deletions | additions       diff --git a/sectionComparison_of.tex b/sectionComparison_of.tex  index 01fba7d..a89ee1b 100644  --- a/sectionComparison_of.tex  +++ b/sectionComparison_of.tex 
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\begin{align}  s^{*}\left(b,k\right)  = & \int\nolimits _{0}^{1}\left[\pi\left(x^{\ast}\left(\delta\right),y^{*}\right)\left(b-\delta\right)-x^{\ast}\left(\delta\right)-y^{\ast}\right]d\delta\nonumber _{0}^{1}  \left[  \pi\left(x^{\ast}\left(\delta\right),y^{*}\right)\left(b-\delta\right)  -x^{\ast}\left(\delta\right)-y^{\ast}  \right]  d\delta\nonumber  \\ = & \left(b-\frac{1}{2}\right)+\frac{\left(1-k\right)^{2}}{2\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:s*}  \end{alignat} \left(b-\frac{1}{2}\right)  +\frac{\left(1-k\right)^{2}}{2\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:s*}  \end{align}  where we define for all $0\leq k/b\leq1$  $$  f\left(\frac{k}{b}\right)= 
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\bigskip{}  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}  \end{equation}  The purpose of the remaining of this section is to derive conditions such that this inequality holds true. \bigskip{}  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{alignat*}  {1} \begin{align*}  \lim_{k\rightarrow0}s^{*}\left(b,k\right)= & \frac{2b}{1+2b}\left(b-1\right)<0.  \end{alignat*} \end{align*}  As condition (\ref{eq:s*>max}) is false, the cost-benefit analysis  is a better decision scheme.\bigskip{}