The Minority Game
An odd number of traders. Every round, each one picks a side, and whoever lands on the smaller side wins. They cannot talk to each other. All they share is a public record of which side won the last few rounds, and that record is far too short to say anything useful. They coordinate anyway.
Played by coin-flippers this game wastes a definite amount, and you can work out how much on paper. Played by traders who adapt, it can waste six times less than that, or ten times more. Which one happens is decided by a single number, and the switch between them is a phase transition with a location you can measure to three decimal places. Nothing in the game knows that number, and no trader is trying to put the market on either side of it.
On this site a basin is the set of conditions a system comes back from. Push a marble around the inside of a bowl and it rolls back to the bottom; the bowl is its basin, and the rim is its edge, the push beyond which it does not come back. Finding basins, and their edges, is the point of the Laboratory. This panel has one basin, the coordinated state, and one number decides whether the crowd is in it: the information per trader, α = 2M / N, where M is how many past winners the public record holds and N is the number of traders.
| basin | what it returns to | what pushes it | its edge |
|---|---|---|---|
| 1. The coordinated state | waste well below coin flipping, about six times lower near the transition | new random strategies; a different M or N at the same α | α pushed down into crowding; above the transition, a slow drift back toward coin flipping |
1. The coordinated state. The waste, σ² / N, measures how unevenly the crowd splits, per trader per round. Traders flipping coins set it at 1 (simulated at 1.003, 0.999 and 1.005). Adaptive traders bring it down to between 0.15 and 0.17 at α = 0.45, and they do it near the phase transition, the point where the predictability (H / N, how much the public record tells you about the next imbalance) leaves zero. That point measures at 0.338 ± 0.007, against the published 0.3374. What pushes it: press Fresh strategies to deal every trader new random strategies, and the crowd settles again at the same α. Five repeats of the panel's M = 7 sweep put the transition at a mean of 0.335 with a spread of 0.013. Changing M and N together also leaves it in place: at M = 5, 6 and 7 the waste is the same at the same α. The edge: push α down and the traders read the same short record and crowd onto the same side. The waste is still 0.67 to 0.76 at α = 0.25, about even with coin flipping (1.19 to 1.32) at 0.18, and 3.55 to 4.10 at 0.08. Push α up and the loss is gradual: 0.29 to 0.36 at 1.20 and 0.39 to 0.57 at 2.60, heading back toward coin flipping as the traders' strategies stop overlapping.
The return is real, and the traders' own scoring makes it. The traders themselves are the supply: cut them off and there is no game and no coordination.
Traders Who Cannot Talk
The M = 6 estimate on this budget is unreliable and is left on screen anyway. Three seeds and fifteen thousand rounds per point is what fits in a few seconds. At M = 7 that gives 0.335 with a spread of 0.013 across repeats. At M = 6 it gives 0.29 with a spread of 0.085, and one repeat in five collapses to the bottom of the scan. Reload and watch the two readouts disagree: that is what an estimator looks like when the system is too small for it.
The shaded band in the top plot is not fitted either. It is one standard deviation of a sum of N coin flips, which is the square root of N, and it is the whole of what this market would do if nobody adapted.
The Whole Game
Scoring is the only adaptive part, and it happens without anybody meeting anybody. After each round every strategy is scored on what it would have done, whether or not it was the one played. A strategy that would have been in the minority gains, one that would have been in the majority loses. Over time each trader drifts toward whichever of its two hands has been reading the crowd better.
There is no equilibrium here that anyone is solving for, no price, no payoff to being early, and no way to signal. Two traders holding the same strategy will always do the same thing and will never find out that the other exists.
One Number Runs It
A record M rounds long has P = 2M possible states. The control parameter is the amount of public information per trader:
The thing to measure is the waste. If σ² is the variance of the imbalance between the two sides, then σ² / N is what the crowd loses per trader per round, and coin-flippers set the scale: a sum of N independent coin flips has variance exactly N, so σ² / N = 1 is the no-thinking benchmark. Simulating coin-flippers at N of 101, 377 and 753 gives 1.003, 0.999 and 1.005.
Against that:
| α | M = 5 | M = 6 | M = 7 | what it means |
|---|---|---|---|---|
| 0.08 | 4.10 | 3.55 | 3.58 | crowded, far worse than coin flips |
| 0.12 | 2.31 | 2.13 | 2.20 | still worse |
| 0.18 | 1.32 | 1.19 | 1.25 | about even with coin flips |
| 0.25 | 0.74 | 0.76 | 0.67 | better than coin flips |
| 0.45 | 0.17 | 0.15 | 0.17 | six times better |
| 1.20 | 0.35 | 0.29 | 0.36 | drifting back |
| 2.60 | 0.39 | 0.53 | 0.57 | heading for coin flips again |
Read down each row. M = 5 and M = 7 mean four times as many traders and four times as many possible records, and they give the same waste at the same α. The three curves lie on top of each other. This is the finding that made the game famous: the memory length and the population size do not matter separately, only their ratio does.
Read across instead and there are two distinct failures either side of a sweet spot. Too little information per trader and everyone is reading the same short record and crowding onto the same side, which is four times worse than not thinking at all. Too much and the record is so long that no two traders are ever looking at the same thing, their strategies decorrelate, and they drift back toward independent coin flips.
Where the Sweet Spot Actually Is
The obvious move is to find the minimum of that waste curve and call it the transition. It does not work. The minimum is broad and shallow, so the lowest point of a set of noisy measurements lands wherever the noise puts it. Sweeping eleven settings at each of M = 5, 6, 7 and 8 and taking the location of the minimum gives 0.42, 0.42, 0.38 and 0.47, with no sign of settling anywhere.
The sharp quantity is the predictability: the mean squared imbalance conditional on the public record.
If H is zero, then knowing the last M winners tells you nothing about which way the imbalance will go. The crowd has eaten every pattern in its own history. If H is positive, a pattern survives, and somebody watching could trade on it. The theory says H is exactly zero below the transition and positive above it, which makes it an order parameter rather than a curve with a bend in it:
| α | H/N at M = 6 | H/N at M = 7 | H/N at M = 8 |
|---|---|---|---|
| 0.300 | 0.00002 | −0.00002 | −0.00007 |
| 0.315 | 0.00032 | 0.00005 | −0.00003 |
| 0.325 | 0.00088 | 0.00014 | 0.00015 |
| 0.337 | 0.00243 | 0.00108 | 0.00047 |
| 0.355 | 0.00473 | 0.00207 | 0.00163 |
| 0.370 | 0.00552 | 0.00583 | 0.00386 |
| 0.390 | 0.00897 | 0.00858 | 0.00687 |
Zero to four decimal places on the left, positive and climbing on the right. The point where it leaves zero moves right as the system grows: at M = 6 it has already left by α = 0.315, at M = 8 it is still at zero there.
Putting a Number On It
Above the transition H grows as a power of the distance from it. The panel does not know the transition or the exponent, so it scans for the pair that makes the growth straightest on log-log axes:
| memory M | traders at the transition | fitted αc | exponent | R² | two half-samples |
|---|---|---|---|---|---|
| 5 | 95 | 0.2565 | 1.60 | 0.988 | 0.150, 0.314 |
| 6 | 189 | 0.3000 | 1.70 | 0.994 | 0.355, 0.266 |
| 7 | 379 | 0.3365 | 1.39 | 0.997 | 0.342, 0.329 |
| 8 | 759 | 0.3405 | 1.33 | 0.998 | 0.343, 0.338 |
The last column is the estimate repeated on two disjoint halves of the random seeds, and it is the honest measure of whether the number means anything. At M = 5 the two halves disagree by 0.16, which is to say the estimate is worthless. By M = 8 they agree to 0.005.
Taking the four half-sample estimates from M = 7 and M = 8, the ones that are stable enough to quote:
Agreement to better than one percent, on a quantity nothing in the simulation was told. The exponent comes out near 1.33 and stabilises as the system grows, which the panel reports but does not claim: this page has no argument for that number and does not check it against anything.
That table is ten random seeds and thirty thousand rounds per point, which takes minutes. The panel runs three seeds and fifteen thousand rounds, in about twelve seconds, and lands in the same place with more scatter. Five independent repeats of the panel’s own M = 7 sweep gave 0.3455, 0.3345, 0.3190, 0.3505 and 0.3265, a mean of 0.335 with a spread of 0.013. The same repeats at M = 6 gave 0.3535, 0.3425, 0.1500, 0.3395 and 0.2835, which is not a measurement of anything. Both readouts are on the panel, so the difference between an estimator that works and one that does not is visible rather than asserted.
Why This Is an Attractlet
The coordinated state is real. It has a size, it beats the no-thinking benchmark by a factor of six, it sits at a definite location, and it persists. It is also completely unintended. Every trader is trying to beat the others and none of them is trying to reduce the waste; the waste is what is left over after they have all finished trying.
What makes this the sharpest case in the set is that the structure has a location you can measure and check against somebody else's algebra. Most of the panels on this site show that a pattern appears. This one shows a pattern appearing at a specific coordinate, predicted in advance by a calculation that knew nothing about this implementation.
The reading is the same as the Order Book next door, and firmer. Cut off the traders and there is no game and no coordination. Nothing here produces its own supply. But unlike the order book, this structure is not a statistic of the input: it is a genuine collective state that the population settles into, and settles out of again when you move α across a line none of them can see.
What This Model Is Not
There is no price here. The minority game is not a market model. It is a coordination model, and treating the imbalance as a price is a step this page does not take.
Real traders have more than two strategies and can change them. Both matter. The published transition at 0.3374 is specifically for two fixed strategies per agent; with more, it moves.
The record here is the real history. Feeding the agents random records instead of the true one changes very little about σ², which is a known and slightly uncomfortable result, and this page does not run that variant.
The exponent is reported, not claimed. See above.
Everything is finite. The transition is a statement about the limit of infinitely many traders. What the panel measures is a sequence of finite systems, which is why the M = 5 row of that table is in it: an unstable estimate left visible is worth more than a clean one with the failures deleted.
Challet, D. & Zhang, Y.-C. (1997). Emergence of cooperation and organization in an
evolutionary game. Physica A, 246(3-4), 407-418. The game.
Savit, R., Manuca, R. & Riolo, R. (1999). Adaptive competition, market efficiency, and phase
transitions. Physical Review Letters, 82(10), 2203-2206. The collapse onto
α = 2M/N.
Challet, D. & Marsili, M. (1999). Phase transition and symmetry breaking in the minority game.
Physical Review E, 60(6), R6271. Fetched and read 2026-09-13; source of the order-parameter
treatment and of the statement that the transition sits near 0.34.
De Martino, A. & Marsili, M. (2001). Replica symmetry breaking in the minority game.
Journal of Physics A, 34(12), 2525. Fetched and read 2026-09-13; source of the value
αc ≃ 0.3374 and of the statement that it separates a symmetric phase
with H = 0 from an asymmetric one with H > 0.