📈 economics

Revealing information -- or not -- in a social network of traders

This paper demonstrates that in a social network of traders, a privately informed agent may strategically choose to share information with positive probability, leading to equilibrium prices that are not fully revealing and significantly altering the distribution of social surplus, even when communication links are endogenously formed.

Original authors: Patrick Allmis, Paolo Pin, Fernando Vega Redondo

Published 2026-06-10

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Patrick Allmis, Paolo Pin, Fernando Vega Redondo

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 group of high-stakes gamblers sitting around a table, betting on the future price of a mysterious box. Inside the box is an asset that will be worth a certain amount tomorrow, but no one knows exactly what that amount is yet.

In this paper, the authors set up a game to figure out: If one person secretly finds out the true value of the box, will they tell their friends at the table, or keep it to themselves?

Here is the story of what they found, explained simply.

The Setup: The "Secret Signal"

Think of the market as a crowded room.

The Box: An asset (like a government bond) with a future value that is currently a mystery.
The Crowd: There are many small, confused people who buy and sell based on general trends.
The Insiders: A small group of smart, strategic traders who know how to play the game.
The Secret: Nature (the universe) secretly whispers the exact future value of the box to one of these insiders.

The big question is: Does this insider shout the secret to their friends (who are also insiders), or do they stay silent to make a bigger profit alone?

The Big Surprise: "Good News" is a Lie, "Bad News" is a Secret

Conventional wisdom suggests that if you have a winning secret, you keep it. If you know a stock is going to skyrocket, you buy it all yourself and don't tell anyone, so you can make all the money.

The paper finds the opposite is true in many cases.

The insider will actually tell their friends the truth, but only if the news is "just right" (moderate).

If the news is AMAZING (The box is worth a fortune): The insider stays silent. If they told everyone, everyone would rush to buy, driving the price up before the insider can buy in. The insider wants to buy cheap, so they hide the good news.
If the news is TERRIBLE (The box is worthless): The insider stays silent. If they told everyone, everyone would panic and sell, crashing the price. The insider might want to sell short (bet against it), but if everyone knows it's bad, the price drops too fast, and the insider can't get a good deal.
If the news is "Meh" (The box is worth a decent, average amount): The insider tells everyone. Why? Because if everyone knows it's just "okay," they won't get too greedy or too scared. They will trade calmly. This keeps the price stable and low, which helps the insider make a steady profit.

The Analogy: Imagine you are a chef selling a special soup.

If you know the soup is perfect, you don't tell anyone, or the line will get too long and you'll run out of ingredients before you can sell your own portion.
If you know the soup is poisonous, you don't tell anyone, or the whole restaurant will close down.
If you know the soup is just okay, you tell your friends. They won't fight over it, and you can sell your portion at a fair price without a panic.

The Result: The Price Hides the Truth

In many financial theories, the price of an asset is supposed to be a "truth-teller." If you look at the price, you should be able to figure out what the asset is actually worth.

This paper says: No, not anymore.

Because the insider is selectively sharing information (telling the truth only when it's "meh"), the price becomes confusing.

If the price is low, is it because the asset is actually worthless? Or is it because the insider knew the value was "meh," told everyone, and everyone traded calmly?
An outsider looking at the price cannot tell the difference. The market is no longer "efficient" at revealing the truth because the insider is strategically hiding the extremes.

The "Friendship Network" (Who talks to whom?)

The paper also asks: How do these traders decide who to be friends with?

They imagine a game where traders build their own social network before the game starts. They want to connect to people who are likely to get the secret first.

The most "lucky" trader (the one most likely to get the secret) wants to connect to as many people as possible, but not too many, or they lose their advantage.
The result? The network forms into cliques (tight-knit groups where everyone talks to everyone).
The "best" traders (those most likely to get the secret) form a clique with each other. The "less lucky" traders form their own separate cliques. It's like a high school where the popular kids hang out together, and the less popular kids hang out together, because they all want to maximize their chances of getting the inside scoop.

The Bottom Line

Traders don't always hide secrets. Sometimes, sharing "average" news is actually the best way to make money because it calms down the competition.
Prices lie. Because traders only share "average" news and hide "extreme" news, the market price stops being a perfect mirror of reality. You can't always tell what an asset is worth just by looking at its price.
Friendships matter. Traders naturally group themselves into tight circles based on how likely they are to get information first.

In short, in a world of strategic traders, silence is golden for extreme news, but sharing is caring for boring news. And because of this, the market price becomes a puzzle rather than a clear answer.

Technical Summary: Revealing Information – or not – in a social network of traders

Problem Statement
This paper investigates the strategic incentives for information sharing among traders in a financial market characterized by asymmetric information. While the literature on rational expectations (e.g., Grossman and Stiglitz, 1980; Kyle, 1985) has extensively analyzed how equilibrium prices aggregate dispersed private information, it typically assumes that agents rely solely on prices to infer others' information, ruling out pre-trade communication. Conversely, the literature on oligopoly (e.g., Vives, 1984; Gal-Or, 1985) has studied information sharing among firms, often finding that firms withhold information to maintain competitive advantages.

The authors address a gap in the financial markets literature by asking: In a market with strategic traders, does an agent privately informed of an asset's future return have an incentive to share that information with neighbors in a social network? The paper challenges the conventional wisdom that private information is always too valuable to share, exploring conditions under which revelation occurs and its implications for market efficiency and surplus distribution.

Methodology
The authors construct a micro-founded, multi-period game based on the Kyle (1985) framework, modified to include endogenous information sharing via a social network.

Market Structure: The model features a secondary market for an asset with a fixed supply $Q$ and a random ex-post return $\tilde{V}$ . The market includes:
- A finite set of $N$ strategic traders.
- A continuum of non-strategic (noise) traders with a linear demand function $H - KP$.
- The market clears such that the price $P$ is determined by aggregate demand.
Information and Network:
- Before trading, Nature selects at most one trader $i$ to receive a perfectly revealing signal $r$ regarding the asset's return (specifically, a transformed variable $\tilde{r} = K\tilde{V} - H + Q$ ).
- Traders are connected via a directed social network $\Gamma$ . If trader $i$ is informed, she can choose to reveal $r$ to all her out-neighbors or to none.
- The model assumes that if $i$ is informed, all other traders observe that $i$ has been informed, even if they do not observe the content of the signal.
Game Sequence:
- Stage 1 (Information): Nature informs trader $i$ . Trader $i$ decides whether to reveal $r$ to her out-neighbors.
- Stage 2 (Trading): All strategic traders simultaneously choose their net positions ( $x_j$ ) based on their information sets (whether they are the informed agent, an informed neighbor, or uninformed).
- Payoffs: Profits are determined by $\pi_j = x_j(V - P)$ .
Equilibrium Concept: The authors analyze the Perfect Bayesian Equilibrium (PBE) of the full game. They also extend the analysis to an endogenous network formation game using the concept of Bilateral Equilibrium (BE), where links are formed through mutual consent prior to the trading stage.

Key Contributions and Results

1. Existence of Partial Revelation
Contrary to the intuition that private information is always withheld, the paper demonstrates that in the unique PBE, an informed trader reveals her information with positive probability.

The Revelation Interval: Revelation occurs if and only if the realized signal $r$ falls within a specific intermediate interval $(a_i, b_i)$ .
Extreme Signals are Hidden: If the signal is "too good" ( $r > b_i$ $r > b_{i}$ ) or "too bad" ( $r < a_i$ $r < a_{i}$ ), the informed trader conceals it.
- High signals: Revealing good news would stimulate aggressive buying by neighbors, driving up the price and reducing the informed trader's profit.
- Low signals: Revealing very bad news would cause neighbors to short aggressively, driving the price down so significantly that the informed trader's profit suffers disproportionately.
Intermediate Signals: When the signal is moderate, the benefit of mitigating competition (by aligning neighbors' expectations and reducing their aggressive trading) outweighs the cost of information leakage.

2. Market Efficiency and Price Revelation
The paper establishes that the possibility of strategic revelation destroys informational efficiency, even in a setting where prices would otherwise be fully revealing (as in the standard Kyle benchmark with common knowledge of non-strategic demand).

Non-Revealing Prices: Because the revelation decision is endogenous and depends on $r$ , the mapping from the signal $r$ to the equilibrium price is not injective.
Discontinuity: A small increase in $r$ that crosses the threshold into the revelation interval causes a discontinuous drop in the price. This occurs because the informed trader switches from concealing to revealing, causing her neighbors to drastically reduce their trading aggression.
Observational Ambiguity: An outside observer (or uninformed trader) observing a specific price cannot uniquely determine the underlying signal $r$ , as the same price could result from a low signal (concealed) or a moderate signal (revealed).

3. Distributional Implications
The game is ex-post zero-sum. The paper finds that whenever information is revealed, the equilibrium price falls. Consequently, the original owners of the asset (the sellers) experience a payoff loss, while the strategic traders (buyers) benefit. The non-strategic sector's demand is fixed, so the surplus redistribution occurs between the strategic traders and the asset owners.

4. Endogenous Network Formation
In the final section, the authors study how the communication network forms when links are determined by bilateral agreement.

Clique Structure: Under Bilateral Equilibrium, the unique network structure consists of cliques (completely connected sub-networks).
Probability Ordering: Traders are partitioned into cliques based on their ex-ante probability of being informed ( $p_i$ ). Traders with higher probabilities of being informed form cliques with other high-probability traders.
Mechanism: A trader with a high probability of being informed seeks to connect with others to increase the chance of receiving information (indirectly). However, they limit the size of their clique to avoid excessive information leakage when they are the ones who become informed. This trade-off leads to a stable configuration of disjoint cliques.

Significance and Claims
The paper claims to provide a novel synthesis of financial market theory and oligopoly theory regarding information sharing:

Contrast with Oligopoly: In standard Cournot oligopoly models with homogeneous goods and common-value uncertainty, firms typically do not share information. The authors show that in their financial trading model, sharing does occur with positive probability. The divergence arises because information revelation in their model creates a negative externality for the seller (the informed trader) but a positive externality for the buyers (the neighbors), whereas in Cournot models, revelation typically reduces uncertainty and increases aggregate output, harming all firms.
Contrast with Rational Expectations: The paper demonstrates that the mere possibility of strategic communication is sufficient to render equilibrium prices non-revealing, even if the underlying market structure (without communication) would support full revelation.
Robustness: The results on partial revelation and market inefficiency hold even when the model is generalized to allow traders to reveal information to subsets of neighbors or when information is relayed sequentially through the network.

The authors conclude that strategic information sharing in financial markets is a nuanced phenomenon driven by the trade-off between reducing competition and avoiding information leakage, leading to equilibrium outcomes where prices fail to fully aggregate private information.