Homework Discussion Read Pages 48 – 62 Page 72: 1 – 4, 6 TEST 1 ON THURSDAY FEBRUARY 8 –The test will cover sections 1.1 – 1.6, and 2.1 – 2.3 in the textbook as well as any other items covered in the class notes and homework assignments.

describes a weighted voting system where q is the quota and are the respective votes of the players. The weights are normally written in numerical order from the highest to the lowest. Example 1. Chapter 2: Weighted Voting Goal: Construct an algorithm to determine the “power” of each player.

coalition - Any set of players that might join forces to vote together is called a coalition. (single player coalitions are allowed) weight of the coalition - The total number of votes controlled by a coalition is called the weight of the coalition. A coalition with enough votes to win is called a winning coalition. A coalition without enough votes to win is called a losing coalition. The coalition consisting of all the players is called the grand coalition. Definitions (page 54)

critical player - A player whose desertion turns a winning coalition into a losing coalition is said to be a critical player for that coalition.

FINDING THE BANZHAF POWER INDEX OF PLAYER P Step 1. Make a list of all possible coalitions. Step 2. Determine which of them are winning coalitions. Step 3. In each of the winning coalitions, determine which of the players are critical players. Step 4. Count the total number of times player P is critical. (Let’s call this number B.). Step 5. Count the total number of times all players are critical. (Let’s call this number T.) The Banzhaf power index of player P is B/T. It represents the proportion of times that player P is critical out of all the times that players are critical. Page 56

Objective 6: Constructing and applying algorithms

Example: [4:3,2,1]

Example. [101:99,98,3]

Example. [6:4,3,2,1]

Example 6. dictator - a player is a dictator if the player’s weight is bigger than or equal to the quota. veto power - A player that is not a dictator, but that can single-handedly prevent the rest of the players from passing a motion, is said to have veto power. dummy - a player without power is called a dummy. Example 7.

Nassau County Board of Supervisors in 1960: Quota is the majority of votes DistrictWeight Hemstead #131 Hemstead #231 Oyster Bay28 North Hempstead 21 Long Beach2 Glen Cove2

Homework Page 72: 11, 14, 17, 20, 55a, 56a, 66a TEST 1 ON THURSDAY FEBRUARY 8 –The test will cover sections 1.1 – 1.6, and 2.1 – 2.3 in the textbook as well as any other items covered in the class notes and homework assignments.

Test Information Textbook sections: 1.1 – 1.6 Aarow’s Impossibility Theorem Preference Schedules Voting Methods –Plurality –Borda Count –Plurality with Elimination –Pairwise Comparison Extended Rankings Recursive Rankings Criterion for a fair election –Majority Criteria –Condorcet Criteria –Monotonicity Criteria –Independence of Irrelevant Alternative Criteria

Test Information Textbook sections: 2.1 – 2.3 Players, Weights, Quota, Coalitions, Critical Player in winning coalitions Dictator, Veto Power, Dummy, Banzhaf Power Index