Sequential private-value all-pay contest

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.