📈 economics

No Trade Under Verifiable Information

This paper contributes to the no trade literature by analyzing conditions under which agents' private information verifies the true value of a security, offering insights into insider trading, market liquidity, and blockchain-based trading.

Original authors: Spyros Galanis

Published 2026-04-23

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Spyros Galanis

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 a bustling marketplace where people are betting on the future value of a mysterious box. Inside the box is a treasure, but no one knows exactly how much gold is inside yet. Some people have clues (private information), and some don't.

Usually, we think people trade because they have different ideas about what's in the box. One person thinks, "It's worth $100!" and another thinks, "It's only worth $10!" So, the first person buys from the second, and they both walk away happy.

But this paper asks a very specific question: What happens if the clues are so good that they actually prove the value of the box?

The author, Spyros Galanis, argues that if the information is "verifiable" in certain ways, no one will trade at all. It's like trying to sell a used car when the seller has already handed the buyer the keys, the engine manual, and a video showing the car runs perfectly. The buyer knows the true value, so they won't pay a penny more than it's worth, and the seller won't sell for less. The deal dies before it starts.

Here is a breakdown of the paper's main ideas using simple analogies:

1. The "Oracle" Problem (The Insiders)

Imagine a group of people betting on the weather.

The Scenario: One person, let's call him "The Oracle," has a magic weather app that tells him exactly what the weather will be tomorrow.
The Result: If everyone knows The Oracle has this app, no one will bet against him. If he says, "It will rain," and offers to sell you a bet that it will not rain, you know he is lying or trying to trick you. You know he knows the truth.
The Paper's Insight: The paper says that even if The Oracle doesn't know the exact amount of rain, but he knows for sure whether it will be "heavy rain" or "light rain" (a Threshold), trade still stops. If you know he knows the difference, you won't gamble.

2. The Three Types of "Proof"

The paper categorizes how much information is needed to stop trading, like different levels of security clearance:

Level 1: Full Verification (The "God Mode")
- Analogy: Someone holds the answer key to the test.
- Result: No trade. If someone knows the exact answer, no one else will bet against them.
Level 2: Max-Min Verification (The "Extreme" Check)
- Analogy: Imagine betting on a new oil well. An expert knows for sure if the well is completely dry (worth $0) or if it's gushing oil (worth $1 million). She doesn't know if it's worth $500k or $600k, but she knows the extremes.
- Result: No trade. If the expert is in the market, she will never bet on the "middle" ground because she knows the risk of the extremes.
Level 3: Threshold Verification (The "Line in the Sand")
- Analogy: A scientist knows if a new drug will cure the disease (Value > $100) or fail (Value < $10). She doesn't know the exact price, but she knows if it crosses a specific "line."
- Result: This is the paper's big discovery. Even this weaker knowledge is enough to stop trade. If you know someone knows where the "line" is, you won't bet on the other side.

3. Why "Collective" Knowledge Doesn't Stop Trade

The paper also looks at a scenario where no single person knows the answer, but if you put all their clues together, you could solve the puzzle.

Analogy: Imagine a jigsaw puzzle. Person A has the top half, Person B has the bottom half. Neither knows the full picture alone.
Result: They will trade! Because neither knows the full truth, they can still gamble on what the other person might know. The paper shows that just because the group could figure it out later, it doesn't stop them from betting now.

4. The Real-World Applications

The author connects these abstract ideas to three real-world situations:

Insider Trading (The Corporate Spy):
Laws often ban insiders from trading if they have "material" info. This paper suggests that even if an insider only knows if a company's stock will be "high" or "low" (crossing a threshold), they shouldn't trade. If they do, the market freezes because everyone else knows the insider has an unfair "line in the sand" advantage.
Market Liquidity (The Frozen Market):
Think of the 1998 financial crisis (LTCM). If market makers (the people who facilitate trading) know exactly which assets are "dead" (worthless) and which are "alive," they will stop buying the "dead" ones. If everyone knows the insiders know this, the market dries up. No one wants to buy a sinking ship if the captain knows it's sinking.
Blockchain and Oracles (The Digital Truth-Tellers):
In the world of crypto, "Oracles" are digital messengers that tell the blockchain what is happening in the real world (e.g., "Did it rain?").
- The Problem: If the Oracle is also a trader, and they know the answer before the contract executes, they will trade against everyone else.
- The Insight: If the Oracle's knowledge is "verifiable" (they know the threshold), the market for that specific bet will collapse. The paper suggests we need to design systems where Oracles can't use their knowledge to gamble, or the market won't work.

5. The Dynamic Twist (Time Matters)

The paper also looks at what happens over time.

Analogy: Imagine a game of "20 Questions."
At the start, no one knows the answer.
As people ask questions and share answers, they narrow down the possibilities.
The paper finds that even if you start with a "safe" situation where trade is possible, as people share information, the situation might change. Suddenly, someone might realize, "Oh, I know the answer is above the line!" and the trading stops instantly.

The Bottom Line

The paper teaches us that information is a double-edged sword.
While we usually think "more information = better markets," this paper shows that if information is too "verifiable" (specifically, if someone knows a specific threshold), it kills the market entirely.

If you know the other guy knows the answer, you won't play the game. And if no one plays, the market has no liquidity, and prices can't reflect the true value of things. To keep markets moving, we sometimes need a little bit of uncertainty—even if that uncertainty feels uncomfortable.

1. Problem Statement

The paper addresses a central puzzle in financial economics: Why do high trading volumes exist in markets despite the "No Trade Theorems" (e.g., Aumann, 1976; Milgrom and Stokey, 1982)?

Standard no-trade theorems suggest that if agents share a common prior and have common knowledge of their beliefs, they cannot agree to trade because any trade implies a disagreement on expected value, which rational agents would not accept. While existing literature explains trade by relaxing assumptions (e.g., allowing for different priors, bounded rationality, or ambiguity aversion), Galanis investigates a different angle: the structure of private information regarding the verifiability of the security's true value.

The core question is: Under what conditions on agents' private information does the true value of a security become "verifiable" (either by a single agent or collectively) such that no trade occurs, even when agents hold different priors? The paper explores this in the context of:

Insider Trading: When an insider knows the value (or a threshold) of a security.
Market Liquidity: How the knowledge of distressed assets by insiders might dry up liquidity.
Blockchain/Oracles: Whether oracle participants knowing the outcome of real-world events prevents trading on decentralized platforms.

2. Methodology and Model

The author employs a formal game-theoretic framework with the following specifications:

Agents & States: A finite set of agents $I$ and a finite set of states of nature $\Omega$ .
Information Structure: Each agent $i$ has a private information partition $\Pi_i$ and a prior $p_i$ . Crucially, common priors are not assumed ( $p_i \neq p_j$ is allowed).
Security: A security $X: \Omega \to \mathbb{R}$ pays a dividend based on the state.
Trade Timing: Trades are agreed upon in an interim stage (after private information is received but before the true state is revealed).

The paper analyzes three distinct trading environments to test the robustness of no-trade conditions:

Static Environment (Aumann, 1976): Agents have common knowledge of their expected values. Trade occurs if these values differ.
Dynamic Environment (Geanakoplos & Polemarchakis, 1982): Agents sequentially announce expected values, updating beliefs until they either agree or reach a steady state of disagreement.
Prediction Market Environment (Ostrovsky, 2012): Agents are myopic and trade using Market Scoring Rules (MSR). The focus is on whether prices converge to the true value (information aggregation).

3. Key Concepts: Verifiability Definitions

The paper introduces four hierarchical definitions of "verifiability" based on agents' private information:

Verifiable: At every state, some agent knows the exact true value of $X$ . (Strong condition).
Maxmin Verifiable: Among the values of $X$ that are common knowledge as possible, there exists an agent who knows whether the maximum or minimum of these possible values has occurred.
Threshold Verifiable: There exists a threshold $x$ and an agent who, at some state, knows whether the value of $X$ is strictly above or below $x$ .
Collectively Verifiable: The conjunction of all agents' private information reveals the true value (i.e., if they pooled their info, they would know $X$ ).

4. Key Results

A. Static Trading: The Necessity and Sufficiency of Threshold Verifiability

Theorem 1 is the paper's central contribution. It establishes that for any set of priors:

No Common Knowledge Trade $\iff$ Threshold Verifiability.
If a security is threshold verifiable, agents cannot agree to trade (no trade occurs).
If a security is not threshold verifiable, there exist priors such that common knowledge trade occurs.

Technical Insight: The proof shows that if threshold verifiability fails, one can construct priors where agents' expected values fall into overlapping intervals that allow for a mutually beneficial trade to be common knowledge. Conversely, if an agent knows the value is above/below a threshold, it breaks the symmetry required for common knowledge of disagreement.

B. Dynamic Trading and Information Aggregation

The paper investigates whether static no-trade conditions hold in dynamic settings where agents update beliefs over time.

Failure of Static Conditions: Maxmin or threshold verifiability at $t=0$ is insufficient to prevent trade in dynamic settings. As agents update information (e.g., by ruling out states), the "verifiable" nature of the security may disappear in subsequent periods (Example 2).
Corollary 1 (Dynamic No Trade): Agents cannot disagree forever (i.e., they eventually agree) if the security is threshold verifiable at the final period $t^*$ (when information updates cease).
Corollary 2 (Information Aggregation): In prediction markets using MSR, if the security is threshold verifiable at $t^*$ , information aggregates, and the price converges to the true value $X(\omega)$ . This holds for all securities, not just separable ones (which is a limitation in previous literature like Ostrovsky, 2021).

C. Multiple Securities

The results are extended to a collection of securities $\{X_i\}$ where $\sum X_i = 0$ .

Proposition 1: A collection of tradable securities admits no common knowledge trade if and only if the collection is threshold verifiable.

5. Significance and Applications

Insider Trading Regulation: The paper provides a theoretical basis for distinguishing between harmful and harmless insider trading. If an insider's knowledge only verifies a threshold (e.g., "the drug will succeed or fail"), trade is precluded, suggesting such trading should be banned to prevent market failure. However, if the insider only knows a range that doesn't verify a threshold, trade may still occur, potentially aiding price discovery.
Market Liquidity: It explains why markets for distressed assets (like during the LTCM crisis) might suffer from a "liquidity freeze." If market makers know that insiders possess threshold-verifiable information about asset distress, rational agents will refuse to trade, causing liquidity to dry up.
Blockchain and Oracles: In Decentralized Finance (DeFi), oracles report real-world data to smart contracts. The paper suggests that if oracles (or agents with oracle access) possess threshold-verifiable information about the outcome of an event, they effectively become "insiders." If this information is common knowledge, it could prevent trading on the blockchain for that specific event, as no rational agent would trade against an oracle that knows the outcome relative to a threshold.
Theoretical Advancement: The paper shifts the focus of no-trade theorems from priors and rationality to the structure of information partitions. It proves that threshold verifiability is the precise boundary condition for the existence of trade, offering a more nuanced understanding than previous "all-or-nothing" verifiability concepts.

Conclusion

Spyros Galanis demonstrates that the ability of agents to verify the true value of a security (specifically via threshold verifiability) is the critical factor preventing trade, even in the absence of common priors. This condition is necessary and sufficient to stop "agreeing to disagree." The findings have profound implications for regulating insider trading, understanding market liquidity crises, and designing decentralized financial systems where information asymmetry is managed by oracles.