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Use the usual continuous-distribution convention $F(0)=0$, $F(1)=1$, and write $f=F^{\prime }$. After observing $b$, the follower can win by matching it, because ties favor the follower. Its best response is to match when $v_2>b$ and choose zero when $v_2<b$; the indifferent equality has probability zero. Thus the leader wins with probability $F(b)$ and a leader of type $v$ maximizesBids above $1$ are dominated by bidding $1$. This is the leader optimization in a sequential private-value all-pay contest. Since $F$ is a concave function, $h_v$ is concave. An interior optimum satisfies $vf(b)=1$, with the usual endpoint conditions when that equation has no interior solution.

For precision, select the smallest maximizer when the leader is indifferent. This defines a Stackelberg equilibrium and supplies the printed strict conditional comparison at the threshold type. Let $q=F^{-1}(1/2)$. The density $f(q)$ is positive: if it vanished there, its nonnegative nonincreasing continuation would force $F$ to remain $1/2$ up to $1$, which is impossible. At the median,If this is positive, every maximizer is strictly greater than $q$, so the leader wins with probability greater than $1/2$. If it is negative, every maximizer is strictly smaller than $q$. If it is zero, $q$ is a maximizer, and the smallest maximizer is at most $q$. Therefore the median-density threshold for the leader in an all-pay contest isStrict concavity of $F$ would make the optimum unique, removing the selection convention. With mere concavity, the printed assertion needs that convention at equality. For example, $F(b)=b$ and $v=1$ make all bids optimal, and choosing $b=3/4$ makes the leader more likely to win even though the strict threshold is not exceeded.

The same issue can occur at an interior type, rather than just an endpoint. A continuously differentiable concave distribution isIts density decreases from $2$ to $5/4$, is constant on $[1/5,2/5]$, and then decreases to $1/24$; integration gives $F(1)=1$. Its median is $31/100$. At $v=4/5=1/f(q)$, every bid in $[1/5,2/5]$ maximizes $h_v$, and choosing $2/5$ gives winning probability $49/80>1/2$. This confirms the genuine best-response selection at a flat leader objective issue. The smallest-maximizer convention avoids it; the threshold type has zero ex ante probability.

The unconditional comparison is valid for every optimal selection. Zero effort guarantees the leader payoff zero, so an optimal bid satisfiesConsequently $F(b(v))\le F(v)$. If $V$ has the continuous distribution $F$, the probability integral transform makes $F(V)$ uniform on $[0,1]$. Taking expectations proves the ex ante follower advantage in a sequential all-pay contest:For strictly increasing atom-free $F$, optimal bids in fact satisfy $b(v)<v$ for almost every $v\in (0,1)$, so the first inequality is strict. Conditional advantage for unusually high leader types is therefore compatible with an unconditional follower advantage.

A sequential model in game theory in which a leader commits to an action before a follower observes it and chooses a best response. The leader anticipates the follower's response when optimizing. With private follower information, the leader maximizes its expected payoff over follower types. The resulting solution is a Stackelberg equilibrium.