ANALYZING THE ELECTORAL COLLEGE

1. ANALYZING THE ELECTORAL COLLEGE Nicholas R. Miller Political Science, UMBC INFORMS Meeting October 14, 2008 http://userpages.umbc.edu/~nmiller/ELECTCOLLEGE.html

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. Problematic Features of the Electoral College • The Voting Power Problem. Does the Electoral College system (as it presently operates) give voters in different states unequal voting power? • If so, voters in which states are favored and which disfavored and by how much? • The Election Reversal Problem. The candidate who wins the most popular votes nationwide may fail to be elected. • The election 2000 provides an example (provided we take the official popular vote in FL at face value). • The Electoral College Deadlock Problem, i.e., the House contingent procedure. • Here I present some analytic results pertaining to the first and second problems of the existing Electoral College as well as variants of the EC.

4. The Voting Power Problem • 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 the apportionment of voting weight, determines voting power).

5. The Apportionment of Electoral Votes • The apportionment of electoral votes is fixed in the Constitution, • except that Congress can by law change the size of the House of Representatives, and Congress can therefore also change • the number of electoral votes, and • the ratio “Senatorial” electoral votes Total electoral votes • which reflects the magnitude of the small-state advantage in apportionment.

6. Chart 1. The Small-State EV Apportionment Advantage

7. The Casting of Electoral Votes • How electoral votes are cast within states is determined by state law. • But, with few exceptions, since about 1836 states have cast their electoral votes on a winner-take-all basis. • By standard voting power calculations, • the winner-take-all practice produces a large-state advantage • that more than balances out the small-state advantage in electoral vote apportionment.

8. A Priori Voting Power • A measure of a priori voting power is 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.). • The standard measure of a priori voting power is the Absolute Banzhaf (or Penrose) Measure. • Dan Felsenthal and Moshe Machover, The Measure of Voting Power: Theory and Practice, Problems and Paradoxes, 1998 • A voter’s absolute Banzhaf voting power is • the probability that the voter’s vote is decisive (i.e., determines the outcome the election), • given that all other voters vote by independently flipping fair coins (i.e., given a Bernoulli probability space producing a Bernoulli election).

9. A Priori Individual Voting Power • In a simple one person, one vote majority rule election with n voters, • the a priori voting power of an individual voter is the probability that his vote is decisive, i.e., • the probability that the vote is otherwise tied (if n is odd), or • one half the probability the vote is otherwise within one vote of a tie (if n is even). • Provided n is larger than about 25, this probability is very well approximated by √ (2 / πn), • Which implies that that individual voting power is inversely proportional to the square root of the number of voters.

10. Calculating Power Index Values • There are other mathematical formulas and algorithms that for calculating or approximating voting power in weighted voting games, i.e., • in which voters cast (unequal) blocs of votes. • Various website make these algorithms readily available. • One of the best of these is the website created by Dennis Leech (University of Warwick and another VPP Board member): Computer Algorithms for Voting Power Analysis, http://www.warwick.ac.uk/~ecaae/#Progam_List which was used in making most of the calculations that follow.

11. A Priori State Voting Power in the Electoral College (with Winner-Take-All) • A state’s a priori voting power is • the probability that the state’s block of electoral votes is decisive (i.e., determines the outcome the election), • given that all other states cast their blocs of electoral votes by independently flipping fair coins. • For example (using Leech’s website), the a priori voting power of CA (with 55 EV out of 583) = .475 . • This means if every other state’s vote is determined by a flip of a coin, • 52.5% of the time one or other candidate will have at least 270 electoral votes before CA casts its 55 votes, but • 47.5% of the time CA’s 55 votes will determine the outcome.

12. Chart 2. Share of Voting Power by Share of Electoral Votes

13. Chart 3. Share of Voting Power by Share of Population

14. Individual Voting Power in the Electoral College System • The a priori voting power of an individual voter in the Electoral College system (as it works in practice) is the probability that the individual voter is decisive in his state multiplied by the probability that the bloc of votes cast by the voter’s state is decisive in the Electoral College or equivalently individual voting power in the state multiplied by state voting power in the Electoral College

15. The Banzhaf Effect • (1) Individual voting power within each state is (almost exactly) inversely proportional to the square root to the number of voters in the state. • (2) As shown in Chart 2, state voting power in the Electoral College is approximately proportional to its voting weight (number of electoral votes). • (3) As shown in Chart 1, the voting weight of states in turn is approximately (apart from the small-state apportionment advantage) proportional to population (and number voters). • (4) As shown in Chart 3, putting together (2) and (3), state voting power is approximately proportional to population. • (5) So putting together (1) and (4), individual a priori voting power is approximately proportional to the square root of the number of voters in a state. • However this large-state advantage is counterbalanced in some degree by the small-state apportionment advantage, as shown in the Chart 4.

16. Banzhaf Effect in Bernoulli Elections

17. Individual Voting Power Under the Existing EC • The following Chart 4 shows how a priori individual voting power under the existing Electoral College varies by state population. • It also shows: • mean individual voting power nationwide, and • individual voting power under direct popular vote (calculated in the same manner as individual voting power within a state). • Note that 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. Methodological note: in most of the following charts, 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. The number of voters in each state is assumed to be a constant fraction (.4337) of state population.

18. Individual Voting Power By State Population: Existing Electoral College

19. The Interpretation of a Priori Voting Power • Remember that Chart 4 displays individual a priori voting power in states with different populations, • which takes account of the Electoral College voting rules but nothing else. • A priori, a voter in California has about three times the probability of casting a decisive vote than one in New Hampshire. • But if we take account of recent voting patterns, current 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. • But if California and New Hampshire had equal “battleground” status, the California’s a priori advantage would be reflected in its a posteriori voting power as well.

20. Winner’s Margin by State Size

21. Interpretation of A Priori Voting Power (cont.) • If it is only weakly 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.

22. 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 [“House” electoral votes only] [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]

23. Individual Voting Power by State Population:“House Electoral Votes” Only

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

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

26. Individual Voting Power by State Population:Electoral Votes Apportioned Equally Among States

27. Can Electoral Votes Be Apportioned So As To 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.

28. Individual Voting Power when States Have Equal Population (Versus Apportionment Proportional to Actual Population)

29. 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.

30. 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, individual voting power can be equalized by apportioning electoral votes so that state voting power is proportional to the square root of state population. • But such apportionment is tricky, because what must be made proportional to population is • not electoral votes (which is what we directly apportion) but • state voting power (which is a consequence of the apportionment of electoral votes).

31. (Almost) Equalized Individual Voting Power

32. Electoral Vote Apportionment to Equalize Individual Voting Power (cont.) • Under such square-root apportionment rules, the outcome of the 2004 Presidential election 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.

33. 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]

34. Individual Voting Power under Alternative Rules for Casting Electoral Votes • Calculations for the Pure District Plan, Pure Proportional Plan, and the Whole-Number Proportional Plan are straightforward. • Under the Modified District Plan and the National Bonus Plan, each voter casts a single vote that counts two ways: • within the district (or state) and • “at-large” (i.e., within the state or nation). • Calculating individual voting power in such systems is far from straightforward. • I am in the process of working out approximations based on very large samples of Bernoulli elections.

35. Pure District System

36. Modified District System (Approximate)

37. District System Is “Out of Equilibrium” • Given a district system, any state can gain power by unilaterally switching to winner-take-all. • Madison to Monroe (1800): “All agree that an election by districts would be best if it could be general, but while ten states choose either by their legislatures or by a general ticket [i.e., winner-take-all], it is folly or worse for the other six not to follow.” • Virginia switched from districts to winner-take-all in 1800. • If it had not, the Jeffersonian Republicans would almost certainly lost the 1800 election. • Madison’s strategic advice is powerfully confirmed in terms of individual voting power, • though the voting-power rationale for winner-take-all is logically distinct from the party-advantage rationale.

39. Winner-Take-All Is “In Equilibrium” • In the mid-1990s, the Florida state legislature seriously considered switching to the Modified District Plan. • The effect of such a switch on the individual voting power is shown in the following chart. • However, I assume a switch to the Pure District Plan, because this can be directly calculated. • Considering “mechanical” effects only, if Florida had made the switch, Gore would have been elected President (regardless of the statewide vote in Florida). • Although small states are penalizing by the winner-take-all system, they are further penalized if the unilaterally switch to districts. • So even if a district system is universally agreed to be socially superior (as Madison considered it to be), states will not voluntary choose to move that direction. • States are caught in a Prisoner’s Dilemma.

41. (Pure) Pure Proportional System

42. Whole-Number Proportional Plan Similar calculations and chart were produced, independently and earlier, by Claus Beisbart and Luc Bovens, “A Power Analysis of the Amend-ment 36 in Colorado,” University of Konstanz, May 2005, and Public Choice, March 2008.

43. National Bonus Plan(s)

44. Individual Voting Power: Summary Chart

45. The Probability of Election Reversals • Any districted electoral system can produce an election reversal. • That is, the candidate or party that wins the most popular votes nationwide may fail to win the most “districts” (e.g., parliamentary seats or electoral votes) and thereby lose the election). • Such outcomes are actually more common in some parliamentary systems than in U.S. Presidential elections. • First, let’s examine the probability that a two-tier Bernoulli election (i.e., given the probability model used in voting power calculations) results in an election reversal, i.e., • that a majority of individuals voters vote “heads” but the winner based on “electoral votes” is “tails” or vice versa? • Based on very large-scale (n = 1,000,000) simulations, if the number of equally populated districts/states is modestly large (e.g., k > 20), about 20.5% of such elections produce reversals. Feix, Lepelley, Merlin, and Rouet, “The Probability of Conflicts in a U.S. Presidential Type Election,” Economic Theory, 2004

46. 30,000 Bernoulli elections with 45 districts each with 2223 voters (n = 100,035) In a more inclusive sample of 120,000 such elections, 20.36% were reversals.

47. Probability of Election Reversals (cont.) • If the districts are non-uniform (as in the Electoral College), the probability of an election reversal is evidently slightly greater. • Simulations of 32,000 Bernoulli elections for each of three EC variants:

48. The Election Reversal Problem • The U.S. Electoral College has produced three manifest election reversals (though all were very close), • plus one massive election reversal that is not usually recognized as such. ElectionWinnerRunner-upWinner’s 2-P PV 2000 271 [Bush (R)] 267 [Gore (D)] 49.73% 1888 233 [Harrison (R)] 168 [Cleveland (D)] 49.59% 1876 185 [Hayes (R)] 184 [Tilden (D)] 48.47% • The 1876 election was decided (on inauguration eve) by a Electoral Commission that, by a bare majority and on a straight party line vote, awarded all of 20 disputed electoral votes to Hayes. • Unlike Gore and Cleveland, Tilden won an absolute majority (51%) of the total popular vote.

49. The 1860 Election CandidatePartyPop. Vote %EV Lincoln Republican 39.82 180 Douglas Northern Democrat 29.46 12 Breckinridge Southern Democrat 18.09 72 Bell Constitutional Union 12.61 39 Total Democratic Popular Vote 47.55 Total anti-Lincoln Popular Vote 60.16 • Two inconsequential reversals (between Douglas and Breckinridge and between Douglas and Bell) are manifest. • It may appear that Douglas and Breckinridge were spoilers against each other. • Under a direct popular vote system, this would have been true. • But under the Electoral College system, Douglas and Breckinridge were not spoilers against each other.

50. A Counterfactual 1860 Election • Suppose the Democrats could have held their Northern and Southern wings together and won all the votes captured by each wing separately. • Suppose further that it had been a Democratic vs. Republican straight fight and that the Democrats had also won all the votes that went to Constitutional Union party. • And, for good measure, suppose that the Democrats had won all NJ electoral votes (which for peculiar reasons were actually split between Lincoln and Douglas). • Here is the outcome of the counterfactual 1860 election: PartyPop. Vote %EV Republican 39.82 169 Democratic 60.16 134