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\centerline{\bf STANFORD UNIVERSITY}
\centerline{\bf DEPARTMENT OF STATISTICS}
\centerline{\big DEPARTMENTAL SEMINAR}
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\centerline{4:15 p.m., Tuesday, January 11, 2000}
\centerline{Sequoia Hall Rm. 200}
\centerline{(Cookies at 3:45 in 1st Floor Lounge)}

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\centerline{\sl Ingram Olkin}
\centerline{\sl Stanford University}
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\centerline{\bf Apportionment of Seat in Proportional Representation Systems:}
\centerline{\bf A Majorization Comparison of Divisor Methods}
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Proportional representation is applied to such problems as the
apportionment of a number of seats to each party proportionally to the
number of votes received, or the apportionment of a number of seats to each
constituency proportionally to its population. From the inception of the
proportional representation movement it has been an issue whether larger
parties are favored at the cost of smaller parties in one apportionment of
seats as compared to another apportionment. A number of methods have been
proposed and used in countries with a proportional representation system.
These methods exhibit regularity of order that captures the preferential
treatment of larger versus smaller parties. This order, namely
majorization, permits the comparison of seat allocation in two
apportionments. For divisor methods, we show that one method is majorized
by another method if and only if their signpost ratios are increasing. This
criterion is satisfied for the divisor methods with power mean rounding,
and the divisor methods with stationary rounding. Majorization places the
five traditional apportionment methods in the order as they are known to
favor larger parties over smaller parties: Adams, Dean, Hill, Webster, and
Jefferson.

(Joint work with A.W. Marshall and F. Pukelsheim)

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