Weighted Voting Systems

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Weighted Voting Systems
  i.   Coalition any subset of a group of
voters that bands together to either support or
oppose 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 isnt 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.
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Weighted Voting Systems
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Weighted Voting Systems
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 gt w
- q . A dictator automatically has veto
power. xii. Critical Voter in any winning
coalition, he is the voter whose votes are
essential to win.
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Weighted Voting Systems
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Weighted Voting Systems
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Banzhaf Power Index

•    Equivalent Voting Systems       
• 1. Two voting systems are equivalent
  if there is a way for all the voters of the
  1st system to exchange places with the voters
  of the 2nd system and preserve all winning
  coalitions.
•       Minimal Winning Coalition a winning
  coalition in which every member is a critical
  voter.
• 1.  A voting system can be completely
  described by listing the minimal
  winning coalitions. All other coalitions are
  formed by adding voters to minimal winning
  coalitions.
• 2.  If a voter belongs to all
  winning coalitions then that voter has veto
  power.
• 3.  When listing minimal wining
  coalitions of a voting system
• a. If 2 minimal winning
  coalitions are distinct, each must have a voter
  who doesnt belong to the other.
• b. Every pair of minimal
  winning coalitions has to overlap with at least
  one voter in common.

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Shapely-Shubik Power Index
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Shapely-Shubik Power Index
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Shapely-Shubik Power Index
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Credits

• Tannenbaum, Excursions in Modern Mathematics, 5th
  ed