 Ranked winning probabilities in an undiscounted elimination contest

For $m<n$ prizes and ordered distinct valuations, the discounted-equilibrium limit gives $x_1=1-2^{-m}{v}_{m+1}/v_1$ and $x_i=1-2^{-(m-i+2)}{v}_{m+1}/v_i$ for $2\le i\le m$. A player must lose its successive fair nonfinal contests and then the asymmetric final contest to receive no prize. The marginal player's probability is $m-{\sum }_{i=1}^m{x}_i$, and lower players have probability zero.