r): Rehab is a middle-ground option. It's better than nothing for a break, but worse than rest for a sprain. It creates a "gray area" where the doctor might just do rehab regardless of the exact diagnosis. This dampens the need for precise information.

Summary

To make information more valuable to a decision-maker, you generally need to add options that are highly specific (refining actions) or increase the stakes (making the rewards steeper).

Adding a "perfect fit" tool? Good for information.
Adding a "meh" tool that muddies the waters? Bad for information.
Taking away a useful tool? Bad for information.
Making the rewards bigger? Good for information.

The paper essentially tells us that flexibility is the engine of information value. The more precisely you can tailor your actions to the truth, the more you will pay to find out what that truth is.

Technical Summary: Making Information More Valuable

1. Problem Statement and Context

The article is situated within decision theory and the economics of information, building on the foundational work of Blackwell (1951, 1953). While classical literature compares the value of different information structures (which signals are most informative), this article reverses the question: how do modifications to a decision problem affect the value of information for the agent?

More precisely, the author seeks to characterize transformations of a decision problem (adding or removing actions, modifying the utility function) that increase the value of information, independently of the exogenous information structure or the endogenous cost of acquiring information.

2. Theoretical Framework and Methodology

Basic Model:

An agent maximizes expected utility $E[u(a, \theta)]$ facing a state of the world $\theta \in \Theta$ (finite) and a set of actions $A$ .
The agent's value function, $V(\mu)$ , is defined as the maximum expected utility for a belief (posterior) $\mu \in \Delta(\Theta)$ :
$V(\mu) = \max_{a \in A} E_\mu[u(a, \theta)]$
This function is convex by construction.
Information acquisition is modeled either exogenously (distribution of posterior beliefs $\Phi$ ) or endogenously (optimal choice of $\Phi$ under a cost separable by posteriors).

Comparison Criterion:
The author defines that a transformation of the decision problem (from $D$ to $\hat{D}$ ) increases the value of information if, for every prior $\mu_0$ and every information structure $\Phi$ :

The expected gain from information is higher in $\hat{D}$ than in $D$ .
The agent never acquires strictly less information in $\hat{D}$ (in the sense of mean-preserving contraction).

Central Tool: Relative Convexity
The fundamental result (Theorem 3.1) establishes an equivalence:

A transformation increases the value of information if and only if the difference between the new and old value functions, $\hat{V} - V$ , is convex.

This means the agent becomes more sensitive to information (increased responsiveness) and the value of distinguishing between actions increases.

Geometric Representation: Polyhedral Subdivisions
For problems with finite actions, the value function is a polyhedral surface. The author uses the notion of a regular polyhedral subdivision ( $\mathcal{C}$ ), which partitions the belief space into cells where a specific action is optimal.

Extensive Margin: Adding an action makes the agent more responsive if the new subdivision $\hat{\mathcal{C}}$ is a refinement ( $\hat{\mathcal{C}} \succeq \mathcal{C}$ ) of the old one. This means the regions of optimality for the old actions shrink or remain unchanged, but do not expand.
Intensive Margin: The value of distinguishing actions (the "slope" of the value function).

3. Main Results

The article analyzes five classes of transformations:

A. Adding Actions (Increased Flexibility)

Adding a single action: The value of information increases if and only if the added action is affine (refining).
- Definition: An action is affine if it is optimal in a region of beliefs that is a subset of the optimality region of a pre-existing action. It does not "replace" an action over part of its domain of optimality without reducing it elsewhere.
- Geometric Condition: The new subdivision $\hat{\mathcal{C}}$ must be a strict refinement of $\mathcal{C}$ ( $\hat{\mathcal{C}} \succ \mathcal{C}$ ).
Adding multiple actions: The convexity condition for $\hat{V}-V$ $\hat{V} - V$ is more complex.
- A sufficient condition is that the set of new actions is totally affine (each new action is either dominated or affine).
- Generically, the necessary and sufficient condition is that the set is totally strictly affine. If this is not the case, the addition may increase responsiveness (extensive margin) but reduce the value of distinction (intensive margin), canceling out the overall gain.

B. Removing Actions (Reduced Flexibility)

Counter-intuitive result: Removing actions can never increase the value of information, except in trivial cases.
If one removes an action that was not dominated (a "leftover" action), the agent necessarily becomes less responsive to information.
The only exception is if the removal leads to a completely new decision problem (all non-dominated actions are removed) or if the removal is inconsequential (only dominated actions are removed).

C. Affine Transformations of Utility

If utility is transformed by $\hat{u} = k u + s$ $\overset{u}{^} = k u + s$ with $k \ge 1$ $k \geq 1$ :
- Responsiveness (extensive margin) remains unchanged ( $\hat{\mathcal{C}} = \mathcal{C}$ ).
- The value of information increases because the slope of payoffs (intensive margin) increases.
Applications:
- Repeating the decision problem (multiplying gains) increases the value of information.
- For an agent with CARA utility (constant absolute risk aversion), a decrease in wealth (or an increase in aggregate risk) increases the value of information, as this is equivalent to an increase in the risk aversion parameter relative to gains, making decisions more sensitive to information.

D. Changes in Risk Aversion (Non-linear)

Unlike affine transformations, an increase in risk aversion (or risk-loving behavior) does not generically generate a higher value of information.
This depends entirely on the extensive margin: if the change in risk modifies the subdivision (by making dominated actions non-dominated), it can reduce responsiveness. The result is generally negative or ambiguous.

E. Comparison with Fixed Prior (Fixed-Mean)

If the prior $\mu_0$ is fixed, the convexity condition for $\hat{V}-V$ is no longer necessary.
The condition becomes that $\hat{V}$ shift-majorizes $V$ (pointwise dominance after adjustment by an affine function passing through $\mu_0$ ).
This allows for cases where the agent acquires less information in the new problem, but the value of information (for this specific prior) is higher.

4. Applications

Delegation:
- In a framework where a principal delegates to an agent who acquires information, the principal benefits from giving the agent a set of actions that is totally affine (refining).
- Contrary to previous literature (e.g., Szalay, 2005) which suggested that removing intermediate actions could improve incentives for information, this article shows that, in a generic and robust framework (independent of specific costs), removing actions reduces the value of information and thus the principal's welfare.
Selling Information (Monopolistic Screening):
- Application to a screening problem where the agent's private type is their value for information.
- If the difference between value functions across types is convex, the optimal mechanism follows the classic rules of screening theory: no distortion for the "high" type (who receives first-best information) and downward distortion for the "low" type.

5. Significance and Contribution

Theoretical: The article provides a complete and necessary/sufficient characterization of the increase in the value of information via the relative convexity of value functions. It unifies the concepts of responsiveness (extensive margin) and value of distinction (intensive margin).
Geometric: The introduction of the notion of "affine" (refining) actions and the use of polyhedral subdivisions offer a powerful visual and mathematical tool for analyzing structural changes in decision problems.
Practical: It challenges the intuition that restricting choices (regulation, removing options) can systematically improve incentives for information. It demonstrates that, except in very specific cases, increased flexibility (via well-chosen actions) is the only robust way to increase the value of information.
Robustness: The results are formulated to be valid for all priors and all information structures, making them particularly robust for mechanism design and public policy.

In summary, Whitmeyer demonstrates that for a modification of a decision problem to make information more valuable, it must make the agent more responsive to information (by refining their decision map) while increasing the value of distinguishing between actions, a condition rarely satisfied by removing actions or by non-linear changes in risk aversion.