 Backward-induction threshold in an elimination all-pay contest

With ordered values $v_1>\cdots >v_k$ and $r<k$ prizes remaining, define $C_r=(1-\delta ){\sum }_{j=2}^r{\delta }^{j-2}{v}_j+{\delta }^{r-1}{v}_{r+1}$. In the recursively constructed discounted subgame perfect equilibrium, it is the second active player's effective prize and the effort-support upper endpoint. The highest player's utility is $v_1-C_r$. The coefficients form a convex combination, so $C_r\ge v_{r+1}$, allowing lower-player deviations to be bounded.