1. § 2.1 Weighted Voting Systems

2. Weighted Voting  So far we have discussed voting methods in which every individual’s vote is considered equal--these methods were based on the concept of “one voter--one vote.”  Weighted voting, is based on the idea of “one voter--x votes”--in other words, some voters ‘count more’ than others.

3. Weighted Voting Systems  More rigorously stated, any formal voting system arrangement in which the voters are not necessarily equal in terms of the number of votes they control is called a weighted voting system.  For the sake of simplicity we shall only examine motions--votes involving only two choices/candidates.

4. Weighted Voting Systems  Weighted voting systems are comprised of: 1. Players - The groups, or individuals that can cast votes. 2. Weights of the players - The number of votes each player controls. 3. Quota - The smallest number of votes needed to pass a motion.

5. Weighted Voting Systems  Notation: We will use N to refer to the number of players in our system. The players will be denoted P 1, P 2, P 3,..., P N. Their corresponding weights are w 1, w 2, w 3,..., w N. The letter q will be used to represent the quota.

6. Weighted Voting Systems  Using this notation we can represent the entire weighted voting system as: [ q : w 1, w 2, w 3,..., w N ]  Here the quota is listed first and the weights are given in decreasing order.

7. Weighted Voting Systems  The quota, q, must always be larger than half the number of votes and not more than the total number of votes. Stated mathematically, w 1 + w 2 + w 3 +... + w N < q ≤ w 1 + w 2 + w 3 +... + w N 2

8. Example 1: Example 1: Suppose that the board of a small corporation has four shareholders, P 1, P 2 and P 3. P 1 has 8 votes, P 2 has 4 votes, P 3 has 2 votes and P 4 has 1. If at least two-thirds of the votes are needed to pass a motion then describe this system using the ‘bracketed’ notation.

9. Example 2: Example 2: Consider weighted voting system with four players, P 1, P 2, P 3 and P 4. P 1 has three times as many votes as P 2. P 2 has twice as many votes as P 3 and P 4 (which have the same number of votes). If a simple majority is all that is necessary to pass a motion then describe this weighted voting system.

10. Weighted Voting Systems  Notice in the last example that P 1 could pass or block any motion. In such a situation, P 1 would be called a dictator.  In general, a player is a dictator if the player’s weight is bigger than or equal to the quota.  Whenever there is a dictator, all of the other players are irrelevant--such a player with no power is called a dummy.

11. Weighted Voting Systems  Now look back at example 1. You might notice that no motion could pass in that weighted system without the support of P 1, but that P 1 would still need the support of at least one other voter in order to pass a motion.  Any player who is not a dictator, but can block the passing of any motion has what is referred to as veto-power.

12. Example 3: Example 3: (Exercise #10 pg 73) In each of the following weighted voting systems, determine which players, if any, (i) are dictators; (ii) have veto power; (iii) are dummies. (a) [ 27 : 12, 10, 4, 2 ] (b) [ 22 : 10, 8, 7, 2, 1 ] (c) [ 21 : 23, 10, 5, 2 ] (d) [ 15 : 11, 5, 2, 1 ]

13. Example 4: Example 4: The US Senate is currently composed of 55 Republicans, 44 Democrats and 1 Independent (who votes with the Democrats). Suppose 6 Republican senators decided to form their own “Consensus Party” (yes, I know this is even sillier than voting muppets). Further suppose that following such defections each party keeps its members strictly in line.

14. Example 4: Example 4: The US Senate is currently composed of 55 Republicans, 44 Democrats and 1 Independent (who votes with the Democrats). Suppose 6 Republican senators decided to form their own “Consensus Party” (yes, I know this is even sillier than voting muppets). Further suppose that following such defections each party keeps its members strictly in line. We might describe this weighted voting system as: [ 51 : 49, 45, 6 ]

15. Example 4: Example 4: The US Senate is currently composed of 55 Republicans, 44 Democrats and 1 Independent (who votes with the Democrats). Suppose 6 Republican senators decided to form their own “Consensus Party” (yes, I know this is even sillier than voting muppets). Further suppose that following such defections each party keeps its members strictly in line. We might describe this weighted voting system as: [ 51 : 49, 45, 6 ] While it might seem that both the Democrats and Republicans hold more power than the “Consensus Party,” this is not actually the case. Why?