VOTING POWER IN THE U.S. ELECTORAL COLLEGE

Note: this discussion is based on the 2000 apportionment of electoral votes

N. R. Miller

Supplementary slides (not to be discussed in class)

Voting Power and the Electoral College• The Voting Power Problem. Does the Electoral College system

(as it has operated [“winner-take-all”] since the 1830s) give voters in different states unequal voting power?– If so, voters in which states are favored and which disfavored and by

how much?• One obvious answer is that voters in battleground states have

more “voting power” than voters in non-battleground states.– They certainly attract more campaign attention.

• But the classification of states by battleground status is contingent and varies by voting alignments and historical eras.

• Suppose our concern is whether there are intrinsic features of the Electoral College that, from behind a “veil of ignorance” about how voters may “chose up sides,” make some voters more powerful than others.– No features of a national popular vote system that would have this

effect.• This is referred to as a priori voting power.

Voting Power and the Electoral College (cont.)

• The only differences among between voters that are intrinsic to the nature of the Electoral College is whether they live is small or large states.

• With respect to the question of whether voters in small or big states are favored by the EC, directly contradictory claims are commonly expressed.

• This results from the failure many by many commentators to make two related distinctions:– the theoretical distinction between • voting weight and • voting power, and

– the practical distinction between • how electoral votes are apportioned among the states

(which determines their voting weights), and • how electoral votes are cast by states (which influences

their voting power).

There is a significant small-state advantage with respect to the apportionment of electoral votes (voting weight)

Weighted Voting Games

• Given the winner-take-all system of casting electoral votes, the top-tier of the Electoral College is an example of a weighted voting game.– Instead of casting a single vote, each voter (state) casts a

bloc of votes, with some voters (states) 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.)• 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• and its complement is “losing.”

– Such a game can be written as (q : w1,w2,…,wn).

• The (top-tier) of 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.– [Based on 2000 apportionment] EC = (270: 55,34,31,27,…,3).– The Electoral College game is almost “strong,” but not quite (because

there may be a 269-269 tie, in which neither opposing coalition is winning).

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 similar (but not identical) voting 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,– especially with a small number of voters;– 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 similarly apportioned (and cannot be).

• 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 smaller and 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 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.

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– (3:1,1,1,1,1).– (1:1,0,0,0,0).

• 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. – All two-party coalitions are “blocking” (neither winning nor losing).

• Thus, changing the decision rule (or quota) 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.

• A measure of a priori voting power is 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.

The Absolute Banzhaf Measure• Here I use the

Absolute Banzhaf measure of voting power, which can be interpreted as follows:

• Imagine a random (or Bernoulli) election, in which everyone votes by independently flipping a fair coin.

New Yorker, 1937 =>

The Absolute Banzhaf Measure (cont.)• A voter’s absolute Banzhaf power is the probability that his or

her vote is decisive, i.e., will decide the outcome of a random election.– In an unweighted voting system, a vote is decisive when it either

• breaks what would otherwise be a tie, or• creates a tie that (we may suppose) will be broken by the flip of a fair coin.

• In the EC weighted voting system, California’s absolute Banzhaf power of 0.475 means that, if the 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 will be decisive and determine the winner.

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 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

Individual A Priori Voting Power• The full Electoral College system is a two-tier voting system, in

which – in the bottom tier, there are 51 (unweighted) one-person,

one-vote elections (in each state), and– the top tier is the 51-state weighted voting game.

• The overall [absolute Banzhaf] voting power of an individual voter in the two-tier voting system is his probability of his “double decisiveness,” i.e.,

the probability that the voter cast a decisive vote in the state election

timesthe probability that the state casts a decisive bloc of votes

in the Electoral Collegein a random election.

Individual A Priori Voting Power (cont.)

• Clearly a small-state voter has an advantage over a large-state voter in that his votes is more likely to be decisive at the state level,– i.e., the popular vote is more likely to be [essentially] tied in a small

state than a large state.

• On the other hand, a large-state voter has an advantage over a small-state voter in that, if his vote is decisive, he will be “swinging” a larger bloc of electoral votes in the EC.

• The question is how these two factors balance out.

Individual A Priori Voting Power (cont.)• On the one hand, we have seen that the voting power of

states is approximately proportional to their voting weights (i.e., electoral votes),– and therefore is (somewhat more) approximately proportional to their

populations (apart from a relatively small bias in favor of small states).• Probability theory tell us that the probability of an

(essentially) even split between Heads and Tails – is not inversely proportional to the number of flips (i.e., voters in a

state), but rather– is inversely proportional (to very good approximation) to the square

root of the number of voters.• Thus we can conclude that individual voting power under the

Electoral College is approximately proportional to the square root of the population of a voters state,– except that voters in small states are somewhat advantaged relative

to this general rule.

The Small-State Apportionment Advantage is More Than Counterbalanced by the Large-State Advantage Resulting from

“Winner-Take-All”

Absent the Small-State Apportionment Advantage, the Overall Large-State Advantage Would be Far More Extreme.

Individual Voting Power by State Population:Electoral Votes Precisely Proportional to Population

Individual Voting Power by State Population:Electoral Votes Proportional Population, plus Two

Individual Voting Power under Alternative Rules for Casting Electoral Votes

• Calculations for the Pure District Plan are entirely straightforward.

• Calculations for the Pure Proportional Plan and the Whole-Number Proportional Plan are relatively straightforward.

• But under the Modified District Plan and the National Bonus Plan, each voter casts a single vote that counts two ways:

• within the voter’s district (or state) and • “at-large” (i.e., within the voter’s state or the nation as

a whole).– Calculating individual voting power in such systems is far

from straightforward.– I have found it is necessary make approximations based on

large samples of Bernoulli elections.

Pure District System

Modified District (ME and NE) Plan

• In his original work, Banzhaf (in effect) – determined each voter’s probability of double decisiveness • through his/her district and the EC and • through his/her state and the EC, and then• summed these two probabilities.

• His table of results (for the 1960 apportionment) is comparable to the following chart (for the 2000 apportionment).

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• There is a vexing problem: mean individual voting power so

calculated exceeds voting power under direct popular vote.• This is anomalous because Felsenthal and Machover (pp. 58-

59) demonstrate that, within the class of ordinary voting games, mean individual voting power is maximized 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.

• The Banzhaf approach ignores the correlation between district and state votes.– For example, Banzhaf in effect assumes that a state with three

electoral votes might split its vote 2-1.

Problems with Banzhaf’s Analysis (cont.)• In a state with a single House seat, individual voting power under the Modified

District Plan operates in just the same way as under the existing Electoral College.

• In a state with two House seats, the state popular vote winner is guaranteed a majority of the state’s electoral votes (i.e., either 3 or 4) and a 2-2 split cannot occur.

• In a state with three or more House seats, electoral votes may be split in any fashion.

• In a state with five or more House seats, the statewide popular vote winner may win only a minority of the state’s electoral votes;– that is, “election inversions” may occur at the state (as well as the national) level.

• I drew a sample of 120,000 Bernoulli elections, with electoral votes awarded to the candidates on the basis of the Modified District Plan. – This generated a database that can be manipulated to determine frequency

distributions of electoral votes for the focal candidate under specified contingencies with respect to first-tier voting, from which relevant second-tier probabilities can be estimated

Modified District System (Approximate)

(Pure) Pure Proportional System

The Whole-Number Proportional Plan

National Bonus Plan (Bonus = 101)

National Bonus Plan (Varying Bonuses)

Summary: Individual Voting Power Under EC Variants