All regimes with \(\theta\in[\theta^*,\theta^{**})\) choose the same apparent strength \(y^*=\theta^{**}=x^*+\gamma\) and consequently face the same sized attack \(S(y^*)=\Phi(-\sqrt{\alpha}\gamma)\). All regimes that do not manipulate have \(y(\theta)=\theta\) and face the attack \(S(y(\theta))=\Phi(\sqrt{\alpha}(x^*-\theta))\). Observe that the attack is continuous at the upper boundary \(\theta^{**}\).