= Two-player complete-information all-pay equilibrium
{title2=$p_1=1-A_2/(2A_1),\quad p_2=A_2/(2A_1)$}
For effective prizes $A_1\geq A_2>0$ and unit effort costs, equilibrium effort CDFs on $[0,A_2]$ are $G_1(b)=b/A_2$ and $G_2(b)=1-A_2/A_1+b/A_1$. The weaker player has an atom at zero. Incremental utilities are $A_1-A_2$ and zero, and winning probabilities are $1-A_2/(2A_1)$ and $A_2/(2A_1)$. Direct payoff indifference and exclusion of larger bids verify the <Nash equilibrium>. A further player with effective prize at most $A_2$ cannot profit by entering against these distributions.
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