Contest theory

Contest theory studies how prize allocation rules and effort costs influence strategic effort investment, participation, and the resulting Nash equilibria or Bayesian Nash equilibria. Unlike a sale in which only the winner pays, an all-pay auction charges every participant for its effort.

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.

Two players pay their invested efforts regardless of outcome. A leader invests first, and a follower observes that effort before deciding. With ties favoring the follower and unit costs, a follower of value $u$ matches leader effort $b$ when $u>b$, and otherwise declines. For a continuous follower cumulative distribution function $F$, the leader wins with probability $F(b)$ and has payoff $vF(b)-b$.

Every optimal leader bid satisfies $b(v)\le v$, because bidding zero guarantees payoff zero and winning probability is at most one. Hence $F(b(v))\le F(v)$. With identically distributed independent continuous valuations, the probability integral transform gives $\mathbb{E}F(V)=1/2$. The leader's unconditional winning probability is therefore at most $1/2$, even when some high leader types have a conditional advantage. This argument does not depend on selecting a unique optimum.

With follower distribution $F$, a leader of value $v$ chooses $b\in \arg {\max }_{0\le b\le 1}\{vF(b)-b\}$. If $F$ is a concave function, the objective is concave and an interior optimum satisfies $v{F}^{\prime }(b)=1$. Endpoint derivatives handle zero or maximal effort. Zero effort guarantees nonnegative payoff, so every optimum satisfies $b\le vF(b)\le v$.

If a concave leader objective $vF(b)-b$ is constant over an interval, all efforts there are best responses, but they can have different winning probabilities. A flat positive-density interval of $F$ produces this at the type $v=1/F^{\prime }$. Thus strict threshold statements must distinguish strict concavity from a selection such as the smallest maximizing effort. For an atom-free type distribution, the specified threshold type has zero ex ante probability.

For increasing concave $F$, write $q=F^{-1}(1/2)$. Under the smallest-maximizer convention, the leader wins with probability greater than $1/2$ exactly when $v>1/F^{\prime }(q)$. The sign of $v{F}^{\prime }(q)-1$ places the maximizer before or after $q$. At equality, concavity alone permits a flat interval of optima; strict concavity or an explicit selection is needed for the strict comparison at that type.

For positive total effort, a proportional allocation contest awards player $i$ the prize with probability $b_i/{\sum }_j{b}_j$, or gives it that proportion of a divisible prize. A convention is needed at zero total effort. This allocation is different from the discontinuous highest-effort rule of an all-pay auction.

An all-pay contest with allocations $x_i=b_i/({\sum }_j{b}_j+\delta )$ includes fixed outside effort $\delta \ge 0$. A player with unit effort cost has payoff $v_i{x}_i-b_i$. The outside effort represents an ineligible fixed competitor and may leave some prize probability unallocated to eligible players. At equilibrium, a player is active exactly when its value exceeds total eligible effort plus outside effort.

For ordered positive valuations, rank $j$ is active exactly when $\delta <v_j(2-j+{\sum }_{i<j}{v}_j/v_i)$. These ranks form a prefix, and the number active is the largest qualifying rank, or zero if none qualifies. This follows by evaluating the decreasing equilibrium equation ${\sum }_i(1-S/v_i{)}_{+}-1+\delta /S=0$ at $S=v_j$. Strict inequality excludes zero-effort boundary players.

With $k>0$ active players of harmonic mean valuation ${\bar{v}}_k$, unit-cost Nash equilibrium effort satisfies $R+\delta =\frac{{\bar{v}}_k}{2k}[(k-1)+\sqrt{(k-1{)}^2+4k\delta /{\bar{v}}_k}]$. Sum the active-player first-order equations $b_i=(R+\delta )(1-(R+\delta )/v_i)$ to obtain the quadratic. If $\delta$ is at least the largest valuation, no one is active and $R=0$. With no outside effort, at least two players must be active.

With values $v_i>0$ and effort costs $b_i^2$, the unique pure Nash equilibrium has the efforts displayed above. Interior first-order conditions give $v_i{b}_j/(b_1+b_2{)}^2=2b_i$, hence $b_1/b_2=\sqrt{v_1/v_2}$ and $(b_1+b_2{)}^2=\sqrt{v_1{v}_2}/2$. Against a positive rival effort, each payoff is a strictly concave function, so these conditions identify global best responses. On the axes, lowering a positive uncontested effort or adding a sufficiently small effort at the zero tie rules out equilibrium.

Players choose efforts in several all-pay contests at the same time. With additive quasilinear utility and no shared effort budget, a player's expected payoff is the sum of its per-contest payoffs. An entry restriction couples the choices of contests even though the subsequent effort optimization is separable.

Consider $n\ge 2$ identical players, prizes $w_j>0$, unit effort costs, and exactly two entries per player. The symmetric equilibrium omission probabilities $q_j$ sum to one. The common per-contest payoff is $u=({\sum }_j{w}_j^{-1/(n-1)}{)}^{-(n-1)}$, and the unconditional rival distribution on $0\le b\le w_j-u$ is $G_j(b)=((b+u)/w_j{)}^{1/(n-1)}$. Conditional on entry, its distribution function is $(G_j(b)-q_j)/(1-q_j)$. Choosing the omitted contest according to $q$ and sampling the two active efforts independently gives a symmetric equilibrium with expected payoff $2u$. These formulas determine the marginal laws; they do not determine the dependence between a player's two efforts.

If contest rewards and costs add, a fixed deviation's expected payoff depends on each rival's per-contest marginal distributions, rather than on dependence between that rival's efforts across contests. Replacing the conditional independent sampling of two active efforts by another copula with the same conditional marginals therefore preserves every deviation payoff. Consequently it preserves the Nash equilibrium, even when the new joint strategy distribution is different. This explains why unique participation probabilities and bid marginals need not imply a unique full mixed strategy.

For a prize of value $w$, let $G(b)$ be the probability that a rival is absent or enters with effort at most $b$. When rivals act independently and have no atoms at positive efforts, effort $b>0$ wins with probability $G(b{)}^{n-1}$. If a player mixes over an interval of best responses, its payoff on that interval is constant, so $G(b)=((b+u)/w{)}^{1/(n-1)}$. If the absence probability is $q$ and the active effort support starts at zero, then $u=w{q}^{n-1}$.

A rank-order contest awards a sequence of prizes according to the ranks of efforts, while every player incurs its own effort cost. With an independent private values model, an increasing symmetric bidding function orders efforts in the same way as valuations.

Suppose values are nonnegative, the rank-order expected prize allocation $a$ is increasing and differentiable, and $a(\underline{v})=0$. With unit effort cost and zero effort at the lowest type, the symmetric equilibrium effort is $b(v)=va(v)-{\int }_{\underline{v}}^{v}a(t)dt$. A true type $v$ imitating type $z$ receives utility $va(z)-b(z)$, whose derivative in $z$ is $(v-z)a^{\prime }(z)$. It increases up to $v$ and decreases afterwards, proving the best response property. This is the interim payment identity specialized to an all-pay contest.

Let $V_{[1]}\ge \cdots \ge V_{[n]}$ be descending order statistics, with $w_n=0$ and nonnegative decreasing prizes. Decompose the prize vector into awards of $w_k-w_{k+1}$ to each of the best $k$ players. The corresponding truthful multi-unit auction charges each winner the next value $V_{[k+1]}$. Its total payment is $k(w_k-w_{k+1})V_{[k+1]}$. Revenue equivalence transfers the expected payment to the all-pay contest, because the interim allocations and lowest-type utilities coincide. Summing the layers proves the formula.

With $n$ independent uniform types, $m<n$ equal prizes of scale $w(m/n)$ and unit effort costs, a symmetric Bayesian Nash equilibrium has total expected effort $R_m=m(n-m)w(m/n)/(n+1)$. The all-pay effort identity yields $b(v)=w(m/n){\int }_0^{v}t{q}_m^{\prime }(t)dt$. The allocation derivative is a Beta distribution density with parameters $n-m,m$, so integration gives the formula.

For prize scale $w(x)=x^{-\alpha }$, total effort is proportional to $h(x)=x^{1-\alpha }(1-x)$ on the feasible grid $x=m/n$. If $\alpha \ge 1$, one prize is optimal. For $0<\alpha <1$, $h$ increases up to $x_{\ast }=(1-\alpha )/(2-\alpha )$ and then decreases, so the optimal feasible integer is among $\lfloor n{x}_{\ast }\rfloor$ and $\lfloor n{x}_{\ast }\rfloor +1$. Remove infeasible candidates and compare their objective values. No definition at $x=0$ is needed.

For $n$ players with independent valuations having a continuous distribution function $F$, a type $v$ occupies rank $k$ when exactly $k-1$ rivals have higher values. Multiplying this binomial distribution by the rank prize $w_k$ and summing gives $a(v)$. Ties occur only on zero-probability events when the valuation law is atomless.

A prize allocation rule assigns winning probabilities or divisible prize shares to a profile of efforts. In an all-pay auction, the greatest effort wins; a proportional allocation contest instead assigns shares continuously according to relative effort. The rule is essential data of a contest model.

A winning probability is the probability that a specified participant receives the prize, conditional on the submitted efforts and the allocation rule's randomization.