Glosten–Milgrom Model for Prediction Markets: Adverse Selection Spread

Q: What is the Glosten–Milgrom model?

A sequential-trade model of a dealer market in which some traders are informed. A competitive, risk-neutral dealer sets the ask equal to the expected asset value given a buy and the bid equal to the expected value given a sell, which produces a positive spread even with zero profit and no inventory cost.

Q: Why is the bid-ask spread widest at 50 cents in prediction markets?

The adverse-selection spread for a binary contract is 4μp(1−p)/(1−μ²(2p−1)²), which is maximised at p = 0.5, where uncertainty about the outcome is greatest. Inventory risk, proportional to √(p(1−p)), also peaks there.

Q: How can you estimate informed trading from a prediction market spread?

In the binary Glosten–Milgrom model the zero-profit spread at 50¢ equals the share of informed traders exactly. A competitive 2¢ spread on a coin-flip contract therefore implies market makers price in at most about 2% informed flow.

AbstractAn event contract is the asset Glosten and Milgrom (1985) wrote down: it is worth exactly 0 or 1. Solving their zero-profit dealer problem for this case gives a closed-form spread,

S (p) = \frac{4 μ p ( 1 - p )}{1 - μ ^{2} ( 2 p - 1 ) ^{2}}

, which equals the informed share

μ

exactly at 50¢ and scales with

p (1 - p)

rather than its square root. Simulating 20,000 trade sequences per setting, we find that markets with few informed traders discover the truth more slowly and, in total, transfer more money from uninformed to informed traders: $9.14 per unit of size at μ = 0.05, against $0.70 at μ = 0.5.

The model, specialised to a binary payoff#

Glosten and Milgrom's sequential-trade model has a risk-neutral dealer facing a stream of traders who arrive one at a time and trade one unit. A fraction $μ$ of them know the asset's true value; the rest trade for reasons unrelated to it. Competition forces the dealer to zero expected profit, so every quote must equal the expected value of the asset conditional on the trade it would trigger.

In equities this is a stylisation: a share is not worth one of two numbers. An event contract is. It pays $V \in {0, 1}$ , and the dealer's belief before the next trade is a single number, $p = Pr (V = 1)$ . Assume:

with probability $μ$ the next trader is informed: they buy if $V = 1$ and sell if $V = 0$ ;
with probability $1 - μ$ they are uninformed and buy or sell with probability ½ each;
one unit per trade, no fees, no inventory costs, and the dealer updates by Bayes' rule after every trade.

Deriving the bid, the ask and the spread#

The probability of a buy given each state is $Pr (buy ∣ V = 1) = μ + \frac{1 - μ}{2} = \frac{1 + μ}{2}$ and $Pr (buy ∣ V = 0) = \frac{1 - μ}{2}$ . The zero-profit ask is the posterior after a buy:

a (p) = E [V ∣ buy] = \frac{p ( 1 + μ )}{p ( 1 + μ ) + ( 1 - p ) ( 1 - μ )} = \frac{p ( 1 + μ )}{1 + μ ( 2 p - 1 )}

(1)

and symmetrically the bid is

b (p) = E [V ∣ sell] = \frac{p ( 1 - μ )}{1 - μ ( 2 p - 1 )}

(2)

Subtracting, with $d = 2 p - 1$ , the numerator collapses to $2 μ p (1 - d) = 4 μ p (1 - p)$ :

S (p) = a (p) - b (p) = \frac{4 μ p ( 1 - p )}{1 - μ ^{2} ( 2 p - 1 ) ^{2}}

(3)

Two consequences are worth stating plainly.

At 50¢ the spread is the informed share. Setting $p = 1/2$ gives $S = μ$ exactly. A zero-profit dealer quoting 48/52 on a coin-flip contract is implicitly asserting that no more than 4% of takers know the answer. Read the other way, a persistently wide spread at 50¢ in a competitive market is a direct estimate of how much informed flow the market makers believe they face.

The adverse-selection spread scales with p(1−p), not its square root. For small $μ$ , $S (p) \approx 4 μ p (1 - p)$ . The risk of holding the contract, by contrast, has standard deviation $p (1 - p)$ . Toward the tails the adverse-selection component shrinks faster than the inventory-risk component, which is one reason inventory management, not information, dominates quoting decisions on contracts priced below 10¢ or above 90¢.

The spread surface in numbers#

Table 1. Zero-profit Glosten–Milgrom spread (ask − bid, in cents) for a binary contract, by current price p and informed share μ. Computed exactly from equation (3).
Price	μ = 0.05	μ = 0.1	μ = 0.2	μ = 0.3	μ = 0.5
2¢	0.39¢	0.79¢	1.63¢	2.56¢	5.09¢
5¢	0.95¢	1.92¢	3.93¢	6.15¢	11.91¢
10¢	1.80¢	3.62¢	7.39¢	11.46¢	21.43¢
20¢	3.20¢	6.42¢	12.99¢	19.84¢	35.16¢
30¢	4.20¢	8.41¢	16.91¢	25.57¢	43.75¢
40¢	4.80¢	9.60¢	19.23¢	28.90¢	48.48¢
50¢	5.00¢	10.00¢	20.00¢	30.00¢	50.00¢
60¢	4.80¢	9.60¢	19.23¢	28.90¢	48.48¢
70¢	4.20¢	8.41¢	16.91¢	25.57¢	43.75¢
80¢	3.20¢	6.42¢	12.99¢	19.84¢	35.16¢
90¢	1.80¢	3.62¢	7.39¢	11.46¢	21.43¢
95¢	0.95¢	1.92¢	3.93¢	6.15¢	11.91¢
98¢	0.39¢	0.79¢	1.63¢	2.56¢	5.09¢

With 10% informed takers the break-even spread is 10¢ at the midpoint but under 2¢ at 5¢ and 95¢. A market that shows 2¢ spreads across the whole range is either facing much less informed flow than 10%, or its market makers are being compensated some other way: rebates, incentive programmes, or edge on their own fair value.

μ = 0.1μ = 0.2μ = 0.3μ = 0.5

Figure 1. Equation (3) across the price range. The curves are not parabolas: the denominator widens them slightly toward the tails when μ is large.

Price discovery, simulated#

The quotes above are for a single trade. The more interesting question is dynamic: starting from $p_{0} = 0.5$ , how many trades does it take for the market to learn the outcome, and who pays for that learning? We simulated 20,000 independent trade sequences for each informed share, drawing the true outcome by a fair coin, and tracked the dealer's posterior and quotes trade by trade.

Table 2. Simulated price discovery in the binary Glosten–Milgrom model (20,000 paths per row, seed 20261030). Losses are per unit of trade size; uninformed losses equal informed gains because the dealer breaks even.
Informed share μ	Median trades to spread < 1¢	90th percentile	Uninformed loss per uninformed trade	Total uninformed loss until spread < 1¢
0.05	413	1017	1.86¢	$9.14
0.1	149	339	3.24¢	$5.29
0.2	45	97	5.88¢	$2.52
0.3	22	46	8.36¢	$1.55
0.5	10	20	11.70¢	$0.70

μ = 0.05μ = 0.1μ = 0.2μ = 0.5

Figure 2. Mean quoted spread across 20,000 paths as trades arrive. High informed share: wide initial spread, fast collapse. Low informed share: narrow spread that stays open for hundreds of trades.

The result that surprises most people is in the last column. When only 5% of takers are informed, each uninformed trade costs less (1.86¢ against 11.7¢ at μ = 0.5), but discovery takes about forty times as many trades, and in aggregate the uninformed hand over thirteen times as much money before the spread closes. A thinly informed market is cheap per trade and expensive in total.

What it means for venues and market makers#

Spreads at 50¢ are a diagnostic. Because $S (1/2) = μ$ , the midpoint spread on liquid coin-flip contracts is an upper bound on the informed share that market makers price in. Comparing it across categories (macro releases versus long-dated politics, for example) is a cheap way to see where professional liquidity providers believe the informed flow is.

Tail spreads should be tight, and are. Equation (3) implies that a contract at 3¢ needs roughly an eighth of the adverse-selection spread of the same contract at 50¢. If tail spreads are nonetheless wide, the cause is usually inventory risk or capital lock-up, not information.

Speed of discovery is a design choice. Venues that attract informed participants early, by listing with clear resolution sources and allowing larger size, buy faster convergence. The simulation says the alternative is not "fairer" to retail participants: it is slower and costs them more in aggregate.

Liquidity incentives subsidise the uninformed. A maker rebate lets market makers quote inside the zero-profit spread. In this model that transfer flows directly to uninformed takers, which is a reasonable economic justification for liquidity programmes on new markets.

Assumptions that matter#

The model is deliberately austere, and three of its assumptions bias the numbers in known directions. Informed traders here know $V$ exactly; real informed flow knows something partial (a better nowcast, a faster read of a release), which lowers effective $μ$ and slows discovery further. There is no news: in a real event market public information arrives continuously and does much of the work that trades do here. And trades are of unit size; when informed traders can choose size, the dealer must also learn from quantities, which is the territory of Kyle (1985).

None of this changes the two structural results: the spread at 50¢ equals the informed share for a zero-profit dealer, and adverse selection scales with $p (1 - p)$ .

# research/glosten_milgrom.py (excerpt)
def ask(p, mu):
    return p * (1 + mu) / (1 + mu * (2 * p - 1))

def bid(p, mu):
    return p * (1 - mu) / (1 - mu * (2 * p - 1))

# spread(0.5, mu) == mu for every mu
assert abs((ask(.5, .1) - bid(.5, .1)) - .1) < 1e-12

Questions#

What is the Glosten–Milgrom model?

A sequential-trade model of a dealer market in which some traders are informed. A competitive, risk-neutral dealer sets the ask equal to the expected asset value given a buy and the bid equal to the expected value given a sell, which produces a positive spread even with zero profit and no inventory cost.

Why is the bid-ask spread widest at 50 cents in prediction markets?

The adverse-selection spread for a binary contract is 4μp(1−p)/(1−μ²(2p−1)²), which is maximised at p = 0.5, where uncertainty about the outcome is greatest. Inventory risk, proportional to √(p(1−p)), also peaks there.

How can you estimate informed trading from a prediction market spread?

In the binary Glosten–Milgrom model the zero-profit spread at 50¢ equals the share of informed traders exactly. A competitive 2¢ spread on a coin-flip contract therefore implies market makers price in at most about 2% informed flow.

References#

Glosten, L. R. and Milgrom, P. R. (1985). Bid, ask and transaction prices in a specialist market with heterogeneously informed traders. Journal of Financial Economics, 14(1), 71–100.
Kyle, A. S. (1985). Continuous auctions and insider trading. Econometrica, 53(6), 1315–1335.
O'Hara, M. (1995). Market Microstructure Theory. Blackwell.

DisclosureThis note is research and education. It is not investment advice or an offer to trade, and nothing in it describes the firm's own results. Code and outputs: research/glosten_milgrom.py in our research repository, reproducible with a fixed seed.