Weighted Voting Systems Quick Summary Weighted Voting System KEY POINT TERMINOLOGY A motion is any vote involving only two alternatives. A player is a voter in a weighted voting system. (The N players are: P_{1} , P_{2} , P_{3} ,…, P_{N} .) The weight of each player is the number of votes he/she controls. (The weights of the N players are (in order): w_{1} , w_{2} , w_{3} ,…, w_{N} .) The quota is the minimum number votes needed to
pass a motion. A dictator is a player who controls enough votes to
pass any motion single-handedly. A dummy is a player with no power. A player that is not a dictator but can single-handedly prevent any group of players from passing a motion is said to have veto power. A coalition is any set of players joining forces to vote together. The total number of votes controlled by a coalition is called the weight of the coalition. A winning coalition is one with enough votes to win. A losing coalition is one without enough votes to win. A coalition consisting of all players is often called a grand coalition. With N players, there are 2^{N} -1 possible coalitions. KEY POINT NOTATION If player P_{1} has w_{1 }votes, player P_{2} has w_{2 }votes, P_{3} has w_{3 }votes,…, and player P_{N} has w_{N }votes and the quota is q, then we will write { q : w_{1 }, w_{2 }, w_{3 },…, w_{N }}. Two measures of POWER: Banzaf Power Index (BPI) A player whose desertion of a winning coalition turns it into a losing one is called a critical player. Assumption: Players can enter and leave coalitions freely. Assumption: A player’s power is proportional to the number of times the player is critical. Shapley-Shubik Power Index (SSPI) A sequential coalition is one in which the players are listed in the order that they entered the coalition. There are N! sequential coalitions containing all N players. A pivotal player is the player in a sequential coalition who changes the coalition from a losing to a winning one. Assumption: Coalitions are formed sequentially and players are NOT free to leave a coalition once in it. Assumption: A player’s power is proportional to the number of times the player is pivotal. |