A PRIORI VOTING POWER

1. A PRIORI VOTING POWER UNDER THE ELECTORAL COLLEGE AND ALTERNATIVE INSTITUTIONS Nicholas R. Miller Revised August 2007 The previous results for the Modified District Plan and the National Bonus Plan were incorrect. Thanks to Claus Beisbart, Dan Felsenthal, and Moshé Machover for comments and suggestions.

2. 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.

3. 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.)

4. 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 poweranalysis using the various power measures.

5. 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.

8. 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.

9. Overview (cont.) • A measure of a priori voting power means a measure that takes account of the structure of the voting rulesbut 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).

10. 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.

11. 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

12. 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).

13. 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.

14. The Small-State Advantage (and Whole Number Effect) in the Apportionment of Electoral Votes

15. Putting Both Log Variables into Percent Shares Straightens the Line of Proportionality

16. Another View of the Small-State Advantage

17. The Apportionment of House Seats (and Electoral Votes) to Smaller States Unavoidably Entails Substantial “Rounding Error”

18. 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.

19. 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).

20. 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).

21. 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).

22. 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 votingpower (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.

23. 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).

24. 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.

25. 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.

26. Weighted Voting Example (cont.) • Suppose at the next election the vote and seat shares change a bit: BeforeNow Party A: 27 Party A: 30 Party B: 25 Party B: 29 Party C: 24 Party C: 22 Party D: 24 Party D: 19 • While 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.

27. Weighted Voting Example (cont.) • In fact, these are the only possible strong simple games with 4 players: • (3:2,1,1,1); • (2:1,1,1,0); and • (1:1,0,0,0), i.e., the “inessential” game in which one party holds a majority of seats (making all other parties dummies), so that no winning (governing) coalition [in the ordinary sense of two or more parties] needs to be formed. • Expanding the number of players to five produces these additional possibilities: • (5:3,2,2,1,1); and • (4:3,1,1,1,1); and • (4:2,2,1,1,1); and • (3:,1,1,1,1). • With six or more players, coalition possibilities become considerably more numerous and complex.

28. 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 votingpower, 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).

29. 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.

30. 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

31. The Shapley-Shubik Index The Shapley-Shubik power index works as follows. Using the previous four-party example, consider every possible ordering (or permutation) of the parties A, B, C, D (e.g., every possible order in which they might line up to form a winning coalition). Given n voters, there n! (n factorial) such orderings. Given 4 voters, there are 4! = 1 x 2 x 3 x 4 = 24 possible orderings:

32. 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 (<=).

33. 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: VoterSS 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.

34. 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 powerAbBz(i) is Number of winning coalitions for which i is critical Total number of coalitions to which i belongs. • Remember, there are 2ncoalitions and i belongs to half of them, i.e., to 2n-1 of them.

35. 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

36. 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 .

37. The Banzhaf Measure (cont.) • Note that exactly one voter is pivotal in each ordering (permutation) of voters, so • the S-S values of all voters necessarily add up to 1 • In contrast, several voters or none of the voters may be critical to a given winning coalition (combination) of voters, so • the AbBz values do not add up to 1 (except in special cases). • However, if we are interested in the “relative” power of voters (i.e., in power values that add up to 1, like the S-S index), we can derive a (relative) Banzhaf index value RBz(i) for voter i that is simply his share of the "total power," so RBz(A) = .75/1.5 = 1/2; and RBz(B) = RBz(C) = RBz(D) = .25/1.5 = 1/6.

38. 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).

39. 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.

40. The Measurement of Voting Power (cont.) • F&M also observe that Banzhaf’s essential ideas • had been laid out twenty years earlier by L.S. Penrose, and • were subsequently and independently rediscovered by Coleman. Felsenthal, Dan S., and Moshé Machover, The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, 1988 Felsenthal, Dan S., and Moshé Machover, “Voting Power Measurement: A Story of Misreinvention,” Social Choice and Welfare, 25, 2005 Penrose, L. S., “The Elementary Statistics of Majority Voting,” Journal of the Royal Statistical Society, 109, 1946 Coleman, James S., “Control of Collectivities and the Power of a Collectivity to Act.” In Bernhardt Lieberman, ed., Social Choice, 1971 • F&M observe that • the Absolute Banzhaf measure can be transformed into the relative Banzhaf index, but • there is rarely good reason to do this.

41. 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 prioriprobability 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.

42. 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.

43. 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.

44. 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.

45. 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.

47. 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 powerfor each state.

49. 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.

50. 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).