Sequential elimination all-pay contest

A finite sequence of all-pay contests in which each stage awards one prize and its winner leaves. Losing players remain eligible for later stages. A player can receive at most one prize, while its effort costs are incurred whenever it participates. Subgame perfect equilibrium accounts for both the immediate prize and the value of remaining eligible.

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 $C_r\ge v_{r+1}$, allowing lower-player deviations to be bounded.

As $\delta \uparrow 1$, 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.

For $m<n$ prizes and ordered distinct valuations, the discounted-equilibrium limit gives $x_1=1-2^{-m}{v}_{m+1}/v_1$ and $x_i=1-2^{-(m-i+2)}{v}_{m+1}/v_i$ for $2\le i\le m$. A player must lose its successive fair nonfinal contests and then the asymmetric final contest to receive no prize. The marginal player's probability is $m-{\sum }_{i=1}^m{x}_i$, and lower players have probability zero.

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.