TERMINOLOGY

Motion
A motion is any vote involving only two alternatives.

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Player
A player is a voter in a weighted voting system.

• The N players are: P₁ , P₂ , P₃ ,…, P_N .

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Weight
The weight of each player is the number of votes he/she controls.

• The weights of the N players are (in order): w₁ , w₂ , w₃ ,…, w_N .

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Quota
The quota is the minimum number votes needed to pass a motion.

• We will use the letter q to stand for quota.
• q must be at least a majority but not more than the total number of votes.

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Dictator
A dictator is a player who has enough votes to pass any motion single-handedly.

• A dictator’s weight is always greater than or equal to the quota .
• Whenever there is a dictator, all other players have NO power.

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Dummy
A dummy is a player with no power.

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Veto 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.

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Coalition
A coalition is any group of players that join 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.

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Critical Player
A player whose desertion of a winning coalition turns it into a losing one is called a critical player.

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Sequential Coalition
A sequential coalition is one in which the players are listed in the order that they entered the coalition.

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Pivotal Player
A pivotal player is the player in a sequential coalition who changes the coalition from a losing to a winning one.

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N!
N! = N(N-1)(N-2) ...(2)(1)

Examples:

6! = 6(5)(4)(3)(2)(1) = 720

4! = 4(3)(2)(1) = 24

and, by definition,

0! = 1

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