A profile of strategies where no player can gain by changing their own move alone. The machine below reads a 2×2 game as two payoff matrices, computes best responses, and finds every pure and mixed equilibrium — each one a fixed point of mutual best-response. Rendered, not quoted.
A finite game: players 1,2, each with a set of pure actions, and a payoff to each player for every action profile. A player may randomize — a mixed strategy is a probability distribution over their actions.
Write ui(σ) for player i's expected payoff at profile σ. Profile σ* is a Nash equilibrium iff for every player i and every alternative strategy si:
ui(σ*i, σ*−i) ≥ ui(si, σ*−i)
Nash proved every finite game has at least one such point (via Brouwer/Kakutani). Finding it is what the engine does — for 2×2 games, in closed form.
In the-bellman-equation a value is the best response to the world — a fixed point of one agent optimizing against a fixed environment.
Nash turns the environment into another mind. The equilibrium is the fixed point of mutual best-response: each player is optimal given the other, who is optimal given them. Bellman's V=max TV becomes a joint fixed point σ=BR(σ). One-player optimality → strategic coherence.
Live re-check. Recomputes the Matching-Pennies value, the Prisoner's-Dilemma equilibrium, and confirms the currently claimed profile is a genuine best-response fixed point. Flips red the instant the Red Team's tamper injects a false claim.
A 2×2 bimatrix. Blue = player 1 (row), red = player 2 (column). Load a classic or edit any cell.
| Left | Right | |
|---|---|---|
| Top | ||
| Bottom |
each cell = (u1, u2)
Underline = a player's best response to the column/row it faces. A cell best for both is a pure Nash equilibrium (boxed green). The mixed equilibrium, if any, makes each player indifferent.
“Existence is not selection.” Nash proves an equilibrium exists; it does not say which one gets played, nor that it is unique. Battle of the Sexes has three. A game can have exponentially many. The theorem hands you a fixed point, not a prediction — the equilibrium-selection problem is left wide open.
Worse: computing one equilibrium in general games is PPAD-complete (Daskalakis–Goldberg–Papadimitriou, 2009). Closed form here works only because 2×2 is small.
“Rational players always reach the good outcome.”
→ The Prisoner's Dilemma's only equilibrium is (Defect, Defect) — jointly worse than (Cooperate, Cooperate), which is not stable. Individual rationality can converge on a bad point.
“A Nash equilibrium is a payoff maximum.”
→ It maximizes no joint quantity. It is only a mutual no-regret condition: no player gains by moving alone. Coalitions or side-payments can break it.
“Mixed strategies are unstable coin-flips.”
→ At the mixed point each pure action in the support pays exactly the same — that indifference is what makes it an equilibrium, not fragility.
The disclosed planted void. Force the engine to claim that (Cooperate, Cooperate) is the Prisoner's-Dilemma equilibrium — a profile from which either player gains +2 by defecting. The Witness (7) recomputes the best-response condition and must catch it.