Weighted Voting. Lecture 13 Section 2.1. Robb T. Koether. Hampden-Sydney College. Mon, Feb 12, 2018
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1 Weighted Voting Lecture 13 Section 2.1 Robb T. Koether Hampden-Sydney College Mon, Feb 12, 2018 Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
2 1 Introductory Example 2 Definitions 3 Votes vs. Power 4 Assignment Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
3 Outline 1 Introductory Example 2 Definitions 3 Votes vs. Power 4 Assignment Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
4 Introduction Normally, every voter gets one vote. Would it ever be fair to give one voter more votes than another voter? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
5 Introduction Normally, every voter gets one vote. Would it ever be fair to give one voter more votes than another voter? Yes. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
6 An Example An Example Four partners start a business. They raise $210,000 by issuing 21 shares at $10,000 per share. Andy buys 9 shares. Bob buys 8 shares. Chuck buys 3 shares. Dave buys 1 shares. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
7 An Example An Example Four partners start a business. They raise $210,000 by issuing 21 shares at $10,000 per share. Andy buys 9 shares. Bob buys 8 shares. Chuck buys 3 shares. Dave buys 1 shares. Each partner gets one vote for each of his shares. They agree that 14 of the 21 votes are required to approve a proposal. (That is, a 2/3 majority is required.) Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
8 An Example An Example Four partners start a business. They raise $210,000 by issuing 21 shares at $10,000 per share. Andy buys 9 shares. Bob buys 8 shares. Chuck buys 3 shares. Dave buys 1 shares. Each partner gets one vote for each of his shares. They agree that 14 of the 21 votes are required to approve a proposal. (That is, a 2/3 majority is required.) How much influence does each partner have? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
9 An Example An Example Four partners start a business. They raise $210,000 by issuing 21 shares at $10,000 per share. Andy buys 9 shares. Bob buys 8 shares. Chuck buys 3 shares. Dave buys 1 shares. Each partner gets one vote for each of his shares. They agree that 14 of the 21 votes are required to approve a proposal. (That is, a 2/3 majority is required.) How much influence does each partner have? What if decisions are made by a simple majority (11 votes)? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
10 Outline 1 Introductory Example 2 Definitions 3 Votes vs. Power 4 Assignment Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
11 Definitions Definition (The Players) The players are the same as the voters. Let N denote the number of players. Definition (The Weights) The weight of a player is the number of votes that he may cast. The weights are denoted w 1, w 2, w 3,..., w N. The total of the weights is V = w 1 + w 2 + w w N. Definition (The Quota) The quota q is the number of votes needed to win. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
12 Definitions Definition (The Quota) The quota, denoted q, is the number of votes needed to pass a motion. We represent the voting system as [q : w 1, w 2,..., w N ]. The previous examples the voting systems were [14 : 9, 8, 3, 1] and [11 : 9, 8, 3, 1]. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
13 Anarchy Example (Anarchy) Change the quota to 10: [10 : 9, 8, 3, 1]. Now we have so-called anarchy. How come? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
14 Anarchy Example (Anarchy) Change the quota to 10: [10 : 9, 8, 3, 1]. Now we have so-called anarchy. How come? What if Andy and Dave vote yes and Bob and Chuck vote no? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
15 Anarchy Example (Anarchy) Change the quota to 10: [10 : 9, 8, 3, 1]. Now we have so-called anarchy. How come? What if Andy and Dave vote yes and Bob and Chuck vote no? Definition (Anarchy) Anarchy occurs when q V /2. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
16 Anarchy Example (Anarchy) Change the quota to 10: [10 : 9, 8, 3, 1]. Now we have so-called anarchy. How come? What if Andy and Dave vote yes and Bob and Chuck vote no? Definition (Anarchy) Anarchy occurs when q V /2. Thus, we want q > V /2. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
17 Anarchy Example (Anarchy) Change the quota to 10: [10 : 9, 8, 3, 1]. Now we have so-called anarchy. How come? What if Andy and Dave vote yes and Bob and Chuck vote no? Definition (Anarchy) Anarchy occurs when q V /2. Thus, we want q > V /2. Might there be a good reason to set q V /2? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
18 Gridlock Example (Gridlock) Change the quota to 22: [22 : 9, 8, 3, 1]. Now we have gridlock. How come? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
19 Gridlock Example (Gridlock) Change the quota to 22: [22 : 9, 8, 3, 1]. Now we have gridlock. How come? What if they all vote yes? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
20 Gridlock Example (Gridlock) Change the quota to 22: [22 : 9, 8, 3, 1]. Now we have gridlock. How come? What if they all vote yes? Definition (Gridlock) Gridlock occurs when q > V. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
21 Gridlock Example (Gridlock) Change the quota to 22: [22 : 9, 8, 3, 1]. Now we have gridlock. How come? What if they all vote yes? Definition (Gridlock) Gridlock occurs when q > V. Thus, we always want V /2 < q V. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
22 Dictators Example (Dictators) If Andy buys 5 shares from Bob, then the situation becomes [14 : 14, 3, 3, 1] and Andy becomes a dictator. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
23 Dictators Example (Dictators) If Andy buys 5 shares from Bob, then the situation becomes [14 : 14, 3, 3, 1] and Andy becomes a dictator. Definition (Dictator) A dictator is a player whose weight is greater than or equal to q. He can pass a motion by himself. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
24 Avoid Dictators To avoid dictators, we need w i < q for every i. Equivalently, q > w i for every i. That is, no single voter s weight is enough to pass a motion. This is accomplished if q is greater than the greatest weight. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
25 Veto Power Example (Veto Power) In the original situation [14 : 9, 8, 3, 1], both Andy and Bob have veto power. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
26 Veto Power Example (Veto Power) In the original situation [14 : 9, 8, 3, 1], both Andy and Bob have veto power. Definition (Veto Power) A player has veto power if the sum of all other votes is less than q. That is V w i < q. In such a case, no motion can pass unless that player votes for it. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
27 Avoid Veto Power To avoid veto power, we need V w i q for every i. Equivalently, q V w i for every i. That is, no single voter s weight is so much that no coalition can pass a motion without his vote. This is accomplished if q V largest weight. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
28 Dictators and Veto Power Example In the voting system [q : 10, 7, 6, 5, 3], What values of q will avoid anarchy? What values of q will avoid gridlock? What values of q will prevent dictators? What values of q will avoid veto power? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
29 Outline 1 Introductory Example 2 Definitions 3 Votes vs. Power 4 Assignment Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
30 Example (Few Votes, Much Power) Consider the situation [19 : 8, 7, 3, 2]. It might as well be [4 : 1, 1, 1, 1]. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
31 Example (Many Votes, Little Power) Consider the situation [18 : 6, 6, 6, 5]. How much influence does Dave (last guy) have? Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
32 Outline 1 Introductory Example 2 Definitions 3 Votes vs. Power 4 Assignment Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
33 Assignment Assignment Ch. 2: Exercises 1, 2, 3, 4, 5, 6, 7, 8. Robb T. Koether (Hampden-Sydney College) Weighted Voting Mon, Feb 12, / 20
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