Weighted Voting Systems

Important Vocabulary

• Weighted voting system

• Motions• The players• The weights• The quota• Dictator• Dummy

• Veto Power• Coalitions• Grand Coalition• Winning Coalition• Losing Coalition• Critical Players

What is a weighted voting system?

Any formal arrangement in which voters are not necessarily equal in terms of the number of votes they control.

Motions

To keep things simple, we will only consider yes-no votes, generally known as motions ( a vote between two candidates or alternatives can be rephrased as a yes-no vote and thus is a motion).

Every voting system is characterized by three elements:

• The players- may be individuals, institutions, or even countries. N denotes the number of players and P1, P2, …, PN the names of players.

• The weights – indicates the number of votes each player controls. w1, w2, …, wN represent the weight of player 1, player 2 and so on.

• The quota – the minimum number of votes needed to pass a motion. May be more than a simple majority. q is used to denote the quota.

q must be more than 50% of the total number of votes.q may not be more than 100% of the votes.

Notation and Examples

[q : w1, w2, . . . , wN] is the notation we use to indicate we are dealing with a weighted voting system. Inside the brackets we always list the quota first, followed by a colon and then the respective weights of the individual players. Note:It is customary to list the weights in numerical order, starting with the highest.

Weighted voting in a partnership

Four partners – P1, P2, P3, and P4 –decide to start a new business. In order to raise the $200,000 needed as startup money, they issue 20 shares worth $10,000 each. Suppose that P1 owns 8 shares, P2 owns 7 shares, P3 owns 3 shares, and P4 owns 2 shares and that each share is equal to one vote in the partnership.

They set up the rules of the partnership so that two-thirds of the partners’ votes are needed to pass any motion.

This can be described using the following:

[14: 8, 7, 3, 2]

Note: 14 is the quota because it is the first integer larger than two-thirds of 20.

The quota can’t be too small

Imagine the same partnership with the only difference being that the quota is changed to 10 votes. We might be tempted to think that this arrangement can be described by [10: 8, 7, 3, 2], but this is not a viable weighted voting system. The problem here is that the quota is not big enough to allow for any type of decision making.(Mathematical Anarchy!!)

Or Too Large

Once again look at the partnership, but this time with a quota q = 21. Given that there are only 20 votes to go around, even if every partner were to vote Yes, a motion is doomed to fail.(Mathematical Gridlock)

One Partner-One Vote?Suppose we look again at the partnership. This time the quota is set to be q = 19. [19: 8, 7, 3, 2] What’s interesting about this weighted voting system is that the only way a motion can pass is by the unanimous support of all the players.In a practical sense this is equivalent to [4: 1, 1, 1, 1]. Just looking at the number of votes a player has can be very deceptive!

Now work on the Introduction to weighted voting follow up questions.

The Making of a Dictator

Consider the weighted voting system [11: 12, 5, 4]What do you notice?P1 owns enough votes to carry a motion singlehandedly. In this situation P1 is in complete control -if P1 is in favor of a motion it will pass; if P1 is against it, it will fail.

Dictator:

In general, a player is a dictator if the player’s weight is bigger than or equal to the quota.

There can only be one dictator. Why is this?

What about the other players?

When P1 is a dictator, all the other players, regardless of their weights, have absolutely no say in the outcome of the voting – there is never a time when any of their votes are needed.

A player that never has a say in the outcome of the voting is a player that has no power and is called a dummy.

The Curse of the Dummy

Four college students decide to go into business together. Three of them invest $10,000 in the business, and each gets 10 votes in the partnership. The fourth student is a little short on cash, so he invests only $9,000 and thus gets 9 votes. Suppose the quota is set at 30 (don’t ask why). Under these assumptions the partnership can be described as the weighted voting system [30: 10, 10, 10, 9]. Everything seems right, right?

Wrong!!

In this weighted voting system the fourth student turns out to be a dummy!Why?Notice that a motion can only pass if the first three are for it, and then it makes no difference whether the fourth studentis for or against it.

The Power of the VetoIn the weighted voting system [12: 9, 5, 4, 2], P1 plays an interesting role – while not having enough votes to be a dictator, he has the power to obstruct by preventing any motion from passing. This happens because without P1’s votes, a motion cannot pass – even if all the remaining players were to vote in favor of the motion. In a situation like this we say that P1 has veto power.

Determining Power

In weighted voting, a player’s weight does not always tell the full story of how much power the player holds. Sometimes a player with lots of votes can have little or no power (think about the curse of the dummy), and conversely, a player with just a couple of votes can have a lot of power (think about one partner – one vote?). What does having “power” mean?

Coalitions

We will use the term coalition to describe any set of players that might join forces and vote the same way.In principle we might have a coalition with as few as one player or as many as all players.The coalition consisting of all players is called the grand coalition.

We use set notation to describe coalitions. For example the coalition consisting of players 1 , 2, and 3 can be written as { P1, P2, P3} or { P3, P2, P1} or { P2, P3, P1}, etc. – the order in which the members of the coalition are listed is irrelevant.

Winning CoalitionsSome coalitions have enough votes to win and some don’t . Naturally, we call the former winning coalitions and the latter losing coalitions. A single player coalition can be a winning coalition only when that player is a dictator. So under the assumption that there are no dictators in our weighted voting systems (dictators are boring!), a winning coalition must have at least two players. A grand coalition is always a winning coalition.

Critical Players

In a winning coalition, a player is said to be a critical player for the coalition if the coalition must have that player’s votes to win.In other words, when we subtract a critical player’s weight from the total weight of the coalition, the total of the remaining votes drops below the quota.

The Weirdness of Parliamentary Politics

The Parliament of Icelandia has 200 members, divided among three political parties – the Red Party (P1), the Blue Party (P2), and the Green Party (P3). The Red Party has 99 seats in the Parliament, the Blue Party has 98, and the Green Party has only 3. Decisions are made by majority vote – which in this case requires 101 out of a total of 200 votes.

Since in Icelandia members of the Parliament always vote along party lines (voting against the party line is extremely rare in parliamentary governments), we can think of Icelandia’s Parliament as the weighted voting system [101: 99, 98, 3].

Coalition Votes Critical Players

{P1, P2} 197 P1 and P2

{P1, P3} 102 P1 and P3

{P2, P3} 101 P2 and P3

{P1, P2, P3} 200 None

Winning Coalitions in [101: 99, 98, 3]

The Banzhaf Power Index

A player should be measured by how often that player is critical.To measure the power, follow these guidelines:Step 1. List and then count the total number of winning coalitions.Step 2. Within each winning coalitiondetermine which are the critical players.

Step 3: Count the number of times that P1 is critical. Call this number B1. Repeat for each of the other players to find B2, B3, . . ., BN.

Step 4: Find the total number of times all players are critical. This total is given by T = B1 + B2 + . . .+BN.

Step 5: Find the ratio β1 = B1/T. This gives the Banzhaf power index of P1. Repeat for each of the other players to find β2, β3,. . . βN.

Banzhaf Power in [4: 3, 2, 1]

Step 1: There are three winning coalitions in this voting system. What are they?

Step 2: The critical players in each winning coalition are:

Step 3: B1 = 3; B2 = 1 and B3 = 1.Step 4: T = 3 + 1 + 1 = 5

Step 5: β1 = B1/T = 3/5; β2 = B2/T = 1/5; β3 = B3/T = 1/5. Another way to express these is in percents: β1 = 60%; β2 = 20%; β3 = 20%