A PRIORI VOTING POWER
UNDER THE ELECTORAL
COLLEGE AND ALTERNATIVE
INSTITUTIONS
Nicholas R. Miller
Revised August 2007
The previous results for the Modified District Plan and the
National Bonus Plan were incorrect. Thanks to Claus Beisbart,
Dan Felsenthal, and Moshé Machover for comments and
suggestions.
Preface
• Polsby’s Law: What’s bad for the political system
is good for political science, and vice versa.
• George C. Edwards, WHY THE ELECTORAL
COLLEGE IS BAD FOR AMERICA (Yale, 2004)
• Deduction: The Electoral College is good for
Political Science.
What This Analysis Does and Does Not Take
Account Of?
• This analysis takes account of the following empirical
data only:
– the 2000 U.S. Census “apportionment population” of the 50
states, and
– the 2000 U.S. Census population of the District of Columbia.
• It also takes account of
– the provisions of the Constitution pertaining to the apportionment
of House seats and electoral votes, and
– the laws passed by Congress that
• fix the size of the House at 435 members, and
• prescribe the “Method of Equal Proportions” for apportioning House
seats.
• It does not take account of other demographic data,
historical voting trends, polling or survey data, or actual
election results.
– This indicates the sense in which the analysis is “a priori” (or
from “the original position,” “behind a veil of ignorance,” etc.)
Overview
• As we all know, the President is elected, not by a direct
national popular vote, but by an Electoral College system
in which separate state popular votes are [in almost
universal practice] aggregated by adding up electoral
votes awarded on a winner-take-all basis to the plurality
winner in each state.
• State electoral votes vary with population and at present
range from 3 to 55.
• The Electoral College therefore generates a weighted
voting game susceptible a priori voting power analysis
using the various power measures.
Overview (cont.)
• With such a measure, we can address questions that
arise with respect to voting power in the Electoral
College, in particular:
– How much (if any) does individual voting power vary
from state to state?
– In so far as it does vary, are voters in larger or smaller
states favored?
– How would voting power change under various
alternatives to the existing Electoral College?
• With respect to the second question, directly contradictory claims are commonly expressed.
Overview (cont.)
• Partly because the Electoral College is viewed by some
as favoring small states and by others as favoring large
states, it is commonly asserted that a constitutional
amending modifying or abolishing the Electoral College
can never by ratified by the required 38 states.
– Hence the National Popular Vote Plan (to use an
interstate compact bypassing the constitutional
amendment process), to which Maryland recently
signed on.
• But voting power analysis suggest that prospects for a
successful constitutional amendment need not be so
hopeless.
Overview (cont.)
• A measure of a priori voting power means a measure
that takes account of the structure of the voting rules but
of nothing else (e.g., demographics, historic voting
patterns, ideology, poll results, etc.).
• As a first step, we need to distinguish between
– voting weight and voting power.
• We also need to distinguish between two distinct issues:
– how electoral votes are apportioned among the states
(which determines voting weight), and
– how electoral votes are cast within states (which, in
conjunction with voting weight, determines voting
power).
The Apportionment of Electoral Votes
• Apportionment deals with voting weights.
• The Constitution provides that states have electoral
votes equal in number to their total representation in
Congress.
• Each state has two Senators and is guaranteed at least
one Representative.
– At least given the present population profiles of the states and the
existing apportionment method, this guarantee is redundant, in that
every state would receive at least one House without the guarantee.
• In any event every state, regardless of population, has a
guaranteed floor of at least three electoral votes.
• Additional Representatives (and electoral votes) beyond
this floor are apportioned (in whole numbers) among the
states on the basis of population.
The Small-State Apportionment Advantage
• The guarantee of three electoral votes produces a
systematic and substantial small-state advantage in the
apportionment of electoral votes (relative to the
apportionment population).
– This is the basis of the argument that the Electoral College
advantages voters in smaller (rural, etc.) states.
• The magnitude of this small-state apportionment
advantage is not fixed in the Constitution.
– It varies (inversely) with the size of the House (relative to the
Senate), which determined by Congress.
– If the House had been sufficiently larger than 435, Gore would
have won the 2000 election (even while losing Florida).
M. G. Neubauer and J. Zeitlin, “Outcomes of Presidential Elections and House
Size,” PS: Politics and Political Science, October 2003
The Apportionment of Electoral Votes
• In practice, Congress has kept House size approximately constant,
relative to the number of states (and the size of the Senate), for the
last 150 years (at about 4 to 4.5 Representatives per Senator).
The Apportionment of Electoral Votes (cont.)
• Since 1912 House size has been fixed at 435, since
1959 there have been 50 states, and since 1964 the
23rd Amendment has given three electoral votes to the
District of Columbia.
– So the total number of electoral votes at present is
435 + 100 + 3 = 538, with 270 votes required for
election.
• Note that a 269-269 electoral vote tie is possible.
• Even apart from the small-state apportionment
advantage, apportionment fails to be precisely proportional to population, because the Constitution requires
apportionment into whole numbers.
The Small-State Advantage (and Whole Number
Effect) in the Apportionment of Electoral Votes
Putting Both Log Variables into Percent Shares
Straightens the Line of Proportionality
Another View of the Small-State Advantage
The Apportionment of House Seats (and Electoral
Votes) to Smaller States Unavoidably Entails
Substantial “Rounding Error”
Selection of Electors and Casting of Electoral Votes
• The Constitution leaves the mode of selection of
Presidential electors up to each state to decide.
• Since the mid-1830s, the almost universal state practice
has been that
– each party nominates a slate of elector candidates, equal in
number to the state’s electoral votes and pledged to vote for the
party’s presidential (and vice-presidential) candidate(s), between
which voters choose; and
– the slate that wins the most votes is elected and casts its bloc of
electoral votes as pledged.
• This winner-take-all (or unit-rule) practice produces the
weighted voting game noted at the outset.
• Many have believed that this practice produces a largestate advantage in voting power that counteracts (in
some degree) the small-state advantage in
apportionment.
– This is one basis for the argument that the Electoral College
gives disproportionate voting power to voters in larger (urban,
etc.) states.
Weighted Voting Games
• As noted, the Electoral College is an example of a
weighted voting game.
– Instead of casting a single vote, each voter casts a
bloc of votes, with some voters casting larger blocs
and others casting smaller.
– Other examples:
• voting by disciplined party groups in multi-party parliaments;
• 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 is commonly described in terms of the
language of “simple games.”
• A [proper] simple game is a (voting or similar) situation
in which every potential coalition (i.e., subset of players
or voters) can be deemed either winning or losing.
– There is a set of n voters and a voting rule that
specifies a set of winning coalitions such that:
• if a coalition S is winning, all more inclusive coalitions
(supersets of S) are also winning;
• if a coalition is winning, its complement is losing; and
• the “grand coalition” of all voters is winning.
– A simple game is strong if it is also true that, if a
coalition is losing, its complement winning.
• Any n-player game has 2n possible coalitions, and each
player/voter belongs to half of them (2n-1).
Weighted Voting Games (cont.)
• A weighted voting game is a simple game in which
– each player is assigned some weight (e.g., a [typically whole]
number of votes); and
– a coalition is winning if and only if its total weight meets or
exceeds some quota.
– Such a game can be written as (q : w1,w2,…,wn).
• In a proper game, q > Σw/2.
• In a strong game, Σw is odd and q = (Σw+1)/2.
• The Electoral College is a weighted voting game in which:
– the states are the voters, so n = 51;
– electoral votes are the weights;
– total weight is 538, and
– the quota is 270.
– Today’s EC = (270: 55,34,31,…,3).
– The Electoral College game is almost strong, but not quite
(because there may be a 269-269 tie).
Weighted Voting Games (cont.)
• With respect to weighted voting games, the fundamental
analytical 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.
– However, it is true that
• two voters with equal weight have equal power, and
• a voter with less weight has no more voting power than one with
greater weight.
• Generally, it is impossible to apportion voting power (as
opposed to voting weights) in a “refined” fashion,
– though as n increases, the possibility of refinement increases.
– As we shall see, n = 51 allows a high degree of refinement.
Weighted Voting Example:
Parliamentary Coalition Formation
• 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 proportional representation formula. In
this case, the apportionment of seats is straightforward:
– Party A: 27 seats
Party C: 24 seats
– Party B: 25 seats
Party D: 24 seats
• While seats (voting weights) have been apportioned in a
way that is precisely proportional to vote support, voting
power has not been proportionally 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 reflects its opportunity to create
(or destroy) winning (governing) coalitions.
• But, with a small number of parties, coalition possibilities
-- and therefore different patterns in the distribution of
voting power -- are highly restricted.
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; so
– 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 above (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, i.e., (3:2,1,1,1)
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
proportional to their vote shares).
• Note: while we have determined that Party A has effectively
twice the weight of the others, we still haven’t evaluated the
voting power of the parties.
Weighted Voting Example (cont.)
• Suppose at the next election the vote and seat shares change a bit:
Before
Party A: 27
Party B: 25
Party C: 24
Party D: 24
Now
Party A:
Party B:
Party C:
Party D:
30
29
22
19
• While seats shares have changed only slightly, 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, i.e., (2:1,1,1,0)
• 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, these are the only possible strong simple games with 4
players:
– (3:2,1,1,1);
– (2:1,1,1,0); and
– (1:1,0,0,0), i.e., the “inessential” game in which one party holds a
majority of seats (making all other parties dummies), so that no winning
(governing) coalition [in the ordinary sense of two or more parties]
needs to be formed.
• Expanding the number of players to five produces these additional
possibilities:
–
–
–
–
(5:3,2,2,1,1); and
(4:3,1,1,1,1); and
(4:2,2,1,1,1); and
(3:,1,1,1,1).
• With six or more players, coalition possibilities become considerably
more numerous and complex.
Weighted Voting Example (cont.)
• Returning to the four-party example, voting power
changes further if the parliamentary decision rule is
changed from simple majority to (say) 2/3 majority (i.e., if
the quota is increased).
• Under 2/3 majority rule, both before and after the
election, all three-party coalitions, and no smaller
coalitions, are winning, so all four parties are equally
powerful, i.e., (3:1,1,1,1)
– 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.
• Making the decision rule more demanding tends to
equalize voting power.
– In the limit, weighted voting is impossible under unanimity rule.
• However, in the Electoral College the decision rule is
fixed at (essentially) simple majority rule (quota = 270).
Voting Power Indices
• Several power indices have been developed that
quantify the (share of) power held by voters in weighted
(and other) voting games.
• These particularly include:
– the Shapley-Shubik voting power index; and
– the Banzhaf voting power measure.
• These power indices provide precise formulas for
measuring the a priori voting power of players in
weighted (and other) voting games.
• Remember, a measure of a priori voting power means
one that takes account of the structure of the voting rules
but of nothing else.
Shapley-Shubik and Banzhaf
• Lloyd Shapley and Martin Shubik are academics (a game theorist
and a mathematical economist, respectively).
Lloyd Shapley and Martin Shubik, “A Method for Evaluating the Distribution of
Power in a Committee System,” American Political Science Review, September
1954.
• John F. Banzhaf is an activist lawyer with a background in
mathematics (B.S. in Electrical Engineering from M.I.T.).
– The mathematics in his law review articles is understandably
rather informal, focused on the practical issues at hand.
– Academics have subjected his ideas to rigorous analysis.
John F. Banzhaf, “Weighted Voting Doesn’t Work,” Rutgers Law Review, Winter
1965; “Multi-Member Districts: Do They Violate the ‘One-man, One, Vote’
Principle?” Yale Law Journal, July 1966; and “One Man, 3.312 Votes: A
Mathematical Analysis of the Electoral College,” Villanova Law Review, Winter
1968.
Pradeep Dubey and Lloyd S. Shapley, “Mathematical Properties of the Banzhaf
Power Index,” Mathematics of Operations Research, May 1979
The Shapley-Shubik Index
The Shapley-Shubik power index works as follows.
Using the previous four-party example, consider every
possible ordering (or permutation) of the parties A, B, C,
D (e.g., every possible order in which they might line up
to form a winning coalition). Given n voters, there n! (n
factorial) such orderings. Given 4 voters, there are
4! = 1 x 2 x 3 x 4 = 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, because the “grand
coalition” {A,B,C,D} is certainly winning.
• For each ordering, we identify the pivotal voter who, when added to
the players already in the coalition, converts a losing coalition into a
winning coalition.
• Given the pre-election seat shares of parties A, B, C, and D, 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
– Note: this “queue model” of voting is intended to provide an intuitive
understanding of how the S-S Index is calculated, not a theory of how voting
coalitions may actually form.
• Clearly the 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 (which has effectively
twice the weight of each other party) has has three times the voting
power of each other party.
The Banzhaf Measure
• While Shapley-Shubik focus on permutations of voters,
Banzhaf focus on combinations of voters, i.e., coalitions.
• The Banzhaf power measure 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 AbBz(i) is
Number of winning coalitions for which i is critical
Total number of coalitions to which i belongs.
– Remember, there are 2n coalitions and i belongs to
half of them, i.e., to 2n-1 of them.
The Banzhaf Measure (cont.)
• Given the pre-election seat shares, 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:
AbBz(A) = 6/8 = .75
The Banzhaf Measure (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 two of these
coalitions only:
– {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: AbBz(B) = AbBz(C) = AbBz(D) = 2/8 = .25.
• The "total absolute Banzhaf power" of all four voters:
= .75 + .25 + .25 + .25 = 1.5 .
The Banzhaf Measure (cont.)
• Note that exactly one voter is pivotal in each ordering
(permutation) of voters, so
– the S-S values of all voters necessarily add up to 1
• In contrast, several voters or none of the voters may be
critical to a given winning coalition (combination) of
voters, so
– the AbBz values do not add up to 1 (except in special
cases).
• However, if we are interested in the “relative” power of
voters (i.e., in power values that add up to 1, like the S-S
index), we can derive a (relative) Banzhaf index value
RBz(i) for voter i that is simply his share of the "total
power," so
RBz(A) = .75/1.5 = 1/2; and
RBz(B) = RBz(C) = RBz(D) = .25/1.5 = 1/6.
Shapley-Shubik vs. Banzhaf
• We see that in this simple 4-voter case, Shapley-Shubik
and Banzhaf evaluate voting power in the same way,
– i.e., they both say that Party A has three times the voting power
of the other parties.
• S-S and RBz values are often identical in small-n
situations like this.
• Rather typically, S-S and RBz values, while not identical,
are quite similar.
• But particular kinds of situations, the indices evaluate the
power of players in radically different ways.
– For example, if there is single large stockholder while all other
holding are highly dispersed.
– It is even possible that the two indices may rank players with
respect to power in different ways (but this cannot occur in
weighted voting games).
Felsenthal and Machover,
The Measurement of Voting Power
• In this book (and related papers), Dan Felsenthal and
Moshé Machover present the most conclusive study of
voting power measures.
• They conclude that
– the fundamental rationale for the S-S Index is based on
cooperative game theory, in that
– it assumes that players seek to form a winning coalition whose
members divide up some fixed pot of spoils (what they call PPower [where P is for “Prize”]), which hardly describes the
Electoral College or most other voting games.
• They conclude, in contrast, that
– the fundamental rationale for the Banzhaf measure (and its
variants) is probabilistic (not game-theoretic), and
– that Banzhaf is the appropriate measure for analyzing typical
voting rules (what they call I-Power [where I is for “Influence”]),
including the Electoral College.
The Measurement of Voting Power (cont.)
• F&M also observe that Banzhaf’s essential ideas
– had been laid out twenty years earlier by L.S. Penrose, and
– were subsequently and independently rediscovered by Coleman.
Felsenthal, Dan S., and Moshé Machover, The Measurement of Voting Power:
Theory and Practice, Problems and Paradoxes, 1988
Felsenthal, Dan S., and Moshé Machover, “Voting Power Measurement: A Story
of Misreinvention,” Social Choice and Welfare, 25, 2005
Penrose, L. S., “The Elementary Statistics of Majority Voting,” Journal of the
Royal Statistical Society, 109, 1946
Coleman, James S., “Control of Collectivities and the Power of a Collectivity to
Act.” In Bernhardt Lieberman, ed., Social Choice, 1971
• F&M observe that
– the Absolute Banzhaf measure can be transformed into the
relative Banzhaf index, but
– there is rarely good reason to do this.
Bernoulli Elections
• Unlike the relative Banzhaf index, the absolute Banzhaf
value that has a probabilistic interpretation that is directly
meaningful and useful:
– AbBz(i) is voter i’s a priori probability of casting a decisive vote,
i.e., one that determines the outcome of an election (for
example, breaking what otherwise would be a tie).
• In this context, “a priori probability” means, in effect,
given that all voters vote randomly, i.e., vote for either
candidate with a probability p = .5 (as if they
independently flip fair coins), so that every point in the
“Bernoulli space” (every combination [coalition] of voters)
is equally likely to occur.
– We call such a two-candidate elections Bernoulli elections.
Bernoulli Elections (cont.)
• Given Bernoulli elections
– the expected vote for either candidate is 50%;
– the probability that either candidate wins is .5; and
– the standard deviation in either candidate’s absolute
vote (over repeated elections) is .5√n, where n is the
number of voters.
– the probability that voter i votes for the winning
candidate is .5 plus half of i’s absolute Banzhaf power
value.
Bernoulli Elections (cont.)
• The distribution of Bernoulli election outcomes looks quite different
from empirical election data.
– For a Presidential candidate to win as much as 50.1% of the national
popular vote would be a landslide of fantastically rare probability.
– Not only is the national popular vote essentially always a virtual tie, but
so are all state (and district) popular votes.
• The rationale of the Bernoulli election concept
– is not to provide an empirical model of elections; but
– is to reflect the a priori condition (i.e., the total absence of empirical
knowledge or assumptions, and derived from the “principle of
insufficient reason”).
• If p were anything even slightly different from p = .5,
– the probabilities that follow would be quite different, and in particular
– the probability that anyone would cast a decisive vote would be
essentially zero (far smaller than the generally small probabilities that
result from p = .5).
• Unlike AbBz(i), RBz(i) has no natural interpretation.
– Focus on the relative, rather than absolute, Banzhaf measure has
produced considerable confusion in discussions of voting power.
Calculating Power Index Values
• Even today it remains impossible to apply these measures
(especially Shapley-Shubik) directly to weighted voting games with
even the rather modest number of voters (states) in the Electoral
College.
– S-S requires the examination of 51! ~ 1.55 x 1066 permutations
of the 51 states.
– Bz requires the examination of 251 ~ 2.25 x 1015 combinations of
the 51 states.
– Such enumerations are well beyond the practical computing
power of even today’s super-computers.
• But by the late 1950s Monte Carlo computer simulations (based on
random samples of permutations) provided good estimates of state
S-S voting power.
– Surprisingly, these estimates indicated that the widely expected
large-state advantage (relative to voting weights) in voting power
was quite modest.
Mann, Irwin, and L. S. Shapley (1964). “The A Priori Voting Strength of the Electoral
College.” In Martin Shubik, ed., Game Theory and Related Approaches to Social Behavior.
John Wiley & Sons.
Calculating Power Index Values
• In recent decades, mathematical techniques have been
developed that quite accurately calculate or estimate
voting power values, even for very large weighted voting
games.
• Computer algorithms have been developed to implement
these techniques.
• Various website make these algorithms readily available.
• One of the best of these is the website created by
Dennis Leech (University of Warwick): Computer
Algorithms for Voting Power Analysis.
http://www.warwick.ac.uk/~ecaae/ .
• This site was used in making the calculations that follow.
State Voting Power in the Existing
Electoral College
• For each state in the present Electoral College,
the following table shows:
– EV PROP: its share of electoral votes (share of voting
weights);
– S-S INDEX: its share of voting power, according to
the Shapley-Shubik index;
– BANZHAF INDEX: its share of voting power,
according to the (relative) Banzhaf index; and
– ABSOLUTE BANZHAF: the absolute Banzhaf power
for each state.
State Voting Power in the Existing EC(cont.)
• It is apparent that
– Shapley-Shubik and Banzhaf provide very similar
estimates of state voting power, and
– state voting power is in fact closely proportional to
electoral votes, though
– the largest states — especially the largest of all
(California) — are somewhat advantaged.
• The second point is consistent with what F&M call the
Penrose Limit Theorem, which asserts that
– as the number of voters increases, and provided the distribution of
voting weights is not “too unequal,” voting power tends to become
proportional to voting weight.
– The “theorem” is a actually conjecture that has been proved in important
special cases and is supported in a wide range of simulations.
State Voting Power in the Existing EC (cont.)
• It is worth noting California’s AbBz value of .475
• Remember what this means:
– if states were repeatedly to cast their electoral votes by
independently flipping coins, almost half [.475] of the time the
other 49 states plus DC would split their 483 votes sufficiently
equally that California’s 55 votes would be decisive (i.e., would
determine the winner).
– It also means, under the same assumptions, that California
would be on the winning side almost three-quarters of the time
[i.e., .5 + .475/2 = .7375].
• Despite its outlier status, California’s voting power is
somewhat reduced by the substantial variability of the
weights of the remaining states.
– While on average the other states have 9.66 electoral votes, they
range from 3 to 34.
– If they all had 8 or 9, California’s AbBz voting power would be
considerably greater (about .55).
State Voting Power in the Existing EC (cont.)
• It is evident from the following charts that
– only California’s share of voting power substantially
deviates from (and exceeds) its share of electoral
votes;
– the modest large-state advantage in voting power
(relative to voting weight) is not sufficient to balance
out the small-state advantage in apportionment;
indeed,
– even California’s distinctive advantage in terms of
voting power (relative to voting weight) is not sufficient
to give it voting power proportional to its population.
Share of Voting Power by Share of Electoral Votes
Share of Voting Power by Share of Population
The 125 Million Two-Tier Voting Game
• But this 51-state Electoral College weighted voting game
is a chimera, since states are not players but just vote
counting devices.
• A U.S. Presidential election really is a two-tier voting
system, in which the casting of electoral votes is
determined by popular vote pluralities within each state.
– Since we assume two-candidate elections, these
pluralities are also majorities.
Double Decisiveness in the 125 Million TwoTier Voting Game
• In such a two-tier system, individual a priori voting power
is the probability of “double decisiveness,” i.e.,
– that a voter casts a decisive vote within his or her
state (i.e., that there is tie among the other voters in
the state),
and
– that his or her state casts a decisive bloc of electoral
votes (i.e., that neither candidate wins 270 electoral
votes from the other states).
Banzhaf Power in the Two-Tier Voting Game
• Put otherwise (and given the probabilistic interpretation
of Banzhaf power), individual AbBz two-tier voting power
is equal to
– individual AbBz voting power in the unweighted (but
large number) majority voting game within the state
times
– the state’s AbBz voting power in the 51-state
weighted voting game.
• We have already determined the second term in this
product [as shown in the last column of the previous
table].
The Within-State Unweighted Voting Game
• Individual AbBz voting power in the unweighted large-n
majority voting game within the state
– is clearly inversely related to the number of voters in the state,
that is,
– the larger the number of voters, the lower the chance of casting
a decisive vote.
• However, probability theory and the properties of
Banzhaf measure tell us that the individual AbBz voting
power:
– is not inversely proportional to the number of voters in the state;
but rather
– it is inversely proportional (to excellent approximation once n ~>
25) to the square root of the number of voters.
The Within-State Unweighted Voting Game (cont.)
Furthermore,
using Stirling’s
approximation
for n!, it follows
that, for n ~>
25, AbBz =
Prob(Tie) can
be very well
approximated
by the following
formula:
AbBz = √[2/πn].
The Two-Tier Voting Game
• Remember that individual AbBz voting power in the 2-tier
EC voting game is
– individual AbBz voting power in the unweighted
majority voting game within the state
times
– the state’s AbBz voting power in the 51-state Electoral
College weighted voting game.
• We have seen that a state’s AbBz power is approximately proportional to its electoral vote and in turn to its
population.
• We have now seen that individual AbBz power within the
state is approximately inversely proportional to the
square root of number of voters in the state.
The Two-Tier Voting Game (cont.)
• Putting these two considerations together, it follows that
individual a priori voting power in the two-tier system
– increases with the population of the voter’s state, and
– is approximately proportional to the square root of the number of voters
in the voter’s state.
• This effect (first noted with explicit reference to the
Electoral College by Banzhaf) may be dubbed the
Banzhaf effect.
– It had been noted in a more general context twenty years earlier by
Penrose, and it is a consequence of what Felsenthal and Machover call
the Penrose square-root rule.
• With specific respect to the Electoral College, the
Banzhaf effect
– is in some degree counterbalanced by the (quite substantial) small-state
advantage in apportionment,
– in small degree reinforced by the (quite weak) large-state advantage in
voting power, and
– among the smallest states is largely hidden by the unavoidable
“rounding error” in apportioning House seats into small whole numbers.
The Two-Tier Voting Game (cont.)
•
The following table shows data relevant to individual a priori voting power in
selected states under the present Electoral College.
– ELECT SIZE: the size of the electorate in each state, which is taken to
be 43.37% of its 2000 apportionment population
• 43.37% is the 2004 Presidential vote as a percent of the 2000
apportionment population.
• A priori, we have no reason to expect that the percent of the
population that is eligible to vote, or the percent of eligible voters
who actually do vote, varies by state (though, empirically and a
posteriori, we know there is considerable variation in both respects).
– Using a different (fixed) percent of population to determine
electorate size would (slightly) affect the following estimates of
individual absolute voting power but comparisons across
states.
– IND VP: individual voting power within each state [as calculated by the
√[2/(π x ELECT)] formula]
– EV: state electoral vote
– STATE VP: state absolute Banzhaf voting power [taken from the
previous table]
– IND 2-T VP: individual voting power in the two-tier voting game [= IND
VP x STATE VP]
– REL VP: individual voting power in the two-tier voting game, rescaled so
that the voting power of the least favored voters [in Montana] is 1.0000.
The Two-Tier Voting Game (cont.)
• Note that voters in the most favored state [California]
have almost three and half times the voting power of
voters in the least favored state [Montana].
– Montana is least favored because it the most populous state with the
smallest possible number of (three) electoral votes.
• The following chart plots the relationship between state
population and rescaled individual a priori voting power
over all 51 states.
• The underlying square-root rule is displayed in the chart,
which makes evident:
– the small-state apportionment advantage,
– the scattering that results as small states fall just above or below
House seat thresholds, and
– California’s advantage in second-tier voting power.
Individual Voting Power By State Population:
Existing Electoral College
The Two-Tier Voting Game (cont.)
• The chart also shows mean individual voting power nationwide.
– Note that the individual voting power values plotted in the chart
must be weighted by ELECT SIZE in order to determine mean
voting power nationwide.
• The chart also shows individual voting power under direct popular
vote (calculated in the same manner as individual voting power
within a state).
– This is necessarily uniform over the nation.
– It is substantially greater than mean individual voting power
under the Electoral College.
• Indeed, it is greater than individual voting power in every
state except California.
– By the criterion of a priori voting power, only voters in California
would be hurt if the existing Electoral College were replaced by a
direct popular vote.
Individual Voting Power By State Population:
Existing Electoral College
Alternatives to the Existing Electoral College
• We now consider three categories of alternatives to the
existing Electoral College:
– those that keep the state-level winner-take all practice
but use a different formula for apportioning electoral
votes among states,
– those that keep the existing apportionment of
electoral votes but use something other than winnertake-all for the casting of state electoral votes, and
– variants of the so-called National Bonus Plan.
• Almost all Electoral College “reform plans” (and certainly
all reforms that can be implemented at the state level)
fall in the second category.
Alternative EV Apportionment Rules
• Keep the winner-take all practice [in 2000, Bush 271,
Gore 267; in 2004, Bush 286, Kerry, 252] but use a
different formula for apportioning electoral votes among
states.
– Apportion electoral votes [in whole numbers] on basis
of population only [no “constant two’] [Bush 211,
Gore 225; Bush 224, Kerry 212]
• Apportion electoral votes [fractionally] to be precisely
proportional to population [Bush 268.96092, Gore
269.03908; Bush 275.67188, Kerry 262.32812]
• Apportion electoral votes [fractionally] to be precisely
proportional to population but then add back the
“constant two” [Bush 277.968, Gore 260.032; Bush
285.40695, Kerry 252.59305]
• Apportion electoral votes equally among the states [in
the manner of the House contingent procedure]
[Bush 30, Gore 21; Bush 31, Kerry 20]
Remove the Small State Advantage
• Apportion electoral votes [in whole numbers] on basis of
population only, i.e., get rid of the “constant two’:
• for example, EV = House seats.
– This removes the systematic small-state advantage
with respect to apportionment.
– Clearly, as small states lose voting weight, small
states and the voters in them lose voting power, while
large states and the voters in them gain.
• Indeed, voters in California now have a a better
than 10 to 1 voting power advantage over those in
Montana.
Methodological note: in each chart, individual voting power is scaled so that
the voters in the least favored state have a value of 1.000, so
• numerical values are not comparable from chart to chart, and
• the scaled value of individual voting power under direct popular vote
changes from chart to chart.
Individual Voting Power by State Population:
“House Electoral Votes” Only
Apportion Precisely Proportional to Population
• Apportion electoral votes so they are precisely proportional to population (with the result that states have
fractional electoral votes, implying that the office of
elector must be abolished).
– The effect on state and individual voting power is
essentially the same as the previous EC variation, but
fractional apportionment smoothes out irregularities of
whole-number apportionment.
– The least favored state is now the one with the
smallest population (WY), rather than the most
populous state with only 3 electoral votes (MT).
• Voters in California now have a slightly less than a 10 to 1
advantage in voting power over those in Wyoming.
– Voters are rank-ordered in terms of their voting power
precisely according to the rank-ordering of their states
by population.
– The Penrose square root rule is fully revealed.
Individual Voting Power by State Population:
Electoral Votes Precisely Proportional to Population
Precise Proportionality with Small-State
Advantage Restored
• Apportion electoral votes [fractionally] to be precisely
proportional to population, but then add back in the
“constant two.”
– This produces a smooth “hockey stick” graph.
– The least favored state is now Idaho, the largest state
with two House seats.
– The ratio of advantage for California voters relative to
those of Idaho is reduced to 3 to 1.
Individual Voting Power by State Population:
Electoral Votes Proportional Population, plus Two
Apportion Electoral Votes Equally
Among States
• Apportion electoral votes equally among the states [in
the manner of the Articles of Confederation or the House
contingent procedure in the event of an Electoral College
deadlock].
– Clearly this reverses the relative advantages of large
and small states.
– Voters in Wyoming are most advantaged, those in
California least, with a ratio of advantage of better
than 8 to 1.
– Voters are rank-ordered in terms of their voting power
precisely according to the reverse of the rankordering of their states by population.
Individual Voting Power by State Population:
Electoral Votes Apportioned Equally Among States
Can Electoral Vote Apportionment Equalize
Individual Voting Power?
• The question arises of whether electoral votes can be
apportioned so that (even while retaining the winnertake-all practice) the voting power of individuals is
equalized across states?
• One obvious (but constitutionally impermissible)
possibility is to redraw state boundaries so that all states
have the same number of voters (and electoral votes).
– This creates a system of uniform representation.
Methodological Note: since the following chart compares voting power under
different apportionments, voting power must be expressed in absolute (rather
than rescaled) terms.
Individual Voting Power when States Have Equal Population
(Versus Apportionment Proportional to Actual Population)
Uniform Representation
• Note that equalizing state populations not only:
– equalizes individual voting power across states, but also
– raises mean individual voting power, relative to that under
apportionment based on the actual unequal populations.
• While this pattern appears to be typically true, it is not invariably
true,
– e.g., if state populations are uniformly distributed over a wide range.
•
However, individual voting power still falls below that under direct popular
vote.
– So the fact that mean individual voting power under the Electoral
College falls below that under direct popular vote is
• not due to the fact that states are unequal in population and
electoral votes, and
• is evidently intrinsic to a two-tier system.
Van Kolpin, “Voting Power Under Uniform Representation,” Economics
Bulletin, 2003.
Individual Voting Power
under Uniform Representation
• One “state” and 122,294,000 “states” (one for each
voter) represent the logical extremes of uniform
representation, and both are equivalent to direct popular
vote.
– Individual voting power as a function of the number of equally
populated “states” forms a U curve.
– Minimum individual voting power occurs when there are 11,059
“states” each with 11,059 voters, i.e., when both are equal to the
square root of the size of the national electorate, and the U curve
is symmetric.
– This minimum individual voting power is about 80% of that under
direct popular vote, so the U curve is quite shallow.
– However, this minimum power is closely approached with just a
few dozen states (or states with just a few dozen voters), so the
U curve is virtually rectangular.
Individual Voting Power in Equally Populated States
Electoral Vote Apportionment to Equalize
Individual Voting Power (cont.)
• Given that state boundaries are immutable, can we apportion
electoral votes so that (without changing state populations and with
the winner-take-all practice preserved) the voting power of
individuals is equalized across states?
• Yes, electoral votes can be apportioned by applying the Penrose
Square Root Rule in reverse (as an engineering principle, rather
than as a descriptive law):
– Individual voting power is equalized when electoral votes are
apportioned so that state voting power is proportional to the
square root of state population.
• But such Penrose Apportionment is tricky, because what must be
made proportional to population is not electoral votes (what we
directly apportion) but state voting power (a consequence of the
apportionment of electoral votes).
Penrose Apportionment
• However, in the case of the Electoral College we can readily
come up with a good approximation and then refine it further
if desired.
• This is because each state’s share of voting power in the
Electoral College is close to its share of population,
– because n = 51 is large enough, and the distribution of state
populations is “regular” enough, that the Penrose limit theorem holds
to good approximation.
• We can initially apportion electoral votes to be precisely
proportional (i.e., allowing fractional electoral votes) to
square root of the population (or number of voters) in each
state.
– How close does this come to making voting power of each state
equal to its share of electoral votes?
State Voting Power When Electoral Votes are
Apportioned on the Basis of SQRT of Population
Penrose Apportionment (cont.)
• This first approximation comes very close to making
voting power proportional to electoral votes.
– Calfornia’s share of voting power deviates from its share of
electoral votes by only about +0.2%.
– Iowa’s share of voting power deviates from its share of electoral
votes by less than -0.02%.
– All other state deviations fall between IW and CA.
• We adjust CA’s electoral votes by -0.2% and likewise for
other states and then recalculate voting power for all
states.
– This actually overcompensates for the previous deviations: CA
now deviates by about -0.02%, and likewise for other states.
– By repeating such adjustments, we can bring voting electoral
votes as close into line with voting power as desired.
• But further refinement seems unnecessary, especially as
we probably must apportion electoral votes into whole
numbers anyway.
(Approximately) Equalized Individual Voting Power
EV Apportionment to Equalize VP (cont.)
• These two methods of apportionment that equalize
individual voting power equalize it at (essentially) the
same level, namely
– 0.00005785 (vs. 0.00007215 for direct popular vote).
• If the Penrose square root rule is used but apportionment of electoral votes must be in whole numbers,
– individual voting power is imperfectly equalized
(especially among small state voters), and
– mean individual voting power is reduced ever so
slightly (to 0.00005784).
State Voting Power When EVs are Apportioned on
the Basis of SQRT of Population (cont.)
• Under such square-root apportionment rules, the
outcome of the 2004 Presidential elections would be
–
–
–
–
Fractional Apportionment: Bush 307.688, Kerry 230.312.
Whole-Number Apportionment: Bush 307, Kerry 231
Actual Apportionment: Bush 286, Kerry 252
Electoral Votes proportional to popular vote: Bush 275.695,
Kerry 262.305
• Clearly equalizing individual voting power is not the
same thing as making the electoral vote (more)
proportional to the popular vote.
Alternative Rules for Casting Electoral Votes
• Apportion electoral votes as at present but use something other than
winner-take-all for casting state electoral votes.
– (Pure) Proportional Plan: electoral votes are cast [fractionally] in
precise proportion to state popular vote. [Bush 259.2868, Gore
258.3364, Nader 14.8100, Buchanan 2.4563, Other 3.1105;
Bush 277.857, Kerry 260.143]
– Whole Number Proportional Plan [e.g., Colorado Prop. 36]:
electoral votes are cast in whole numbers on basis of some
apportionment formula applied to state popular vote. [Bush 263,
Gore 269, Nader 6, or Bush 269, Gore 269; Bush 280, Kerry
258]
– Pure District Plan: electoral votes cast by single-vote districts.
– Modified District Plan: two electoral votes cast for statewide
winner, others by district [present NE and ME practice]. [Bush
289, Gore 249, if CDs are used; no data for 2004]
– National Bonus Plan: 538 electoral votes are apportioned and
cast as at present but an additional 100 electoral votes are
awarded on a winner-take-all basis to the national popular vote
winner. [Bush 271, Gore 367; Bush 386, Kerry 252]
Alternative Rules for Casting
Electoral Votes (cont.)
• Calculations for the Proportional Plan, the WholeNumber Proportional Plan, and the Pure District Plans
are straightforward.
• Calculations for the Modified District Plan and the
National Bonus Plans, under which each voter casts a
single voter than counts in two separate upper-tier, are
considerably less straightforward and need to be set out
in more detail.
The (Pure) Proportional Plan
• Electoral votes are retained but the office of elector is
abolished.
• A state’s electoral votes are cast [fractionally] in precise
proportion to the candidates’ shares of the popular vote
in the state (which is why the office of elector must be
abolished).
• If proportionality is sufficiently refined, such a system
becomes a 122-million single-tier weighted voting game,
– where the weights applied to individual votes reflect
their state’s electoral votes per voter.
• As proportionality becomes less refined, such a system
begins to resemble a whole-number proportional system
(considered next).
The (Pure) Proportional Plan (cont.)
• The constitutional amendment for a proportional system
[“Lodge-Gossett Plan] proposed in the 1950s specified
that candidates would be credited with fractional
electoral votes calculated out to four decimal places.
• The following chart assumes that proportionality is
sufficiently refined to create a single-tier weighted voting
game.
• We can invoke the Penrose Limit Theorem to justify the
assumption that voting weight = voting power in this
single-tier very large-n weighted voting game.
Individual Voting Power by State Population:
(Pure) Pure Proportional System
(Pure) Proportional Plan (cont.)
• The small-state apportionment advantage carries
through very strongly to voting power.
– Voters in Wyoming in have almost four times the
voting power of voters in California.
• Nevertheless, voters in Montana (the largest state with
only three electoral voters) have less voting power than
voters in Rhode Island (the smallest state with four
electoral votes).
– Close inspection of the chart shows that similar but less striking
inversions exist with respect to a number of larger states.
• Since sufficiently refined proportionality creates what is
effectively a (weighted) single-tier voting game, mean
individual voting power is equal to individual voting
power under a direct (unweighted) popular vote.
The Whole-Number Proportional Plan
• Such a plan was proposed by Colorado’s Proposition 36
in 2004.
• This plan divides a state’s electoral votes between (or
among) the candidates in a way that is as close to
proportional to the candidates’ state popular vote shares
as possible, given that the apportionment must be in
whole numbers.
– Unlike the (Pure) Proportional Plan, whole-number
proportionality allows for the retention of electors.
– Accordingly, it is the only proportional plan that can be
implemented at the state level (as Colorado Prop. 36
proposed).
The Whole-Number Proportional Plan (cont.)
• In principle, there are as many such plans as there are
apportionment formulas.
– In addition, candidates might be required to meet some vote
threshold in order to win any electoral votes.
– Colorado Proposition 36
• had no explicit vote threshold, and
• used a distinctly ad hoc apportionment formula
– that was overtly biased toward the strongest candidate and
against the weakest candidates.
• But, in the event there are just two candidates (as we
assume here), all apportionment formulas work in the
same straightforward way:
– multiply each candidate’s share of the popular vote by the state’s
number of electoral votes and then round off in the normal
manner (to the nearest whole number of electoral votes).
Whole-Number Proportional Plan (cont.)
• In this two-tier system, individual a priori voting power is the
probability that
– the voter casts a decisive vote within his or her state in the sense
that
• other votes are so divided that the individual’s vote
determines whether a candidate gets k or k+1 electoral votes
from the state,
and
• this single electoral vote is decisive in the Electoral College.
where, as usual, these probabilities result from the Bernoulli
elections.
• The following chart shows that this plan produces a distinctly odd
distribution of a priori individual voting power.
Note. I have discovered that similar calculations and chart have been produced,
independently and earlier, by Claus Beisbart and Luc Bovens, “A Power
Analysis of the Amendment 36 in Colorado,” University of Konstanz, May 2005,
and Public Choice, forthcoming.
Individual Voting Power by State Population:
Whole-Number Proportional Plan
Whole-Number Proportional Plan (cont.)
• As can be seen, this plan produces a truly bizarre
allocation of voting power.
– Voters in a large number (17) of states are rendered
(essentially) powerless.
– These are voters in states with an even number of
electoral votes.
• Here’s why this happens.
– Given Bernoulli elections and any fairly large number
of voters, the vote essentially always is almost equally
divided between the two candidates.
• As previously noted, the expected (i.e., the mean over
repeated coin flipping elections) vote share for each
candidate is .5, and
• the standard deviation of the distribution of vote shares is
.5√n .
Whole-Number Proportional Plan (cont.)
• Consider a state with 4 electoral votes.
– For its electoral votes to be divided otherwise than 2 to 2, one
candidate must receive at least 62.5% of the vote,
• because 0.625 x 4 EV = 2.5 EV, rounding to 3 EV.
– Such a state has about 500,000 voters.
• The a priori expected vote share for either candidate is
250,000 with a standard deviation of about .5 x √500,000 =
354 votes.
• So a candidate has to receive 125,000 votes (about 350
SDs) above the expected vote share in order for anyone to
cast a decisive vote.
– So, it is essentially guaranteed in Bernoulli elections that the
electoral will be split 2-2, and all voters in such a state are
rendered (essentially) powerless.
Whole-Number Proportional Plan (cont.)
• As the even number of electoral votes increases, two
things happen.
– The vote margin required to produce anything other
than a even split of electoral votes slowly decreases.
• For example, in state with 50 electoral votes, a candidate
needs to get only about 52.3% of the votes to win 26 of them.
– At the same time, the standard deviation of the a
priori expected vote percent (the “sampling error”)
decreases with the square root of electorate size.
• Overall, the gap between the required margin and 50%
relative to the SD diminishes somewhat with electorate
size, but not nearly fast enough to give voters any
measurable a priori voting power in even the largest
states.
Ratio of Margin to SD by Electorate Size
Whole-Number Proportional Plan (cont.)
• With respect to the 34 states with odd numbers of
electoral votes, the results are only slightly less bizarre.
– For (appropriate) example, Colorado has 9 electoral
votes.
• For the electoral vote to be divided anything other
than 5 to 4, one candidate must receive a bit over
61% of the vote.
– Even in state with 55 electoral votes (i.e., California),
one candidate must to win a bit over 51.8% of the
votes to win more than 28 of them.
• By the same considerations that applied in the even
electoral vote case, the probability of achieving such
margins given Bernoulli elections is essentially zero.
Whole-Number Proportional Plan (cont.)
• Thus in each state with an odd number of electoral
votes, effectively only one electoral vote is at stake.
– So among such states, the distribution of voting
power is effectively the same as if electoral votes
were equally apportioned among the states, thereby
giving a huge advantage to voters in smaller states
with odd numbers of electoral votes.
• Note. As Beisbart and Bovens demonstrate, if Proposition 36 had
passed and Colorado unilaterally adopted Whole Number
Proportionality, the 2-tier voting power of Colorado voters would be
reduced to about 1/9 of what is presently is [i.e., as if Colorado had
a single electoral vote], with essentially no impact on voting power in
other states.
– Colorado voters would have had about 1/7 of the voting power of the
currently least favored voters in Montana.
Equal Apportionment vs. Whole-Number Proportional
Voting Power Under Whole-Number Proportionality
When All States Have an Even Number
of Electoral Votes
• Suppose the Whole Number Proportional Plan were in use and it
happened that every state had an even number of electoral votes.
– Could it be that all voters in all states would then have
(essentially) zero a priori voting power?
– Yes:
• in Bernoulli elections, each candidate gets just about 50% of
the vote in each state, so
• the electoral vote in every state would be equally divided, so
• the electoral vote nationwide would be equally divided (269269),
– The Electoral College would deadlock, and the election would be
thrown into House.
• So no voter would ever cast a decisive vote in such a setup,
and all voters would have zero a priori voting power.
All States Have an Even Number
of Electoral Votes (cont.)
• Suppose every state had an even number of electoral
votes but DC still had only three (this would require an
odd number of electoral votes nationwide, in turn
requiring an even-number sized House):
– DC voters would have all the voting power (which
might make up for their not having a voting House
member and having no say in the House contingent
election in the prior scenario).
•
Note. The Shapley-Shubik index does not have the same bizarre
implications for a priori voting power under the Whole Number Proportional
Plan.
– But, since it does reflect probabilities of decisiveness, it is not clear that
it is legitimate to use Shapley-Shubik in two-tier voting games in the first
place.
The Pure District Plan
• All electors are elected by popular vote in single-member
districts, so “winner-take-all” is applied at the district,
rather than state, level.
– Historical Note 1: this is how the most of the framers thought
electors would (and should) be selected.
– Historical Note 2: this mode of selection was prescribed in early
drafts of what became the 12th Amendment.
• Accordingly, there are 538 electoral districts and the
popular vote winner in each district is awarded one
electoral vote.
– All districts are subdivisions of states.
– All districts in the same state have equal numbers of voters.
– Districts in smaller states have smaller numbers of voters than
those in larger states, because of the small-state advantage in
the apportionment of electoral votes.
The Pure District Plan (cont.)
• Individual voting power within states is equal,
– because the number of voters in each district is equal.
• All districts have equal voting power in the Electoral
College,
– because they have equal weight, i.e., 1 EV, and the
second tier voting game entails 538 equally weighted
voters.
• Individual voting power across states is not equal,
– because districts in different states have unequal
numbers of voters.
Individual Voting Power by State Population:
Pure District System
The Pure District Plan (cont.)
• We see that the small-state advantage in apportionment
carries through in terms of voting power.
– Voters in Wyoming have an almost 2 to 1 voting power
advantage over voters in California.
– While Wyoming districts have only about ¼ as many voters as
California districts, the Banzhaf effect means that California
voters have about ½ the voting power as Wyoming voters.
• Nevertheless, voters in Montana (the largest state with
only three electoral voters) have less voting power than
voters in Rhode Island (the smallest state with four
electoral votes).
– Close inspection of the chart shows that similar but less striking
inversions exist with respect to a number of larger states.
The Modified District System
• Within each state, two electors are elected at-large and
the others by districts.
– Use of Congressional Districts is typically proposed.
– Maine and Nebraska actually use this system at present (though
it has never caused their electoral votes to be split).
• Individual voting power within states is equal,
– because the number of voters in each district is equal.
• All districts have equal voting power in the Electoral
College,
– because they have equal weight, i.e., 1 EV.
• All states have equal voting power in the Electoral
College,
– because they have equal weight, i.e., 2 EVs.
The Modified District System (cont.)
• However, individual voting power across states is not
equal because:
– districts in different states have (somewhat) unequal numbers of
voters, and
– states with (sometimes hugely) different populations have equal
electoral votes.
• This two-tier voting game presents difficult analytical
problems.
– Votes count in two parallel upper tiers.
– The two upper tiers are combined in a 487 player weighted
voting game with
• 51 voters, i.e., the states, with weight 2, and
• 436 voters, i.e., the districts, with weight 1.
Banzhaf’s Analysis
• In his original work, Banzhaf (in effect)
– determined each voter’s probability of double decisiveness
• through his district and
• through his state, and then
• simply summed these two probabilities.
• However, Banzhaf reported only
– individual 2-tier voting power rescaled in the manner we have
done in most charts (so that the voting power of voters in the least
favored voters set to 1.000),
– as opposed to absolute individual 2-tier voting power.
• His table of results is similar to the following chart (which,
however, also shows voting power under direct popular
vote).
John F. Banzhaf, “One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,”
Villanova Law Review, Winter 1968.
Banzhaf-Style Calculations for Modified District Plan
Problems with Banzhaf’s Analysis
• The distribution of voting power among voters in different states is
rather similar that under the Pure District Plan.
– However, the small-state advantage is further enhanced (though
not enough to boost Montana above Rhode Island).
• But there is a vexing anomaly: mean individual voting power
exceeds (by a considerable margin) individual voting power under
direct popular vote.
– This anomaly was not evident in Banzhaf’s original analysis,
because
• he reported only rescaled voting power values, and
• he made no voting power comparison with direct popular
vote (or with other Electoral College variants).
– Recalculation of Banzhaf’s results (using 1960 apportionment
populations) shows that the same anomaly exists in that data.
• This is anomalous because Felsenthal and Machover demonstrate
that, within the class of ordinary voting games, (mean) individual
voting power is maximized under direct popular vote.
Problems with Banzhaf’s Analysis (cont.)
• This anomaly occurs because Banzhaf (implicitly) assumes that
each voter casts separate and independent votes at the district and
state levels.
– Assuming voters have two or more separate and independent
votes
• takes us beyond the class of ordinary voting games covered
in F&M’s work, and
• in the direction of recent work by Paul Edelman and
extended Claus Beisbart (to be discussed next in conjunction
with the National Bonus Plan).
– In any event, under the Modified District Plan each voter casts a
single vote for one candidate and that single vote counts at both
the district and state levels.
Paul Edelman, “Voting Power and At-Large Representation,” Mathematical
Social Science, 2004.
Claus Beisbart, “One Man, Several Votes,” University of Dortmund, June 2007.
Problems with Banzhaf’s Analysis (cont.)
• There are additional analytical problems with the
Modified District Plan that Banzhaf’s seemingly does not
take account of.
• A voter may cast a doubly decisive vote in the tiered
voting game in three distinct ways:
– by casting a vote that is decisive in the voter’s district and the
district’s one electoral vote is decisive in the Electoral College;
– by casting a vote that is decisive in the voter’s state and the
state’s two electoral votes are decisive in the Electoral College;
and
– by casting a vote that is decisive in both the voter’s district and
state and the combined three electoral votes are decisive in the
Electoral College (though one or two votes would not be).
• Note that the first two possibilities are not mutually
exclusive.
– Moreover, it turn out the number of voter combinations in the
overlap of the first two possibilities is equal to the number of
combinations in the third. (Beisbart)
Problems with Banzhaf’s Analysis (cont.)
• Under most circumstances, the probability that the
district and state votes are both tied is remote, even
compared with the already small probabilities that either
one of them is tied.
• However, the fact that voters cast a single vote that
counts the same way in both district and state increases
this probability somewhat.
– Put otherwise, a single vote (unlike two separate and independent votes) in Bernoulli (and other) elections creates some
degree of correlation between the vote in any district and the
state as a whole.
• Therefore (with respect at least to the probabilistic
interpretation of Banzhaf power) it is not legitimate to
treat the district and state voting games separately and
simply combine the results, because
– vote distributions at the state level are not independent of vote
distributions at the district level.
The Modified District System (cont.)
• This is most conclusively true in the event the voter’s
state has only three electoral votes (and only one
district).
– Voter decisiveness in district and state necessarily coincide, so
– district and state electoral votes are perfectly correlated, with the
result that
• the state’s electoral votes cannot be divided.
– Put otherwise, the Modified District Plan in no way changes the
nature of Presidential voting within the (at present eight) smallest
states with only three electoral votes.
– As a side point, we also note that election reversals cannot occur
in these smallest states --- that is
• the candidate who wins the most districts always wins the statewide vote as well.
The Modified District System (cont.)
• Therefore, we can refine Banzhaf’s analysis of the
Modified District Plan a bit by making the Electoral
College a weighted voting game with 479 voters:
– 8 states with 3 electoral votes each;
– 43 states with 2 electoral votes each; and
– 428 districts (in the 43 states) with 1 electoral vote each.
• Furthermore,
– voters in the 8 smallest states are participating in a single 2-tier
voting game; while
– voters in the 43 other states are participating in two parallel 2-tier
voting games.
• The voting power of voters in the 8 states with 3 electoral
votes is calculated in the normal manner.
• The voting power of voters in the other states is
calculated as the sum of their voting powers in the
parallel district and state tiers (in the manner of
Banzhaf’s analysis for voters in all states).
Refined Calculations for Modified District Plan
The Modified District System (cont.)
• Since the smallest states now cast the largest blocs of (three)
electoral votes, the relative voting power of voters in these states is
further enhanced (beyond their apportionment advantage that shows
up in the Pure District Plan and Banzhaf’s analysis of the Modified
District Plan).
– Voters in Wyoming have a better than 3.5 to 1 advantage over voters in
California.
– Indeed, even Montana pulls well ahead of Rhode Island.
• However, while mean voting power still slightly exceeds that under
direct popular vote, mean absolute individual voting power declines
considerably relative to the Banzhaf-style calculations.
• Moreover, the following chart shows that, by singling out the
smallest states for special analytic treatment (by recognizing that
they exhibit a perfect correlation between district and state votes),
we have created a “discontinuity” (speaking loosely, since the
independent variable is discrete) as state electoral votes increase
from 3 to 4.
Modified District System: Banzhaf vs. Refined Analysis
The Modified District System (cont.)
• The “discontinuity” is artificial, resulting from the fact that we have
treated states with only three electoral in a manner qualitatively
different from our treatment of the other states.
• The substantive problem is that a positive correlation between each
district vote and the state-wide vote remains even for larger states,
though it diminishes as the number of districts increases.
• The methodological problem is that
– intuition does not suggest how substantial this effect may be, and
– measuring this effect is analytically complicated (if not intractable).
• However, simulation techniques allow us to generate quickly a very
large number of Bernoulli elections to get a handle on this (and
related) problems.
Note. These (and subsequent) simulations start at the district (or state) level
and aggregate to the state (or national) level. The simulations that provide the
data for the following chart assume 99,999 voters per district. Thus the vote in
each district is generated as a normally distributed random variable with a
mean of 49,999.5 and a standard deviation of .5 x √99,999 = 158.1123 .
Interdependency Between District and
State Votes
• For various state sizes (number of districts), the following
chart shows:
– the probability (i.e., frequency) that a the winner of a
given district is also the state-wide winner;
– the (Pearson) correlation between district votes and
the state-wide vote [not yet displayed]; and
– The [Spearman] correlation between each district
winner and the state-wide winner.
• It is evident that interdependencies between district and
state-wide votes/outcomes are not trivial for states with
more than three electoral votes.
Interdependency Between District and State Votes
in Bernoulli Elections
Interdependency Between District and
State Votes (cont.)
• Proceeding further, we note that in a state with four electoral votes:
– the electoral vote cannot be equally divided,
– the electoral vote splits 4-0, 3-1, 1-3, and 0-4 each occur with
probability .25 (given Bernoulli elections), and
– [strict] election reversals cannot occur at the state level.
• More generally (given many Bernoulli elections),
– under the Pure District Plan, electoral votes won by a candidate
in a state with k electoral are binomially (and approximately
normally) distributed about a mean of k/2, but
– under the Modified District Plan,
• equal [k/2 – k/2], or almost equal [(k+1)/2 – (k-1)/2], electoral votes
splits between the candidate are less probable than some less
equal splits, as they entail (evidently somewhat unlikely) election
reversals within the state (such that the candidate who carries the
most districts loses the statewide vote), and therefore
• the electoral votes won by a candidate are distributed in a
noticeably bimodal fashion.
– The following charts illustrate this bimodal effect for several state
sizes.
The Bimodal Effect
The Bimodal Effect Aggregated to the
National Level
• While the Modified District Plan produces a clear
bimodal effect on the distribution of state electoral votes
for a candidate aggregated over many (Bernoulli)
elections, it is unlikely to produce a bimodal effect on the
distribution of national electoral votes for a candidate.
• But we can expect that the distribution will be different
from that resulting from the Pure District Plan and, in
particular, that it will be more spread out with a lower
peak in the vicinity of the mean.
• This expectation is clearly borne out in a sample of 7680
Bernoulli elections.
The distribution of district electoral votes won by a candidate in the
simulated elections closely matches the expected normal distribution.
The distribution of states won by a candidate in the simulated elections
likewise closely matches the expected normal distribution.
But the distribution of district + state electoral votes is distinctly different (in the
expected manner) from the sum of the two separate normal distributions (that
would result from either the Pure District Plan or the Modified District Plan with
separate and independent votes for district and state electors).
The Bimodal Effect Aggregated to the
National Level (cont.)
• In the following chart, we
– convert the absolute frequencies of elections into
relative frequencies (which in turn are equal to
estimated probabilities expressed as percentages),
and
– zoom into the center of the electoral vote distribution.
Note: each fit line is a quadratic (a good approximation to a normal
curve close to its peak) fitted only to the data points that show up
in the chart.
The Bimodal Effect Aggregated (cont.)
The Bimodal Effect Aggregated (cont.)
In preparation for
subsequent use, we
slide the blue
(Modified District
Plan) curve so that is
centered on 269 (as
we know it should be)
and label
– the horizontal axis
in terms of
deviations from
the mean of 269,
and
– the vertical axis in
terms of
probabilities.
The Bimodal Effect Aggregated (cont.)
• Note that in the preceding chart, almost all of the plotted point in the
266-272 range lie on or above the fitted blue curve.
• There is also some suggestion that the true line representing the
probability of different electoral vote splits in this central interval has
an essentially flat plateau.
• So we might also fit the plotted points to a horizontal line equal to
the local mean, which lies entirely above the curved line fitted over a
somewhat wider central interval (see following chart).
• We can alternatively use these estimates in calculating voting power
under the Modified District Plan.
The Bimodal Effect Aggregated (cont.)
Calculating Voting Power under the
Modified District Plan
• We can now outline what I believe is a legitimate and
essentially accurate (apart from the sampling error
entailed by simulations) method for calculating individual
2-tier voting power under the Modified District Plan,
when we take account of the fact that voters cast a
single vote that counts at both the district and state
levels.
– We will illustrate the procedure by focusing on a voter
in the state of Maryland (with 8 districts and 10
electoral votes).
– A similar procedure is will be used for calculating
voting power under the National Bonus Plan.
Calculations for Modified District Plan
•
Maryland has n = 2,302,057 votes equally divided into 8 districts with n =
287,757 voters each.
– The probability that a MD voter casts a vote that is decisive at the
district level (i.e., that the district vote is otherwise tied) is √[2/πn] =
.0014874.
– The probability that a MD voter casts a vote that is decisive at the state
level (i.e., that the state vote is otherwise tied) is √[2/πn] = .00052587.
– If these were separate and independent votes, the probability that a MD
voter casts a vote that is simultaneously decisive at both the district and
state levels would be .0014874 x .00052587 = .000000782 .
– But the votes are not separate and independent. Given that the vote in
the voter’s district is tied, the probability that the state vote is also tied is
√[2/πn] = .00056218, where n = 2,302,057 - 287,757 = 2,014,300.
– Therefore the probability that the district and state votes are both tied
(Contingency 1) is .00056218 x .0015197 = .000000836 (somewhat
larger than if the votes were separate and independent).
– By subtraction,
• the probability that the state vote is tied while the district vote is not
tied (Contingency 2) is .000557864, and
• The probability that the district vote is tied while the state vote is not
tied (Contingency 3) is .00148740.
– This is summarized in the table that follows.
First-Tier Probabilities for a MD Voter
SV Tied
SV Not
Tied
Total
DV Tied
DV Not Tied
Total
.00056218
Contingency 1
.00000083619
Contingency 2
.000557864
.00052587
.99943782
Contingency 3
.00148656
.99947413
1.000000
.00148740
.9985126
1.000000
Calculations for Modified District Plan (cont.)
• In each contingency, voter i is doubly decisive if and only
if the remaining electoral votes are divided sufficiently
equally that neither candidate has won 270 (the quota for
Electoral College victory).
• The following calculations apply to voters in any state
(not just Maryland).
Methodological Note. The Banzhaf voting power (probability of decisiveness) of a voter i in a voting game n(n/2 + 1; 1,…,1) with n even is the
same as the voting power of a voter in the game n+1([n-1]/2; 1,…,1).
(However, the former game is not strong, i.e., ties may occur, and i has
greater “power to initiate action” than “power to prevent action,” i’s Bz
power being equal to the harmonic mean of the two; the latter game is
strong and both aspects of power are equal to i’s Bz power.) In this
sense, the Banzhaf measure takes account of possible ties in even
numbered voting bodies.
Calculations for Modified District Plan (cont.)
•
In Contingency 3, voter i tips 1 electoral vote one way or the other and so is
doubly decisive if and only if the leading candidate has won precisely 269 of
the other 537 electoral votes (electing the leading candidate or creating a
tie).
– By the preceding note, the probability of this event is equal to the
probability of an equal 269-269 split with 538 votes, i.e., approximately
• .0259 using the fitted curve, or
• .02662 using the local mean line.
– but in any case substantially less than the .0344 using the red
normal curve
•
In Contingency 2, voter i tips 2 electoral votes one way or the other and so is
doubly decisive if and only if the leading candidate has won 268 (tied with
the other candidates) or 269 (2 ahead of the other candidate) of the other
536 electoral votes (electing one or other candidate or creating a tie).
– By the preceding note, the probability of this event is equal to the
probability of a 269-269 or 270-268 split with 538 votes, i.e.,
approximately
• .0259 + .0258 = .0517 using the fitted curve, or
• 2 x .02662 = .05324 using the local mean line,
– but in any case substantially less than the .06865 using the red
normal curve.
Calculations for Modified District Plan (cont.)
• In Contingency 1, voter i tips 3 electoral votes one way
or the other, and so is doubly decisive if and only if the
leading candidate has won 268 or 269 of the remaining
535 electoral vote is split 266-269 or 269-266 (electing
one or other candidate).
– By the preceding note, the probability of such a split
occurring is equal to the probability of a 269-269, 270268, or 271-267 split occurring with 538 electoral
votes, i.e., approximately
• .0259 + .0258 + .02565 = .07735 using the fitted
curve, or
• 3 x .02662 = .07986 using the local mean line,
– But in any case substantially less than the .10245
using the red normal curve.
Calculations for Modified District Plan (cont.)
• Returning now specifically to Maryland voters, we
determine overall voting power by multiplying the
probability of each contingency (i.e., the probability of
each type of decisiveness in the first tier) by the
probability of decisiveness in the second-tier given that
contingency and sum the products
• Using the fitted curve, we get:
– Contingency 1: .00000083619 x .07735 = .0000000647
– Contingency 2: .000557864 x .0517 =
.000028841
– Contingency 3: .00148656 x .0259 =
.000038502
.000067408
Calculations for Modified District Plan (cont.)
• Using the local mean line, we get:
– Contingency 1: .00000083619 x .07986 = .0000000668
– Contingency 2: .000557864 x .05324 = .000029701
– Contingency 3: .00148656 x .02662 =
.000039572
.000069340
• Finally if we estimate probability of second-tier decisiveness from
the red normal curve curve, we get approximately:
– Contingency 1: .00000083619 x .10245 = .0000000857
– Contingency 2: .000557864 x .06865 = .000038297
– Contingency 3: .00148656 x .0344 =
.000051138
.000089521
– This correspondents to voting power under Banzhaf style analysis.
National Bonus Plan
• A total of 538 electoral votes are apportioned and cast as at present.
– An additional 100 electoral votes are awarded on a winner-takeall basis to the national popular vote winner. [Bush 271, Gore
367; Bush 386, Kerry 252]
• The obvious purpose of the National Bonus Plan is to reduce (to
almost but not quite zero) the probability of a “reversal of winners”
outcome (as it would have easily accomplished in 2000).
• Much as under the Modified District Plan, a voter may cast a
decisive vote in the tiered voting game in three distinct ways:
– the voter casts a vote that is decisive in the voter’s state and the
state’s bloc of (3 to 55) electoral votes is decisive in the Electoral
College;
– the voter casts a vote that is decisive in the national popular
voter and the bonus bloc of 100 electoral votes is decisive in the
Electoral College; and
– the voter casts a vote that is decisive in both his state and in the
national popular vote and the combined bloc of (103 to 155)
electoral votes is decisive in the Electoral College.
• This last probability is sufficiently small that it can be neglected.
National Bonus Plan (cont.)
• Given Bernoulli elections (or real elections, for that
matter), the 100 electoral vote bonus is almost always
decisive, which implies that individual voting power
should be
– almost as high under direct popular vote; and
– much more equal than without the bonus.
• The National Bonus Plan falls with the category of
Edelman’s voting systems that mix district and at-large
representation.
– According to Edelman-style analysis, the National Bonus Plan,
with two parallel upper-tiers, may increase mean individual
voting power beyond that under direct popular vote.
– This is in fact true, as shown in the following chart.
National Bonus (100 EV) Plan
National Bonus Plan (cont.)
• Effects on voting power of the National Bonus Plan:
– Mean individual voting power is considerably higher than under
direct popular vote.
– The ratio of advantage in voting power of California voters over
Montana voters is more than cut in half, relative to the existing
Electoral College.
– Only voters smaller states (not including excluding the smallest
[Wyoming]) would have greater voting power under direct
popular vote.
• We can extend this analysis by considering a national
bonus B varying
– from B = 0 (equivalent to the existing Electoral College) to
– to B = 534 (equivalent to direct popular vote).
• Note. A candidate who wins the national bonus must win at least
one state with at least 3 electoral votes and 537 is bare majority of
538 + 534 = 1072 electoral votes.
National Bonus Plans: Varying Bonuses
National Bonus Plans (cont.)
• The preceding chart resembles the earlier chart
illustrating the Edelman at-large effect, given 100,000
voters and 51 “seats” (e.g., electoral votes).
• But two differences should be noted:
– the earlier chart
• entailed uniform representation at the district level,
and
• fixed the total number of seats (at 51);
– the present chart
• is based on the current Electoral College (with
non-uniform representation), and
• fixes the number of district seats [at 538] (and
allows the total number of “seats” to increase as
the bonus increases).
National Bonus Plans (cont.)
• It is interesting to note (but probably is a coincidence)
that the actually proposed national bonus of 100
electoral votes comes close to maximizing mean
individual voting power.
• Clearly individual voting power is equalized as the size of
the national bonus increases.
• The following two charts demonstrate this with respect to
– the extremes of individual voting power in California
and Montana, and
– the distribution of voting power over all states.
Individual Voting Power in California vs. Montana
under National Bonus Plans
Distribution of Individual Voting Power over All
States Under National Bonus Plans
Overview of Voting Power under the Electoral
College and Alternative Institutions
• Equal individual voting power:
• direct popular vote;
• uniform two-tier representation; and
• Penrose (fractional) apportionment of electoral votes.
• Unequal individual voting power [ratio of most/least favored]:
– Large-state advantage:
• the existing Electoral College [3.41];
• electoral votes apportioned in whole numbers as close as possible to
population [10.13];
• electoral votes apportioned precisely proportional to population [9.51];
• electoral votes apportioned precisely proportional to population plus
constant two [3.02]; and
• National Bonus (100 EV) Plan [1.41].
– Small-state advantage:
•
•
•
•
•
electoral votes apportioned equally among states [8.28];
pure district system [1.93];
modified district system [2.69];
pure proportional system [3.74]; and
whole-number proportional system [infinite].
Overview (cont.)
• The alternative institutions also vary with respect to
mean absolute individual voting power,
– and thus also with respect to total Banzhaf power –
what Felsenthal and Machover call sensitivity.
• In this respect, the alternative institutions sorts
themselves pretty clear into four categories:
– the Whole-Number Proportional Plan scores strikingly
low,
– the National Bonus Plan scores strikingly high,
– the Proportional and Modified Districts Plans score,
respectively, the same as, and slightly higher than,
Direct Popular Vote, and
– and all other institutions score about the same and
somewhat lower than Direct Popular Vote.
Mean Absolute Individual Voting Power
•
•
•
•
•
•
•
•
•
•
•
Whole-Number Proportional
Equal EV Apportionment
EV Proportional to Population (Whole Number)
EV Precisely Proportional to Population
Existing Electoral College
EV Precisely Proportional to Population + 2
Pure District Plan
Uniform Representation = Penrose Apportionment
Proportional Plan = Direct Popular Vote
Modified District Plan
National Bonus (100 EV) Plan
= .00004221
= .00005133
= .00005284
= .00005295
= .00005488
= .00005498
= .00005707
= .00005785
= .00007215
= .00007264
= .00008257
Why Does Two-Tier Voting Reduce
Mean Individual Voting Power?
Why Does Two-Tier Voting Reduce
Voting Power? (cont.)
• Districting means that
– a voter is sometimes decisive even when the
remaining votes are not equally split, but also
– a voter is sometimes not decisive even when the
remaining votes are equally split.
• The net effect of districting is to reduce the
number of ways in which a voter is decisive.
– Thus, if all the ways the other voters can split are
equally likely (random voting with p = .5), districting
reduces the probability of being decisive.
Critiques of A Priori Voting Power
• Critiques of the Banzhaf effect rest fundamentally on the
(indisputable) claim that the Bernoulli random voting
model is in no way representative of empirical voting
patterns.
Howard Margolis, “The Banzhaf Fallacy,” American Journal of Political
Science, 1983.
Andrew Gelman, Jonathan N. Katz, and Francis Tuerlinckx, “The Mathematics
and Statistics of Voting Power,” Statistical Science, 2002.
Andrew Gelman, Jonathan N. Katz, and Joseph Bafumi, “Standard Voting
Power Indexes Do Not Work: An Empirical Analysis,” British Journal of
Political Science, 2004.
Jonathan N. Katz, Andrew Geman, and Gary King, “Empirically Evaluating the
Electoral College,” in Ann N. Crigler, Marion R. Just, and Edward J.
McCaffrey, Rethinking the Vote, Oxford University Press, 2004.
Critiques of A Priori Voting Power (cont.)
• Such critiques overlook the fact that the Banzhaf (and
Shapley-Shubik) measures pertain to a priori voting
power, measuring the power of states — and, in the twotier version, of individual voters — in a way that takes
account of the Electoral College voting rules but nothing
else.
– A priori, a voter in California has three times the probability of
casting a decisive vote than one in New Hampshire.
– But if we take account of recent voting patterns, poll results, and
other information, a voter in New Hampshire may have a greater
empirical (or a posteriori) probability of decisiveness in the
upcoming election, and accordingly get more attention from the
candidates and party organizations, than one in California.
– In like manner, the members of the U.S. Supreme Court have
equal a priori voting power, but lawyers arguing before the Court
may have had good reason to address their persuasive
arguments especially to Justice O-Conner (and perhaps now
Justice Kennedy).
Relevance of A Priori Voting Power
• If it is hardly related to empirical voting power in any particular
election, the question arises of whether a priori voting power and the
Banzhaf effect should be of concern to political science and practice.
• Constitution-makers arguably should — and to some extent must —
design political institutions from behind a “veil of ignorance”
concerning future political trends.
• Accordingly they should — and to some extent must — be
concerned with how the institutions they are designing allocate a
priori, rather than empirical, voting power.
– The framers of the U.S. Constitution did not require or expect electoral
votes to be cast en bloc by states.
– However, at least one delegate [Luther Martin] expected that state
delegations in the House of Representatives would vote en bloc, which
he thought would give large states a Banzhaf-like advantage.
William H. Riker, “The First Power Index.” Social Choice and Welfare, 1986.
Relevance of A Priori Voting Power (cont.)
– While party politicians within states initially
manipulated the rules for selecting Presidential
electors for immediate partisan advantage, in due
course almost all states moved to the winner-take-all
rule that gives rise to the Banzhaf effect.
– This dominant trend suggests that legislators in large
states, operating behind a “veil of ignorance”
concerning its long-term partisan implications,
understood that doing so would enhance (in so far as
other states had not done the same) or restore (in so
far as other states had done the same) the influence
of their state and its voters in Presidential elections.
– Legislators in small states then had little choice but do
the same.