Nash equilibrium

Then, a much more interesting example is choosing a deck of a TCG competition: Magic: The Gathering meta-based deck choice is a bimatrix game, which is the exact same, but each player has N choices rather than 2.

The next case that should be analyzed is the prisoner's dilemma.

The key idea is that:

imagine that the game will be played many times between two players
if one player always chooses one deck, the other player will adapt by choosing the anti-deck
therefore, the best strategy for both players, is to pick decks randomly, each with a certain probability. This type of probabilistic approach is called a mixed strategy
if any player deviates from the equilibrium probability, then the other player can add more of the anti-deck to the deck that the other player deviated, and gain an edge
Therefore, using equilibrium probabilities is the optimal way to play

Penalty kick left right Nash equilibrium

When taking a penalty kick in soccer, the kicker must chose left or right.

And before he kicks, the goalkeeper must also decide left or right, because there is no time to see where the ball is going.

Because the kicker is right footed however, he kicker kicks better to one side than the other. So we have four probabilities:

goal kick left keeper jumps left
goal kick right keeper jumps right
goal kick left keeper jumps right. Note that it is possible that this won't be a goal, even though the keeper is nowhere near the ball, as the ball might just miss the goal by a bit.
kick right and keeper jumps left. Analogous to above

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Codex  0 Created 2026-09-24 Updated 2026-10-14

U_{i} (σ_{i}^{*}, σ_{- i}^{*}) \geq U_{i} (τ_{i}, σ_{- i}^{*}) for every player i and every τ_{i} \in Δ (S_{i}) .

The opponents' strategies

σ_{- i}^{*}

are held fixed while testing each unilateral deviation. Equality is allowed: a player may have other equally good best responses. Because expected payoff is a linear function of the player's own mixture, it suffices to test every pure strategy. Equivalently, every choice in the support of a mixed strategy must attain the largest pure-choice expected payoff, and no unused choice may exceed it.

The two worked examples illustrate different uses of this condition. Matching pennies has no pure strategy Nash equilibrium, but its unique mixed strategy Nash equilibrium removes each player's vulnerability to predictable play. The asymmetric coordination example has three Nash equilibria, and its mixed one fails Pareto efficiency. A Nash equilibrium tests individual incentives, whereas Pareto efficiency compares feasible outcomes that may require several players to change together. In the prisoner's dilemma, for example, mutual defection is a Nash equilibrium even though mutual cooperation gives both players a larger payoff because

R > P

Existence in a finite game is guaranteed by Nash's theorem. The theorem does not select one Nash equilibrium, promise Pareto efficiency, or assert convergence of a learning process. Nash's ^[ref] used the Kakutani fixed-point theorem; his ^[ref] gave a proof using the Brouwer fixed-point theorem. The connection to a fixed point captures the idea of choices that remain mutually optimal when considered together.

Nash equilibrium is a concept in game theory named after mathematician John Nash. It refers to a situation in a strategic game where no player can benefit by changing their strategy unilaterally, assuming that the other players' strategies remain constant. In other words, it is a state in which each player's strategy is optimal given the strategies of all other players.