VOTING POWER

1. VOTING POWER IN THE ELECTORAL COLLEGE

2. Weighted Voting Games • The Electoral College is an example of a weighted voting game. • Instead of each voter casting a single vote, each voter casts a block of votes, with some voters casting larger blocks and others casting smaller blocks. • Other examples: • voting by disciplined party groups in multi-party parliaments (probably elected on the basis of proportional representa-tion); • 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).

3. Weighted Voting Games (cont.) • Weighted voting can be analyzed using the theory of [simple] games. • A simple game is a (voting or similar) situation in which every potential coalition (set of players/votes) can be deemed to be either winning or losing. • With respect to lawmaking power of the United States, the winning coalitions are: • All coalitions including 218 House members, 51 Senators*, and the President, and also • All coalitions including 290 House members and 67 Senators. *Including the Vice President, or 60 Senators [excluding the Vice President] in so far as filibustering is an option.

4. Weighted Voting Games (cont.) • With respect to weighted voting games, the most basic finding is that voting power is not the same as (and is not proportional to) voting weight; in particular • voters with very similar (but not identical) weights may have very different voting power; and • voters with quite different voting weights may have identical voting power. • In general, it is impossible to apportion votingpower (as opposed to voting weights) in a “refined” fashion, especially within small groups.

5. Weighted Voting Example • Consider a country that uses proportional representation to elect members of parliament. • Duverger’s law implies that the country will have a multi-party system. • Hotelling-Downs implies that the parties will be spread over the ideological spectrum and may receive rather similar vote shares.

6. Weighted Voting Example • Suppose that four parties receive these vote shares: Party A, 27%; Party B, 25%; Party C, 24%; Party 24%. • Seats are apportioned in a 100-seat parliament according some apportionment formula. In this case, the apportionment of seats is straight-forward: • Party A: 27 seats Party C: 24 seats • Party B: 25 seats Party D: 24 seats • Seats (voting weights) have been apportioned in a way that is precisely proportional to vote support, but voting power has not been so apportioned (and cannot be).

7. Weighted Voting Example (cont.) • Since no party controls a majority of 51 seats, a governing coalition of two or more parties must be formed. • A party’s voting power is based on its oppor-tunities to help create (or destroy) winning (governing) coalitions. • But, with a small number of parties, coalition possibilities -- and therefore differences in voting power -- are highly limited.

8. Weighted Voting Example (cont.) A: 27 seats; B: 25 seats; C: 24 seats; D: 24 seats • Once the parties start negotiating, they will find that Party A has voting power that greatly exceeds its slight advantage in seats. This is because: • Party A can form a winning coalition with any one of the other parties; and • the only way to exclude Party A from a winning coalition is for Parties B, C, and D to form a three-party coalition. • The seat allocation (totaling 100 seats) is strategically equivalent to this much simpler allocation (totaling 5 seats): • Party A: 2 seats; • Parties B, C, and D: 1 seat each; • Total of 5 seats, so a winning coalition requires 3 seats. • So the original seat allocation is strategically equivalent to one in which Party A has twice the weight of each of the other parties (which is not at all proportional to their vote shares).

9. 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 the seats shares have only slightly changed, the strategic situation has changed fundamentally. • Party A can no longer form a winning coalition with Party D. • Parties B and C can now form a winning coalition by themselves. • The seat allocation is equivalent to this much simpler allocation: • Parties A, B, and D: 1 seat each; • Party D: 0 seats • Total of 3 seats, so a winning coalition requires 2 seats. • Party A has lost voting power, despite gaining seats. • Party C has gained voting power, despite losing seats. • Party D has become powerless (a so-called dummy), despite retaining a substantial number of seats.

10. Weighted Voting Example (cont.) • In fact, the only possible strong simple games with 4 players are these two with simplified weights of (2,1,1,1) and (1,1,1,0), • plus the “inessential” game (1,0,0,0), in which one party holds a majority of seats (making all other parties dummies, so no coalition need be formed. • Expanding the number of players to five produces these additional possibilities: (3,2,2,1,1), (3,1,1,1,1), (2,2,1,1), (1,1,1,1,1). • With 6 or more players coalition possibilities become much more numerous and complex.

11. Weighted Voting Example (cont.) • Returning to the four-party example, voting power would change further if the parliamentary decision rule were to change from simple majority to (say) 2/3 majority (like the old nominating rule in Democratic National Conven-tions). • Both before and after the election, all three-party coalitions, and no smaller coalitions, are winning coalitions (so all four parties are equally powerful). • In particular, under 2/3 majority rule, Party D is no longer a dummy after the election. • Thus, changing the decision rule reallocates voting power, even as voting weights (seats) remain the same. • Generally making the decision rule more demanding tends to equalize voting power. • In the limit, weighted voting is impossible under unanimity rule.

12. Weighted Voting Example (cont.) • A simple weighted majority voting game (with no ties) is a strong simple game, i.e., • Given any complementary pair of coalitions, one is winning and the other is losing. • A (for example) 2/3 weighted majority voting game is no longer strong, i.e., • Given some complementary pairs of coalitions, neither may be winning (both are blocking).

13. Power Indices • Several power indices have been pro-posed to quantify the share of power held by each player in simple games. • These particularly include: • the Shapley-Shubik power index; and • the Banzhaf power index. • Such power indices provide precise formulas for ascertaining the voting power of players in weighted voting games.

14. The Shapley-Shubik Index The Shapley-Shubik power index works as follows. Consider every possible ordering of the players A, B, C, D (e.g., every possible order in which they might line up to support a proposal put before a voting body). Given 4 voters, there are 4! = 4 x 3 x 2 x 1 = 24 possible orderings:

15. The Shapley-Shubik Index (cont.) • Suppose coalition formation starts at the top of each ordering, moving downward to form coalitions of increasing size. • At some point a winning coalition formed. • The “grand coalition” {A,B,C,D} is certainly winning. • For each ordering, identify the pivotal player who, when added to the players already in the coalition, converts a losing coalition into a winning coalition. • Given the seat shares of parties A, B, C, and D before the election, the pivotal player in each ordering is identified by the arrow (<=).

16. The Shapley-Shubik Index (cont.) • Voter i’s Shapley-Shubik power index value SS(i) is simply: Number of orderings in which the voter i is pivotal Total number of orderings • Clearly such power index values of all voters add up to 1. • Counting up, we see that A is pivotal in 12 orderings and each of B, C, and D is pivotal in 4 orderings. Thus: 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 has 3 times the voting power of each other party. Lloyd Shapley and Martin Shubik, American Political Science Review, 1955.

17. The Banzhaf Index • The Banzhaf power index works as follows: • A player i is critical to a winning coalition if • i belongs to the coalition, and • the coalition would no longer be winning if i defected from it. • Voter i’s absolute Banzhaf powerBz(i) is Number of winning coalitions for which i is critical Total number of coalitions to which i belongs. • Bz(i) is equivalent to voter i’s a priori probability of casting a decisive vote, e.g., breaking what otherwise would be a tie. • In this context, a priori probability means, in effect, given that all other voters vote randomly (i.e., by flipping coins).

18. The Banzhaf Index (cont.) • Given the seat shares before the election, and looking first at all the coalitions to which A belongs, we identify: {A},{A,B},{A,C}, {A,D}, {A,B,C}, {A,B,D}, {A,C,D}, (A,B,C,D}. • Checking further we see that A is critical to all but two of these coalitions, namely • {A} (because it is not winning); and • {A,B,C,D} (because {B,C,D} can win without A). • Thus: Bz(A) = 6/8 = .75

19. The Banzhaf Index (cont.) • Looking at the coalitions to which B belongs, we identify: {B},{A,B}, {B,C}, {B,D}, {A,B,C}, {A,B,D}, {B,C,D}, (A,B,C,D}. • Checking further we see that B is critical to only two of these coalitions: • {B}, {B,C}, {B,D} are not winning; and • {A,B,C}, {A,B,D}, and {A,B,C,D} are winning even if B defects. • The positions of C and D are equivalent to that of B. • Thus: Bz(B) = Bz(C) = Bz(D) = 2/8 = .25

20. The Banzhaf Index (cont.) • The "total absolute Banzhaf power" of all four voters: = .75 + .25 + .25 + .25 = 1.5 . • Voter i's Banzhaf index power values BP(i) is his share of the "total power," so BP(A) = .75/1.5 = 1/2; and BP(B) = BP(C) = BP(D) = .25/1.5 = 1/6. John F. Banzhaf, “Weighted Voting Doesn’t Work,” Rutgers Law Review, Winter, 1965, and “One Man, 3.312 Votes: A Mathematical Analysis of the Electoral College,” Villanova Law Review, Winter 1968.

21. Shapley-Shubik vs. Banzhaf • In this simple 4-voter case, the two indices evaluate power in the same way. • This is often true in simple examples, but it is not true more generally. • In particular kinds of situations the indices evaluate the power of players in radically different ways. • For example, if there is single large stockholder while along other holding are highly dispersed. • It is even possible that the two indices may rank players with respect to power in different ways (but not in weighted voting games). • It is now generally believed that the Banzhaf is more appropriate to apply to the analysis of voting institutions, including the Electoral College.

22. Do Constitution Makers Understand Voting Power? • Luther Martin (delegate to the Constitutional Convention but an opponent of its proposal): " ...even if the States who had most inhabitants ought to have a greater number of delegates, yet the number of delegates ought not to be in exact proportion to the number of inhabitants, because the influence and power of those States whose delegates are numerous, will be greater when compared to the influence and power of the other States, than the proportion which the numbers of delegates bear to each other….” • Application to House of Representatives vs. Electoral College.

23. Do Constitution Makers Understand Voting Power? (cont.) • The allocation of voting power in the original (six-member) European Common Market Council of Ministers made the smallest member (Luxembourg) a dummy. • The recent Nice Treaty expanding and reallocating voting power in the EU had effects on voting power that almost certainly were not intended. Dan S Felsenthal and Moshé Machover, “The Treaty of Nice and Qualified Majority Voting,” Social Choice and Welfare, 2001.

24. Evaluating Voting Power in the Electoral College • Let’s first review the apportionment of voting weights (electoral votes) in the Electoral College, in relation to states’ share of the U.S. population. • We know that: • there is a small-state bias in this apportionment; and • there is the problem of apportionment into whole numbers that is most significant among small states. • The following chart shows the relationship between electoral vote and population shares following the 2000 Census.

26. Voting Power in the Electoral College (cont.) • When the Shapley-Shubik index was first developed in the early 1950s, it seemed apparent that that • the voting power of the weightiest players is typically greater than their weights, while • the voting power of less weighty players is typically less than their weights. • This seemed consistent with intuition (going back to Luther Martin and the move to the general ticket system) that large states are substantially advantaged in the Electoral College, even though the small are favored by the apportionment of voting weights. • However, when the first (Shapley-Shubik approxima-tions, using “Monte Carlo” methods) evaluations of voting power in the Electoral College were done, this tendency manifested itself only very weakly.

27. Voting Power in the Electoral College (cont.) • Clearly we cannot apply the formulas sketched above for calculating power index values in the Electoral College (let alone larger weighted voting bodies): • There are 51! ~ 1.55 x 1066 ways 51 voters might line up to vote. • Indeed, such calculations are well beyond the practical computing power even of super-computers. • Fortunately quite accurate indirect methods of calculation exist • There are websites that can make power index calculations. • The best is: http://www.warwick.ac.uk/~ecaae/ • Also see: http://www.lse.ac.uk/collections/VPP/ • Note that Electoral College Voting Game is not quite strong, i.e., a 269-269 tie is possible.

28. Voting Power in the Electoral College

31. The Electoral College as a Two-Stage Voting Game • With respect to the Electoral College as a (one-stage) weighted voting game, the conjecture that large states are greatly advantaged by the winner-take-all practice is not supported. • But the one-stage Electoral College voting game is a chimera: • each state is a mere agent of the popular voting majority within the state. • We should expand the application of the Banzhaf index to include individual voters with each state. • Within each state, we have an (unweighted) majority voting game that determines how that state's bloc of electoral votes is to be cast in the weighted majority EC game.

32. The Electoral College as a Two-Stage Voting Game (cont.) • If there are nk voters in state k, clearly (on the basis of common sense and either power index) each voter has 1/nk of the voting power within the state. • Since (as we have seen in the prior chart) each nk is approximately proportional to the state's voting power Bz(k) in the Electoral College, it appears that the voting power (in the full 100,000,000-person Presidential election game) of all voters throughout the country is just about the same. • This appears to follow because the ratio Bx(k) / nk [state voting power to population] is approximately constant over all states.

33. The Electoral College as a Two-Stage Voting Game (cont.) • But closer analysis of the properties of the Banzhaf index shows that this apparent uniformity of individual voting power across the states does not hold after all. • While all voters in the same state have equal voting power in determining the allocation of their state’s electoral votes, voters in different states clearly do not have the same voting power in determining the allocation of their respective state’s electoral voters.

34. The Electoral College as a Two-Stage Voting Game (cont.) • Recall that Banzhaf power is equivalent to a voter’s a priori probability of casting a decisive vote, e.g., breaking what otherwise would be a tie (when other voters vote randomly. • Compare the “voting power” of a voter in Wyoming and a voter in California. • One the one hand, the Wyoming voter has the greater chance of casting a decisive vote in the first stage of the voting game, simply because there are fewer other voters in Wyoming and the voter has a larger (though still very small) of breaking what would otherwise be a tie in the Wyoming popular vote. • On the other hand, while the voter in California has a smaller chance of casting a decisive vote, if that voter does cast a decisive vote, it is much more likely to determine the outcome of the Presidential election, because it will tip 55 (rather than 3) electoral votes into one candidate’s column or the other’s.

35. The Electoral College as a Two-Stage Voting Game (cont.) • Put informally, voters in small states have a bigger chance of determining the winner of a smaller prize, while voters in large states have a smaller chance of determining the winner of a bigger prize. • The question is how these relative advantages and disadvantages balance out. • We have seen the value (Banzhaf power) of the “prizes” are approximately proportional to weights (number of electoral votes).

36. The Electoral College as a Two-Stage Voting Game (cont.) • However, statistical theory tells us that, while the probability of casting a decisive vote is inversely related to the number of voters, it is inversely proportional -- not to the number of voters -- but to the square root of this number. • This is an approximation that becomes very good once the number of voters reaches a few dozen. • Now we can put the probabilities for the two stages together.

37. The Electoral College as a Two-Stage Voting Game (cont.) • California has about 68 times the population of Wyoming. • Therefore a voter in Wyoming has a greater probability of casting a decisive vote. • But the Wyoming voter does not have 68 times the probability -- but rather about √68 = 8.25 times the probability. • Meanwhile, California has 18.33 times more electoral votes than Wyoming and about 21 times the probability of being decisive in the Electoral College. • Suppose the Wyoming voter’s probability of casting a decisive vote is p; then the California voter’s probability is about 21p/8.33 = 2.5p. • In sum, the California voter has a considerably larger probability of casting a decisive vote in the two-stage Presidential election voting game than the Wyoming voter. • Banzhaf’s reference to “One Man, 3.312 Votes” compares DC with NY in the 1960s. • DC actually had a larger population than some states with 4 electoral votes. • See language of the 23rd Amendment.

38. The Electoral College as a Two-Stage Voting Game (cont.) • Bear in mind that all these probabilities are a priori, on the assumption of random voting. • That is, they factor out any empirical data or assumptions about actual voting patterns. • Some political scientists have critiqued this conclusion as “the Banzhaf fallacy.” • Probably what they mean is not really that the Banzhaf analysis is fallacious but that it is not especially relevant in practice. • Certainly the status of states as actual or potential “battlegrounds” is likely to be more relevant to the amount of attention that voters in different states get in Presidential elections than is their Banzhaf power. Howard Margolis, “The Banzhaf Fallacy,” American Jounal of Political Science, 1983; Andrew Gellman, Jonathan Katz, and Gary King, various recent papers and articles.