MODEL GAME THEORY & SAPPHONS · GAME THEORY · EVOLUTION · under KEEPER ERIS

Hawk–Dove · Evolutionarily Stable Strategies

Animals fighting over a resource of value V can escalate (Hawk) or display (Dove); escalating risks an injury costing C. When C > V, no pure strategy is stable: the population settles at a fraction V/C of hawks. Maynard Smith and Price called this an evolutionarily stable strategy.

Dave (t252): "i love that set up can you build some pages looking into game theory and those sappons you just built, see if you can build some folded stars and stuff"

LIT John Maynard Smith & George Price, “The logic of animal conflict”, Nature 246:15 (1973) defined the ESS: a strategy that, once nearly universal, can’t be invaded by a rare mutant. Hawk–Dove payoffs: H vs H (V − C)/2, H vs D V, D vs H 0, D vs D V/2.

LIT If V < C, the mixed ESS has hawk frequency p* = V/C, where hawks and doves earn the same. If V ≥ C, Hawk itself is the ESS.

LIT Taylor & Jonker, Math Biosci 40:145 (1978): the replicator equation ṗᵢ = pᵢ(fᵢ − f̄) grows strategies that beat the population average. A regular ESS is a stable rest point of it. For rock–paper–scissors the product of the three shares stays constant in continuous time, so the population cycles forever.

AMBER Euler steps (used on the VM) are an approximation. Taylor and Jonker note that discrete-time dynamics can overshoot even when the continuous ones converge. The small drift of r·p·s in the engine check shows this.

MODEL Our two agents play Hawk–Dove as two separate players, not one population. Two learners usually end in an asymmetric pure equilibrium (one hawk, one dove), unlike the single-population mixed ESS. That difference is itself a lesson.

MODEL Two agents, as Dave set for every world: ANIMAL A and ANIMAL B play Hawk–Dove (V = 2, C = 3) over and over by regret matching (Hart & Mas-Colell 2000), each shifting toward the moves it regrets not having played. Their score is 1 − NashConv/2.5 of their empirical average play (1 = an exact Nash equilibrium), and the I-13 VM re-computes NashConv from the raw averages every time the score improves. The centrepiece is a folded star {12/5} whose fold rises with the score. Three.js r160 from a pinned CDN, devicePixelRatio capped at 2. Drag to orbit, scroll to zoom.

LIT on load: the seal is recomputed, the page’s facts are re-checked as arithmetic (throws on mismatch), 200 seeded rounds of the two agents must improve their score, the workspace loop logs 5 entries, and the OBJ and SVG exports must be consistent. Then the engine boots, its SHA-256 and regression lab are checked, the VM recomputes the agents’ NashConv, and the page’s VM results are compared with independent JS.

⏸ Pause Reseed Export training + workspace log (.json) Export star + trajectory (.svg) Export folded star (.obj)

Gallery of kept moments (WebGL snapshots):

Shared board MODEL ANIMAL A ↔ ANIMAL B

Background art: MODEL a generative field seeded from this page’s seal.

proving…

Seal sha256(name|slug|blurb): ce65d171b06485890a72ede7665b95a9d6e4ce008dc9dde0af3445d4cbfc72c1

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