Social Choice_Lecture 10_Apportionment

Peyton young that the only apportionment methods that

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Unformatted text preview: onment 11 / 15 A Glimpse of Impossibility There is a result due to Michael Balinski and H. Peyton Young that the only apportionment methods that satisfy the population property are the divisor methods. But divisor methods, it turns out, are never guaranteed to satisfy the quota property. Thus, we have an impossibility theorem. The Balinski-Young result is complicated, but the following weaker theorem gives us a taste of the impossibility. Theorem There is no apportionment method that satis…es the monotonicity property, the quota condition, and the population property. Van Essen (U of A) Apportionment 12 / 15 A Glimpse of Impossibility Proof. Assume that we have an apportionment scheme that satis…es monotonicity and the quota condition. We’ show that it fails to satisfy the population ll property. Suppose we have 7 seats, 4 states, and a total population of 4200 distributed State Population Ideal Quota A 3003 5.005 B 400 0.667 C 399 0.665 D 398 0.663 Van Essen (U of A) Apportionment 13 / 15 A Glimpse of Impossibility Proof Cont. Because of the quota condition and the monotonicity condition, the only possible apportionments are (5,1,1,0) and (6,1,0,0) (why?). Speci…cally, A gets at least 5 seats and state D gets no seats. Now suppose after the next census there are 1100 additional people such that: State Population Ideal Quota A 3004 (+1) 3.96 B 1503 (+1103) 1.985 C 396 (-3) 0.523 D 397 (-1) 0.524 Van Essen (U of A) Apportionment 14 / 15 A Glimpse of Impossibility Proof Cont. Because of the Quota and Monotonicity conditions, the only possible apportionments are (4,2,0,1), (4,1,1,1), and (3,2,1,1). Now state A gets at most 4 seats and state D gets at least 1 seat. Thus, State A has increased in population and lost a seat whereas State D decreased in population and gained a seat. Van Essen (U of A) Apportionment 15 / 15...
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This note was uploaded on 04/08/2014 for the course ECON 497 taught by Professor Vanessen during the Spring '12 term at Alabama.

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