THE NASH EQUILIBRIUM — a dart

THE NASH EQUILIBRIUM nobody gains by moving alone

A Nash equilibrium is a set of strategies where no player can do better by changing theirs alone — everyone is already best-responding to everyone else. Nash proved that every finite game has at least one (in mixed strategies). In the prisoner’s dilemma it is mutual defection: individually rational, collectively worse than cooperating. The knot at the heart of why rational agents get stuck.

THE TECHNIQUE no profitable unilateral deviation

The demo checks the prisoner’s dilemma: (defect, defect) is a Nash equilibrium, (cooperate, cooperate) is not: live demo

HISTORY & CREDIT Nash · 1950

“The Nash equilibrium is the best outcome for the players.” — it is stable, not optimal; the prisoner’s dilemma’s equilibrium is worse for both than cooperating. cited

best response · each player’s strategy is optimal given the others’.
the trap · defect dominates in the one-shot dilemma — (D,D) is stable though (C,C) pays both more.
1950 · John Nash proved existence for every finite game (Nobel 1994).

Stable is not the same as good. theorem

RECOMMEND FOR I-13 the equilibrium, on the compiler

On i-13, with payoffs T>R>P>S, (D,D) has no profitable deviation while (C,C) does:

$ i13 run gm_nash-equilibrium.i13 RUN OK · 32 step(s) nash_DD = 1 -- at (D,D), deviating to C pays S

R dominant_defect = 1

Recommend as a NULL — a fixed point (B43) + existence theorem (B39). An equilibrium is a fixed point of the best-response map, pinned by the payoffs; existence is Nash’s theorem. Fixed-points auto-NULL. NULL — nobody gains by moving alone.