### Weighted Voting Systems

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

Motion
Players
Weights
Quota
Dictator
Dummy
Veto Power
Coalition
-the weight of a coalition
-winning coalitions
-losing coalitions
-grand coalition

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 }.

Example:
The three stockholders in a small company form a Board of Directors to oversee the company. John (
P1 ) has 5 votes as the largest stockholder, Ginny (P2 ) has 3 votes, and Ann (P3 ) has 2 votes. The quota is 7; that is, it takes 7 or more votes to pass a motion. This weighted voting system is represented mathematically as { 7: 5, 3, 2 }.

ANOTHER KEY POINT

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

For instance, it is possible for a weighted voting system to actually reduce to a one-person, one-vote situation in which case all players have the same POWER even though they don’t all have the same weight. For example, for the weighted voting system { 5: 4, 3, 2 } none of the players can pass a motion alone and any two players can join together to pass a motion; so, although the players have different weights, each player has the same amount of power.

We will discuss two ways of defining POWER.

Banzhaf Power Index

Key Points

• A player whose desertion of a winning coalition turns it into a losing one is called a critical player.
• A player’s power is proportional to the number of times the player is critical.

Banzhaf Power Index for Player P: BPI(P)

Computing the BPI for Player P:

STEP 1: Determine all WINNING coalitions.
(If you can list just the winning coalitions, then there is no need to list all of the 2N -1 coalitions.)

STEP 2: Determine the critical players in each winning coalition.

STEP 3: Find the number of times all players are critical.

STEP 4: Find the number of times Player P is critical.

STEP 5: BPI(P) is the smaller # (the one from STEP 4) divided by the larger number (the one from STEP 3)

DO THIS FOR EACH PLAYER AND YOU HAVE THE BANZHAF POWER DISTRIBUTION.

Banzhaf Power Index example

Underlying Assumptions:

• Players can enter and leave coalitions freely.
• A player’s power is proportional to the number of times the player is critical.

Shapley-Shubik Power Index

Key Points

• A sequential coalition is one in which the players are listed in the order that they entered the coalition.
• A pivotal player is the player in a sequential coalition who changes the coalition from a losing to a winning one.
• There are N! sequential coalitions containing all N players.

Shapley-Shubik Power Index for Player P: SSPI(P)

Computing the SSPI for Player P:

STEP 1: Make a list of all N! sequential coalitions containing N players.

STEP 2: In each of these coalitions, determine the pivotal player.
(There is always exactly one in each of these coalitions.)

STEP 3: Determine the number of times Player P is pivotal.

STEP 4: SSPI(P) is the smaller # (the one from STEP 3) divided by the larger number, N! (the one from STEP 1)

DO THIS FOR EACH PLAYER AND YOU HAVE THE SHAPLEY-SHUBIK POWER DISTRIBUTION.

Shapley-Shubik Power Index example

Underlying Assumptions:

• Players enter coalitions in a certain order.
• Players are NOT free to leave a coalition once in it.
• A player’s power is proportional to the number of times the player is pivotal.

Looking at Power from a Banzhaf and Shapley-Shubik Point of View

Example
{4: 3, 2, 2} Which player has the most power?

• from a Banzaf point of view:

Look at all the winning coalitions.

{P1}
{P2}
{P3}
{P1, P2} (winning)
{P1, P3} (winning)
{P2, P3} (winning)
{P1, P2, P3} (winning)

Find the critical players in each.

 winning coalition{P1, P2} {P1, P3} {P2, P3} {P1, P2, P3} critical playersP1 & P2 P1 & P3 P2 & P3 none

BPI(P)
= (# of times player P is critical) / (# times all are critical)

BPI(P1 ) = 2/6 = 1/3 ; BPI(P2 ) = 2/6 = 1/3 ; BPI(P3 ) = 2/6 = 1/3

• from a Shapley-Shubik point of view:

Find all sequential coalitions containing all players (N! of them).

{P1, P2, P3}
{P1, P3, P2}
{P2, P1, P3}
{P2, P3, P1}
{P3, P2, P2}
{P3, P2, P1}

Find the pivotal player in each.

 sequential coalition{P1, P2, P3} {P1, P3, P2} {P2, P1, P3} {P2, P3, P1} {P3, P1, P2} {P3, P2, P1} pivotal playerP2 P3 P1 P3 P2 P2

SSPI(P) = (# of times player P is pivotal) / (# times all are pivotal)

SSPI(P1 ) = 1/6 ; SSPI(P2 ) = 3/6 = 1/2 ; SSPI(P3) = 2/6 = 1/3

Example:
{5: 3, 2, 2} Which player has the most power?

• from a Banzaf point of view:

Look at all the winning coalitions.

{P1}
{P2}
{P3}
{P1, P2}(winning)
{P1, P3} (winning)
{P2, P3}
{P1, P2, P3} (winning)

Find the critical players in each.

 winning coalitions{P1, P2} {P1, P3} {P1, P2, P3} critical playersP1& P2 P1& P3 P1

BPI(P) = (# of times player P is critical) / (# times all are critical)

BPI(P1) = 3/5 ; BPI(P2) = 1/5 ; BPI(P3) = 1/5

• from a Shapley-Shubik point of view:

Find all sequential coalitions containing all players (N! of them).

{P1, P2, P3}
{P1, P3, P2}
{P2, P1, P3}
{P2, P3, P1}
{P3, P1 , P2}
{P3, P2, P1}

Find the pivotal player in each.

 sequential coalitions{P1, P2, P3} {P1, P3, P2} {P2, P1, P3} {P2, P3, P1} {P3, P1, P2} {P3, P2, P1} pivotal playerP2 P3 P1 P1 P1 P1

SSPI(P) = (# of times player P is pivotal) / (# times all are pivotal)

SSPI(P1) = 4/6 = 2/3 ; SSPI(P2) = 1/6 ; SSPI(P1) = 1/6

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