VOTING POWER
IN THE ELECTORAL COLLEGE
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 representation);
• 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).
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.
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.
Weighted Voting Example
• Consider a country that uses proportional
representation to elect members of parliament.
• Duverger’s 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.
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).
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 party’s voting power is based on its opportunities 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.
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).
Weighted Voting Example (cont.)
• Suppose at the next election the vote and seat shares change a bit:
Before
Party A:
Party B:
Party C:
Party D:
27
25
24
24
Now
Party A:
Party B:
Party C:
Party D:
30
29
22
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.
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.
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 Conventions).
• 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.
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).
Power Indices
• Several power indices have been proposed 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.
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:
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 (<=).
The Shapley-Shubik Index (cont.)
• Voter i’s 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
A
B
C
D
SS Power
1/2 = .500
1/6 = .167
1/6 = .167
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.
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 i’s 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 i’s 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).
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
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
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 Doesn’t Work,” Rutgers Law Review, Winter,
1965, and “One Man, 3.312 Votes: A Mathematical Analysis of the Electoral
College,” Villanova Law Review, Winter 1968.
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.
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.
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.
Evaluating Voting Power in the
Electoral College
• Let’s 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.
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 approximations, using “Monte Carlo” methods) evaluations of
voting power in the Electoral College were done, this
tendency manifested itself only very weakly.
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 supercomputers.
• 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.
Voting Power in the Electoral College
The Electoral College as a TwoStage 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.
The Electoral College as a TwoStage 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.
The Electoral College as a TwoStage 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 state’s electoral votes, voters in different
states clearly do not have the same voting
power in determining the allocation of their
respective state’s electoral voters.
The Electoral College as a TwoStage Voting Game (cont.)
• Recall that Banzhaf power is equivalent to a voter’s 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 candidate’s
column or the other’s.
The Electoral College as a TwoStage 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).
The Electoral College as a TwoStage 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.
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 √68 = 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 voter’s probability of casting a decisive vote
is p; then the California voter’s 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.
– Banzhaf’s 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.
The Electoral College as a TwoStage 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.