 Proportional contest with outside effort

An all-pay contest with allocations $x_i=b_i/({\sum }_j{b}_j+\delta )$ includes fixed outside effort $\delta \ge 0$. A player with unit effort cost has payoff $v_i{x}_i-b_i$. 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.