Discounted continuation value in an elimination contest ($\delta J_i(S\setminus \{j\},r-1)$)

If $J_i(S,r)$ is a player's net expected utility with remaining players $S$ and $r$ prizes, losing the current stage to player $j$ gives baseline $\delta J_i(S\setminus \{j\},r-1)$. This baseline includes later prize values and later effort costs. Subtracting it from the immediate prize value gives an effective prize in a sequential contest. Discounting must be applied to utilities, rather than to winning probabilities.

Relative to losing against a particular rival, the gain from winning now is the current value minus discounted continuation utility. In an elimination contest this is $A_i=v_i-\delta J_i(S\setminus \{j\},r-1)$. When the relevant losing baseline is fixed, current effort incentives reduce to a two-player complete-information all-pay equilibrium. In general, the baseline can depend on which rival wins; that dependence must be verified before using a single effective prize.