
Alabama Paradox
The Alabama Paradox occurs when, (under the same number of states and the same state populations), an increase in the House size causes a state to lose a seat. It can happen because increasing the number of seats increases the fractional part of the standard quota at a faster rate for the larger populated states than for the smaller populated states.

The Population Paradox
occurs when two states have populations increasing at different rates and, (under the same House size and the same number of states), the state with faster growth loses a seat to the state with slower growth.

The New State Paradox
occurs when adding new states (keeping the House size and the state populations unchanged), increases the number of Representatives of some original states. Because of the additional state and the same number of seats, it is expected that states would get either the same or a smaller number of Representatives, but no more than originally.

THE QUOTA CONDITION
Another desirable property of apportionment methods is to always yield for each state a number of seats equal to either the state’s lower quota or upper quota. This is called the quota condition. All the divisor methods fail to satisfy the quota condition.

BIAS
An apportionment method is said to be biased if, when used time after time, it tends to benefit the large populated states (against low population states) or benefits the low populated states (against large populated states).

Jefferson’s Method is bias
favor of large populated states.

how is Webster’s Method biased
shows little bias (it shows very little bias in favor of large populated states).

Adams method is biased
in favor of low populated states.

HillHuntington’s Method is also biased
in favor of low populated states, but it shows a lot less bias than Adams Method.

Hamilton’s Method is biased how
Of the methods discussed in class, its purely unbiased

Fair Division methods:
Is methods intended for the distribution of a finite set of goods (or items) among a finite group of individuals in such a way that each individual receives a “fair share” of the whole. The individuals are called players, members or participants. In most cases, the goods (or items that must be distributed are physically indivisible).

Discrete
set of goods (or items that are not physically indivisible). Property

Continuous
Continuous: set of goods (or items are infinitely divisible). Bank accounts

mixed cases
of discrete and continuous. Like divorce, inheritance, chores.

proportionality
if a set of goods (or items) is to be divided among N individuals, then a proportional method guarantees to each member at least the fraction 1/n of the whole. We must realize that the fraction 1/n is measured from each participant’s point of view (or perception

envyfreeness
a fair division method must be envy free in the sense that since each participant receives at least the fraction 1/n of the whole, then no one will be envious of another person’s share.

equitability
a fair division method is said to be equitable if and only if each participant receives exactly the same share.

Equitability
a fair division method is said to be equitable if and only if each participant receives exactly the same share.

Pareto Optimality
(or efficiency) a fair division method is said to be Pareto optimal (or pareto efficient ) if and only if there does not exist another allocation that can make someone better off without making another person worse off.

