The genus, Plurality

The following started as a further comment to propagate some ideas already planted beneath the post on Palestinian exit polls. It seemed appropriate to give it a new planting of its own. It is specifically in response to the point raised by Kao Hsien Chih (welcome back to the orchard, by the way!) regarding coordination failure. And a tip of the hat to Wilfred for inspiring the idea of plurality as a genus. Why hadn’t I though of that?

[Correction, Jan. 29. The definition below of the limited vote has been fixed. Initially, Word Press took my attempt at defining mathematically “1 is less than w” as an html tag and failed to render a whole block of rather important text!]

On coordination failure, all forms of nominal plurality voting are vulnerable to it one way or another. (When I say ‘nominal’ I mean a vote given to candidates by name, and in this context mean that votes are counted only for candidates, and not pooled onto a party list. In the case of Palestine this is true for 66 district seats. There are also party-list PR seats, but they are allocated completeley separately, or in ‘parallel’ and based on a separate list vote.)

Of all the plurality species, the single nontransferable vote (SNTV) is the most prone to coordination failure, because it is not clear to a party how many candidates to run, optimally, and if it rus more than one, then the party has to find a way to spread its voters out across the candidates to ensure that they don’t pile up a lot of votes on just one. Under multi-seat plurality or block vote* (as in Palestine’s nominal tier), the nomination strategy is clear: run as many as there are seats in a district. The problem then is to ensure that your voters vote the full slate. If they do not, then you have a problem.

The “genus” plurality (which forms its own orchard block here at F&V) includes FPTP, multi-seat plurality (a.k.a. block vote), SNTV (like Taiwan’s nominal tier through 2004, and Japan formerly), and other systems.

What this genus has as its identifying trait is that all its species establish that the winners are the M candidates with the M highest individual vote shares, where M is the district magnitude (the number of seats elected in the district). Where the sepcies vary is in two things: First, of course, M. Secondly, in whether the voter has M nominal votes or less than M. Let w be the number of votes per voter.** Then the systems are:

    FPTP, M=1, w=M=1
    SNTV, M>1, w=1
    Block vote or multi-seat plurality, M>1, w=M
    Limited vote, M>1, 1< w < M (SNTV is actually a special case of limited vote, so perhaps we should say that limited vote is the species and SNTV a variety)

If some voters do not use all their votes in multi-seat plurality, then they are treating the system as though it were limited vote. This may have happened to Fatah. It certainly happened to nearly all parties in the Liberian 2-seat-plurality senate races.

If we also allow w>1 and the voter to cast more than one vote per candidate, then we have cumulative vote also as part of the same family. (However, rationally the voter ought to cumulate all on one candidate, in which case it becomes SNTV.) And if we allow w=c, where c is the number of candidates, then approval vote also is part of the family.

All of these systems provide both parties and voters with dilemmas about how to coordinate their support across multiple candidates in ways to ensure that the desired candidates and not some others are elected. In the case of FPTP, of course, the problem is not usually that a party has more than one candidate in a district; it is that multiple parties lead to coordination problems and may result in the candidate of a majority-disapproved party winning.

* Actually, the term, multi-seat plurality, is not precise, and block vote, while well established, is even less so. Consider the typical rule used in US states for the electoral college: the candidate with the plurality of votes wins the full slate of electors. That’s multi-seat plurality, and given that the votes go as a block (i.e. the voter isn’t voting for individual electors and thus cannot partially abstain or split her ticket), it could be called block vote. I prefer to call systems on the order of the US electoral college list plurality, because the list with the plurality is elected as a whole. For the type of system used in Palestine’s district races, I would prefer multiple nontransferable vote (MNTV), because that is far more descriptive of a system in which the voter has more than one vote which need not be cast as a block, and each is strictly nominal and cannot be transferred or pooled to other candidates. MNTV would be identifable as a sub-class of both the broader category of multi-seat plurality (which includes list plurality) and the broader category of nominal plurality (which includes FPTP, SNTV, etc.).

**w rather than v, as we like to use v to refer to the total number of votes cast in an election, rather than the number cast by an individual voter.

0 thoughts on “The genus, Plurality

  1. I notice people talking like there’s a problem here. Which part is the problem?

    It looks to me like MNTV should be pretty good. For each race, you vote for the candidate you like best. Then the candidate with the most votes wins. What’s unfair about that? Of course I’d prefer something like IRV so that there’s at least a chance that the winner gets a majority. But if you only get to vote for the candidate you prefer, what’s wrong with the most votes getting the win?

    What outcome should they be aiming for, if not that the candidates who get the most votes should win?

  2. To J. Thomas: There is nothing wrong with MNTV. That is, if you care not at all about the party affiliations of the candidates or the proportionality of the representation of groups in the electorate (whether parties or any other group affiliation or identity) within the elected delegation.

    But if you care at all such things, then MNTV (block vote) is one of the worst of all electoral systems. I refer you to the threads on Palestine and Liberia for a further discussion.

  3. Professor Shugart, maybe I’ve misunderstood your terminology. I thought what is meant by MNTV is that you have a voting district with multiple representatives, and they each get elected independently. If there are 6 representatives then each voter gets 6 votes, one for each race. They each vote for the particular candidate in that race which they like the best, and in each individual race the candidate with the most votes wins.

    [MSS: Allow me to insert my own clarification here within J.’s comment. Under what is conventionally called “block vote” (as I noted, it is not a very good name for it) and I am calling MNTV, there are not individual races. There are M candidates elected, voter may cast up to M votes, and the winners are the candiadtes with the M highest vote totals.]

    This does not look like block vote to me at all. I don’t see that there’s anything in particular wrong with it, but it would be better with IRV. The way it is you will almost always get a winner that the majority of voters voted against. But that’s true for most alternatives too.

    As for worse systems when you care about parties, the obvious worst to me is one where you get one vote for the party you like most, and the party that gets the most votes gets all the representatives.

    It looks to me like what you dislike about this one is that Hamas got too many wins. And the reason they got so many wins is that more people voted for all-Hamas representatives than voted for all-Fatah representatives. I don’ t see anything wrong with that. If the candidates you happen to like best are all in one party, that’s what you like. If somebody else likes the best candidates from one party and the best candidates from another party, that’s fine too. He’s likely to get the winners he wants — if enough people agree with him that those are the best people, then those candidates win.

    [MSS again: What I see wrong with it is that Hamas was the choice of well under a majority of voters and yet won a huge majority. In fact, Hamas won almost 70% of the seats elected by MNTV. If there is any point in using MNTV, it would be in nonpartisan races. Even for those there are many better alternatives. But for partisan races, where voters might be expected to give, on average, some number approximating M of their votes to candidates of the same party, then the result will be mostly clean sweeps of the districts by one party.]

    Hmm. Maybe I misread your complaint. Is your problem that MNTV doesn’t give political parties as much control as they want? That it results in popular candidates winning and not giving enough support to unpopular candidates from the same party? In that case I have to agree, it doesn’t make parties as strong as they could be. For that, some sort of winner-takes-all approach would be best. Maybe, let each party run the single candidate they think will be the most popular, and then if he wins he gets a ceremonial position where his popularity counts, and mostly-anonymous party members do all the legislation etc.

    Pleas pardon me if I have misunderstood you.

  4. Thank you! So it wasn’t individual races. In an area that had 6 winners, each voter would get six votes. And the six most-popular candidates would win.

    And your problem with this was that the winners didn’t get votes from the most people, but merely got the most votes.


    Suppose it was separate races? Would you expect the result to be any different? If Hamas candidates had the plurality, wouldn’t they win? To my mind the solution to that is something like IRV. Rank all the candidates you find acceptable, and the guy with the plurality might not win — a majority might prefer somebody else over him.

    How else could this be solved? They could divide up the voting districts to be smaller, so everybody gets one vote for one candidate. Then Hamas still wins in each district where it gets a plurality, and the way to reduce Hamas’s share would be to gerrymander. This doesn’t look like an improvement to me.

    Well, if party is the only important thing you could vote once for the party of your choice, and the various parties would get a number of representatives proportional to the votes. Then you’d know your vote counted for something — because of you and a few other people, your party got 38 representatives instead of 37. You made that difference. But when you voted you wouldn’t know which member of the list you were voting for! It would make party pretty much the only important thing. Absurd.

    When the guy with the plurality wins, the natural response is to look at the guy you think has the plurality in your district and if you don’t like him, vote for the runner-up. This is the method used in the USA, where voting for any third party is considered throwing your vote away.

    I really don’t see anything wrong with the most popular candidates winning. Hamas candidates happened to be the most popular. The problem is that there was no accounting for second-choices. Possibly if people had given their second and third choices etc that someone else would have won more of the elections. Hamas candidates were the most popular but perhaps a majority of voters would have preferred somebody else, given sufficient choices. We’ll never know since the experiment was not done.

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