THE MODEL
Symmetric two-player contest. Each animal plays Hawk (escalate) or Dove (display, retreat if attacked). Expected payoffs:
E(H,H) = (V−C)/2 both risk injury
E(H,D) = V Hawk takes it all
E(D,H) = 0 Dove yields
E(D,D) = V/2 share the resource
A strategy S is an ESS if, for every mutant T≠S:
E(S,S) > E(T,S) OR
[ E(S,S) = E(T,S) AND E(S,T) > E(T,T) ].
First clause = strict best reply to itself. Second = when a mutant ties on the incumbent, the incumbent must beat the mutant against the mutant.
THE LINEAGE
the-nash-equilibrium, hardened against mutation. Nash asks only that no player gains by unilateral switch. Maynard Smith's biology adds a second-order test — a tie against the incumbent is not enough; the incumbent must out-earn the invader inside the invaded pool.
The mixed ESS is exactly the interior Nash equilibrium, and it is the asymptotically stable rest point of the-replicator-dynamics: perturb the population share and it flows back to Hawk-fraction V/C.
THE WITNESS
Live re-check. Re-derives the ground truth with the tamper flag forced OFF, compares it to the engine's current live verdict, and confirms the closed-form invasion margin. Flips red the instant window 6 plants its void.
witness idle
DATA IN in ▼
Set the resource value and the injury cost. The mixed case needs V<C.
THE PANEL LIT
| vs Hawk | vs Dove | |
|---|---|---|
| Hawk | ||
| Dove |
Mixed ESS S* plays Hawk with probability = V/C.
Invasion test of incumbent S* vs mutant q (Hawk-share):
E(S,S)= E(T,S)=
E(S,T)= E(T,T)=
margin E(S,T)−E(T,T) = (closed form (C/2)(p*−q)²)
Pure all-Hawk verdict (this is what window 6 attacks):
DATA OUT out ▼
Proven result: when V<C the unique ESS is the mix S* = (Hawk with prob V/C). All-Hawk is a Nash-candidate that fails invasion resistance; the interior mix passes.
booting…
THE ADVERSARY WALL
"ESS = Nash. Solve the equilibrium and you're done." False. Nash permits ties; a neutral or better mutant can drift in. ESS demands the strict second-order margin. The game [[1,0],[0,0]] has a Nash (play B) that any A-mutant erodes — Nash-not-ESS is real and window 0 exhibits it.
"The pure aggressive type wins." Only if V≥C. With V<C an all-Hawk world earns (V−C)/2 < 0 per contest; a single Dove scoring 0 beats it and spreads.
THE GRAVEYARD
"An ESS is any strategy no one wants to deviate from."
→ that is Nash. ESS adds: incumbent must out-earn a tying mutant against the mutant.
"The mixed ESS plays Hawk half the time."
→ only when C=2V. In general the Hawk-share is exactly V/C.
"Every game has an ESS."
→ AMBER: some symmetric games have none (e.g. rock–paper–scissors payoff cycles admit no ESS, only a Nash mix).
THE TAMPER
Planted void (disclosed): force the engine to declare pure all-Hawk an ESS while V<C. A Dove mutant earns 0 against the Hawk pool while Hawk earns (V−C)/2<0, so invasion-resistance is false — the witness in window 7 must catch it.