ESS Game Theory Calculator
Calculate fitness payoffs and evolutionary stable strategies for hawk-dove and other games.
Why the Equilibrium Sits Where It Does
An evolutionarily stable strategy is one that, once common, cannot be invaded by any rare alternative. In the Hawk–Dove game the answer depends entirely on whether the cost of fighting exceeds the value of the resource.
| Condition | ESS | Interpretation |
|---|---|---|
| V > C | All Hawk | Resource worth more than injury risk |
| V < C | Mixed, p = V/C | Neither pure strategy is stable |
The mixed equilibrium is frequency-dependent, and that is what makes it stable. When Hawks are rare they mostly meet Doves and do well, so their frequency rises. When Hawks are common they mostly meet each other and pay the injury cost, so their frequency falls. The population settles where the two payoffs are equal.
Note the equilibrium can be reached two ways: a population of individuals each playing Hawk with probability V/C, or a stable polymorphism of pure Hawks and pure Doves in that ratio. The game theory cannot distinguish them.
Why Real Contests Are Rarely Lethal
The model predicts that as injury cost rises relative to resource value, the equilibrium proportion of escalated fighting falls. This matches observation: most animal contests are settled by display rather than combat.
| Refinement | What it adds |
|---|---|
| Bourgeois strategy | Respect ownership — often an ESS itself |
| Assessor strategy | Fight only if larger than the opponent |
| War of attrition | Contests over persistence rather than injury |
| Asymmetric contests | Resource value differs between individuals |
The Bourgeois strategy — play Hawk when the owner, Dove when the intruder — is stable and needs no assessment of fighting ability, only a convention. Territorial residency effects in butterflies and spiders match this closely.
The model's limits are worth stating: it assumes fitness payoffs are additive and contests are pairwise and independent, and it ignores kin selection, learning and reputation, all of which shape real animal conflict substantially.
Worked Examples
Common Mistakes
That formula only holds when cost exceeds value. When V > C the ESS is pure Hawk, and V/C would exceed 1 which is meaningless as a frequency.
An ESS is uninvadable, not efficient. The Hawk–Dove equilibrium has lower mean fitness than an all-Dove population would — which is precisely why all-Dove is unstable.
The same equilibrium arises from every individual randomising or from a stable polymorphism of pure types. Game theory cannot distinguish them.
Resource value and injury cost vary with condition, hunger and ownership. Asymmetric versions of the game capture this and predict different outcomes.