Am I mistaken or is there a problem with strategic voting issue in The Wasted Vote Refund method? It seems voters are incentivized to be "loosers" when they predict that their party will most likely win anyway to gain extra votes next time. Could make the method instable.
Making it concrete:
All 75 residents of Abesville like party A and all 25 residents of Boon County like party B. For the first election cycle everyone has one vote. You live in Abesville and are considering strategic voting.
If you vote A, then A wins 75-25, B voters get refunded their wasted votes, A voters can pay 75 votes and need 100 to win the election, so the system credits B voters the extra 25 votes. Each B voter is up two votes, the next election everyone gets another vote, there will be 75 votes in Abesville and 75 votes in Boon County.
If you vote B then A wins 74-26 and the same thing happens, A voters spend all their votes and B voters are each up two votes. You are one of those B voters and take your two votes back to Abesville, intent on voting party A next time. The next election there will be 77 votes in Abesville and 75 votes in Boon County.
It looks like strategic voting can print extra votes for the next election. Seems bad. I'll also note that borrowing is a misleading term for what the system does, as I would take it to imply some kind of conservation of votes principle which is being violated here.
i tried to work it out today. using a naive strategy where the B bloc always votes honestly, and the A bloc votes only honestly enough to eke out a victory, it actually worked out that candidate B won 25% of the time. so i agree that some gamesmanship is possible, but (at least to my cursory look), it seems that it doesn't undermine the outcomes.
(i was just copying numbers in a spreadsheet during some spare time, so i am not very certain of my result here.)
First of all, many thanks to everyone in this thread: Gunnar for raising the issue, Ninety-Three for the concrete example, and kbear for testing whether it made a difference in long term outcomes.
I wrote some Python code to simulate three different situations:
a) Wasted Vote Refund with no strategic voting,
b) Wasted Vote Refund with one member of the majority strategically voting, and
c) Wasted Vote Refund with as many members of the majority strategically voting as possible.
I then ran these simulations for every possibility of a 100 voter community for 1000 elections.
Here are the results: (a) gave the right proportion every time. For (b), the only difference was for a 99 person majority, which won 991 elections instead of 990. For (c), there was no difference up to a majority of 55, a one election difference up to 85, a two election difference up to 90, and so forth, all the way up to the 99 person majority winning all 1000 elections (when they should have only won 99).
I admit this surprised me, because I guess I had been thinking that a two party situation with strategic voting would be akin to a three party situation (with the strategic voters as the additional party), and we know that in case, each of the three parties wins only the expected percentage of the time. But that analogy only really serves as a lower bound, since it ignores the fact that the strategic voters can merge their votes with the non-strategic voters when it is in their interest.
I also ran one additional simulation: (d) Wasted Vote Refund with maximal strategic voting but no "borrowing". Without borrowing, winners of an election will often have to go into "debt" and carry over negative vote totals into the next election(s). The good news is that this eliminates any advantage to strategic voting. The bad news is that negative vote totals come with a bunch of headaches in terms of real world administration and perception. So we are stuck in a pick your poison situation, and honestly, a small 1% advantage to strategic voting might be the more palatable option.
Interesting! but probably not applicable to any human jurisdiction.
1) the population isn't static - both the individuals and their preferences change over time.
2) in the US, at least, we don't vote for parties, we vote for individuals. and THEY change over time as well.
3) representation value is time-dependent. It's FAR more valuable to be in power earlier, as you can (imperfectly) commit the people who replace you.
I apologize if my exposition was unclear: both the random voter and the Wasted Vote Refund are totally applicable to dynamic populations where there are people coming and going and changing their opinions. I only spend a bunch of time talking about the static case because it is easier to analyze (both in terms of saying what properties it should have and in proving them).
"Normal" (non-temporal) proportional representation is typically party based -- each party puts up a list of candidates -- and that's what people vote for. But temporal proportional representation isn't. You vote for your preferred dogcatcher, and if they don't win, you have more votes in the next dogcatcher election, and it's fine if that election has a completely different set of people running.
Re: 3, this could theoretically be solved by paying interest on refunded votes. It's a fascinating question of how to estimate the p such that the average person would be indifferent between receiving 1 vote today versus 1+p votes a year from now ...
I had a different thought in reaction to the title/opening, before I'd read further: proportional representation by splitting the duration of the term in office between candidates in proportion to the number of votes. If the vote for a 4-year office splits 60/40 then put the first candidate in for 2.4 years, and the other candidate for 1.6 years.
But there would be questions to resolve about who gets to go first (probably either rank order by number of votes received, or "largest share of the vote entitles you to choose the order"), and also a sensible minimum required threshold (does receiving 10 votes mean you get to be POTUS for a period of about 8.4 seconds?)
I like the way you think! I actually discuss this version of temporal proportional representation in this older post of mine: https://whirledofideas.blogspot.com/2015/05/auction-powers.html
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 integer 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.