 Solution

Let $w_m=w(m/n)$, so a type $v$ values a prize at $v{w}_m$. In a symmetric increasing Bayes-Nash equilibrium of this rank-order contest, a player reporting type $t$ wins when at most $m-1$ opponents have larger types. Its winning probability isHere the count of opponents above $t$ has a binomial distribution. Differentiation, or the associated order statistic density, givesDefine the effort by the all-pay effort identityThis also verifies equilibrium globally. The derivative of a type $v$'s payoff from reporting $t$ is $w_m(v-t)q_m^{\prime }(t)$, positive before $v$ and negative after $v$. Thus truthful reporting is a best response. Bidding above the maximal equilibrium effort gains no additional winning probability. Type zero chooses zero effort.

By exchanging the two integrations, the expected value of total effort isThe last integral is the moment $\mathbb{E}[X(1-X)]$ for a Beta distribution with parameters $n-m$ and $m$, namely $m(n-m)/(n(n+1))$. This is the uniform-value multi-prize all-pay effort formula.

Put $x=m/n$ and $h(x)=w(x)x(1-x)$. The positive constant $n^2/(n+1)$ multiplying $h(x)$ does not affect the maximizing $m$. Sincethe assumed inequality makes $h$ nonincreasing on the feasible interval. Hence one prize maximizes expected total effort.

For the power family, the PDF gives $w(x)=x^{-\alpha }$. ThenIf $\alpha \ge 1$, this derivative is strictly negative for $0<x<1$: the bracket is affine and its values at the endpoints are $1-\alpha \le 0$ and $-1$. Thus $m=1$ is optimal.

If $0<\alpha <1$, the unique continuous maximizer isThe objective strictly increases before $x_{\ast }$ and decreases after it. Therefore its discrete maximizer lies amongCompare the surviving candidates using $m^{1-\alpha }(n-m)$, since $R_m=n^{\alpha }{m}^{1-\alpha }(n-m)/(n+1)$. If $n{x}_{\ast }$ is an integer, that integer is the unique maximizer; if its floor is zero, the only feasible candidate is $1$. This proves the discrete prize-count optimization for a power-valued contest including the endpoint cases.

The sufficient condition need only hold on $[1/n,(n-1)/n]$. The power family is undefined at zero, so the printed endpoint $x=0$ cannot apply to it. More generally, a positive finite value $w(0)$ would make the displayed inequality fail at zero. The design argument uses only positive feasible prize fractions and needs no value there.