All-Pay Contest Break-Even Effort

Two equally skilled quants enter an all-pay contest for a prize worth $V$. A contestant who spends effort $e$ while the rival spends $e_0$ wins with probability $e/(e+e_0)$, and everyone pays their own effort regardless of outcome. Holding $e_0$ fixed, write the payoff of effort $e$ and solve the equation that defines break-even effort.