Weighted Voting Systems - Terms by malj


									              Weighted Voting
• When we try to make collective decisions, it is
  only natural to consider how things are done in
• We are familiar with voting for class president –
  one for per person, winner is one with most votes.
• The electoral college seeks to give more power to
  states having more population
• We want to know who has the most power – as
  that could influence whose mind we try to change
  or whether the system is fair.
           Weighted Voting
• We are trying to decide whether or not a
  measure passes.
• The coalition is the group of people that can
  get the measure passed.
• Not everyone has the same ―clout‖.
  Modeled as some having multiple votes.

• A weighted voting system is characterized
  by three things — the players, the weights
  and the quota. The voters are the players (P1
  , P2, . . ., PN). N denotes the total number of
  players. A player's weight (w) is the number
  of votes he controls. The quota (q) is the
  minimum number of votes required to pass a
  motion. [q:w1,w2…wn]. Normally we
  require that q  2 w  w  ... w 
                     1   2    n

   as we can say more interesting things that
  way.                                            3
• A player's power is defined as that player's ability to influence
• The power of a coalition is not simply determined by its size.
• Consider the voting system [6: 5, 3, 2].
• Notice that a motion can only be passed with the support of
  P1. In this situation, P1 has veto power. A player is said to
  have veto power if a motion cannot pass without the support
  of that player. This does not mean a motion is guaranteed to
  pass with the support of that player – as player 1 doesn’t have
  enough votes by himself.
• Who has the most power? How is power divided between the
  players with 3 and 2?
        Weighted Voting Systems – Terms
 i. Coalition: any subset of a group of voters that bands
      together to either support a measure.

ii. Winning/Losing Coalition: a coalition that has
      enough votes to pass a measure is a winning
      coalition, otherwise it is a losing coalition.

iii. Dummy: a voter in a winning coalition whose vote
       isn’t needed to pass the measure.

iv. Voters Weight: the number of votes each voter has.

 v. Quota: the number of votes, q, necessary to pass
     a measure.

          Weighted Voting Systems - Terms
vi. Notation for voting system: [q : w1 , w2 ,..., wn ] where q is
   the quota, wi are the individual weights of the voters,
   and n is the number of voters.

vii. Requirements:
       1.                           as otherwise definition of
          q  w1  w2  ...  wn  dictator is problematic
       2. q  w  w  ...  w  as no point is having q larger
                 1    2       n

 viii. Changing q affects the way power is distributed.

 ix. Blocking Coalition: subset of voters opposing a
   motion with enough votes to defeat it. Any coalition
   with weight  w  q. w  w1  w2  ...  wn 

         Weighted Voting Systems – Terms

x. Dictator: voter whose voting weight meets or exceeds
       the quota for passing a measure. All other voters
       are dummies.

xi. Veto Power: a voter who has enough votes to block a
       measure is said to have veto power. A voter with
       weight  w  q . A dictator automatically has veto

xii. Critical Voter: in any winning coalition, he is the
      voter whose votes are essential to win.

• Now let us look at the weighted voting system [10: 11, 6,
  3]. With 11 votes, P1 is called a dictator. A player is typically
  considered a dictator if his weight is equal to or greater than the quota.
  The difference between a dictator and a player with veto power is that a
  motion is guaranteed to pass if the dictator votes in favor of it.
• The dictator has veto power. The measure passes if and only if he votes
  for it. Since the quota must be more than half the total, a dictator
  always has veto power.
• A dummy is any player, regardless of his weight, who has
  no say in the outcome of the election. A player without any
  say in the outcome is a player without power. Dummies
  always appear in weighted voting systems that have a
  dictator (provided the quota is more than half total) but also
  occur in other weighted voting systems
• Consider the voting system [8: 5, 3, 2]. Which are
  dictators? have veto power? are dummies?
  5 and 3 have veto power. 2 is a dummy
• Consider the voting system [8: 9, 3, 2]. Which are
  dictators? have veto power? are dummies?
• Consider the voting system [20:10,10,9]. Which are
  dictators? have veto power? are dummies?
• Consider the voting system [7:4,2,1]. Which are dictators?
  have veto power? are dummies?

           Banzhaf power index
             (sometimes called Penrose-Banzhaf index)
• Designed to quantify the power a voter has
• defined by the probability of changing an
  outcome of a vote
• To calculate the power of a voter using the Banzhaf index,
  list all the winning coalitions, then count the critical voters.
  A critical voter is a voter who, if he changed his vote from
  yes to no, would cause the measure to fail. A voter's power
  is measured as the fraction of all swing votes that he could
• Warning: in our electoral college of 50 states, there are
  51,476,301,254,318 winning coalitions!
An example Game Theory and Strategy by Phillip
                     D. Straffin:

• [6; 4, 3, 2, 1]
• The winning groups, with underlined critical voters,
  are as follows:
• Notice we assume that we only worry about what
  ONE player does in each case.
• There are 12 total critical votes, so by the Banzhaf
  index, power is divided thus.
• A = 5/12 B = 3/12 C = 3/12 D = 1/12
    Consider the U.S. Electoral College.
• There are a total of 538 electoral votes. A majority vote is considered
  270 votes. The Banzhaf Power Index would be a mathematical
  representation of how likely a single state would be able to swing the
  vote. For a state such as California, which is allocated 55 electoral
  votes, they would be more likely to swing the vote than a state such as
  Montana, which only has 3 electoral votes.
• Example: The United States is having a presidential election between a
  Republican and a Democrat. For simplicity, suppose that only three
  states are participating: California (55 electoral votes), Texas (34
  electoral votes), and New York (31 electoral votes).

Consider having republicans win. The democrats winning is
similar. Need 61 votes to win.

Calif Texas   NY     R     D     States that could
(55) (34)     (31)   votes votes swing the vote
R R           R        120     0 none
R R           D         89 31 California, Texas
R D           R         86 34 California , New York
D R           R         65 55 Texas , New York

 Power, each state has 1/3
     Consider a different set of states
            Need 55 to win
California   Texas   Ohio          States that could
(55)         (34)    (20)   R D swing
R            R       R      109 0 California
R            R       D       89 20 California
R            D       R       75 34 California
R            D       D       55 54 California

factor twenty

      Shapley-Shubik Power Index:
i. Shapley-Shubik Power Index:
    a. Permutation: total number of ways n things can
           be taken r at a time . P n     n!     .
                                        (n  r )!

           Order is important in a permutation.

    b. Pn
              n!   is used to find the number of ways to
                     order n elements in a set.

2. 1st voter in a permutation whose vote would make
     the coalition a winning coalition is called a pivotal

3. Shapley-Shubik Power Index is fraction of
    permutations in which a voter is pivotal.

          Shapley-Shubik Power Index:
  Given a voting system [ q : w1 , w2 ,..., wn ] create a Shapley-
  Shubik table: For this example use {A,B,C} with the voting system
  P33  3!  6 Pivotal voter is underlined.
AC                                                           4 1 1
ABC                                                           , , 
                                                             6 6 6

1. Count number of times A,B, and C are pivotal voters. Divide each
   value by 6 to get the Shapley-Shubik Power Index:
              4 1
2. Voter A is 6  6  4 times more powerful than B or C. Voter A
   has 4/6 or 66.67% of the power in this voting system.

factor twenty
Quite similar.
Factor of 4.1-4.3

              Try this one
• Suppose decisions are made by majority
  rule in a body consisting of A, B, C, D, who
  have 3, 2, 1 and 1 votes, respectively. The
  majority vote threshold is 4. There are 24
  possible orders for these members to vote:

              A=1/2, B=C=D = 1/6





B has no more power than C or D
            Shapley-Shubik Power Index:
   1. Sometimes permutations are too large to list all of them
       so we do it by grouping.

Consider the voting system [5:3,1,1,1,1,1,1]. 7! = 5040
      GSSSSSS 3456789                 SSSSGSS 1234789
      SGSSSSS 1456789                 SSSSSGS 1234589
      SSGSSSS 1256789                 SSSSSSG 1234569
      SSSGSSS 1236789
G is pivotal 3/7 of the time. S is pivotal in (4/7)/6=2/21 of the
  time. Therefore, the Shapley-Shubik Power Index is
                     3 2 2 2 2 2 2 
                      , , , , , , 
                      7 21 21 21 21 21 21 


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