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Weighted voting systems: Show #1 of 4

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WEIGHTED VOTING SYSTEMS

• sets of voters (slide #3--17)

• the power of coalitions (#18--23)

• power of individual voters(#24--26)

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WEIGHTED VOTING SYSTEMSExample:

• A certain small business has 4 shareholders.

ALICE owns 45 shares of stock

BRADLEY owns 30 shares

CHARLES owns 15 shares

DOROTHY owns 10 shares

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• ALICE owns 45 shares of stock

• BRADLEY owns 30 shares

• CHARLES owns 15 shares

• DOROTHY owns 10 shares

Each stockholder attends monthly meetings, at which they are able to vote on matters that affect the business.

We will look at 3 ways that the voting might be accomplished, and discuss the “fairness” of each of the voting systems considered.

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WEIGHTED VOTING SYSTEMSWVS NOTATION: A wvs with “n” voters...

[q: w1, w2, w3, … wn]

where q = quota (number of votes needed to pass a motion)

and wn = the weight of voter n (number of

votes that particular voter has)

• NOTE: We will say that the wvs is valid as long as the QUOTA is at least a *simple majority and the QUOTA is not more than the total number of votes possible.

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WEIGHTED VOTING SYSTEMSWVS NOTATION: A wvs with “n” voters...

[q: w1, w2, w3, … wn] where q = quota & wn = the weight of voter n

•NOTE: We will say that the wvs is valid as long as the QUOTA is at least a simple majority and the QUOTA is not more than the total number of votes possible.

NOTE also…• The voting system is made up of the actual

voters…• (ex) {Alice, Brad, Chuck, Dorothy}• We will not allow the voters to simply be “named”

by their # of votes.• i.e. If Alice has 45 votes, we won’t call her “voter

45” … we’ll still call her Alice!!!

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WEIGHTED VOTING SYSTEMS•ALICE owns 45 shares of

stock•BRADLEY owns 30 shares•CHARLES owns 15 shares•DOROTHY owns 10 shares

We will look at 3 ways that the voting might be accomplished, and discuss the “fairness” of each of the voting systems considered.

• Wvs#1

The number of votes each person has (weight of each voter) is equal to the number of shares of stock (s)he owns.

And the number of votes needed to pass a measure (quota) is the simple majority.

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WEIGHTED VOTING SYSTEMS•ALICE owns 45 shares of stock

•BRADLEY owns 30 shares•CHARLES owns 15 shares•DOROTHY owns 10 shares

• Wvs#1

FIND the simple majority: 45+30+15+10 = 100 total votes.

The simple majority is the first integer that is more than half of 100 (the total # of votes possible).

That’s 51 votes!(note: a vote of 50 to 50 would be a tie…

a vote of 51 to 49 would produce a winner & loser)

Wvs#1The number of votes each person has (weight of each voter) is equal to the number of shares of stock (s)he owns.

And the number of votes needed to pass a measure (quota) is the simple majority.

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WVS #1

shorthand notation:

[51: 45, 30, 15, 10]

• Let’s use some shorthand notation for the voters’ too… so that we can take a look at the whole system, and how the voters might vote on certain issues.

• We’ll look at the set of voters as follows:

{A, B, C, D}

ALICE owns 45 shares of stockBRADLEY owns 30 sharesCHARLES owns 15 sharesDOROTHY owns 10 shares

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WEIGHTED VOTING SYSTEMSHow the voting could work...

• The next set of slides will deal with looking at all of the possibilities for a vote on a motion.

• We will assume that no voter is allowed to abstain from voting.

• When voters join together on one side of an issue (FOR or AGAINST), they will form a subset of voters called a COALITION.

• After you have done this look at the workings of a weighted voting system, you will NOT have to do this for any other problem.

• This is the BIG PICTURE… so you get a feel for the possibilities.

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Let’s do an exhaustive look at all of the possible ways voting could work here. (**handout**)

Coalition voting FOR the motion

Coalition voting AGAINST the motionweight status weight status

Motion PASS or FAIL

BD ____ __ __ __ __

A BCD45 lose 55 block FAIL

B ACD30 lose 70 block FAIL

C ABD15 lose 85 block FAIL

D ABC10 lose 90 block FAIL

AB CD75 win 25 lose PASS

AC ____ __ __ __ __

AD ____ __ __ __ __

BC ____ __ __ __ __

ABC ____ __ __ __ __

CD ____ __ __ __ __

BCD ____ __ __ __ __

ABD ____ __ __ __ __

ACD ____ __ __ __ __

none ABCD0 lose 100 block FAIL

ABCD ____ __ __ __ __

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Motion PASS or FAIL

BD ____ __ __ __ __






AC ____ __ __ __ __

AD ____ __ __ __ __

BC ____ __ __ __ __

ABC ____ __ __ __ __

CD ____ __ __ __ __

BCD ____ __ __ __ __

ABD ____ __ __ __ __

ACD ____ __ __ __ __


ABCD ____ __ __ __ __

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WEIGHTED VOTING SYSTEMS[51: 45,30,15,10]

BD AC40 lose 60 block FAIL

AC BD60 win 40 lose PASS

AD BC55 win 45 lose PASS

BC AD45 lose 55 block FAIL

ABC D90 win 10 lose PASS

CD AB25 lose 75 block FAIL

BCD A55 win 45 lose PASS

ABD C85 win 15 lose PASS

ACD B70 win 30 lose PASS



Motion PASS or FAIL







ABCD none100 win 0 lose PASS

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WEIGHTED VOTING SYSTEMS• Notice that there are 8

winning coalitions;• And that ALICE is in 7 out

of the 8 winning coalitions,

• While the other three voters are each in 5 out of 8 winning coalitions.

• Brad might not be too happy about that!

• He’s got a lot more votes than Chuck and Dot…

• Maybe if the QUOTA was higher, he would gain some “power?”

• So, for WVS#2, let’s make the QUOTA higher than just a simple majority of the votes…

• And for WVS#3, we’ll try something different...

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•WVS#2:

The number of votes will be the same as in WVS#1,

but the QUOTA will be 70 votes. (70 votes needed to pass a motion).

• WVS#2:

[70: 45, 30, 15, 10]

ALICE owns 45 shares BRADLEY owns 30 sharesCHARLES owns 15 sharesDOROTHY owns 10 shares

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•WVS#3:

The number votes will be the tens digit of the # of shares of stock,

and the QUOTA will be the simple majority.

• WVS#3:

[?: 4, 3, 1, 1]• the simple majority

will be the first integer that is more than half of:

4+3+1+1 = 9• 9/2 = 4.5• 5 votes will be the

simple majority.(notice that a vote of 5 to 4

would have a winner and loser.)

[5: 4, 3, 1, 1]

ALICE owns 45 shares BRADLEY owns 30 sharesCHARLES owns 15 sharesDOROTHY owns 10 shares

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Complete wvs charts• If available, use the

handout(s) and complete a chart that looks at the way each of these wvs work.

• If a handout is not available, create the charts on your own.

• Notice the similarities and differences in the STATUS and PASS/FAIL columns

• You should see that in Method #1 and Method #3, the STATUS columns are the same for corresponding coalitions.

• These are called EQUIVALENT wvs

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WVS power

• Notice that a coalition of voters can either:

• WIN• LOSE or• *BLOCK

when voting on a motion

• Winning Coalitions have a weight that is at least the quota. They pass a motion.

• Losing Coalitions have a weight that is less than the quota.

• *Blocking Coalitions have enough votes to keep their opponents from passing a motion.

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WVS power

• Winning Coalitions have a weight that is at least the quota.

• Losing Coalitions have a weight that is less than the quota.

• *Blocking Coalitions have enough votes to keep their opponents from winning.

How many votes are needed to BLOCK a measure??

• If you know the TOTAL # of votes in the system

• and you know the QUOTA• It is always true that a

LOSING coalition will have a weight that is less than or equal to the

TOTAL - QUOTA• So, in order to BLOCK, a

coalition must have MORE THAN the (TOTAL - QUOTA)

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WVS powerHow many votes are

needed to BLOCK a measure??

• In order to BLOCK, a coalition must have

MORE THAN the

(TOTAL - QUOTA)

(ex) In WVS#1, • the total # votes = 100• the quota = 51

A coalition with (100 - 51) = 49 votes is a losing

coalition(in fact, any coalition with less

than 49 votes loses, too!)

Any coalition (voting AGAINST a motion) that

has MORE THAN 49 votes can BLOCK a motion from being

passed.

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WVS power(ex) In WVS#1,

Any coalition with MORE THAN 49 votes can BLOCK a motion from being passed.

REMEMBER…

A BLOCKING COALITION is voting AGAINST the motion under consideration.

This coalition might have a weight equal to the quota, but it might not!

It just depends on how many votes the coalition voting FOR the motion has!

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WVS powerIn WVS#2, how many votes are needed to BLOCK a measure from being passed?

•The quota is 70•The total # of votes is 100

To BLOCK a motion, a coalition must have

MORE THAN (100 - 70)

that is MORE THAN 30 votes.(Look at the coalitions you’ve marked BLOCK to confirm this)

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WVS power• Your text does not mention BLOCKING

coalitions• However, in order to thoroughly discuss

POWER in a wvs, it is important to remember that whatever coalition has enough votes to attain their goal, has some type of power.

• A WINNING COALITION: has the power to PASS a measure.

• A BLOCKING COALITION: has the power to prohibit a measure from being passed (blocking power.)

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WVS power• Winning or Blocking Power is related to the

status of the entire coalition.• We must also discuss the POWER OF

INDIVIDUAL VOTERS in the system and within their coalitions.

• A voter whose weight is greater than or equal to the quota is called a DICTATOR.

• A DICTATOR is a one-person winning coalition.

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WVS power• A voter whose weight is greater than or equal to the

quota is called a DICTATOR. • A DICTATOR is a one-person winning coalition.

• A voter has VETO POWER if (s)he can prevent a measure from being passed even if all others vote for the measure.

• A voter with VETO POWER is a one-person blocking coalition.

• A voter whose vote is NEVER necessary in any winning coalition is called a DUMMY VOTER (if that voter leaves, the remaining

members will still make up a winning coalition) .

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WVS: The power of a voter...• In any wvs, there will

always be the argument about “fairness.”

• Do some voters have too much power?

• Are there any voters who have no power at all?

• How can we describe the “power of a voter” in a system?

• We will look at TWO measures of power.

• The first is called the BANZHAF POWER INDEX.

• This index measures the power of a voter by describing how many winning coalitions the voter is critical for the success of.

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End of show #1

Going on?...

Weighted voting system:

Show #2: Banzhaf Power Distribution

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