Weighted Voting Systems Quick Summary

Weighted Voting System
A weighted voting system is one in which the preferences of some voters carry more weight than the preferences of other voters. (Here we will consider only those situations in which the voting is between 2 alternatives.)

KEY POINT
When a vote deals with only two alternatives, all reasonable voting methods have the same outcome as "majority rule." For this reason, our major interest here will not be comparing voting systems but rather, the concept of power: Who has it and how much do they have?


TERMINOLOGY

A motion is any vote involving only two alternatives.

A player is a voter in a weighted voting system. (The N players are: P1 , P2 , P3 ,…, PN .)

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 ,…, wN .)

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 simple majority but not more than the total number
of votes.)

A dictator is a player who controls 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.)

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 2N -1 possible coalitions.

KEY POINT
The weight of a player is not a good measure of a player’s POWER.


NOTATION

If player P1 has w1 votes, player P2 has w2 votes, P3 has w3 votes,…, and player PN has wN votes and the quota is q, then we will write { q : w1 , w2 , w3 ,…, wN }.


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.

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