deletions | additions       diff --git a/sectionComparison_of.tex b/sectionComparison_of.tex  index a89ee1b..b973e5b 100644  --- a/sectionComparison_of.tex  +++ b/sectionComparison_of.tex 
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Clearly, from the social point of view, the contest is preferable if and only if  \begin{equation}  s^{*}\left(b,k\right)>\max\left(0,b-1/2\right).\label{eq:s*>max} s^{*}\left(b,k\right)>\max\left(0,b-1/2\right).  \label{eq:s*>max}  \end{equation}  The purpose of the remaining of this section is to derive conditions such that this inequality holds true. 
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Let us first consider the situation where the level of assets of lobby $I$'s is small (i.e., $k\rightarrow0$). Then, we can calculate that  \begin{align*} $$  \lim_{k\rightarrow0}s^{*}\left(b,k\right)= & \frac{2b}{1+2b}\left(b-1\right)<0.  \end{align*} $$  As condition (\ref{eq:s*>max}) is false, the cost-benefit analysis is a better decision scheme.\bigskip{} scheme.  \bigskip{}  Let us now consider the situation where the level of assets of lobby $I$'s is large relative to his benefit (i.e., $k\rightarrow b$). Then, we can show that \begin{alignat*}  {1} $$  \lim_{k\rightarrow b}x^{\ast}\left(\delta\right)+y^{*} & =0  \end{alignat*}  \begin{alignat*}  {1} $$  $$  \lim_{k\rightarrow b}\pi\left(x^{\ast}\left(\delta\right),y^{*}\right)= b}\pi\left(x^{\ast}\left(\delta\right),y^{*}\right)  =  & \begin{cases} 1\mbox{,} & \text{if }\delta  \end{alignat*} $$  In other words, the contest costlessly implements the first-best decision rule. Of course, condition (\ref{eq:s*>max}) is then true, meaning that the Tullock contest is a better decision scheme.\footnote{We show that the expected social welfare resulting from the contest game equals $\lim_{k\rightarrow b}s^{*}\left(b,k\right)=b^{2}/2$.}\bigskip{}  Intuitively, these findings suggest that the second-best decision  scheme will depend on the fraction of his benefit that the industrial  lobby can lose in court. Proposition 1 below rigorously confirms this.  \bigskip{}  \[ \bigskip{}  a\left(b\right)\in\left(0,1\right)\mbox{,} Intuitively, these findings suggest that the second-best decision scheme will depend on the fraction of his benefit that the industrial lobby can lose in court. Proposition 1 below rigorously confirms this.  \] \bigskip{}  \textbf{\textit{Proposition 1.}} For all $b$, there exists  $$  a\left(b\right)\in\left(0,1\right)\mbox{,}  $$  such that  $$  s^{*}\left(b,k\right)>\max\left(0,b-1/2\right)  \mbox{ if and only if }  k/b>a\left(b\right)\mbox{.}  $$  \[ \bigskip{}  s^{*}\left(b,k\right)>\max\left(0,b-1/2\right)\mbox{ To rephrase it, Proposition 1 says that  if and only if }k/b>a\left(b\right)\mbox{.} the industrial lobby risks in court a sufficiently large fraction of his benefit, in the sense that the ratio of his level of assets with respect to his benefit, $k/b$, is larger than a given threshold, $a\left(b\right)$, then it is socially preferable, \textit{ex ante}, that the government decides  to approve, or to ban, the new product according to the Tullock contest, rather than according to the cost-benefit analysis (and reciprocally).  \] \bigskip{}  such Intuitively, this result can be explained as follows. On the one hand, as the lobbyists divert resources from productive activities, the contest is a costly mechanism. This favors a decision according to the cost-benefit analysis. On the other hand, as lobby $I$ is held liable for damage, the larger the damage is, the less effort lobby  $I$ has incentive to expend in the contest, as long as he can afford to pay for it in court (i.e., when $\delta  that the contest performs very badly when $k\rightarrow0$ and costlessly implements the first-best decision rule when $k\rightarrow b$.  \bigskip{}  To rephrase it, Proposition 1 says that if the industrial lobby risks  in court a sufficiently large fraction of his benefit, in the sense  that the ratio of his level of assets with respect to his benefit,  $k/b$, is larger than a given threshold, $a\left(b\right)$, then  it is socially preferable, \textit{ex ante}, that the government decides  to approve, or to ban, the new product according to the Tullock contest,  rather than according to the cost-benefit analysis (and reciprocally).\bigskip{}  Intuitively, this result can be explained as follows. On the one hand,  as the lobbyists divert resources from productive activities, the  contest is a costly mechanism. This favors a decision according to  the cost-benefit analysis. On the other hand, as lobby $I$ is held  liable for damage, the larger the damage is, the less effort lobby  $I$ has incentive to expend in the contest, as long as he can afford  to pay for it in court (i.e., when $\delta  probability of approving the new product is thus decreasing in the  harm. This favors a decision using the contest. Proposition 1 holds  true, simultaneously because increasing $k$ reduces the lobbying  efforts and enlarges the region where the probability of approving  the new product is negatively correlated to the harm. The picture  is complete recalling that the contest performs very badly when $k\rightarrow0$  and costlessly implements the first-best decision rule when $k\rightarrow b$.\bigskip{} Finally, for the sake of succinctness, Proposition 1 focuses on the existence of the threshold $a\left(b\right)$ but remains silent about its value.\footnote{In fact, we show in the proof of Proposition 1 that $a\left(b\right)$ is bounded below by $\tau\left(b\right)=\left(1/b\right)\max\left\{ 2\sqrt{1-b}(1-\sqrt{1-b}),(2\sqrt{b}-1)\right\} $\tau\left(b\right)=\left(1/b\right)\max\left\{2\sqrt{1-b}(1-\sqrt{1-b}),(2\sqrt{b}-1)\right\}  $.} Of course, in practice, it would be useful to be more precise, in order to better identify the situations where to use the contest. Figure \ref{Fig1} below plots the frontier $a\left(b\right)$. It shows emphasizes  that the use of the contest should be limited to situations where the industrial lobby risks in court a large fraction of his benefit.\bigskip{}