All Substitution Is Local

Imagine you are trying to make a big decision, like choosing which restaurant to go to, which job offer to accept, or which medical treatment to try. You have a few "information sources" you can consult: maybe a friend's opinion, a review website, or a doctor's test.

The big question this paper asks is: Does getting one piece of information make the next piece of information more useful, or less useful?

Sometimes, two pieces of info work together like a team (Complements). Other times, they step on each other's toes like rivals (Substitutes).

The authors, Nidhish Shah, Shaurjya Mandal, and Asfandyar Azhar, discovered a surprising rule: Substitution is a local phenomenon. In plain English, information only becomes a "rival" when you are standing right on the edge of making a decision. Everywhere else, information is a helpful teammate.

Here is the breakdown using simple analogies:

1. The Map and the Decision Lines

Imagine your mind is a map (a "belief simplex"). On this map, there are different zones.

Zone A: You decide to eat Pizza.
Zone B: You decide to eat Sushi.
Zone C: You decide to eat Salad.

The lines separating these zones are Decision Boundaries.

If you are deep inside the "Pizza Zone," you are 100% sure you want pizza.
If you are right on the line between Pizza and Sushi, you are on the fence.

2. The Two Forces at Play

When you get a new piece of information (like a friend telling you "The pizza place is closed"), two things happen simultaneously:

The "Helper" Force (Complement): This force tries to sharpen your focus. It helps you see the map more clearly, making the next piece of info more valuable. It's like a flashlight that makes the next step easier to see.
The "Spoiler" Force (Substitute): This force tries to solve the problem for you. If the new info tells you "Pizza is closed," and you were on the fence, you might immediately switch to Sushi. The problem is solved! Because the decision is already made, the next piece of info (like a review of the Sushi place) becomes less valuable. You don't need it anymore.

3. The Golden Rule: "Substitution is Local"

The paper proves a beautiful geometric truth:

The "Spoiler" Force only works if you cross a line.

Scenario A: You are deep in the middle of a zone.
Imagine you are deep in the Pizza Zone. Your friend says, "Actually, the pizza place is definitely open." You were already going to eat pizza, so this news doesn't change your mind.
- Result: The "Spoiler" force is zero. You didn't cross a line.
- Outcome: The "Helper" force wins. Because your friend clarified the situation, your next piece of info (maybe a review of the Sushi place to see if you should switch) becomes more valuable. You are now more confident in your current choice, but you are also better prepared if you do need to switch. They are teammates.
Scenario B: You are standing on the line.
Imagine you are exactly on the line between Pizza and Sushi. Your friend says, "Pizza is closed."
- Result: You instantly jump to the Sushi Zone. The decision is made!
- Outcome: The "Spoiler" force is huge. Because you already switched to Sushi, a review of the Sushi place is now useless (you already know you're going there). They are rivals.

4. The "Crossing" Nuance

The paper adds a cool twist: Just crossing a line doesn't guarantee they are rivals.

Imagine you are on the line between Pizza and Sushi.

Info Source 1 tells you "Pizza is closed" (pushing you to Sushi).
Info Source 2 tells you "Sushi is out of fish" (pushing you to Salad).

Even though Source 1 crossed a line, it didn't solve the specific problem Source 2 is trying to solve. Source 1 moved you to Sushi, but Source 2 is still needed to decide between Sushi and Salad. In this case, they are still teammates, even though a line was crossed.

Substitution only happens when both sources are trying to solve the exact same boundary.

5. Why This Matters in Real Life

This isn't just about math; it's about how we gather information in the real world.

Medical Testing: If a doctor runs a test that rules out a specific disease (crossing a boundary), a second test for that same disease becomes useless. But a test for a different disease becomes more valuable because you've narrowed down the possibilities.
Hiring: If you interview a candidate and decide they are "too junior" (crossing a boundary), asking them more questions about their junior skills is a waste of time (Substitution). But asking about their leadership skills (a different boundary) might still be very valuable (Complement).
Machine Learning: When training AI, you don't need to feed it redundant data that solves the same problem. You want data that helps it cross new boundaries.

The Big Takeaway

Information is rarely a rival unless it solves the exact same immediate problem you are struggling with.

If you are deep in your decision-making process, every new piece of data helps the next one. It's only when you are standing on the edge of a decision, and two sources are both trying to push you over that same edge, that they start competing.

"Substitution is local; everywhere else, information cooperates."

1. Problem Statement

The paper addresses a fundamental question in information economics and Bayesian decision theory: When does consulting one information source increase the value of another (complementarity), and when does it decrease it (substitutability)?

While previous work (e.g., Chen and Waggoner [2016]) established global conditions for when information sources are complements or substitutes across all possible beliefs, this paper seeks to localize these conditions. The authors aim to characterize, for a fixed decision problem and a fixed pair of information channels, exactly which beliefs lead to complementarity and which lead to substitution.

The core setting involves a Bayesian decision-maker facing a finite set of actions ( $A$ ) and states ( $S$ ). The agent holds a prior belief $b$ and can observe signals from information channels $i$ and $j$ . The interaction is measured by the second-order interaction term, $\Delta \text{VoI}(j | i, b)$ , which represents the change in the Value of Information (VoI) of channel $j$ after observing channel $i$ .

2. Methodology and Framework

The authors employ a geometric and convex-analytic approach to decompose the interaction between information channels.

Decision Problem: The value function $V(b)$ is defined as the maximum expected reward over actions. Since the action set is finite, $V(b)$ is Piecewise-Linear and Convex (PWLC). The belief simplex is partitioned into convex polytopes (decision regions $R_a$ ), within which $V(b)$ is linear.
Value of Information (VoI): The VoI of a channel is the expected gain in value from updating beliefs, calculated as the difference between the expected posterior value and the prior value.
Bregman Decomposition: The authors define two auxiliary convex functions:
1. $g(b) = \mathbb{E}_{o_j}[V(\hat{b}(j, o_j))]$ : The expected value after observing channel $j$ .
2. $h(b) = V(b)$ : The current value function.
Using the Martingale Identity (which relates the Jensen gap of a convex function to the expected Bregman divergence), they decompose the interaction term $\Delta \text{VoI}(j | i, b)$ into two opposing forces:
$\Delta \text{VoI}(j | i, b) = \underbrace{\mathbb{E}[D_g(\hat{b}, b)]}_{\text{Complement Force}} - \underbrace{\mathbb{E}[D_h(\hat{b}, b)]}_{\text{Substitute Force}}$
Where $D_\phi$ is the Bregman divergence associated with convex function $\phi$ , and $\hat{b}$ represents the posterior beliefs after observing channel $i$ .

3. Key Contributions and Results

The paper establishes three primary theoretical results that characterize the "localization principle" of information substitution.

A. Bregman Decomposition (Proposition 2)

The interaction between two channels is the difference between two non-negative terms:

Complement Force ( $\mathbb{E}[D_g]$ ): Measures how channel $i$ moves beliefs to regions where channel $j$ becomes more useful (enhancing future relevance).
Substitute Force ( $\mathbb{E}[D_h]$ ): Measures how much channel $i$ resolves the current decision uncertainty. This force is non-zero only if the posterior beliefs of channel $i$ straddle a decision boundary (where $V$ is non-linear).

B. Interior Complementarity (Theorem 4)

Result: If all posterior beliefs generated by channel $i$ remain within the same decision region as the prior belief $b$ , then $\Delta \text{VoI}(j | i, b) \geq 0$ .

Implication: Inside a decision region, the value function $V$ is linear. Therefore, the substitute force vanishes ( $\mathbb{E}[D_h] = 0$ ) because the "Jensen gap" is zero for linear functions.
Significance: Even if channel $i$ provides no immediate decision value (i.e., it doesn't change the optimal action), it weakly amplifies the value of channel $j$ . Sources are guaranteed to be complements as long as they do not cross a decision boundary.

C. Converse: Substitution Requires Boundary Crossing (Theorem 6)

Result: If $\Delta \text{VoI}(j | i, b) < 0$ (substitution occurs), then at least one posterior of channel $i$ must lie in a different decision region from the prior $b$ .

Implication: Substitution is strictly a boundary phenomenon. For information to be substitutive, the first source must resolve the specific uncertainty that the second source was intended to address by pushing the belief across a decision threshold.
Nuance: Crossing a boundary is necessary but not sufficient for substitution. If the complement force (diffuse across the simplex) dominates the substitute force (concentrated at boundaries), the channels can still be complements even if a boundary is crossed.

4. Illustrative Example

The authors illustrate these concepts using a 3-state, 3-action problem:

Interior Complementarity: When the prior is deep inside a decision region, channel $i$ does not change the action. The substitute force is zero. Channel $i$ refines the belief in a way that makes channel $j$ more effective, resulting in pure complementarity.
Boundary Complementarity: When the prior is near a boundary and channel $i$ crosses a boundary, the substitute force becomes positive. However, if channel $i$ and channel $j$ target different boundaries (orthogonal uncertainties), the complement force can still dominate, resulting in net complementarity.
Substitution: When channel $i$ crosses the same boundary that channel $j$ targets, the substitute force (regret reduction) dominates. Channel $i$ resolves the decision that channel $j$ was relevant for, rendering channel $j$ worthless.

5. Significance and Applications

Geometric Criterion for Redundancy: The paper provides a precise geometric rule: Information sources substitute only if they resolve the same decision boundary. Sources resolving different boundaries complement each other.
Sequential Decision Making: In sequential acquisition (e.g., clinical trials, active learning), the results offer a heuristic: after observing a source that resolves a specific boundary, the agent should prioritize sources targeting different boundaries (which retain or increase in value) and de-prioritize sources targeting the same boundary (which lose value).
Refinement of Global Conditions: It moves beyond global "Blackwell" orderings to a pointwise analysis, showing that the nature of information interaction is highly dependent on the agent's current position in the belief space relative to decision boundaries.
Formal Verification: The results are formalized in Lean 4, ensuring mathematical rigor.

6. Limitations

Finite Actions: The results rely on the Piecewise-Linear Convex (PWLC) structure of the value function, which arises from finite actions. Continuous action spaces yield smooth, strictly convex value functions without flat regions, meaning the "interior complementarity" mechanism does not directly apply.
Conditional Independence: While the decomposition holds for correlated channels, the direct economic interpretation of the forces as "complement" and "substitute" relies on the channels being conditionally independent given the state.
Bayesian Rationality: The derivation assumes the Bayesian martingale property ( $\mathbb{E}[\hat{b}] = b$ ). Agents with miscalibrated updates would break the identity.

Conclusion

The paper's central thesis is that "Substitution is local." Information sources compete (substitute) only in the "thin boundaries" where decisions change. Everywhere else, in the interior of decision regions, information sources cooperate (complement). This localization principle offers a powerful tool for designing optimal information acquisition strategies in complex, multi-source environments.