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<blockquote>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)>a(b-c)</math> since <math>1<a<0/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.]</blockquote>
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