Weighted Voting Systems

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Weighted Voting Systems
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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

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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.

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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).

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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.

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• 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.
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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.

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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.

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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.
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The quota cant 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!!)
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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)

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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 Whats 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!

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

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Dictator

• In general, a player is a dictator if the
  players weight is bigger than or equal to the
  quota.
• There can only be one dictator. Why is this?

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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.

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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
(dont ask why). Under these assumptions the
partnership can be described as the weighted
voting system 30 10, 10, 10, 9. Everything
seems right, right?
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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 student is for or
against it.
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The Power of the Veto
In 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 P1s
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.
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Determining Power

• In weighted voting, a players 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?

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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.

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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.
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Winning Coalitions

• Some coalitions have enough votes to win and some
  dont . 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.

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Critical Players

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

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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.

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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 Icelandias
Parliament as the weighted voting system 101
99, 98, 3.
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Winning Coalitions in 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
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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 coalition
• determine which are the critical
• players.

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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 ß1
B1/T. This gives the Banzhaf power index of
P1. Repeat for each of the other players to find
ß2, ß3,. . . ßN.
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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

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Step 5 ß1 B1/T 3/5 ß2 B2/T 1/5 ß3
B3/T 1/5. Another way to express these is in
percents ß1 60 ß2 20 ß3 20