Minimum Votes to Win the Electoral College

With a Republican once again winning the electoral college despite losing the popular vote, I was wondering, what, mathematically, is the fewest number of ballots needed to win the electoral college?

My first approach was to rank the states in order of votes per electoral vote, and assign the states with the fewest ballots per EV to one candidate until that candidate has enough electoral votes to win.

I found an interesting source of data in the United States Elections Project: They have already tallied in one spreadsheet not just the population of each state or the number of registered voters, but the projected actual number of ballots counted, including outstanding provisional / mail-in ballots. Since the goal is to compare electoral votes to popular vote outcomes, the number of votes cast by state seems the most relevant data to answer the original question. So, I used their estimated 2016 November General Election Turnout data, retrieved the morning of November 12th.

I put those in a spreadsheet which contains the number of electoral votes by state, added a calcualted column Ballots per EV = Ballots / Electoral Votes, sorted by Ballots per EV in ascending order, and marked the first 32 states as belonging to the electoral vote winner.

In the winner's states (marked as Republican [red] in the map below), I assigned the winner 50.01% of the vote, and the loser the remainder. In the rest of the states, I assigned the winner 0% of the vote, and the loser 100% of the vote. This of course is an impossible outcome in real life, but this is math, not real life!

This resulted in 30 million votes going to the electoral college winner out of 133 million votes cast: The winner won with only 22.5% of the votes.

Map of states assigned to winner by Knapsack method

Maps generated at 270 To Win.

Given that the intent is to produce the most electoral votes from the fewest total votes, I found it interesting that this scenario included in the winner's column many larger states that are typically thought of as underrepresented, including 3 of the 5 most populous states: California, Texas, and New York. But that intuition is based on a measure of population per electoral vote. The many factors that prevent residents from becoming voters - citizenship status, age, and voter enthusiasm to bother voting - reduce the total number of ballots cast in these states by more than most, and moved the voters in these states to in fact be slightly over-represented in the critical metric of ballots per electoral vote. I imagine enthusiasm is significantly driven by partisan makeup of the state. Florida, a swing state, gives voters far more reason to believe their vote might tip the national election, and has far more political operatives dragging voters to the polls.

But I also noticed that this solution produced a winner wih 284 electoral votes - a non-trivial amount more than the fewest electoral votes required to win. That New York was the tipping point state demonstrated that it might be necessary to un-select one state before adding the next state. I wanted to know if we could do better.

This seemed like a textbook case where the optimal solution needs to be found from among all possible ways to reach the current state, not just the most obvious one. So I whipped up a knapsack-style dynamic programming solution. For each number of electoral votes, it computers from each fewer number of electoral votes (including zero) which state, if any, can most efficiently reach the curent number of electoral votes. That is:

For EV_count in 1 to 538
    Mark ev_count as having no solution
    For subproblem in 0 to EV_count - 1 
        ev_gap  = EV_count - subproblem
        possible_states = states with ev_gap votes not in solution(subproblem)
        best_state = state in possible_states with fewest ballots per electoral vote
        If ballots(solution(subproblem) + best_state) < ballots(solution(ev_count))
            let solution(ev_count) = solution(subproblem) + best_state

Again assigning 50.01% of the winner's states' votes to the winner, and 100% of the loser's states' vote to the loser, the solution this found required about 2.5 million votes fewer to win: 27.5 million out of 133 million cast, or 20.7% of the total ballots cast. It did this by substituting the 22 electoral votes of Arizona and Indiana with Louisiana's 8 electoral votes, creating a solution with exactly 270 electoral votes. While an impossible scenario in any political reality (let alone the one we exist in), it is impressive just how unrepresentative of public preference the electoral college could get.

Map of states assigned to winner by Knapsack method

Map generated at 270 To Win.

Is it inevitable that the best winning solution would have exactly 270 votes? Perhaps not. I also printed out the minimum ballots to reach every other electoral vote total, and there were in fact three non-trivial such totals that required fewer ballots than the optimal solution for one fewer electoral vote (at 489, 498, and 508 electoral votes).

For this exercise, I broke out the states who assign electoral votes based on congressional district into separate "states" for the at-large and congressional-district based electoral votes by dividing the at-large ballots in half and assigning the other half of those ballots to the congressional-district based 'states' proportionally to the number of votes cast in those districts. I did not attempt to prevent any 'silly' cases such as an at-large winner who did not win any congressional districts, but fortunately the optimal solution did not require me to do so.

A list of the states assigned to the winner:

  • Alabama
  • Alaska
  • Arkansas
  • California
  • Connecticut
  • Delaware
  • District of Columbia
  • Hawaii
  • Idaho
  • Kansas
  • Kentucky
  • Louisiana
  • Maine (At-Large and Congressional District-based)
  • Mississippi
  • Montana
  • Nebraska (At-Large and Congressional District-based)
  • Nevada
  • New Hampshire
  • New Mexico
  • New York
  • North Dakota
  • Oklahoma
  • Rhode Island
  • South Carolina
  • South Dakota
  • Tennessee
  • Texas
  • Utah
  • Vermont
  • West Virginia
  • Wyoming

Resources:

  • evpack.pl, a knapsack implementation of the problem in Perl. No, it's not especially elegant code.
  • states.txt, state data I used, in tab-delimited format
  • MinVote.numbers, a spreadsheet used for the first pass, with vote assignments modified for the results of the second pass.

But it's not as bad as it looks!

A little bit of bonus content for those who dare to click the Comments link.

Interestingly, if you eliminate the supposition that the winner gets 0% of the vote in any state they lose, and instead allocate 49.99% of the vote to them (the same percentage that the election loser won in the states the election winner wins), even with this optimized electoral vote allocation, the winner needs 49.6% of the popular vote - a mere 4/10ths of a percent short of a majority.

So, that does demonstrate that the electoral college's purpose — to reward broad support over deep support — continues to be achieved centuries later. And any commentary on who is selected is really commentary about the people who cast their ballots, not the system in which our elections operate.

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