VOTING POWER

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VOTING POWER

• IN THE ELECTORAL COLLEGE

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Weighted Voting Games

• The Electoral College is an example of a weighted
  voting game.
• Instead of each voter casting a single vote, each
  voter casts a block of votes, with some voters
  casting larger blocks and others casting smaller
  blocks.
• Other examples
• voting by disciplined party groups in multi-party
  parliaments (probably elected on the basis of
  proportional representa-tion)
• balloting in old-style U.S. party nominating
  conventions under the unit rule
• voting in the EU Council of Ministers, IMF
  council, etc.
• voting by stockholders (holding varying amounts
  of stock).

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Weighted Voting Games (cont.)

• Weighted voting can be analyzed using the theory
  of simple games.
• A simple game is a (voting or similar) situation
  in which every potential coalition (set of
  players/votes) can be deemed to be either winning
  or losing.
• With respect to lawmaking power of the United
  States, the winning coalitions are
• All coalitions including 218 House members, 51
  Senators, and the President, and also
• All coalitions including 290 House members and 67
  Senators.
• Including the Vice President, or 60 Senators
  excluding the Vice President in so far as
  filibustering is an option.

4
Weighted Voting Games (cont.)

• With respect to weighted voting games, the most
  basic finding is that voting power is not the
  same as (and is not proportional to) voting
  weight in particular
• voters with very similar (but not identical)
  weights may have very different voting power and
• voters with quite different voting weights may
  have identical voting power.
• In general, it is impossible to apportion voting
  power (as opposed to voting weights) in a
  refined fashion, especially within small
  groups.

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Weighted Voting Example

• Consider a country that uses proportional
  representation to elect members of parliament.
• Duvergers law implies that the country will have
  a multi-party system.
• Hotelling-Downs implies that the parties will be
  spread over the ideological spectrum and may
  receive rather similar vote shares.

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Weighted Voting Example

• Suppose that four parties receive these vote
  shares Party A, 27 Party B, 25 Party C, 24
  Party 24.
• Seats are apportioned in a 100-seat parliament
  according some apportionment formula. In this
  case, the apportionment of seats is
  straight-forward
• Party A 27 seats Party C 24 seats
• Party B 25 seats Party D 24 seats
• Seats (voting weights) have been apportioned in a
  way that is precisely proportional to vote
  support, but voting power has not been so
  apportioned (and cannot be).

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Weighted Voting Example (cont.)

• Since no party controls a majority of 51 seats, a
  governing coalition of two or more parties must
  be formed.
• A partys voting power is based on its
  oppor-tunities to help create (or destroy)
  winning (governing) coalitions.
• But, with a small number of parties, coalition
  possibilities -- and therefore differences in
  voting power -- are highly limited.

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Weighted Voting Example (cont.)

• A 27 seats B 25 seats C 24 seats D 24
  seats
• Once the parties start negotiating, they will
  find that Party A has voting power that greatly
  exceeds its slight advantage in seats. This is
  because
• Party A can form a winning coalition with any one
  of the other parties and
• the only way to exclude Party A from a winning
  coalition is for Parties B, C, and D to form a
  three-party coalition.
• The seat allocation (totaling 100 seats) is
  strategically equivalent to this much simpler
  allocation (totaling 5 seats)
• Party A 2 seats
• Parties B, C, and D 1 seat each
• Total of 5 seats, so a winning coalition requires
  3 seats.
• So the original seat allocation is strategically
  equivalent to one in which Party A has twice the
  weight of each of the other parties (which is not
  at all proportional to their vote shares).

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Weighted Voting Example (cont.)

• Suppose at the next election the vote and seat
  shares change a bit
• Before Now
• Party A 27 Party A 30
• Party B 25 Party B 29
• Party C 24 Party C 22
• Party D 24 Party D 19
• While the seats shares have only slightly
  changed, the strategic situation has changed
  fundamentally.
• Party A can no longer form a winning coalition
  with Party D.
• Parties B and C can now form a winning coalition
  by themselves.
• The seat allocation is equivalent to this much
  simpler allocation
• Parties A, B, and D 1 seat each
• Party D 0 seats
• Total of 3 seats, so a winning coalition requires
  2 seats.
• Party A has lost voting power, despite gaining
  seats.
• Party C has gained voting power, despite losing
  seats.
• Party D has become powerless (a so-called dummy),
  despite retaining a substantial number of seats.

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Weighted Voting Example (cont.)

• In fact, the only possible strong simple games
  with 4 players are these two with simplified
  weights of (2,1,1,1) and (1,1,1,0),
• plus the inessential game (1,0,0,0), in which
  one party holds a majority of seats (making all
  other parties dummies, so no coalition need be
  formed.
• Expanding the number of players to five produces
  these additional possibilities (3,2,2,1,1),
  (3,1,1,1,1), (2,2,1,1), (1,1,1,1,1).
• With 6 or more players coalition possibilities
  become much more numerous and complex.

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Weighted Voting Example (cont.)

• Returning to the four-party example, voting power
  would change further if the parliamentary
  decision rule were to change from simple majority
  to (say) 2/3 majority (like the old nominating
  rule in Democratic National Conven-tions).
• Both before and after the election, all
  three-party coalitions, and no smaller
  coalitions, are winning coalitions (so all four
  parties are equally powerful).
• In particular, under 2/3 majority rule, Party D
  is no longer a dummy after the election.
• Thus, changing the decision rule reallocates
  voting power, even as voting weights (seats)
  remain the same.
• Generally making the decision rule more demanding
  tends to equalize voting power.
• In the limit, weighted voting is impossible under
  unanimity rule.

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Weighted Voting Example (cont.)

• A simple weighted majority voting game (with no
  ties) is a strong simple game, i.e.,
• Given any complementary pair of coalitions, one
  is winning and the other is losing.
• A (for example) 2/3 weighted majority voting game
  is no longer strong, i.e.,
• Given some complementary pairs of coalitions,
  neither may be winning (both are blocking).

13
Power Indices

• Several power indices have been pro-posed to
  quantify the share of power held by each player
  in simple games.
• These particularly include
• the Shapley-Shubik power index and
• the Banzhaf power index.
• Such power indices provide precise formulas for
  ascertaining the voting power of players in
  weighted voting games.

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The Shapley-Shubik Index

• The Shapley-Shubik power index works as follows.
  Consider every possible ordering of the players
  A, B, C, D (e.g., every possible order in which
  they might line up to support a proposal put
  before a voting body). Given 4 voters, there are
  4! 4 x 3 x 2 x 1 24 possible orderings

15
The Shapley-Shubik Index (cont.)

• Suppose coalition formation starts at the top of
  each ordering, moving downward to form coalitions
  of increasing size.
• At some point a winning coalition formed.
• The grand coalition A,B,C,D is certainly
  winning.
• For each ordering, identify the pivotal player
  who, when added to the players already in the
  coalition, converts a losing coalition into a
  winning coalition.
• Given the seat shares of parties A, B, C, and D
  before the election, the pivotal player in each
  ordering is identified by the arrow (lt).

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The Shapley-Shubik Index (cont.)

• Voter is Shapley-Shubik power index value SS(i)
  is simply
• Number of orderings in which the voter i is
  pivotal
• Total number of orderings
• Clearly such power index values of all voters add
  up to 1.
• Counting up, we see that A is pivotal in 12
  orderings and each of B, C, and D is pivotal in 4
  orderings. Thus
• Voter SS Power
• A 1/2 .500
• B 1/6 .167
• C 1/6 .167
• D 1/6 .167
• So according to the Shapley-Shubik index, Party A
  has 3 times the voting power of each other party.
• Lloyd Shapley and Martin Shubik, American
  Political Science Review, 1955.

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The Banzhaf Index

• The Banzhaf power index works as follows
• A player i is critical to a winning coalition if
• i belongs to the coalition, and
• the coalition would no longer be winning if i
  defected from it.
• Voter is absolute Banzhaf power Bz(i) is
• Number of winning coalitions for which i is
  critical
• Total number of coalitions to which i belongs.
• Bz(i) is equivalent to voter is a priori
  probability of casting a decisive vote, e.g.,
  breaking what otherwise would be a tie.
• In this context, a priori probability means, in
  effect, given that all other voters vote randomly
  (i.e., by flipping coins).

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The Banzhaf Index (cont.)

• Given the seat shares before the election, and
  looking first at all the coalitions to which A
  belongs, we identify
• A,A,B,A,C, A,D, A,B,C, A,B,D,
  A,C,D, (A,B,C,D.
• Checking further we see that A is critical to all
  but two of these coalitions, namely
• A (because it is not winning) and
• A,B,C,D (because B,C,D can win without A).
• Thus Bz(A) 6/8 .75

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The Banzhaf Index (cont.)

• Looking at the coalitions to which B belongs, we
  identify
• B,A,B, B,C, B,D, A,B,C, A,B,D,
  B,C,D, (A,B,C,D.
• Checking further we see that B is critical to
  only two of these coalitions
• B, B,C, B,D are not winning and
• A,B,C, A,B,D, and A,B,C,D are winning even
  if B defects.
• The positions of C and D are equivalent to that
  of B.
• Thus Bz(B) Bz(C) Bz(D) 2/8 .25

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The Banzhaf Index (cont.)

• The "total absolute Banzhaf power" of all four
  voters
• .75 .25 .25 .25 1.5 .
• Voter i's Banzhaf index power values BP(i) is his
  share of the "total power," so
• BP(A) .75/1.5 1/2 and
• BP(B) BP(C) BP(D) .25/1.5 1/6.
• John F. Banzhaf, Weighted Voting Doesnt Work,
  Rutgers Law Review, Winter, 1965, and One Man,
  3.312 Votes A Mathematical Analysis of the
  Electoral College, Villanova Law Review, Winter
  1968.

21
Shapley-Shubik vs. Banzhaf

• In this simple 4-voter case, the two indices
  evaluate power in the same way.
• This is often true in simple examples, but it is
  not true more generally.
• In particular kinds of situations the indices
  evaluate the power of players in radically
  different ways.
• For example, if there is single large stockholder
  while along other holding are highly dispersed.
• It is even possible that the two indices may rank
  players with respect to power in different ways
  (but not in weighted voting games).
• It is now generally believed that the Banzhaf is
  more appropriate to apply to the analysis of
  voting institutions, including the Electoral
  College.

22
Do Constitution Makers Understand Voting Power?

• Luther Martin (delegate to the Constitutional
  Convention but an opponent of its proposal)
• " ...even if the States who had most inhabitants
  ought to have a greater number of delegates, yet
  the number of delegates ought not to be in exact
  proportion to the number of inhabitants, because
  the influence and power of those States whose
  delegates are numerous, will be greater when
  compared to the influence and power of the other
  States, than the proportion which the numbers of
  delegates bear to each other.
• Application to House of Representatives vs.
  Electoral College.

23
Do Constitution Makers Understand Voting Power?
(cont.)

• The allocation of voting power in the original
  (six-member) European Common Market Council of
  Ministers made the smallest member (Luxembourg) a
  dummy.
• The recent Nice Treaty expanding and reallocating
  voting power in the EU had effects on voting
  power that almost certainly were not intended.
• Dan S Felsenthal and Moshé Machover, The Treaty
  of Nice and Qualified Majority Voting, Social
  Choice and Welfare, 2001.

24
Evaluating Voting Power in the Electoral College

• Lets first review the apportionment of voting
  weights (electoral votes) in the Electoral
  College, in relation to states share of the U.S.
  population.
• We know that
• there is a small-state bias in this
  apportionment and
• there is the problem of apportionment into whole
  numbers that is most significant among small
  states.
• The following chart shows the relationship
  between electoral vote and population shares
  following the 2000 Census.

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Voting Power in the Electoral College (cont.)

• When the Shapley-Shubik index was first developed
  in the early 1950s, it seemed apparent that that
• the voting power of the weightiest players is
  typically greater than their weights, while
• the voting power of less weighty players is
  typically less than their weights.
• This seemed consistent with intuition (going back
  to Luther Martin and the move to the general
  ticket system) that large states are
  substantially advantaged in the Electoral
  College, even though the small are favored by the
  apportionment of voting weights.
• However, when the first (Shapley-Shubik
  approxima-tions, using Monte Carlo methods)
  evaluations of voting power in the Electoral
  College were done, this tendency manifested
  itself only very weakly.

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Voting Power in the Electoral College (cont.)

• Clearly we cannot apply the formulas sketched
  above for calculating power index values in the
  Electoral College (let alone larger weighted
  voting bodies)
• There are 51! 1.55 x 1066 ways 51 voters might
  line up to vote.
• Indeed, such calculations are well beyond the
  practical computing power even of
  super-computers.
• Fortunately quite accurate indirect methods of
  calculation exist
• There are websites that can make power index
  calculations.
• The best is http//www.warwick.ac.uk/ecaae/
• Also see http//www.lse.ac.uk/collections/VPP/
• Note that Electoral College Voting Game is not
  quite strong, i.e., a 269-269 tie is possible.

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Voting Power in the Electoral College
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The Electoral College as a Two-Stage Voting Game

• With respect to the Electoral College as a
  (one-stage) weighted voting game, the conjecture
  that large states are greatly advantaged by the
  winner-take-all practice is not supported.
• But the one-stage Electoral College voting game
  is a chimera
• each state is a mere agent of the popular voting
  majority within the state.
• We should expand the application of the Banzhaf
  index to include individual voters with each
  state.
• Within each state, we have an (unweighted)
  majority voting game that determines how that
  state's bloc of electoral votes is to be cast in
  the weighted majority EC game.

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The Electoral College as a Two-Stage Voting Game
(cont.)

• If there are nk voters in state k, clearly (on
  the basis of common sense and either power index)
  each voter has 1/nk of the voting power within
  the state.
• Since (as we have seen in the prior chart) each
  nk is approximately proportional to the state's
  voting power Bz(k) in the Electoral College, it
  appears that the voting power (in the full
  100,000,000-person Presidential election game) of
  all voters throughout the country is just about
  the same.
• This appears to follow because the ratio Bx(k) /
  nk state voting power to population is
  approximately constant over all states.

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The Electoral College as a Two-Stage Voting Game
(cont.)

• But closer analysis of the properties of the
  Banzhaf index shows that this apparent uniformity
  of individual voting power across the states does
  not hold after all.
• While all voters in the same state have equal
  voting power in determining the allocation of
  their states electoral votes, voters in
  different states clearly do not have the same
  voting power in determining the allocation of
  their respective states electoral voters.

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The Electoral College as a Two-Stage Voting Game
(cont.)

• Recall that Banzhaf power is equivalent to a
  voters a priori probability of casting a
  decisive vote, e.g., breaking what otherwise
  would be a tie (when other voters vote randomly.
• Compare the voting power of a voter in Wyoming
  and a voter in California.
• One the one hand, the Wyoming voter has the
  greater chance of casting a decisive vote in the
  first stage of the voting game, simply because
  there are fewer other voters in Wyoming and the
  voter has a larger (though still very small) of
  breaking what would otherwise be a tie in the
  Wyoming popular vote.
• On the other hand, while the voter in California
  has a smaller chance of casting a decisive vote,
  if that voter does cast a decisive vote, it is
  much more likely to determine the outcome of the
  Presidential election, because it will tip 55
  (rather than 3) electoral votes into one
  candidates column or the others.

35
The Electoral College as a Two-Stage Voting Game
(cont.)

• Put informally, voters in small states have a
  bigger chance of determining the winner of a
  smaller prize, while voters in large states have
  a smaller chance of determining the winner of a
  bigger prize.
• The question is how these relative advantages and
  disadvantages balance out.
• We have seen the value (Banzhaf power) of the
  prizes are approximately proportional to
  weights (number of electoral votes).

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The Electoral College as a Two-Stage Voting Game
(cont.)

• However, statistical theory tells us that, while
  the probability of casting a decisive vote is
  inversely related to the number of voters, it is
  inversely proportional -- not to the number of
  voters -- but to the square root of this number.
• This is an approximation that becomes very good
  once the number of voters reaches a few dozen.
• Now we can put the probabilities for the two
  stages together.

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The Electoral College as a Two-Stage Voting Game
(cont.)

• California has about 68 times the population of
  Wyoming.
• Therefore a voter in Wyoming has a greater
  probability of casting a decisive vote.
• But the Wyoming voter does not have 68 times the
  probability -- but rather about v68 8.25 times
  the probability.
• Meanwhile, California has 18.33 times more
  electoral votes than Wyoming and about 21 times
  the probability of being decisive in the
  Electoral College.
• Suppose the Wyoming voters probability of
  casting a decisive vote is p then the California
  voters probability is about 21p/8.33 2.5p.
• In sum, the California voter has a considerably
  larger probability of casting a decisive vote in
  the two-stage Presidential election voting game
  than the Wyoming voter.
• Banzhafs reference to One Man, 3.312 Votes
  compares DC with NY in the 1960s.
• DC actually had a larger population than some
  states with 4 electoral votes.
• See language of the 23rd Amendment.

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The Electoral College as a Two-Stage Voting Game
(cont.)

• Bear in mind that all these probabilities are a
  priori, on the assumption of random voting.
• That is, they factor out any empirical data or
  assumptions about actual voting patterns.
• Some political scientists have critiqued this
  conclusion as the Banzhaf fallacy.
• Probably what they mean is not really that the
  Banzhaf analysis is fallacious but that it is not
  especially relevant in practice.
• Certainly the status of states as actual or
  potential battlegrounds is likely to be more
  relevant to the amount of attention that voters
  in different states get in Presidential elections
  than is their Banzhaf power.
• Howard Margolis, The Banzhaf Fallacy, American
  Jounal of Political Science, 1983 Andrew
  Gellman, Jonathan Katz, and Gary King, various
  recent papers and articles.