demo · you against 50 subjects
Ten Little Treasures
In 2001 Jacob Goeree and Charles Holt ran a set of games chosen so that game theory's predictions were either exactly right or spectacularly wrong, often between two versions of the same game. This page puts you in front of one of them before you see the answer. Every experimental number here is theirs.
One payoff changes. Nash does not move.
A two-by-two game. The Row player's mixing probability in Nash equilibrium is fixed entirely by the Column player's payoffs — and those are identical in all three versions below. So Nash says Row plays Top exactly half the time, whatever happens to Row's own numbers.
| Row action | Left | Right |
|---|---|---|
| Top | 0.80, 0.40 | 0.40, 0.80 |
| Bottom | 0.40, 0.80 | 0.80, 0.40 |
In the basic game the subjects obliged: 48% chose Top against a Nash prediction of 50%. That is the treasure. The contradiction is on the next widget.
Show the maths
Row's equilibrium probability p solves Column's indifference: p(0.40 − 0.80) + (1 − p)(0.80 − 0.40) = 0, so p = 0.5 in every treatment, because Column's four payoffs never change. Column's equilibrium probability q solves Row's indifference and therefore does move: q = 0.5, 0.125 and 0.909 in the three treatments.
Source: Goeree & Holt (2001), “Ten Little Treasures of Game Theory and Ten Intuitive Contradictions”, American Economic Review 91(5), 1402–1422, matching-pennies treatments. 50 subjects in five cohorts of ten, one-shot with random pairing. Every experimental number on this page is from that paper; none is simulated, illustrative or reconstructed.
Guess what the subjects did
Now Row's payoff in the top-left cell is raised from $0.80 to $3.20. Column's payoffs are untouched, so the Nash prediction for Row is still exactly 50%. Drag the bar to where you think the actual subjects landed, then lock it in.
| Row action | Left | Right |
|---|---|---|
| Top | 3.20, 0.40 (the changed cell) | 0.40, 0.80 |
| Bottom | 0.40, 0.80 | 0.80, 0.40 |
Your guess is not scored against theory. It is scored against 50 people who actually played this game.
Show the maths
Observed Row choices: 48% Top in the basic game, 96% Top with the $3.20 cell, 8% Top with the $0.44 cell. Column choices moved in the direction Nash predicts for Column — 48%, 16%, 80% Left against Nash values of 50%, 12.5%, 90.9% — which is why the paper calls the Row result an own-payoff effect rather than confusion.
Goeree & Holt (2001), AER 91(5), 1402–1422. n = 50 subjects, one-shot, random pairing, payoffs in dollars exactly as printed above.
One dial moves the prediction Nash cannot move
Give the players a precision λ instead of perfect maximisation and the equilibrium condition changes shape: Row's own payoffs now enter Row's own behaviour. Drag the λ line and watch the three predictions separate from the flat Nash line at 50%.
Solid is the $3.20 treatment, long-dashed the basic game, short-dashed the $0.44 treatment. Each treatment's finely-dotted horizontal line is what the subjects actually did; the heavy dashed line across the middle is Nash, which is 50% in all three at every λ.
At λ = 3.00 the model puts Row on Top 85% of the time in the $3.20 treatment, closing 75% of the distance from Nash to the data. Nash closes none of it, at any parameter value, because it has none to spend.
Show the maths
The curves are the logit equilibrium of each 2×2 game, solved by bisection on the column player's probability rather than by damped iteration, because matching pennies cycles under iteration at every λ worth plotting. Payoffs are in dollars, so λ carries units of 1/dollar.
Least squares over the three Row frequencies puts the best fit at λ = 4.79, giving predicted Top of 50% / 82% / 36% against observed 48% / 96% / 8%. Logit alone therefore captures the direction and roughly two-thirds of the $3.20 effect and about a third of the $0.44 effect — it does not land on the data. Goeree, Holt and Palfrey added risk aversion to close the rest (Games and Economic Behavior, 2003); this page shows the plain logit fit and does not pretend otherwise.
The peak of the $3.20 curve is 0.847 at λ ≈ 3.03, and every curve returns to 0.5 as λ → ∞ because that is the Nash point. The own-payoff effect is a property of the interior of the λ range, not of its limit.
The same shape, in a different game
In the traveler's dilemma two players claim an integer between 180 and 300; the lower claim is paid to both, with a reward R added to the low claimer and taken from the high one. Iterated deletion leaves 180 as the only Nash equilibrium — for every R.
Claims run from 180 to 300. The ▏ mark is the Nash equilibrium — 180 in both rows — and the bar is the observed average claim.
With R = 180 the subjects played the Nash equilibrium: average claim 201. With R = 5 — same game, same equilibrium — the average claim was 280, at the opposite end of the range.
Show the maths
Goeree & Holt (2001), AER 91(5), 1402–1422, traveler's dilemma treatments: 50 subjects (25 pairs) played R = 180 and then a matched R = 5 treatment. “Close to 80 percent of all the subjects chose the Nash equilibrium strategy, with an average claim of 201”; in the low-R treatment “roughly the same fraction chose the highest possible claim … for which the average was 280”.
No model with a parameter is fitted to these two numbers here. They are on the page because they are the cleanest statement of the pattern: the equilibrium is a fixed point of the payoff structure, and behaviour is a function of the payoff gradients the structure leaves behind.