View source for Snyder (1991) - On Buying Legislatures

[http://www.edegan.com/wiki/images/d/da/On_Buying_Legislatures_by_James_Snyder.pdf pdf of paper]

==Abstract==

This paper analyzes a simple spatial voting model that includes lobbyists who 
are able to buy votes  on bills to  change the status q u o .  T h e  key  results a r e :  
(i) if  lobbyists can discriminate across legislators when buying votes, then 
they will pay the largest bribes  to legislators wh o  a r e  slightly opposed t o  the 
proposed change, rather than t o  legislators who strongly support o r  strongly 
oppose  the change;  (ii)  equilibrium  policies exist,  and  with  q u a d r a t i c  
preferences  these equilibria always lie between t h e  average of t h e  lobbyists’ 
ideal  points  a n d   the median  of  the legislators’  ideal  points;  a n d   (iii) 
“moderate”  lobbyists,  whose  positions  on  a  policy issue  a r e  close t o  the 
median of the legislators’ ideal points, will prefer the issue t o  be salient, while 
more extreme  lobbyists  will  generally prefer  the  issue t o  be  obscure.

==Model  Setup==

The model examines a policy space <math>x\in[-0.5,0.5]</math> and a lobbyist's efforts to bribe legislators to adopt a policy near his ideal point. The lobbyist's utility function is <math>u(x,B)=-(x-L)^{2}</math>, where x is the policy chosen, L is the lobbyist's ideal point and B is the total number of bribes paid to legislators. The lobbyist is assumed to have an infinite budget.  


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.

==Model Solution with Price Discrimination (1) ==

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. 

===Proposition 1===

Suppose s is the status quo and let <math>x>s</math> be an alternative proposal. If <math>0<(x+s)/2\leq 1/2</math>, then the least cost payment function <math>b_{D}(\dot,x,s)</math> which insures that x ties or defeats s is given by the following: If <math>z\in[0,(s+x)/2], b_{D}(z,x,s)=\alpha(x^2-s^2-2(x-s)z)<math>, and <math>b_{D}(z,x,s)=0</math> otherwise. Proof is in the appendix and is not complicated. Note that the above discusses proposals <math>x</math> only when <math>0<(x+s)/2\leq 1/2</math>, because otherwise the proposal passes without any bribes. 

Note: Highest bribes paid to legislators whose ideal points are close to the median, but close to his side of the median. The lobbyist does not bribe his close supporters, but rather his marginal supporters. Close supporters will vote for a motion even without bribes. 

===Proposition 2===

Now suppose that the lobbyist has some agenda power, and wants to make a proposal <math>x</math> and then bribe the legislators to vote for his 

==Model Solution without Price Discrimination (2) ==

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.