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A Nash equilibrium is a strategy profile in which every player's strategy is a best response to the others. For a finite game and mixed strategies, its precise condition isThe opponents' strategies ${\sigma }_{-i}^{\ast }$ 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.

By the Kakutani fixed-point theorem, choose $(p^{\ast },v^{\ast })\in P\times Z$ with $p^{\ast }\in g(v^{\ast })$ and $v^{\ast }\in z(p^{\ast })$. The maximizing property of $g$ and the assumed Walras law giveEvery coordinate of $v^{\ast }$ is therefore nonpositive. Since $v^{\ast }\in z(p^{\ast })$, we obtainThis step does not yet imply that the entire vector is zero: a zero price can coexist with negative excess demand in its coordinate.

Use the same Kakutani fixed-point theorem pair as in part (b). If any coordinate of $p^{\ast }$ vanished, the additional boundary condition would force some coordinate of every member of $z(p^{\ast })$, including $v^{\ast }$, to be positive. This contradicts $v^{\ast }\le 0$. Hence $p_i^{\ast }>0$ for every $i$.

Now Walras law reads ${\sum }_i{p}_i^{\ast }{v}_i^{\ast }=0$, a sum of nonpositive terms with strictly positive coefficients. Every term must vanish, soThe boundary hypothesis first excludes boundary prices; strict positivity of all prices then turns nonpositive excess demand into exact market clearing.

At any price vector, the total value of aggregate excess demand is zero when consumers exhaust their budgets. After normalization to a price simplex, choosing a price that maximizes the value of a candidate excess-demand vector is a useful Kakutani fixed-point theorem construction.