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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 BalinskiYoung 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.
 Spring '12
 Vanessen

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