Equilibrium manipulation \(\delta^*\) as a function of \(c\) for \(\alpha<4\) (lighter), \(\alpha=4\) and \(\alpha>4\) (darker). For \(\alpha\leq 4\), the manipulation \(\delta^*\) is continuous in \(c\). But for \(\alpha>4\) the manipulation jumps discontinuously at \(c=1\). In the limit as \(\alpha\rightarrow\infty\) the boundaries \(\underline{\delta}(\alpha)\rightarrow 0^+\) and \(\overline{\delta}(\alpha)\rightarrow 1^+\) so that the manipulation jumps by the maximum amount, from \(\delta^*=0\) if \(c<1\) to \(\delta^*=1\) if \(c>1\).