1. Discrete Math CHAPTER TWO 2.1 • Weighted Voting: One voter - x votes. Ex. Electoral College. • Weighted Voting System: Any formal voting arrangement in which the voters are not necessarily equal in terms of the final number of votes they control. • Motion: A yes/no vote. • Players: The voters • Weight: Number of votes the player controls. To be continued…

2. Discrete Math CHAPTER TWO 2.1 (Continued...) • Quota: The minimum number of votes needed to pass a motion. • May be something other than the majority. • Notation: [Q :w1, w2, w3, ...] • Dictator: A player that has a weight equal to or larger than the quota. • Dummy: A player with no power. • Veto Power: A player that is not a dictator, but can single-handedly prevent the rest of the players from passing a motion.

3. Discrete Math CHAPTER TWO 2.2 • Banzhaf Power Index: Method of calculation Power. • Coalition: Any set of players that might join forces to vote together. • Winning Coalition: Has enough votes to win. • Losing Coalition: Does not have enough votes to win. • Critical Player: Players whose desertion turns a winning coalition into a losing coalition. To be continued…

4. Discrete Math CHAPTER TWO 2.2 (Continued...) • Steps: • One: Make a list of all possible coalitions. • Two: Determine which of the above are winning coalitions. • Three: Determine the critical players in each winning coalition. • Four: Count the number of times each player is critical (B). • Five: Count the total of times all players are critical() and create the power for each player as B/T… may write as percentage. To be continued…

5. Discrete Math CHAPTER TWO 2.2 (Continued...) • Number of possible coalitions: 2n-1 • Sometimes we can save ourselves a lot of work by figuring out directly which are the winning coalitions instead of listing all of them. Range of Quota: Sum of weights / 2 < Q <= Sum of weights

6. Discrete Math CHAPTER TWO 2.3 • Shapley-Shubik Power Index: • Sequential Coalition: Difference in Shapley-Shubik and Banzhaf, coalitions are assumed to be formed sequentially. • The Number of Sequential Coalitions: N! • Pivotal Player: The player joins the coalition and causes the coalition to change from losing to winning. To be continued…

7. Discrete Math CHAPTER TWO 2.3 (Continued...) • Steps: • One: Make a list of all sequential coalitions. • Two: In each coalition determine the pivotal player. • Three: Count the total number of times P is pivotal and call this number S. The Shapley-Shubik power index of each player will be calculated as S/N!. (Fraction or Percentage)

8. 2 End of Chapter Discrete Math CHAPTER TWO