THE REPLICATOR EQUATION where game theory becomes biology
Evolution as a game that plays itself: a strategy’s share grows in proportion to how far its fitness beats the population average — ẋi = xi(fi − φ). Above average, you spread; below, you fade. The astonishing payoff (the folk theorem of evolutionary games): every Nash equilibrium is a rest point of this purely biological dynamic, and every stable rest point is a Nash equilibrium. No rational players, no foresight — selection alone finds the equilibrium. This is the bridge from THE GAME to life.
THE TECHNIQUE ẋi = xi(fi − φ) ; Nash ⇒ rest point
The demo grows above-average strategies, shrinks below-average, and shows the rest point is a Nash equilibrium: live demo
HISTORY & CREDIT Taylor & Jonker · 1978
“Evolution needs rational agents to reach equilibrium.” — the replicator dynamic reaches Nash equilibria with no rationality; selection substitutes for reason. cited
the growth law · a strategy above the average fitness φ grows; below, it shrinks. the rest point · where all fitnesses equal φ; every Nash equilibrium is such a rest point, and stable rest points are Nash. 1978 · Taylor & Jonker; the tie to Maynard Smith’s ESS — the bridge from THE GAME (batch 56).
Selection finds the equilibrium reason was supposed to. evolutionary dynamics
RECOMMEND FOR I-13 rest = Nash, on the compiler
On i-13, the above-average strategy grows and the below-average shrinks; equal fitness is the rest point (a Nash equilibrium):
$ i13 run lf_replicator-equation.i13
RUN OK · 75 step(s)
f_a=5 f_b=3 phi=4 : a_grows=+1 b_shrinks=-1
at_rest(4,4)=1 rest_is_nash = 1
Recommend as a NULL — a fixed point / deep equivalence (B43/B39). The rest points are fixed points of the dynamic; the folk-theorem correspondence (every Nash is a rest point; every stable rest point is Nash) is a pinned theorem (WITNESSED), like Sprague-Grundy. A profound bridge, not a same-function DOF. NULL — where game theory becomes biology.