a priori voting power under the electoral college and alternative institutions nicholas r. miller...

A PRIORI VOTING POWERUNDER 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 contra-dictory 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 propor-tional 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 large- state 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.


• Suppose at the next election the vote and seat shares change a bit: Before NowParty A: 27 Party A: 30Party B: 25 Party B: 29Party C: 24 Party C: 22Party D: 24 Party D: 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.


• 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 IndexThe 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 SS Power A 1/2 = .500 B 1/6 = .167

C 1/6 = .167 D 1/6 = .167

• So according to the Shapley-Shubik index, Party A (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; andRBz(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 P-Power [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, 1988Felsenthal, Dan S., and Moshé Machover, “Voting Power Measurement: A Story of Misreinvention,” Social Choice and Welfare, 25, 2005Penrose, L. S., “The Elementary Statistics of Majority Voting,” Journal of the Royal Statistical Society, 109, 1946Coleman, 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 Two-Tier 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 approxi-mately 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 winner-take-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 propor-

tional 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 rank-ordering 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 winner-take-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 apportion-ment 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 Whole-Number 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


• 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 .


• 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.


• 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


• 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.


• 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 (269-

269), – 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.


• 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)


• 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 indepen-

dent 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.


• 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 state-wide vote as well.


• 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 “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.)


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.


• 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.

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

DV Tied DV Not Tied Total

SV Tied .00056218

Contingency 1

.00000083619

Contingency 2

.000557864 .00052587

SV Not

Tied

.99943782

Contingency 3

.00148656 .99947413

Total 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 decisive-ness) 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.


• 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, 270-268, 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.


• 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


• 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-take-all 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 = .00004221

• Equal EV Apportionment = .00005133

• EV Proportional to Population (Whole Number) = .00005284

• EV Precisely Proportional to Population = .00005295

• Existing Electoral College = .00005488

• EV Precisely Proportional to Population + 2 = .00005498

• Pure District Plan = .00005707

• Uniform Representation = Penrose Apportionment = .00005785

• Proportional Plan = Direct Popular Vote = .00007215

• Modified District Plan = .00007264

• National Bonus (100 EV) Plan = .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 two-tier 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.

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