= Effective prize in a sequential contest
{title2=$A_i=v_i-\delta J_i^{\mathrm{lose}}$}
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.
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