Chapter 2 Weighted Voting Systems

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					Chapter 2 Weighted Voting
              2.1 Terminology

   Weighted voting system —any formal voting
    arrangement in which voters are not necessarily
    equal in terms of the number of votes they
   Motions —only consider yes-no votes or
    “motions”—a vote between two candidates or
    alternatives can be rephrased as a yes-no vote.
   Players —voters in a weighted voting system.
            Terminology (cont.)
   The Weights —the weight of the player is the number of
    votes a player controls. Representation: Player 1 (P1)
    holds Weight 1 (w1). The last player is represented by Pn
    with wn weight.
   The Quota (q) —the minimum number of votes to pass a
    motion. 50% < q < 100% Needs to be more than 50%
    to avoid a stalemate. Cannot be more than the total
    number of votes (100%). Quota is stipulated at the
    beginning of an election.
   Representation: [q: w1, w2,…, wn] It is customary to list
    the weights in descending order.,
          Representation of Quota

   Example 2.1: Suppose four partners decide to start a new
    business adventure. There are 20 shares (votes) and P1
    owns 8 shares, P2 owns 7 shares, P3 owns 3 shares,
    and P4 owns 2 shares. Imagine now they set up the rules
    of the partnership so that 2/3 of the partner’s votes are
    needed to pass any motion.
   Use the simplified notation [q: w1, w2,…, wn] to describe
    this partnership.
   Answer: [14: 8, 7, 3, 2] (remember, 14 is the first integer
    larger than 2/3 of 20.
   What if the above quota was changed to [19: 8, 7, 3, 2]?
                More Terminology

   Dictator —the player (can only be one) whose weight is greater
    than or equal to the quota. wd > q
   Ex. [11: 12, 5, 4]
   Dummy —the player whose votes will never matter—all other
    players are dummies when there is a dictator. There can be
    dummies without a dictator.
   Ex. [30: 10, 10, 10, 9] Don’t be a dummy!
   Veto Power —a player has veto power if a motion cannot pass
    without his/her votes—he/she CAN prevent a motion from
    passing—this does NOT mean that whatever the player votes
    for must pass.
   Ex. [12: 9, 5, 4, 2]
Clarifying Example…
          “I can see clearly now…”

   Find the players who are dictators, have veto power, or are
   [19: 9, 7, 5, 3, 1]
    P1 & P2 veto—P5 dummy
   [15: 16, 8, 4, 1]
    P1 dictator—all of the rest are dummies
   [17: 13, 5, 2, 1]
    P1 & P2 veto—P3 & P4 are dummies
   [25: 12, 8, 4, 2]
   All have veto power
   Weighted Voting Worksheet--#1 only
   Classwork for tomorrow Book—Pg. 72: 1-7 odd (Bring Book)
2.2 Preface—More Terminology

   Coalitions —any set of players who might join forces and
    vote the same way. (grand coalition —consisting of all
   Winning and Losing Coalitions –self explanatory. Note:
    a single-player coalition can only be a winning coalition if
    that player is a dictator. Winning coalitions will have at
    least two players for our purposes. A grand coalition is
    always a winning coalition.
   Critical Players –in a winning coalition, a player is said to
    be a critical player if the coalition must have that player’s
    votes to win. So, W – w < q where W is the weight of the
    coalition and w is the weight of the critical player.
                 Coalition Example

   Think about [101: 99, 98, 3]

              Coalition            Votes        Critical
              {P1, P2}              197           P1 and P2
              {P1, P3}              102            P1 and P3
              {P2, P3}              101            P2 and P3
           {P1, P2, P3}             200               None
   Since there are three players, each critical twice, we can
    say that each player holds two out of six, or one-third of
    the power.
    2.2/2.3 The Banzhaf Power Index

   1st count the total number of winning coalitions in which P1 is
    a critical player—call this B1. Do this for the remaining
   2nd find the total number of times all players are critical ( T =
    B1 + B2 + … + Bn).
   3rd find all of the ratios B1/T, B2/T, …, Bn/T these ratios
    should be between 0 and 1 or a percent between 0 and
   The list β1,β2, …, βn is called the Banzhaf power distribution
    (using “beta”)…the sum should total 1 or 100%.
   Ex. 2.9 Pg. 56 Banzhaf Power in [4: 3, 2, 1] & Ex. 2.10 Pg.
    57 NBA Draft
How many Coalitions are there?

   Think of subsets…how many subsets does {a, b, c}
   { }, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}—for our
    purposes, the empty set doesn’t count.
   Number of subsets 2N
   Number of subsets without the empty set
                   2N -1 where N is the number of players
   Weighted Voting Worksheet #1 Parts 6A and 6C
   Classwork/Homework Pg. 73:11-17 odd
2.4 The Shapley-Shubik Power Index

   Coalitions are formed sequentially—order matters! (not so
    with Banzhaf)
   {P1, P2,P3} can form six (3!) different sequential
    coalitions—try to find them all.
   We notate this by using P2, P1, P3 to show that P2
    entered the coalition 1st, then P1, and P3 entered last.
   Pivotal Player —the player who turns a losing coalition
    into a winning one—the player who “tips the scale”. Every
    sequential coalition has one and only one pivotal player.
Computing the Shapley-Shubik Power

   1st make a list of all possible sequential coalitions of the given number
    of players. (let this be T)
   2nd count the number of times P1, P2, …,Pn are pivotal players…label
    this SS1, SS2, …, SSn.
   3rd find the ratio σ1 = SS1/T (sigma) which is the power index for P1—
    use this ratio to find the power index of all players.
   The Shapley-Shubik power distribution is a list of all σ’s—σ1, σ2, …,
   The number of sequential coalitions with N players is found by N!
   Ex. 2.17 Pg. 66 (SS Power in [4: 3, 2, 1])
   Complete Weighted Voting Worksheet
   Classwork/Homework Pg. 74: 23-31 odd
   Complete the “Are you always a dummy?” Worksheet (in class
   Quiz on 2.1-2.3 (includes Shapely-Shubik) Friday
   Chapter 2 Test Tuesday

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