## Nassau County problem: solutions from class

Here are some solutions from our class discussion and previous classes for this problem.

1. Change the threshold to 27 votes) needed to win, with the same vote distribution as the original. This will certainly give the small towns some power (how much), but is likely to lead to deadlock.

2. Change the threshold to 2/3 majority (20 votes) needed to win, with the same vote distribution as the original. This will certainly give the small towns some power (how much), but may often lead to deadlock.

3. Modify the weighted vote as follows:

- Hempstead1 12
- Hempstead2 12
- North Hempstead 10
- Oyster Bay 6
- Glen Cove 4
- Long Beach 4

Since a majority now needs 25 votes, no two of the top three can dominate, as was the case before and the small towns can make a difference.

4. Divide larger towns into districts, so that they have representatives with the following votes. Districts are drawn so that the population is proportional to the number of votes, e.g. Hempstead 1 has 2/15 of the total vote so it should have about 2/15 of the population.

- Hempstead 1 4
- Hempstead 2 4
- Hempstead 3 4
- Hempstead 4 3
- Hempstead 5 3
- North Hempstead 1 4
- North Hempstead 1 3
- Oyster Bay 3
- Glen Cove 1
- Long Beach 1

5. Divide larger towns into districts, so that they have representatives with the following votes. Districts are drawn so that the population is proportional to the number of votes, e.g. Hempstead 1 has 2/15 of the total vote so it should have about 2/15 of the population.

- Hempstead 1 4
- Hempstead 2 4
- Hempstead 3 5
- Hempstead 4 5
- North Hempstead 1 4
- North Hempstead 1 3
- Oyster Bay 3
- Glen Cove 1
- Long Beach 1

6. (This is a modification of the bicameral solution, letting one representative cast votes for both stages.) Two stage voting procedure:

- Vote with the original weights. Simple majority (16 votes) to pass this stage. If not passed at this stage, proposition is defeated
- Proposition that passes first stage is voted with each representative cast a single vote, 3 needed to win. If passed, proposition is approved and the process ends, otherwise it is defeated.

Find the Banzhaf index for each of these methods.