 Best-response selection at a flat leader objective

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.