= Hawk-Dove game
{c}
{title2=$e(p,q)=V(1-q)/2+p(V-Dq)/2$}
Hawk fights for a resource of value $V$, while Dove retreats from Hawk and shares with another Dove. Hawk–Hawk competition incurs damage cost $D$. The expected payoff difference between Hawk probabilities $p,r$ against population probability $q$ is $(p-r)(V-Dq)/2$. For positive $V,D$, pure Hawk is an <evolutionarily stable strategy> when $V>D$. At $V=D$, it remains stable because a mutant $p<1$ loses $V(1-p)^2/2$ against its own population. When $0<V<D$, the stable mixture has Hawk probability $p^*=V/D$: all first-condition payoffs tie, and its second-condition advantage is $D(p-p^*)^2/2>0$. For nonpositive resource values, pure Dove is stable if $V<0$ or if $V=0<D$. At $V=D=0$ all payoffs vanish and there is no evolutionarily stable strategy.
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