Existence of Nash equilibrium with full compensation of damage.

We show that if \(\pi\left(0,0\right)=\alpha<1\), then no equilibrium exists when lobby \(E\) is fully compensated for damage. Indeed, if fully compensated for damage, \(y=0\) is a dominant strategy for lobby \(E\). As a result, lobby \(I\) chooses \(x\) to maximize \(\pi\left(x,0\right)\left(b-\min\left(\delta,k\right)\right)-x\). If \(\alpha<1\), no equilibrium strategy exists for lobby \(I\). Indeed, the strategy \(x=0\) is dominated by any strategy \(x\) such that \(0<x<\left(1-\alpha\right)\left(b-\min\left(\delta,k\right)\right)\); and for all \(x>0\), the expected utility is strictly decreasing in \(x\). \(\blacksquare\)