[EM] Power of Voting Blocs
Alexander Small
asmall at physics.ucsb.edu
Fri Feb 1 18:32:59 PST 2002
When discussing power in the Electoral College we discussed things like a
nation where 3 states have 3 votes and 1 state has 2 votes. The state with
2 votes has NO power because you need 6 votes for a majority, and any 2 of
the other states will suffice. Here's a series of progressively more
complex situations:
Each state has 2, 3, or 4 votes, and the average is 3. I use this
particular example because I understand that the German Parliament uses
this system in its upper house. More generally, many national legislatures
have bottom-heavy upper houses, and this is an example of a bottom-heavy
upper house that isn't flat (like the US Senate).
1) If all but 2 states have 3 votes and the total number of states is odd
then each state has equal power.
Proof: For 2n+1 states there are 6n+3 votes, and you need 3n+2 votes for a
majority. If n-1 states with 3 votes support a measure then 5 more votes
are needed and any 2 states will suffice to make the quota, but not any
single state.
2) If the situation is the same except the number of states is even then
the state with 4 votes has the power of 2 other states combined.
Proof: For 2n states there are 6n votes, and 3n+1 votes are needed for a
majority. If n-1 states (3 votes each) support a measure then 4 more votes
are needed. The state with 4 votes will suffice, or any 2 other states
(including the state with 2 votes).
3) Every state has 2 or 4 votes. A state with 4 votes has twice the power
of a state with 2 votes.
Proof: There are 2n states, 3n+1 votes are needed to back a measure. When
piecing together a majority, adding a state with 4 votes will accomplish
the same end as adding 2 states with 2 votes. Because no state is unique
(in the sense that only one state has that number of votes) it cannot be
meaningfully singled out for its role in completing the majority.
4) Every state except one has 2 or 4 votes. A state with 3 vctes has as
much power as a state with 2 votes.
Proof: There are 2n+1 states, so 3n+2 votes are needed. Before adding the
state with 3 votes to the coalition there will be an even number of votes.
If the number of votes assembled thus far is 3n-2 then a single state with
4 votes will suffice to make the majority, or 2 states with 2 votes, or a
state with 2 votes and a state with 3 votes.
QUESTIONS:
1) Are there holes in my proofs? I'm just a physicist, and we don't like
rigor.
2) What if each category has the same number of states? I don't think
that my proof of statement 3 is very good, so I'm not sure how to
generalize it to this case.
3) Will it suffice to put at least 2 states in each category to make the
power of a state proportional to its size?
NOTE: I defined
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