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
