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Sébastien Rouillon added In_the_rest_of_the__.tex
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In the rest of the paper, we will admit the following conditions.\bigskip{}
\textbf{\textit{Assumption 1. }} The new product may or not be socially worthwhile, but group $I$ always wants to market it. Formally, the parameters $b$ and $k$ are such that $0
\textbf{\textit{Assumption 2. }} The contest success function is such that the regulator approves the new product if lobby $E$ does not object it. Formally, $\pi\left(x,0\right)=1$ for all $x\geq0$ (in particular, $\alpha=1$).
It should be clear that Assumption 1 is the only case of interest. On the one hand, if $b\geq1$, it is always socially optimal to market the product (because $\delta\leq1\leq b$), implying that the regulator can trust his prior beliefs to implement the first-best decision. On the other hand, if $k\geq b$, both decision schemes can implement
the first-best allocation at no cost. This is because lobby $I$ only wants to market the new product if he can earn a profit (i.e., $b\geq\mbox{min}\left(\delta,k\right)$) and will then always afford to pay for damages in court (because $k\geq b$ and $b\geq\mbox{min}\left(\delta,k\right)$ implies $k\geq\delta$).
By Assumption 2, the contest success function used here slightly differs from the literature. The standard assumption, justified by an anonymity condition (Skaperdas, 1996), is that if the players exert identical efforts, their probability of success should be the same. The contest success function (\ref{eq:proba}) satisfies it almost everywhere, except when neither player bids.\footnote{Formally, $\pi\left(x,y\right)=1/2$ for all $x=y>0$, but $\pi\left(x,y\right)=1$ when $x=y=0$.} We justify our choice as follows. Firstly, the condition of anonymity
is not as natural in our application, where the players are not interchangeable. Secondly, the consent of any new product that no one contests may be seen as a rule to ease technical progress. Thirdly, Assumption 2 is necessary to ensure the existence of an equilibrium when lobby $E$ is fully compensated for damage (see the Appendix). Finally,
it does not impact our results, since we show below that the Nash equilibrium of the contest game is interior (i.e., $x>0$ and $y>0$).
From Assumption 1, \textit{ex post}, given $\delta$, the expected utilities of $I$ and $E$ are respectively equal to
\begin{equation}
\pi\left(x,y\right)\left(b-\min\left(\delta,k\right)\right)-x,\label{eq:uI}
\end{equation}
\begin{equation}
\pi\left(x,y\right)\left(\min\left(\delta,k\right)-\delta\right)-y.\label{eq:uE}
\end{equation}
