Chapter 4 study questions   Contemporary Mathematics Fall 2009

1. Apportionment is a type of fair division; we say we have an apportionment problem when the items to be divided up have what properties (think about legislative seats versus cakes; also, think seats versus estates consisting of, say, a boat, a car, and a lamp - how is apportionment different than the cake or the estate problems?)
   
2. If we are given the total population and the number of seats to apportion, how do we find the standard divisor?  The standard quota?
3. What's a lower quota?  An upper quota?  What important property do they have that makes them more useful in apportionment problems than standard quotas? (Look at an apportionment example for a hint).
4. What's the quota rule?  What's an upper quota violation?  A lower quota violation?
5. Make sure, of course, that you can compute the apportionment for a problem by Hamilton's, Jefferson's, Adams' and Webster's methods.
6. Which apportionment methods that we have studied can have quota violations?  (Also, what kind of quota violations can they have?)
1. Why is it impossible for Hamilton's method or Jefferson's method to have lower quota violations?
2. Why is it impossible for Adams' method to have upper quota violations?
7. What are the three apportionment paradoxes we've studied?  Give both their names and their descriptions.
8. Which apportionment methods that we have studied can give rise to which of the paradoxes we've studied?
9. Which apportionment methods that we have studied favor large states?  Small states?
   
10.  (Harder) Why does Jefferson's method favor large states?  Hint:  When the modified divisor goes down, which states' modified quotas rise the fastest?
11. What method of apportionment is currently under use in the US congress (see Historical Note and also the supplement in chapter 4 after the homework problems).  While I won't ask for precise details, say very roughly how it works, building on analogies with the other methods.
12. Is there a "best" apportionment method?  See p. 142 in chapter 4's conclusion.
    

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