📈 economics

Information Aggregation with Costly Information Acquisition

This paper extends the theory of information aggregation by introducing costly information acquisition, demonstrating that a security's ability to aggregate information is characterized by a new " $\kappa$ separability" condition that depends solely on its payoff structure and exhibits a discontinuous transition as acquisition costs decrease.

Original authors: Spyros Galanis, Sergei Mikhalishchev

Published 2026-04-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Spyros Galanis, Sergei Mikhalishchev

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 "Wisdom of the Crowd" Problem

Imagine a group of people trying to guess the weight of a giant pumpkin at a county fair.

The Goal: The group wants to figure out the exact weight.
The Problem: No single person knows the whole truth. Person A knows the pumpkin is heavy because it's green. Person B knows it's light because it's small. But neither knows the full picture.
The Old Theory: In 2012, a researcher named Ostrovsky showed that if the group keeps talking back and forth forever, they will eventually figure out the exact weight—BUT only if the "prize" they are guessing for is structured in a very specific, complex way. If the prize is structured "wrong," they might get stuck guessing the same wrong number forever, even if they have all the clues.

The Catch: To know if the prize is "structured right," you need to know exactly what everyone else knows. In the real world, a market designer (like the person running a prediction market) rarely knows everyone's private secrets. So, they can't be sure the market will work.

The New Twist: Buying Clues

This paper asks a new question: What if people can buy extra clues?

Imagine that before guessing, anyone can pay a small fee to get a better look at the pumpkin (e.g., a magnifying glass or a scale).

The Cost: Looking through the magnifying glass costs money (let's call it $c$ ).
The Question: If the cost of looking is very low, does the market work better? Does the price eventually reveal the true weight, even if the "prize" is structured poorly?

The Three Types of "Prizes" (Securities)

The authors discovered that the answer depends entirely on the structure of the prize (the "security"). They found three distinct categories:

1. The "Magic" Prizes (Always Work)

These are prizes where the payout is unique for every single possible outcome.

Analogy: Imagine a raffle where every single ticket number wins a different amount of money.
Result: These are so unique that the market figures out the truth even if looking for clues costs a fortune. The structure is so clear that people naturally figure it out just by talking.
The Catch: These are rare and not very useful for predicting complex things.

2. The "Broken" Prizes (Never Work)

These are prizes where two different outcomes pay the exact same amount, but the "hidden" details of those outcomes are confusing.

Analogy: Imagine a game where "Rain" and "Snow" both pay you $10, but "Sun" pays $20. If the group is split on whether it's Rain or Snow, they might agree it's worth $10 and stop talking. Even if they could buy a magnifying glass for a penny, they might decide it's not worth the effort to distinguish between Rain and Snow because the payout is the same.
Result: Even if information becomes free, the market might still get stuck. The group agrees on a wrong answer and stops trying to learn more.

3. The "Mostly Good" Prizes (The Sweet Spot)

This is the most important finding. These are prizes that pay differently for almost every outcome (like the "Magic" ones), but they aren't perfect.

Analogy: Imagine a game where every outcome pays a unique amount, except maybe two very similar outcomes pay the same.
The Discovery: If the cost of buying a clue (the magnifying glass) drops just a little bit, something magical happens. The market suddenly snaps into gear.
The "Discontinuous" Jump: The authors found that the market doesn't get slightly better as costs go down. It gets dramatically better.
- High Cost: The market is stuck guessing wrong.
- Slightly Lower Cost: Suddenly, someone buys a clue, reveals a piece of the puzzle, and the whole group realizes the truth. The accuracy jumps from 50% to 100% instantly.

Why Markets Are Better Than Polls

The paper compares a Prediction Market (where people trade back and forth) to a Poll (where everyone guesses once and you take the average).

The Poll: As clues get cheaper, the average guess gets slowly better. It's a smooth, gradual improvement.
The Market: Because people can trade on new information, a small drop in the cost of clues causes a massive leap in accuracy. The market acts like a lever; a tiny push (lower cost) lifts a huge weight (perfect information).

The Takeaway: You don't need information to be free for markets to work perfectly. You just need it to be "cheap enough" to trigger that sudden jump. This makes markets incredibly valuable tools for gathering intelligence, even when information isn't free.

Summary in One Sentence

If the cost of finding the truth is low enough, financial markets can act like a super-efficient machine that instantly aggregates everyone's scattered knowledge, but only if the "game" being played isn't fundamentally broken.

1. Problem Statement

The paper addresses a fundamental limitation in the literature on information aggregation in financial and prediction markets.

The Context: Ostrovsky [2012] established that in dynamic trading models, information aggregates in all Nash equilibria if and only if the traded security is separable. A security is separable if there is no prior distribution under which all traders agree on the expected value of the security despite having private information that implies uncertainty about the true value.
The Limitation: Determining whether a security is separable requires the market designer to know the exact information structure (partitions) of all traders. In practice, a designer often does not know who participates or what their private signals are. Consequently, they cannot guarantee that prices will converge to the true value.
The Question: Can the ability of traders to acquire costly signals during trading restore information aggregation for a broader class of securities, even when the initial information structure is unknown or unfavorable? Specifically, does lowering the cost of information acquisition ( $c$ ) allow markets to aggregate information for "generic" securities?

2. Methodology and Model

The authors extend the dynamic trading model of Ostrovsky [2012] by introducing costly information acquisition.

Trading Environment:
- States: A finite state space $\Omega$ .
- Traders: A finite set of traders $I$ , each with an initial private information partition $\Pi_i$ . The join of all partitions reveals the true state.
- Mechanism: Trading occurs over infinitely many periods using a Market Scoring Rule (MSR). Traders take turns making predictions (announcements) $y_t$ . Payoffs are determined by a strictly proper scoring rule $s(y, x^*)$ , where $x^*$ is the true value of the security.
- Information Acquisition: Before making an announcement, a trader can acquire a signal $R$ at a cost $cK(R)$, where $c > 0$ is the marginal cost and $K$ is a cost function. Signals generate random posteriors.
- Strategies: Traders are strategic (Nash equilibrium), but the concept of "no information acquisition" is defined via a myopic condition: a trader will not acquire a signal if the expected immediate gain from a better prediction does not outweigh the cost.
Key Definitions:
- $\kappa$ -Separability: A security $X$ $X$ is $\kappa$ -separable (where $\kappa = (c, K)$ $κ = (c, K)$ ) if there is no prior $\mu$ $μ$ such that:
  1. $X$ is non-separable at $\mu$ (traders agree on the expected value despite uncertainty).
  2. No trader finds it myopically optimal to acquire any signal at $\mu$ given cost $\kappa$ .
- If such a prior exists, the security is $\kappa$ -non-separable.

3. Key Contributions and Results

A. Characterization of Information Aggregation (Theorem 1)

The paper's primary theoretical contribution is generalizing Ostrovsky's result to environments with costly information.

Result: Information aggregates in all Nash equilibria if and only if the security is $\kappa$ -separable.
Implication: If a security is $\kappa$ -non-separable, there exists a prior and a strategy profile where traders agree on a price (which is incorrect) and no one is willing to pay to acquire information to correct it. Thus, the price fails to converge to the true value.

B. Classification of Securities (Propositions 1, 2, and 3)

The authors provide a complete classification of securities based solely on their payoff structure, independent of the specific information structure of traders. There are three distinct classes:

Always Separable Securities (The "Small" Class):
- These aggregate information even if information acquisition is impossible (cost is infinite).
- Structure: They include constant securities, Arrow-Debreu securities (paying $a$ in one state, $b$ in all others), and a specific 3-value structure (paying $a$ in one state, $d$ in another, and $b$ in all others, where $a < b < d$ ).
- Limitation: These are "uninformative" because they only reveal whether a specific state (or pair of states) occurred, not the full state space.
$\kappa$ -Separable for Some $\kappa$ (The "Generic" Class):
- Structure: Securities that pay a different value in every state ( $X(\omega) \neq X(\omega')$ for all $\omega \neq \omega'$ ).
- Result: For any such security and any information structure, there exists a sufficiently low marginal cost $c$ such that the security becomes $\kappa$ -separable.
- Significance: As information becomes cheap, almost all securities (those with distinct payoffs) will aggregate information, regardless of the traders' initial private information.
$\kappa$ -Non-Separable for All $\kappa$ (The "Failure" Class):
- Structure: Securities that pay the same value $d$ in two distinct states, while paying values strictly higher or strictly lower than $d$ in two other states (e.g., payoff vector $(a, d, b, d)$ where $a, b < d$ or $a, b > d$ ).
- Result: Even as the cost of information $c \to 0$ , these securities remain $\kappa$ -non-separable for some information structures. Information aggregation can fail even with near-zero costs.
- Mechanism: Traders can reach a consensus on the expected value $d$ in a way that makes the marginal benefit of acquiring a signal (to distinguish between the two states paying $d$ ) arbitrarily small compared to the cost, preventing the "break" in the equilibrium.

C. Discontinuous Switch and Market Value (Proposition 4 & Section 5)

Discontinuity: The transition from non-aggregation to aggregation is discontinuous. A small decrease in the cost of information can cause a security to switch from $\kappa$ -non-separable to $\kappa$ -separable, instantly enabling full information aggregation.
Market vs. Poll: The authors compare the prediction market to a "poll" (the average of traders' predictions after they acquire information independently).
- As $c$ decreases, the accuracy of a poll improves smoothly.
- The accuracy of the market jumps discontinuously to 1 (perfect accuracy) once the cost drops below a threshold where $\kappa$ -separability is achieved.
- Conclusion: Markets are strictly more valuable than polls when information is cheap but not free. The market mechanism leverages the strategic interaction to achieve aggregation long before individual traders would find it profitable to acquire full information on their own.

4. Significance and Implications

Robustness of Market Design: The paper suggests that market designers do not need to know the precise information structure of participants to ensure aggregation, provided they trade securities with distinct payoffs and information acquisition costs are sufficiently low.
Role of Technology: The findings support the intuition that advancements in information technology (AI, big data) which lower the cost of acquiring and processing information ( $c \to 0$ ) will significantly enhance the efficiency of prediction markets and financial markets.
Limits of Aggregation: The paper identifies a specific, small class of securities that are inherently prone to information aggregation failure, regardless of how cheap information becomes. This highlights that not all financial instruments are equally effective at revealing truth.
Theoretical Advancement: By endogenizing the perturbation of beliefs through costly information acquisition, the authors resolve an open question from Ostrovsky [2012] regarding mixed-strategy equilibria and generic priors, showing that low-cost information acquisition effectively "genericizes" the environment, ensuring aggregation for almost all securities.

5. Conclusion

The paper demonstrates that costly information acquisition acts as a powerful mechanism for information aggregation. While the classic separability condition depends on the unknown information structure, the introduction of a cost parameter $\kappa$ allows for a classification based purely on payoff structures. For the vast majority of securities (those with distinct payoffs), making information cheap ensures that market prices converge to the true value in all equilibria, often outperforming simple polling mechanisms by a discontinuous margin. However, a specific class of securities with "flat" payoff regions remains resistant to aggregation even in the limit of zero cost.