Changes

For simplicity consider the case where <math>z_m < y < \frac{x+y}{2} </math>. Putting these points on a line divides the line into four regions. The resource provision required to make a legislator indifferent in each region is:

<math>z \in [-\infty,z_m] \quad r_xr^*=0</math><math>z \in (z_m,y] \quad r_xr^*=\alpha (x-y)</math><math>z \in (y,\frac{x+y}{2}] \quad r_xr^*=2 \alpha \left (\frac{x+y}{2} + z \right )</math><math>z \in (\frac{x+y}{2},\infty] \quad r_xr^*=0</math>

Note that the legislators with ideal points <math>z_m > \frac{x+y}{2}</math> always vote for the interest's policy and there is no need to contribute resources to them. Likewise in it unnecessary to contribute to legislators below the median, at least if there is no uncertainty of types and a majority rule is in place (etc). Further more the resources needed are decreasing in <math>z</math> for <math>z \in (z_m,\frac{x+y}{2}]</math>, so interests must provide more resources to more strongly opposed legislators, and are strictly increasing in <math>x</math>.