= Vanishing-discount limit of an elimination all-pay contest
As $\delta\uparrow1$, both active players' effective prizes in every nonfinal subgame tend to the marginal valuation $v_{r+1}$. The <two-player complete-information all-pay equilibrium> then gives each a winning probability $1/2$. In the final stage, actual valuations determine the asymmetric winning probabilities. Taking this limit from discounted equilibria specifies the continuation selection instead of independently choosing an undiscounted game equilibrium.
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