Rank-order contest

A rank-order contest awards a sequence of prizes according to the ranks of efforts, while every player incurs its own effort cost. With an independent private values model, an increasing symmetric bidding function orders efforts in the same way as valuations.

Suppose values are nonnegative, the rank-order expected prize allocation $a$ is increasing and differentiable, and $a(\underline{v})=0$. With unit effort cost and zero effort at the lowest type, the symmetric equilibrium effort is $b(v)=va(v)-{\int }_{\underline{v}}^{v}a(t)dt$. A true type $v$ imitating type $z$ receives utility $va(z)-b(z)$, whose derivative in $z$ is $(v-z)a^{\prime }(z)$. It increases up to $v$ and decreases afterwards, proving the best response property. This is the interim payment identity specialized to an all-pay contest.

Let $V_{[1]}\ge \cdots \ge V_{[n]}$ be descending order statistics, with $w_n=0$ and nonnegative decreasing prizes. Decompose the prize vector into awards of $w_k-w_{k+1}$ to each of the best $k$ players. The corresponding truthful multi-unit auction charges each winner the next value $V_{[k+1]}$. Its total payment is $k(w_k-w_{k+1})V_{[k+1]}$. Revenue equivalence transfers the expected payment to the all-pay contest, because the interim allocations and lowest-type utilities coincide. Summing the layers proves the formula.

With $n$ independent uniform types, $m<n$ equal prizes of scale $w(m/n)$ and unit effort costs, a symmetric Bayesian Nash equilibrium has total expected effort $R_m=m(n-m)w(m/n)/(n+1)$. The all-pay effort identity yields $b(v)=w(m/n){\int }_0^{v}t{q}_m^{\prime }(t)dt$. The allocation derivative is a Beta distribution density with parameters $n-m,m$, so integration gives the formula.

For prize scale $w(x)=x^{-\alpha }$, total effort is proportional to $h(x)=x^{1-\alpha }(1-x)$ on the feasible grid $x=m/n$. If $\alpha \ge 1$, one prize is optimal. For $0<\alpha <1$, $h$ increases up to $x_{\ast }=(1-\alpha )/(2-\alpha )$ and then decreases, so the optimal feasible integer is among $\lfloor n{x}_{\ast }\rfloor$ and $\lfloor n{x}_{\ast }\rfloor +1$. Remove infeasible candidates and compare their objective values. No definition at $x=0$ is needed.

For $n$ players with independent valuations having a continuous distribution function $F$, a type $v$ occupies rank $k$ when exactly $k-1$ rivals have higher values. Multiplying this binomial distribution by the rank prize $w_k$ and summing gives $a(v)$. Ties occur only on zero-probability events when the valuation law is atomless.