Twenty Traders, One of Them Informed, and the Nickel That Empties the Book

A closed market has twenty traders in it. The regulator announces that one of them holds inside information about a stock, and does not say which one. What happens to the volume?

News arriving usually means activity, so the reflex is up. It goes down, and the mechanism is worth more than the answer: nobody has left the market, nobody has learned anything about the stock, and yet every trade in the room has become worse for nineteen out of twenty participants.

Stand on the other side of the trade

The question is easy to answer and impossible to feel from the point of view of a trader. Move to the point of view of whoever is quoting. Let the stock be worth $V = +1$ or $V = -1$, each with probability a half, so the range is two dollars and the mid is zero. Exactly one of the $N = 20$ traders knows $V$. An order is as likely to come from any one of them, so

The informed trader buys when the stock is worth $+1$ and sells when it is worth $-1$. The other nineteen have reasons of their own, none of them related to $V$, so treat them as buying or selling with equal chance.

A buy order is weak evidence, and weak is enough

Count the twenty possible orders in each state instead of writing Bayes' rule, because the counting is what makes it obvious. If the stock is worth $+1$, the informed order is a buy and the other nineteen split evenly: 10.5 buys against 9.5 sells. If it is worth $-1$, the tilt runs the other way. So buys outnumber sells by exactly one order in twenty, and the likelihood ratio of a buy across the two states is $(1+\alpha) : (1-\alpha)$. Hence

The middle step simplifies for every $\alpha$, so the shape of the answer is not an artefact of there being twenty traders. And 52.5 percent is a feeble signal. Anyone would call it noise. It is still enough to move a price, because a market maker who ignores it is offering a free option to whoever knows better.

The honest quote, and where the spread comes from

Applying that to (2),

so the ask sits five cents above the mid, the bid five cents below it by symmetry, and the spread is

ten cents, or five percent of the two-dollar range. Notice what produced it. No inventory, no fees, no risk aversion, no processing cost. A spread appeared out of nothing but the possibility that the person hitting you knows something, which is the reason spreads exist at all in markets where the other three explanations are negligible.

Check the maker's book at that quote and it is flat: the nineteen-in-twenty chance of collecting five cents from an uninformed trader exactly pays for the one-in-twenty chance of losing 95 cents to the informed one. Quote the mid instead and the loss is precisely $-\alpha$ per order, measured at $-0.0518$ over two million simulated orders. The spread is not a profit margin. It is the exact size of the leak.

The nickel has to come from somewhere

Follow the money. An uninformed trader buying at the ask pays $\alpha$ for something whose expected value, conditional on nothing they know, is zero, so their expected loss is five cents per trade. The informed trader buys at the ask when the stock is worth $+1$ and makes $1 - \alpha = 0.95$. With nineteen losers and one winner,

and the books close to the penny. That is not a coincidence at twenty. In general each of the $N-1$ uninformed traders pays

so the per-trade loss to each uninformed trader always equals the probability that any given order is informed. The insider's profit and the crowd's losses are two views of one transfer, and the market maker is a conduit rather than a participant in it.

Volume falls, and this part is not a theorem

Here I want to be careful, because the honest version of this answer is weaker than the confident version and more useful. The model above produces a spread. It does not produce a quantity of trades, because the uninformed order flow was assumed into existence rather than derived from anybody's decision. Nothing in the algebra says how much of it survives.

What the algebra does say is that every uninformed trade now has an expected value of minus five cents, measured at $-0.0482$in the simulation. A trader whose reason for trading is a liquidity need will still trade, because five cents is cheaper than not rebalancing. A trader who was near indifferent now has a reason to wait. So volume falls because the marginal trade stopped being free, which is a comparative static rather than a result, and it is the answer the question is looking for. The extreme version, where every uninformed trader withdraws and the market closes entirely, needs the traders' motives specified before it can be argued either way.

Naming the insider would have helped

This is the sharpest thing in the problem and it is easy to miss. Suppose the regulator had named the trader. Then the other nineteen simply refuse to be that person's counterparty, $\alpha$ drops to zero, equation (4) gives a spread of zero, and the market carries on exactly as before. All the damage comes from the anonymity, not from the information.

Announcing that somebody unnamed is informed is therefore the worst available action: it creates the full adverse selection cost without giving anybody the means to avoid it. A regulator who cannot name the trader is better off saying nothing, and one who can name them should.

What the size of the crowd does

Equation (4) is a hyperbola, so the harm from one insider is $2/N$ and it dies quickly. In a room of 500 the spread is four tenths of a cent on a two-dollar range, invisible next to any real trading friction. Going the other way it gets brutal fast. At $N = 2$ the ask is $+0.50$ and the bid is $-0.50$, so the spread is half the entire range and an uninformed trade costs fifty cents. Push $\alpha$ toward one, which this model cannot reach but a richer one can, and the spread reaches the full range: the ask is the best case and the bid is the worst, and no price exists at which an uninformed trader should deal. That limit is the no-trade result, and twenty traders is already on the uncomfortable part of the curve.

Two assumptions do quiet work throughout. The informed trader gets one order, so the model never asks how somebody with private information would ration their trading to avoid revealing it, which is what the strategic versions of this problem are about. And the stock takes two values, which keeps the posterior to a single line.

Sources and further reading

• The mechanism the whole article is an instance of — Adverse selection
• What equation (4) is a model of — Bid-ask spread
• The field this sequential-quote setup belongs to — Market microstructure
• The step in (2) — Bayes' theorem
• The reasoning style of the last section — Comparative statics

The model is small enough to enumerate, so everything above was computed in exact fractions rather than sampled: the posterior $21/40$, the ask $1/20$, the spread $1/10$, and the transfer identity in (6) checked for every $N$ from 2 to 500. The spread was verified to equal $2\alpha$ on a grid of $\alpha$ values, to vanish at $\alpha = 0$ and to be strictly increasing. A two-million-order simulation then measured the posterior at $0.52607$, the maker's profit at the honest quote at $-0.0018$, which is zero inside four standard errors, and the two traders' books at $-0.0482$ and $+0.95000$.