Temporal Proportional Representation

thomascolthurst20 May 2026 1:39 UTC
10 points
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Ezra Klein has a new podcast episode out discussing proportional representation (PR), so I thought I would point out that there is a little discussed variant of PR that doesn’t require redrawing any districts and can even be used for singleton offices like the President.

As the title of this post suggests, the idea is to achieve proportionality over time. In traditional PR, if one party gets 60% of the votes and another party gets 40%, then the first party gets 60% of the available seats and the other party 40%, both rounded to the nearest seat. In temporal PR, if one party consistently gets 60% of the votes and the other party consistently 40%, then the first party wins control of a singleton office 60% of the time and the other party wins control of it 40% of the time.

One very simple temporal PR system is “let a random voter decide the result of each election”. This system has the right long-term average behavior, but can easily produce results which seem intuitively unfair, like when one of two evenly matched parties wins an election four times in a row (as will happen by chance 12.5% of the time). This method is also vulnerable to real or perceived attacks on the method of choosing the random voter …

“The Wasted Vote Refund” is a slightly more complicated temporal PR system, but with provably minimal variance. In this scheme, voters gain a vote in every election, and that vote does not go away unless their candidate wins. The tricky thing is what happens if you do vote for the winner. In that case, the winners must collectively pay N votes, where N is the total number of voters. If the winners do not have N votes—and they often won’t—they must borrow them from the losers, giving the losers even more votes in the next election. Mathematically, if the winning party has M members with M+S votes (where S are the votes they have stored from previous elections), then when M+S < N, after the election everyone in the winning party has 0 votes, and each of the L losers have an additional 1 + (N - (M+S)) /​ L votes as compared to before the election. In the other case, when M+S >= N, the losers have only an additional 1 vote and each winner has ((M+S)-N)/​M.

Here’s a concrete example for people who like concrete examples more than algebra: Party A and Party B both have 50 members who always vote. At the start of the first election, they all receive a vote and cast them, which results in a tie. So we flip a coin and say Party A wins. As winners, they must pay 50+50=100 votes. But they collectively only have 50 votes, so they must borrow 50 from the members of Party B. That brings the post-election one totals to zero votes for each member of Party A and two (=1 unused + 1 borrowed) for each member of Party B.

In election two, we first give everyone a vote, and then Party B wins, with a vote total of 3 * 50 = 150 versus Party A’s 1*50 = 50. Party B can easily pay for 100 votes out of their 150 votes, so after election two, everyone in Party A has 1 votes and everyone in Party B has 1 vote (because 1 = (150 − 100) /​ 50). In election three, we have another tie (100 vs. 100) which we will again have Party A randomly win. After election three, everyone in Party A has 0 votes and everyone in Party B has 2 votes. And we are back to where we were after election one, so we can stop.

[Sidenote: if you are the sort of person who wants to encourage people to vote in every election, the Wasted Vote Refund does incentivize that behavior by handing out extra votes to some voters. But I also take seriously the economics viewpoint that correctly categorizes thinking about who to vote for as a cost, and suggests that we might be better off if everyone voted in just 1/​k of the elections and thought about their decisions for k-times as long. Should this viewpoint ever prevail—perhaps in dath ilan—the Wasted Vote Refund could easily be modified to hand out its extra votes to everyone, whether they voted in a particular election or not.]

Anyway, here is the theorem and proof I implicitly promised you above:

Theorem In a static population with m political parties, each with a constant number s_i of supporters, the Wasted Vote Refund elects each political party i a fraction of the time given by s_i /​ (\sum_j s_j), and no other temporal proportional representation system with that property achieves a lower variance.

Proof The Wasted Vote Refund in this situation is isomorphic to an digital differential analyzer for drawing a line between the origin and the point (s_1, s_2, …) in m-dimensional space on a device where no diagonal moves are allowed (i.e., only one coordinate may be incremented at each step). Standard techniques show that this digitized line differs from the true line by a distance of at most sqrt(m) /​ 2 and that this is minimal.

thomascolthurst20 May 2026 1:39 UTC
10 points
9 comments3 min readLW link