Effective prize in a sequential contest ($A_i=v_i-\delta J_i^{\mathrm{lose}}$)

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.