Weingast, B. (1979), A Rational Choice Perspective on Congressional Norms
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The author notes that lots of formal models of legislatures suggest that "minimum winning coalitions" should prevail. Ie, the winning coalition will have a size of 50% of the total legislators, plus one. This should happen because the majority should attempt to divide the benefits of a project to as few members as possible while still having enough votes to pass a majority vote on the project.
This never seems to appear in real life: Winning coalitions are often much larger than 50%. The paper's goal is to develop a formal model to explain these larger margins. The author achieves this by modeling the "informal rules" seen in Congress within the formal game theoretic setup. The author shows how legislative rules that lead to large majorities are better for legislators than rules that lead to smaller majorities -- thus suggesting how such rules could come about endogenously.
The paper studies two games: The "Distributive Legislative Game" (DLG) and the "Universalism Legislative Game" (ULG) and compares the expected utility of the two games to the legislators. In both games, a legislator [math]i[/math] proposes a project or program with total benefits [math]b[/math] and costs [math]c\lt b[/math]. The benefits to the [math]i[/math]th project accrue entirely to district [math]i[/math], but the costs are distributed equally to all districts. No side payments possible in either game. Both games are majority rule.
Given this setup: A legislator who proposes his project alone will be rejected by everyone else. Therefore some coalition building and logrolling is necessary: Rather than voting on single projects, legislators will vote on collections of them. If a legislator is part of the winning coalition, she gets the benefits of her own district's projects and pays an equally distributed slice of the costs. If a legislator is NOT part of the winning coalition, she still pays an equally distributed slice of the costs but gets no benefits.
The ULG game requires unanimous consent, and the DLG game is majority rule. In Proposition 1, the authors prove that in a DLG -- the smallest possible majority will prevail (the "minimum winning coalition", or "MWC"). Because the model does not feature committees, seniority, parties or other sources of varying power between legislators -- the model assumes that all possible MWCs are equally likely. As such, each legislator has a [math]a=(N+1)/2N[/math] probability of being part of the prevailing MWC (where N is the number of legislators, assumed to be odd. Proof on page 251). In the ULG -- each member has a probability of being part of the winning coalition equal to 1.
This brings us to Proposition 2, which shows the following: If risk-neutral legislators are trying to maximize the payoff to their constituents, they will prefer the ULG to the DLG.
Proof: I'll first study the expected benefits of being part of the winning coalition. The benefit of being a part of the winning coalition is [math]b[/math], and the benefit of not being part of the coalition is zero. The costs are the same no matter what: Equal to [math]1/N[/math]th of the costs of [math](N+1)/2[/math] projects. This multiplies out to be a cost of [math]c(N+1)/2N=ac[/math]. As such, the expected return to a district is equal to [math]a(b-ac)+(1-a)(-ac)=a(b-c)[/math]
Turning to the ULG: The net benefit is equal to [math]b-c[/math] no matter what. We can easily show that [math](b-c)\gt a(b-c)[/math] since [math]1\lt a[/math]. Therefore the ULG maximizes expected benefits to constituency. [Editorial comment from Bo: This would probably be even moreso if the legislator was risk averse rather than risk-neutral.]
From here the author studies the relaxation of the [math]b\gt c[/math] assumption.
- First he shows the conditions under which legislators will propose their project --assuming that all other legislators are proposing theirs.
- Then the author discusses the notion that [math]b/c[/math] is decreasing over time -- so that the "worthwhile" projects are used early in the game, and the bad ones will continue later. The author notes that at some point, pork is no longer rational for the legislators and voters. However, this happens sooner in the majority rule than the universal rule.