Proportional allocation contest ($x_i(b)=b_i/{\sum }_j{b}_j$)

For positive total effort, a proportional allocation contest awards player $i$ the prize with probability $b_i/{\sum }_j{b}_j$, or gives it that proportion of a divisible prize. A convention is needed at zero total effort. This allocation is different from the discontinuous highest-effort rule of an all-pay auction.

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.

For ordered positive valuations, rank $j$ is active exactly when $\delta <v_j(2-j+{\sum }_{i<j}{v}_j/v_i)$. These ranks form a prefix, and the number active is the largest qualifying rank, or zero if none qualifies. This follows by evaluating the decreasing equilibrium equation ${\sum }_i(1-S/v_i{)}_{+}-1+\delta /S=0$ at $S=v_j$. Strict inequality excludes zero-effort boundary players.

With $k>0$ active players of harmonic mean valuation ${\bar{v}}_k$, unit-cost Nash equilibrium effort satisfies $R+\delta =\frac{{\bar{v}}_k}{2k}[(k-1)+\sqrt{(k-1{)}^2+4k\delta /{\bar{v}}_k}]$. Sum the active-player first-order equations $b_i=(R+\delta )(1-(R+\delta )/v_i)$ to obtain the quadratic. If $\delta$ is at least the largest valuation, no one is active and $R=0$. With no outside effort, at least two players must be active.

With values $v_i>0$ and effort costs $b_i^2$, the unique pure Nash equilibrium has the efforts displayed above. Interior first-order conditions give $v_i{b}_j/(b_1+b_2{)}^2=2b_i$, hence $b_1/b_2=\sqrt{v_1/v_2}$ and $(b_1+b_2{)}^2=\sqrt{v_1{v}_2}/2$. Against a positive rival effort, each payoff is a strictly concave function, so these conditions identify global best responses. On the axes, lowering a positive uncontested effort or adding a sufficiently small effort at the zero tie rules out equilibrium.