Snyder (1991) - On Buying Legislatures

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This paper analyzes a simple spatial voting model that includes lobbyists who are able to buy votes on bills to change the status quo . The key results are: (i) if lobbyists can discriminate across legislators when buying votes, then they will pay the largest bribes to legislators who are slightly opposed to the proposed change, rather than t o legislators who strongly support or strongly oppose the change; (ii) equilibrium policies exist, and with quadratic 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; and (iii) “moderate” lobbyists, whose positions on a policy issue are close to the median of the legislators’ ideal points, will prefer the issue to be salient, while more extreme lobbyists will generally prefer the issue to be obscure.

Model Setup

The model examines a policy space [math]x\in R[/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}\gt 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.

Rui's Notes about what the setup means

  • Median voter has ideal point 0.
  • Bribes conditional on votes, not outcomes.

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\gt s[/math] be an alternative proposal. If [math]0\lt (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\lt (x+s)/2\leq 1/2[/math], because otherwise the proposal passes without any bribes.

There is some discussion of the "truncated" nature of the bribe scheme. This refers to the fact that people whose ideal points are close to the status quo get zero.

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 proposal. Snyder restricts attention to cases where the lobbyist's ideal point [math]L\leq 1/2[/math]. Given this, the lobbyist's optimal bribes always satisfy Proposition 1.

Note that there are a few boring cases in which the lobbyist's strategy is quite dull. For example, if the status quo is equal to the lobbyist's ideal point ([math]s=L[/math] thn the lobbyist does nothing. If [math]s\leq-L[/math], then the the lobbyist can propose [math]x_{D}=L[/math], offer no bribes and win the vote.

The more interesting case is when [math]-L\lt s\lt L[/math]. Here, there exists an [math]s_{D}\in(0,L)[/math] such that (i) if [math]-L\lt s\lt s_{D}[/math], then the lobbyist's optimal proposal [math]x_{D}^{\ast}[/math] is unique, and satisfied [math]\max(s,-s,s_{D})\lt x_{D}^{\ast}\lt L[/math]; 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\lt 0, \partial x_{D}^{\ast}/\partial L\gt 0[/math] and [math]{\partial} x_{D}^{\ast}/\partial \alpha\lt 0[/math].

Rui's Notes

  1. Without bribes, if s<0, the lobbyist can get -s>0.
  2. If [math]s\geq 0[/math], can choose s.
  3. L never chooses [math]x\lt \max\{s,-s\}[/math].
  4. As the status quo s changes, there are three regions. Where [math]s_{D}\lt s\lt L, x=s[/math].
  5. Rui says we "get variation" based on changes on alpha and s.
  6. Risk aversion plays a big role here.
  7. Anchoring effect of s, median.
  8. This provides predictions.


  • Models suggest cut points in models with no vote buying and incomplete OR complete info.
  • For models with vote buying, cut point appears in complete information but not incomplete information.

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.

Rui said this was not as interesting.