= Proportional contest with outside effort
{title2=$x_i=b_i/(R+\delta)$}
An <all-pay contest> with allocations $x_i=b_i/(\sum_jb_j+\delta)$ includes fixed outside effort $\delta\geq0$. A player with unit effort cost has payoff $v_ix_i-b_i$. The outside effort represents an ineligible fixed competitor and may leave some prize probability unallocated to eligible players. At equilibrium, a player is active exactly when its value exceeds total eligible effort plus outside effort.
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