Mixed strategy
Named by 10 essays across one field — 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.
The road that makes everyone later
An equilibrium is a state nobody can improve alone, which is a much weaker thing than a state anybody would choose. Adding a link that costs nothing to use makes every traveller in this network strictly slower, and the arithmetic says by exactly how much.
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.
Two equilibria and no way to choose
A game can have two states nobody wants to leave, one paying more than the other, and the definition of an equilibrium has nothing to say about which happens. The two standard tie-breakers disagree, and the one that wins is usually the worse.
A mixture that is a population
A mixed equilibrium between two choosers is a knife-edge nobody has a reason to stand on. Read the same mixture as a population whose shares grow with how well they do, and it becomes a point every population is carried to — or one every population circles for ever without arriving.
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.
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.
A best reply to the past
Let two players each answer the whole history of the other's play as though it were a fixed plan. In a zero-sum game the history closes in on the equilibrium, as Julia Robinson proved in 1951. In Lloyd Shapley's three-by-three game it never does: play runs round six profiles for ever, each run the one before plus the one three before plus two, so the runs grow by the real root of x³ = x² + 1 and every run is a fixed share of all the history.
Sellers on a street stand back to back
Put two ice-cream sellers on a beach and they end up side by side in the middle, though customers would walk half as far if they stood at the quarter marks. Three sellers never settle at all: whoever is squeezed in the middle leaps outside, for ever. Four stand in pairs, five in pairs with one alone in the centre, six in endlessly many ways. Let them set prices too, with travel costing the square of the distance, and the conclusion reverses: the two sellers move as far apart as the beach allows.
Named alongside it
The objects these essays reach for when they reach for this one.
Nash equilibriumBest replyZero-sum gameEquilibrium selectionMinimaxConvexityCorrelated equilibriumDominant strategyExhaustive searchExistence proofCommitmentConditional probability