Changes

The legislature is infinitely sized and consists of individual legislators whose ideal points <math>z</math> are distributed uniformly over [-0.5,0.5] (so z~<math>U[-0.5,0.5]</math>). Legislators preferences preferences are also negative quadratic. A legislator will choose policy x over policy y iff <math>b_{x}-\alpha(x-z)^{2}>b_{y}-\alpha(y-z)^{2}</math>, where <math>b_{x},b_{y}</math> refer to the amount of bribes offered for voting for position x or y, and z is the legislator's ideal point. The parameter <math>\alpha</math> represents the "intensity" of the legislator's preferences -- ie, how much he cares. One might alternatively think of <math>\alpha</math> as how much his constituents care.

The paper first attempts to solve for equilibrium strategies in which the lobby know the individual legislator's ideal points and can offer bribes that depend on their ideal points.

The more interesting case is when <math>-L<s<L<math>. Here, there exists an <math>s_{D}\in(0,L)</math> such that (i) if <math>-L<s<s_{D}</math>, then the lobbyist's optimal proposal <math>x_{D}^{\ast}</math> is unique, and satisfied <math>\max(s,-s,s_{D})<x_{D}^{\ast}<L; and (ii) if <math>s\geq s_{D}</math>, then the lobbyist does nothing and s remains the policy outcome. In case (i), <math>\lim_{s\rightarrow s_{D}}x_{D}^{\ast}=s_{D}</math> and has comparative statics of <math>\partial x_{D}^{\ast}/partial s<0, \partial x_{D}^{\ast}/partial L>0</math> and <math>{\partial} x_{D}^{\ast}/partial \alpha<0</math>.

The paper continues to solve for equilibrium strategies in which the lobbyist does NOT know the individual legislator's ideal points and must offer all legislators the same bribe.