Chapter 2 Weighted Voting Systems

1. Chapter 2 Weighted Voting Systems

2. 2.1 Terminology • Weighted voting system —any formal voting arrangement in which voters are not necessarily equal in terms of the number of votes they control. • Motions —only consider yes-no votes or “motions”—a vote between two candidates or alternatives can be rephrased as a yes-no vote. • Players —voters in a weighted voting system.

3. Terminology (cont.) • The Weights —the weight of the player is the number of votes a player controls. Representation: Player 1 (P1) holds Weight 1 (w1). The last player is represented by Pn with wnweight. • The Quota(q) —theminimum number of votes to pass a motion. 50% < q < 100% Needs to be more than 50% to avoid a stalemate. Cannot be more than the total number of votes (100%). Quota is stipulated at the beginning of an election. • Representation: [q: w1, w2,…, wn] It is customary to list the weights in descending order.,

4. Representation of Quota • Example 2.1: Suppose four partners decide to start a new business adventure. There are 20 shares (votes) and P1 owns 8 shares, P2 owns 7 shares, P3 owns 3 shares, and P4 owns 2 shares. Imagine now they set up the rules of the partnership so that 2/3 of the partner’s votes are needed to pass any motion. • Use the simplified notation [q: w1, w2,…, wn] to describe this partnership. • Answer: [14: 8, 7, 3, 2] (remember, 14 is the first integer larger than 2/3 of 20. • What if the above quota was changed to [19: 8, 7, 3, 2]?

5. More Terminology • Dictator—the player (can only be one) whose weight is greater than or equal to the quota. wd >q • Ex. [11: 12, 5, 4] • Dummy—the player whose votes will never matter—all other players are dummies when there is a dictator. There can be dummies without a dictator. • Ex. [30: 10, 10, 10, 9]Don’t be a dummy! • Veto Power—a player has veto power if a motion cannot pass without his/her votes—he/she CAN prevent a motion from passing—this does NOT mean that whatever the player votes for must pass. • Ex. [12: 9, 5, 4, 2]

6. Clarifying Example… “I can see clearly now…” • Find the players who are dictators, have veto power, or are dummies… • [19: 9, 7, 5, 3, 1] • P1 & P2 veto—P5 dummy • [15: 16, 8, 4, 1] • P1 dictator—all of the rest are dummies • [17: 13, 5, 2, 1] • P1 & P2 veto—P3 & P4 are dummies • [25: 12, 8, 4, 2] • All have veto power • Weighted Voting Worksheet--#1 only • Classwork for tomorrow Book—Pg. 72: 1-7 odd (Bring Book)

7. 2.2 Preface—More Terminology • Coalitions —any set of players who might join forces and vote the same way. (grand coalition —consisting of all players) • Winning and Losing Coalitions –self explanatory. Note: a single-player coalition can only be a winning coalition if that player is a dictator. Winning coalitions will have at least two players for our purposes. A grand coalition is always a winning coalition. • Critical Players–in a winning coalition, a player is said to be a critical player if the coalition must have that player’s votes to win. So, W – w < q where W is the weight of the coalition and w is the weight of the critical player.

8. Coalition Example • Think about [101: 99, 98, 3] • Since there are three players, each critical twice, we can say that each player holds two out of six, or one-third of the power.

9. 2.2/2.3 The Banzhaf Power Index • 1st count the total number of winning coalitions in which P1 is a critical player—call this B1. Do this for the remaining players. • 2nd find the total number of times all players are critical ( T = B1 + B2 + … + Bn). • 3rd find all of the ratios B1/T, B2/T, …, Bn/T these ratios should be between 0 and 1 or a percent between 0 and 100%. • The list β1,β2, …, βn is called the Banzhaf power distribution (using “beta”)…the sum should total 1 or 100%. • Ex. 2.9 Pg. 56 Banzhaf Power in [4: 3, 2, 1] & Ex. 2.10 Pg. 57 NBA Draft

10. How many Coalitions are there? • Think of subsets…how many subsets does {a, b, c} have? • { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}—for our purposes, the empty set doesn’t count. • Number of subsets 2N • Number of subsets without the empty set 2N -1 where N is the number of players • Weighted Voting Worksheet #1 Parts 6A and 6C • Classwork/Homework Pg. 73:11-17 odd

11. 2.4 The Shapley-Shubik Power Index • Coalitions are formed sequentially—order matters! (not so with Banzhaf) • {P1, P2,P3} can form six (3!) different sequential coalitions—try to find them all. • We notate this by using to show that P2 entered the coalition 1st, then P1, and P3 entered last. • Pivotal Player —the player who turns a losing coalition into a winning one—the player who “tips the scale”. Every sequential coalition has one and only one pivotal player.

12. Computing the Shapley-Shubik Power Distribution • 1st make a list of all possible sequential coalitions of the given number of players. (let this be T) • 2nd count the number of times P1, P2, …,Pn are pivotal players…label this SS1, SS2, …, SSn. • 3rd find the ratio σ1 = SS1/T (sigma)which is the power index for P1—use this ratio to find the power index of all players. • The Shapley-Shubik power distribution is a list of all σ’s—σ1, σ2, …, σn. • The number of sequential coalitions with N players is found by N! • Ex. 2.17 Pg. 66 (SS Power in [4: 3, 2, 1]) • Complete Weighted Voting Worksheet • Classwork/Homework Pg. 74: 23-31 odd • Complete the “Are you always a dummy?” Worksheet (in class tomorrow) • Quiz on 2.1-2.3 (includes Shapely-Shubik) Friday • Chapter 2 Test Tuesday