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\section{Extensions}
Two amendments of the benchmark model are sketched here. The first one introduces the incentive to free-ride within the environmental pressure group. The second one deals with the situation where both lobbies know the damage at the time of the contest game.
\subsection{Free-riding}
We depart from the benchmark model by assuming that the environmental lobby brings together $n>1$ identical individual members, whereas the industrial lobby still represents a single innovative firm. Each member $i$ of lobby $E$ exerts a nonnegative effort $y_{i}$. Given the \textit{aggregate} efforts of lobbies $I$ and $E$, $x$ and $y=\sum_{i=1}^{n}y_{i}$ respectively, the regulator approves the new product with the probability $\pi\left(x,y\right)$. If the product is marketed, each member $i$ of lobby $E$ bears a damage $\left(1/n\right)\delta$ and receives in court a compensation $\mbox{\ensuremath{\left(1/n\right)}min}\left(\delta,k\right)$ from lobby
$I$.\bigskip{} $I$.
\bigskip{}
A Nash equilibrium of the contest game, where lobby $I$ with type $\delta$ plays $x^{n}\left(\delta\right)$ and each member $i$ of lobby $E$ plays $y_{i}^{n}$, can easily be adapted from Section 3, yielding\footnote{There is an infinity of Nash equilibria, each allocating the same aggregate effort within lobby $E$.}
\begin{equation}
x^{n}\left(\delta\right)=\sqrt{\left(b-\min\left(\delta,k\right)\right)y^{n}}-y^{n}\mbox{, x^{n}\left(\delta\right)=
\sqrt{\left(b-\min\left(\delta,k\right)\right)y^{n}}-y^{n}\mbox{, for all
}\delta,\label{eq:xn} }\delta,
\label{eq:xn}
\end{equation}
\begin{equation}
y^{n}=\sum_{i=1}^{n}y_{i}^{n}=\left(\frac{\left(1-k\right)^{2}}{2n\left(b-k\right)+\left(1-k\right)^{2}}\right)^{2}\left(b-k\right).\label{eq:yn} y^{n}=\sum_{i=1}^{n}y_{i}^{n}
=\left(\frac{\left(1-k\right)^{2}}{2n\left(b-k\right)+\left(1-k\right)^{2}}\right)^{2}\left(b-k\right).
\label{eq:yn}
\end{equation}
From this, we can calculate the \textit{ex ante} expected social surplus\footnote{The calculus is not different from the one used in Section 5. See the Appendix. Clearly, we verify that $s^{*}\left(b,k\right)=s^{1}\left(b,k\right)$.}
$$
s^{n}\left(b,k\right)=
\left(b-\frac{1}{2}\right)
+\frac{\left(1-k\right)^{2}}{2n\left(b-k\right)+\left(1-k\right)^{2}}\left(\frac{1}{2}-2b+k+\frac{1}{3}b^{2}f\left(\frac{k}{b}\right)\right).
\label{eq:sn}
$$
\begin{alignat}
{1}
s^{n}\left(b,k\right)= & \left(b-\frac{1}{2}\right)+\frac{\left(1-k\right)^{2}}{2n\left(b-k\right)+\left(1-k\right)^{2}}\left(\frac{1}{2}-2b+k+\frac{1}{3}b^{2}f\left(\frac{k}{b}\right)\right).\label{eq:sn}
\end{alignat}
\bigskip{}
Pursuing the last section, we wish to know whether the incentive to free-ride should prompt a broader use of the contest as a social decision scheme. Comparing (\ref{eq:s*}) and (\ref{eq:sn}), we see that the free-riding behaviors do not harm the efficiency of the contest (i.e., $s^{*}\left(b,k\right)\leq s^{n}\left(b,k\right)$ for all $n$) if
and only if $1/2-2b+k+b^{2}f\left(k/b\right)/3\leq0$. However, this also implies that $s^{*}\left(b,k\right)\leq s^{n}\left(b,k\right)\leq b-1/2\leq\max\left(0,b-1/2\right)$, meaning that the cost-benefit analysis is a better decision scheme in both settings. These findings imply Proposition 2 below.
\bigskip{}
\textbf{\textit{Proposition 2.}} Compared with the benchmark model, the incentive to free-ride within the environmental pressure group should prompt to restrict further the use of the contest as a tool for social decision.
\bigskip{}
We can explain it as follows, using (\ref{eq:xn}) and (\ref{eq:yn}). The free-riding behaviors reduce the socially wasteful efforts of lobbying. The members of lobby $E$ reduce their efforts, in the prospect of benefiting from the other's contributions, and lobby $I$ replies by limiting his own effort in turn. However, the free-riding behaviors
increase the probability of approving the new product, no matter the level of harm. This is because lobby $I$ proportionally reduces his bid less than lobby $E$ does. In general, any of the two effects can dominate the other. Proposition 2 holds true because the former dominates the latter when the cost-benefit analysis is a better instrument.
\subsection{Public information}
Up to here, we have assumed that only the industrial lobby observes the damage \textsl{ex ante}. This is appealing if the latter, having registered a patent, is in a privileged position to conduct the laboratory experiments necessary to assess the detrimental effects of his product. However, the alternative assumption, where both lobbies are informed, is also relevant and should be
considered.\bigskip{} considered.
\bigskip{}
Given that both players observe $\delta$ before playing the contest, a Nash equilibrium of the contest game, where lobby $I$ plays $X^{*}\left(\delta\right)$ and lobby $E$ plays $Y^{*}\left(\delta\right)$, for all $\delta$, is standard to derive
(\citep{Nti1999}), (Nti, 1999), giving
\begin{equation}
X^{*}\left(\delta\right)=\begin{cases} X^{*}\left(\delta\right)=
\begin{cases}
0 & \mbox{if \ensuremath{\delta\leq k,}}\\
\frac{\left(b-k\right)^{2}\left(\delta-k\right)}{\left(b+\delta-2k\right)^{2}}, & \mbox{otherwise}.
\end{cases}\label{eq:X*} \end{cases}
\label{eq:X*}
\end{equation}
\begin{equation}
Y^{*}\left(\delta\right)=\begin{cases} Y^{*}\left(\delta\right)=
\begin{cases}
0 & \mbox{if \ensuremath{\delta\leq k,}}\\
\frac{\left(b-k\right)\left(\delta-k\right)^{2}}{\left(b+\delta-2k\right)^{2}}, & \mbox{otherwise}.
\end{cases}\label{eq:Y*} \end{cases}
\label{eq:Y*}
\end{equation}
From this, we can calculate the\textit{ ex ante} expected social surplus\footnote{The calculus can be found in the Appendix.}
\begin{alignat}
{1} \begin{equation}
S^{*}\left(b,k\right)=
& -2b+2k+3bk-\frac{5}{2}k^{2}+3\left(b-k\right)^{2}\ln\left(1+\frac{1-k}{b-k}\right).\label{eq:S*}
\end{alignat} -2b+2k+3bk-\frac{5}{2}k^{2}+3\left(b-k\right)^{2}\ln\left(1+\frac{1-k}{b-k}\right)
\label{eq:S*}
\end{equation}
\bigskip{}
Again, we wish to know whether the public information about the damage should encourage a larger use of the contest as a social decision scheme. Proposition 3 below states that this is indeed the case. The proof in the Appendix shows that the condition such that the contest is socially better is satisfied on a larger set of the parameters here than in the benchmark model.
\bigskip{}
Again, we wish to know whether the public information about the damage
should encourage a larger use of the contest as a social decision
scheme. Proposition 3 below states that this is indeed the case. The
proof in the Appendix shows that the condition such that the contest
is socially better is satisfied on a larger set of the parameters
here than in the benchmark model. \bigskip{} \textbf{\textit{Proposition 3.}} Compared with the benchmark model, the setting with public information should prompt to expand further the use of the contest as a tool for social
decision.\bigskip{} decision.
\bigskip{}
We can understand this using the equilibrium outcome (\ref{eq:X*}) and (\ref{eq:Y*}). In contrast with the benchmark model, we see here that the pressure groups exert zero effort when the damage is smaller than the level of assets of the industrial lobby (i.e., $\delta\leq k$). Therefore, not only less resources are wasted in lobbying activities,
but also the new product is approved with probability one (which is the socially optimal outcome since $kk$, the equilibrium outcome remains inefficient here.
\subsection{Policy prescription}
The main message that emerges from all settings is that the outcome of the contest is socially preferable than that of the cost-benefit analysis, provided that the industrial lobby risks in court a sufficient fraction of his benefit. Moreover, in the limit case where $k\rightarrow b$, our results even imply that the contest always implements the first-best allocation at no cost (i.e., $x+y\rightarrow0$, for all $\delta$, and $\pi\left(x,y\right)\rightarrow1$ if $\delta
