Authorea

Altogether, this shows that the Nash equilibrium of the contest game is unique and interior.¹¹It is worth noting that this result is not specific to the uniform distribution of damage. In fact, it remains true for any cumulative distribution of damage \(F\left(\delta\right)\) such that \(\int\nolimits_{k}^{1}\left(\delta-k\right)dF\left(\delta\right)>0\). We thank David Malueg for pointing us this point. Solving (\ref{eq:focE}), we obtain that the equilibrium efforts of lobbies \(I\) and \(E\) equal

\begin{equation} \label{eq:x*} \label{eq:x*}x^{\ast}\left(\delta\right)=\sqrt{\left(b-\min\left(\delta,k\right)\right)y^{*}}-y^{*}\mbox{, for all }\delta,\\ \end{equation}

\begin{equation} \label{eq:y*} \label{eq:y*}y^{\ast}=\left(\frac{\left(1-k\right)^{2}}{2\left(b-k\right)+\left(1-k\right)^{2}}\right)^{2}\left(b-k\right).\\ \end{equation}