Two players each hide a penny under their hand, heads or tails up, and lift their hands together.
You want the pennies to match. The other player wants them to differ.
If they match, you win 1 and the other player loses 1. If they differ, it is the other way round.
You are Player A
Watch one round, or choose now.
Player A
or
You
Player A
?
You
Player B
or
The other player
Player B
?
The other player
The other player picks at random here, like a coin toss.
The whole game on one grid
Every pair of choices and what each side gets: yours first, then theirs. Each number is a payoff.
Step 0 of 3
The cards are face down. Turn them, then step through which squares are stable.
What do you think they'll do?
Your best move depends on what you expect from the other player. Drag to set how likely you think it is they pick Heads.
What it teaches
With pure opposed interests there is nothing to settle on, so the only stable play is to be unpredictable.
The usual mistakeReading the fifty-fifty mix as concealment, a way of hiding your intentions, rather than as the frequency that leaves the opponent with nothing to gain by leaning either way.
The checkYour mix is chosen to make the other player indifferent. It is a property of their payoffs, not a disguise for yours.
What people actually do
Aggregate mixing fits, individual sequences do not
The theory says
A mixed-strategy equilibrium requires two things of a player: the actions in the support must yield equal expected payoffs, and the realised sequence of choices must be serially independent, because any pattern in it is exploitable. A change in one player's own payoffs should shift the opponent's mixture and leave that player's own mixture unchanged.
People actually
Aggregate frequencies are often close to the predicted mixture and win rates across actions are close to equal, in the laboratory and in professional sport. Serial independence fares much worse: tennis servers alternate too much, and reanalysis of laboratory zero-sum play rejects independence at the individual level even where the aggregate fits. The comparative static also fails, with players changing behaviour in response to changes in their own payoffs.
Why is still argued over.
In the real world
Fiction: The Princess Bride
The battle of wits
Westley poisons one of two goblets out of Vizzini's sight and lets Vizzini choose which to drink. Vizzini talks himself through several layers of reasoning about where a clever opponent would put the poison, which is exactly the reasoning that cannot terminate: any rule he settles on is a rule Westley could have anticipated. The scene then cheats the game rather than solving it, because Westley had poisoned both goblets and was himself immune.
Two players, one binary choice each, and directly opposed interests in whether the choices coincide, which is why no amount of reasoning about the opponent produces a safe answer and only unpredictability survives.
HistoryContested
Normandy or the Pas de Calais
An invasion of north west France in 1944 had to come ashore somewhere, and the German defence had reserves that could decisively reinforce one sector but not two. The Allies ran an elaborate deception, including a fictitious army group in south east England, to point at the Pas de Calais, and landed in Normandy; German reserves were held back for weeks against a second landing that never came. The defenders' problem was not how strong to be overall but which of two places to be strong in, and the attackers' problem was to be unreadable.
Pure opposed interests over a single binary choice, where one side wins by matching the other's choice and the other by avoiding it, so concealment and unpredictability are the whole of the strategy rather than any allocation across fronts.
Colonel BlottoThe randomisation runs over a continuum of allocations tied together by one budget, so what has to be concealed is where the strength is rather than which of two things will be done. Not on the wall yet.
A zero-sum 2x3 gameOne side has a third strategy that must be deleted before mixing applies. Not on the wall yet.