A Nash equilibrium is a standoff no player wants to break alone: given what everyone else is doing, you can’t do better by changing YOUR move. It is the resting point of strategy — not necessarily the best outcome, just the stable one. Nash proved every finite game has at least one. Slide a player’s strategy and watch the best-response arrows pull back to the equilibrium.
A Nash equilibrium is a strategy profile where each player’s choice is a BEST RESPONSE to the others’ — no one can raise their own payoff by unilaterally deviating. It is the fixed point of the mutual best-response map; John Nash (1950) proved every finite game has one (in mixed strategies). It need not be efficient (the Prisoner’s Dilemma’s is not), only self-enforcing. In this coordination game both matching on the high-payoff option is a Nash equilibrium: deviate and you fall to 0. A fail-loud self-check throws unless the equilibrium cell admits no profitable unilateral deviation. ◆ real game theory, node-verified.
A 2×2 coordination game with pure equilibria (the exact no-profitable-deviation condition); many games have only mixed equilibria and some have several — the best-response fixed-point definition is exact and general.