Zero-sum game
Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.
The value from both sides
Two choosers move at the same instant, and each asks the cautious question — how much can be guaranteed, whatever the other does. With pure choices the two answers are usually different numbers; allow a probability and they are forced to be the same one.
A signal both can see
Two choosers who randomise privately can reach a set of outcomes that is smaller, and worse, than the set they reach when a device draws one cell and whispers each of them their half of it. Nothing is enforced and nobody is bound, and the arrangement is stable anyway.
Worth more for being seen first
Moving first sounds like a disadvantage, since the other side gets to see the move and answer it. When the move is a mixture that is announced and believed, it is never a disadvantage, it is worth exactly nothing in a game of pure conflict, and in other games it is worth more than any equilibrium — sometimes by announcing an action that would never be played in secret.
Patience instead of a contract
Commitment had to assume an announcement binds. Play the same game again tomorrow and the assumption is unnecessary — the future does the binding. What it costs is that nearly every outcome becomes an equilibrium, so a theory that could not choose between two now cannot choose between infinitely many.
A coin that lets the first player win
On a fair coin the second player in Penney's game always has a better pattern than the first, and the first can hold them to no worse than two to one. Bend the coin and every overlap is paid for in the letters it uses: the replies change, the first player's share swings between a third and a half, and past a heads chance of 1/∛2 the first player simply names HHH and wins.
A third of random games have no pure equilibrium
Fill a game's payoff table with random numbers and ask whether some cell is each player's best reply to the other. Exactly one cell is expected to be, at every size, and the chance that none is climbs from one in eight to 1/e — the same constant that counts the shuffles in which nobody gets their own hat back, and for the same reason.
Every random game has an odd number of equilibria
Find every equilibrium of eleven thousand random games — pure and mixed, by trying every pair of strategy sets the players might mix over — and the totals are 1, 3, 5, 7, … and never an even number. The count is a sum of signs: equilibria of index +1 outnumber those of −1 by exactly one in every game. The average grows about 28% with each strategy added, and searching for the game with the most turns up the coordination game's 2ⁿ − 1, a pattern that holds only up to five strategies.
Named alongside it
The objects these essays reach for when they reach for this one.
Nash equilibriumMixed strategyMinimaxBest replyCommitmentConditional probabilityConvexityCorrelated equilibriumDominant strategyEquilibrium selectionExhaustive searchLinear program