A PRIORI VOTING POWER AND THE U.S. ELECTORAL COLLEGE Nicholas R. Miller UMBC 2010 Annual Meeting of the Public Choice Society Monterey, California March 11-14, 2010

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? With respect to this question, directly contradictory claims are commonly expressed as result of 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.

A Priori Voting Power Felsenthal and Machover instruct us that the Absolute Banzhaf Power Measure should be used to assess voting power in the Electoral College and similar institutions. – A voter’s absolute Banzhaf voting power is the probability that he/she casts a decisive vote in a “random” or “Bernoulli” election. ==> – Therefore we can calculate the overall voting power of an individual in a two-tier voting game as the the probability that the voter cast a decisive vote in the state election times the probability that the state casts a decisive (bloc of) votes in the Electoral College, given a Bernoulli election. Dan S. Felsenthal and Moshé Machover, The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, 1998.

A Bernoulli Election (from The New Yorker, 1937)

Dennis and Robert Leech’s Website: Computer Algorithms for Voting Power Analysis

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

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.

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

Uniform Representation Note that equalizing state populations not only: – equalizes individual voting power across states, but also – raises mean individual voting power, relative to that under apportionment based on the actual unequal populations. While this pattern appears to be typically true, it is not invariably true, – e.g., if state populations are uniformly distributed over a wide range. However, individual voting power still falls below that under direct popular vote. – So the fact that mean individual voting power under the Electoral College falls below that under direct popular vote is not due to the fact that states are unequal in population and electoral votes, and is evidently intrinsic to a two-tier system. Van Kolpin, “Voting Power Under Uniform Representation,” Economics Bulletin, 2003.

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).

( Almost) Equalized Individual Voting Power

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 , Kerry – Whole-Number Apportionment: Bush 307, Kerry 231 – Actual Apportionment: Bush 286, Kerry 252 – Electoral Votes proportional to popular vote: Bush , Kerry Clearly equalizing individual voting power is not the same thing as making the electoral vote (more) proportional to the popular vote.

Alternative Rules for Casting Electoral Votes Apportion electoral votes as at present but use something other than winner-take-all for casting state electoral votes. – Pure 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] – (Pure) Proportional Plan: electoral votes are cast [fractionally] in precise proportion to state popular vote. [Bush , Gore , Nader , Buchanan , Other ; Bush , Kerry ] – 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] – 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]

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 entirely straightforward. Calculations for the Pure Proportional Plan are 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 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 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, 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 ) 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

From Claus Beisbart and Luc Bovens, “A Power Analysis of the Amend-ment 36 in Colorado,” University of Konstanz, May 2005; subsequently published in Public Choice, March 2008.

National Bonus Plan (Bonus = 101)

National Bonus Plan (Varying Bonuses)

Summary: Individual Voting Power Under EC Variants

Mean Voting Power Under Electoral College Variants

Ratio of Maximum to Minimum Voting Power under EC Variants

Inequality (SD/Mean) in Voting Power Under EC Variants