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Modeled as an n-person cooperative gameThe paper studies two games: A majority rule game with some special features. The authors call it "Distributive Legislative Game" (DLG) and the "Distributive Universalism Legislative Game" (ULG) and compares the expected utility of the two games to the legislators." In the gameboth games, each representative a legislator <math>i</math> proposes a project or program with total benefits <math>b</math> and costs <math>c<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 possiblein 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=\frac{N+1}{2N}</math> probability of being part of the prevailing MWC (where N is the number of legislators, assumed to be odd). 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. If a legislator is part of the winning coalition, one gets <math>b</math> benefits and costs 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>\frac{(N+1)c}{2N}=ac</math>.  The benefit to a random district in the ULG is equal to: <math>a(bc-ac)</math> .