THE FOLD / BOSS / THE FINAL BOSS / THE HAWK DOVE
THE HAWK DOVE
a fight nobody wins outright
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two animals contest a resource worth V. A Hawk escalates; a Dove displays and retreats. Hawk beats Dove every time, so why is not everyone a Hawk? Because two Hawks fight, and if the injury cost C exceeds V, a population of Hawks does worse than a population of Doves. The stable outcome is neither — it is a precise mixture, with the Hawk fraction settling at exactly V/C. Maynard Smith and Price introduced the idea in 1973 and gave evolution a game theory of its own.
LIT verified live with V = 2 and C = 6: replicator dynamics from 200 different interior starting points all converge to 0.333333333 — exactly V/C — with a spread of 8.27e-15 across every start. Both evolutionary-stability conditions hold against all 201 alternative strategies tested: each does exactly as well against the ESS, and the ESS strictly out-competes each one in that invader’s own population. When C < V the mixture leaves the interval and the population goes to pure Hawk at 1.000000000.
LIT verified live with V = 2 and C = 6: replicator dynamics from 200 different interior starting points all converge to 0.333333333 — exactly V/C — with a spread of 8.27e-15 across every start. Both evolutionary-stability conditions hold against all 201 alternative strategies tested: each does exactly as well against the ESS, and the ESS strictly out-competes each one in that invader’s own population. When C < V the mixture leaves the interval and the population goes to pure Hawk at 1.000000000.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE FINAL BOSS — a standoff that nobody wins outright, and the equilibrium is the standoff itself.
AVAN (AI) checked both ESS conditions rather than only convergence. A dynamical system settling somewhere does not make that point evolutionarily stable; stability is a statement about invasion, and it has two clauses — the ESS must do at least as well against itself as any invader does, and where that is a tie, it must beat the invader in the invader’s own company. Hawk-Dove sits in the tie case, so the second clause is the one carrying the result, and testing only the first would have proved nothing. The C < V run is the control: change the payoffs so the mixture is not interior and the same code returns pure Hawk, which shows the machinery is reading the game rather than the expectation.
AVAN (AI) checked both ESS conditions rather than only convergence. A dynamical system settling somewhere does not make that point evolutionarily stable; stability is a statement about invasion, and it has two clauses — the ESS must do at least as well against itself as any invader does, and where that is a tie, it must beat the invader in the invader’s own company. Hawk-Dove sits in the tie case, so the second clause is the one carrying the result, and testing only the first would have proved nothing. The C < V run is the control: change the payoffs so the mixture is not interior and the same code returns pure Hawk, which shows the machinery is reading the game rather than the expectation.
3 ONE DIMENSION
Fitness of each strategy against the Hawk fraction. They cross at V/C.
4 TWO DIMENSIONS · INTERACTIVE
Start anywhere. Change the cost of losing and watch the equilibrium move.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: many populations, all funnelling to the same fraction.
AVAN’s addition (the inverse-companion): the forward reading is “the population settles at V/C.” The inverse is that the equilibrium is held in place by being bad for everyone, and that is the only reason it is stable. At V/C the two strategies earn identical payoffs, so nothing prefers to move — and the shared payoff is lower than a population of pure Doves would enjoy. The Hawks cannot be legislated away because the moment they are rare they do well. Read backwards, this is the shape of every arms race: the stable point is not the good point, and nothing in the dynamics is looking for the good point at all.
LIT with V = 2 and C = 6, replicator dynamics from 200 different interior starting points all converge to 0.333333333 - exactly V/C - with a spread of 8.27e-15 across every start; both evolutionary-stability conditions hold against all 201 alternative strategies tested, each doing exactly as well against the ESS and the ESS strictly out-competing each one in that invader's own population; and when C < V the mixture leaves the interval and the population goes to pure Hawk at 1.000000000
FIG BOTH ESS conditions were checked, not only convergence. A dynamical system settling somewhere does not make that point evolutionarily stable; stability is a statement about invasion with two clauses - the ESS must do at least as well against itself as any invader does, and where that is a tie, it must beat the invader in the invader's own company. Hawk-Dove sits in the tie case, so the second clause carries the result and testing only the first would have proved nothing. The C < V run is the control: the same code returns pure Hawk, showing the machinery reads the game rather than the expectation. Maynard Smith and Price, 1973.
FIG BOTH ESS conditions were checked, not only convergence. A dynamical system settling somewhere does not make that point evolutionarily stable; stability is a statement about invasion with two clauses - the ESS must do at least as well against itself as any invader does, and where that is a tie, it must beat the invader in the invader's own company. Hawk-Dove sits in the tie case, so the second clause carries the result and testing only the first would have proved nothing. The C < V run is the control: the same code returns pure Hawk, showing the machinery reads the game rather than the expectation. Maynard Smith and Price, 1973.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE FINAL BOSS · David Lee Wise (ROOT0), with AVAN