950 CHAPTER 14 Voting and Apportionment When a new group is added to the apportionment, we must determine how many additional items should be added to the total to be apportioned. In Example 3, we were told that six additional laptops were to be added to the total apportioned when a new library was included in the apportionment. If we are given the new group’s population but are not given the number of additional items to be added to the total to be apportioned, we calculate the number of items to be apportioned to the new group as follows. First determine the new group’s standard quota. The number of additional items would be the new group’s standard quota rounded down to the nearest integer. For example, in Example 3, the new library, library C, served a population of 625. The standard quota for library C would be 625 100 6.25 = Rounding 6.25 down to the nearest integer gives us six laptops to be added to the total to be apportioned, or 106 laptops. As we have discovered in Section 14.3, Hamilton’s method appears to be a fair and reasonable apportionment method, since it satisfies the quota rule. In this section, however, we discovered that Hamilton’s method can produce paradoxes. Jefferson’s, Adams’, and Webster’s methods can all violate the quota rule but do not produce paradoxes. Hamilton’s and Jefferson’s apportionment methods can favor large states, whereas Adams’ and Webster’s apportionment methods can favor small states. Is there a perfect apportionment method that satisfies the quota rule, does not produce any paradoxes, and favors neither large nor small states? In 1980, mathematicians Michel Balinski and H. Payton Young proved that there is no apportionment method that satisfies the quota rule while also avoiding all known paradoxes. Their theorem is called Balinski and Young’s impossibility theorem. Learning Catalytics Keyword: Angel-SOM-14.4 (See Preface for additional details.) Balinski and Young’s Impossibility Theorem There is no perfect apportionment method that satisfies the quota rule and avoids all known paradoxes. Table 14.52 summarizes the four apportionment methods we have discussed in this chapter and indicates which methods may violate the quota rule and which methods may produce the paradoxes we have discussed. Table 14.52 Comparison of Apportionment Methods Apportionment Method Hamilton Jefferson Adams Webster May violate the quota rule (apportionment should always be either upper or lower quota) No Yes Yes Yes May produce the Alabama paradox (an increase in the total number of items results in a loss of an item for a group) Yes No No No May produce the population paradox (group A loses an item to group B although group A’s population grew faster than group B’s population) Yes No No No May produce the new-states paradox (the addition of a new group reduces the apportionment of another group) Yes No No No Apportionment method favors Large states Large states Small states Small states Just as there is no perfect voting method, there is also no perfect apportionment method.
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