Politician’s payoff \(v^*\) as a function of \(\alpha\) for \(c>1\) and \(c<1\). The politician’s payoff is strictly decreasing in \(\alpha\) and \(c\). Hence \(v^*>v_{nm}^*\). For \(c>1\) the politician’s payoff \(v^*\rightarrow 0\) as \(\alpha\rightarrow\infty\). For \(c<1\) the politician’s payoff \(v^*\rightarrow 1-c\) as \(\alpha\rightarrow\infty\). In this sense the politician’s gain from manipulation \(v^*-v_{nm}^*\) is large when \(\alpha\) is large and \(c\) is small.