Consider now the equilibrium behavior of lobby \(E\). Anticipating lobby \(I\)’s effort \(x\left(\delta\right)\), for all \(\delta\), lobby \(E\) chooses \(y\) to maximize his expected utility

Given that \(\min\left(\delta,k\right)-\delta=0\), for all \(\delta\leq k\), and \(\min\left(\delta,k\right)-\delta=k-\delta\), for all \(\delta>k\), it can be rewritten as

Moreover, in equilibrium, every player \(I\) with type \(\delta\geq k\) plays the same strategy, that is, \(x\left(\delta\right)=x\left(k\right)\), for all \(\delta\geq k\), implying that lobby \(E\)’s expected utility can be written as

If \(x\left(k\right)=0\), it is clear that \(y=0\) cannot be optimal. Indeed, since \(\pi\left(0,0\right)=1\) and \(\pi\left(0,y\right)=0\) for all \(y>0\), an infinitesimal bid allows lobby \(E\) to avoid the expected damage \(\int\nolimits_{k}^{1}\left(\delta-k\right)d\delta\). If \(x\left(k\right)>0\), we can differentiate lobby \(E\)’s expected utility with respect to \(y\), to obtain the first-order condition describing lobby \(E\)’s optimal effort

which holds with equality for an interior solution. Then, substitute \(x\left(k\right)=\sqrt{\left(b-k\right)y}-y\) to get

Clearly, it will be positive if \(y\) is small enough and negative if \(y\geq b-k\), meaning that lobby \(E\)’s equilibrium strategy satisfies \(0<y<b-k\). In turn, this implies that lobby \(I\)’s equilibrium strategy satisfies \(x\left(\delta\right)>0\).