= Backward-induction threshold in an elimination all-pay contest
{title2=$C_r=(1-\delta)\sum_{j=2}^{r}\delta^{j-2}v_j+\delta^{r-1}v_{r+1}$}
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 $C_r\geq v_{r+1}$, allowing lower-player deviations to be bounded.
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