A vot­ing the­ory primer for rationalists

What is vot­ing the­ory?

Vot­ing the­ory, also called so­cial choice the­ory, is the study of the design and evau­la­tion of demo­cratic vot­ing meth­ods (that’s the act­iv­ists’ word; game the­or­ists call them “vot­ing mech­an­isms”, en­gin­eers call them “elect­oral al­gorithms”, and polit­ical sci­ent­ists say “elect­oral for­mu­las”). In other words, for a given list of can­did­ates and voters, a vot­ing method spe­cifies a set of valid ways to fill out a bal­lot, and, given a valid bal­lot from each voter, pro­duces an out­come.

(An “elect­oral sys­tem” in­cludes a vot­ing method, but also other im­ple­ment­a­tion de­tails, such as how the can­did­ates and voters are val­id­ated, how of­ten elec­tions hap­pen and for what of­fices, etc. “Vot­ing sys­tem” is an am­bigu­ous term that can refer to a full elect­oral sys­tem, just to the vot­ing method, or even to the ma­chinery for count­ing votes.)

Most vot­ing the­ory lim­its it­self to study­ing “demo­cratic” vot­ing meth­ods. That typ­ic­ally has both em­pir­ical and norm­at­ive im­plic­a­tions. Em­pir­ic­ally, “demo­cratic” means:

  • There are many voters

  • There can be more than two candidates

In or­der to be con­sidered “demo­cratic”, vot­ing meth­ods gen­er­ally should meet vari­ous norm­at­ive cri­teria as well. There are many pos­sible such cri­teria, and on many of them the­or­ists do not agree; but in gen­eral they do agree on this min­imal set:

  • Anonym­ity; per­mut­ing the bal­lots does not change the prob­ab­il­ity of any elec­tion out­come.

  • Neut­ral­ity; per­mut­ing the can­did­ates on all bal­lots does not change the prob­ab­il­ity of any elec­tion out­come.

  • Un­an­im­ity: If voters uni­ver­sally vote a pref­er­ence for a given out­come over all oth­ers, that out­come is se­lec­ted. (This is a weak cri­terion, and is im­plied by many other stronger ones; but those stronger ones are of­ten dis­puted, while this one rarely is.)

  • Meth­ods typ­ic­ally do not dir­ectly in­volve money chan­ging hands or other en­dur­ing state-changes for in­di­vidual voters. (There can be ex­cep­tions to this, but there are good reas­ons to want to un­der­stand “money­less” elec­tions.)

Why is vot­ing the­ory im­port­ant for ra­tion­al­ists?

First off, be­cause demo­cratic pro­cesses in the real world are im­port­ant loci of power. That means that it’s use­ful to un­der­stand the dy­nam­ics of the vot­ing meth­ods used in such real-world elec­tions.

Se­cond, be­cause these real-world demo­cratic pro­cesses have all been cre­ated and/​or evolved in the past, and so there are likely to be op­por­tun­it­ies to re­place, re­form, or add to them in the fu­ture. If you want to make polit­ical change of any kind over a me­dium-to-long time ho­ri­zon, these sys­temic re­forms should prob­ably be part of your agenda. The fact is that FPTP, the vot­ing method we use in most of the Eng­lish-speak­ing world, is ab­so­lutely hor­rible, and there is reason to be­lieve that re­form­ing it would sub­stan­tially (though not of course com­pletely) al­le­vi­ate much polit­ical dys­func­tion and suf­fer­ing.

Third, be­cause un­der­stand­ing so­cial choice the­ory helps cla­rify ideas about how it’s pos­sible and/​or de­sir­able to re­solve value dis­putes between mul­tiple agents. For in­stance, if you be­lieve that su­per­in­tel­li­gences should per­form a “val­ues hand­shake” when meet­ing, re­pla­cing each of their in­di­vidual value func­tions by some com­mon one so as to avoid the dead weight loss of a con­flict, then so­cial choice the­ory sug­gests both ques­tions and an­swers about what that might look like. (Note that the eth­ical and prac­tical im­port­ance of such con­sid­er­a­tions is not at all lim­ited to “post-sin­gu­lar­ity” ex­amples like that one.)

In fact, on that third point: my own ideas of eth­ics and of fun the­ory are deeply in­formed by my dec­ades of in­terest in vot­ing the­ory. To sim­plify into a few words my com­plex thoughts on this, I be­lieve that vot­ing the­ory elu­cid­ates “eth­ical in­com­plete­ness” (that is, that it’s pos­sible to put world-states into eth­ical pref­er­ence or­der par­tially but not fully) and that this in­com­plete­ness is a good thing be­cause it leaves room for fun even in an eth­ic­ally un­sur­passed world.

What are the branches of vot­ing the­ory?

Gen­er­ally, you can di­vide vot­ing meth­ods up into “single-win­ner” and “multi-win­ner”. Single-win­ner meth­ods are use­ful for elect­ing of­fices like pres­id­ent, gov­ernor, and mayor. Multi-win­ner meth­ods are use­ful for di­vid­ing up some fi­nite, but to some ex­tent di­vis­ible, re­source, such as vot­ing power in a le­gis­lature, between vari­ous op­tions. Multi-win­ner meth­ods can be fur­ther sub­divided into seat-based (where a set of sim­ilar “seats” are as­signed one win­ner each) or weighted (where each can­did­ate can be given a dif­fer­ent frac­tion of the vot­ing power).

What are the ba­sics of single-win­ner vot­ing the­ory?

(Note: Some read­ers may wish to skip to the sum­mary be­low, or to read the later sec­tion on multi-win­ner the­ory and pro­por­tional rep­res­ent­a­tion first. Either is valid.)

Some of the earli­est known work in vot­ing the­ory was by Ra­mon Llull be­fore his death in 1315, but most of that was lost un­til re­cently. Per­haps a bet­ter place to start would be in the French Academy in the late 1700s; this al­lows us to couch it as a de­bate (Amer­ican Chop­per meme?) between Jean-Charles de Borda and Nic­olas de Con­dorcet.

Con­dorcet: “Plur­al­ity (or ‘FPTP’, for First Past the Post) elec­tions, where each voter votes for just one can­did­ate and the can­did­ate with the most votes wins, are of­ten spoiled by vote-split­ting.”
Borda: “Bet­ter to have voters rank can­did­ates, give can­did­ates points for fa­vor­able rank­ings, and choose a win­ner based on points.” (Borda Count)
Con­dorcet: “Rank­ing can­did­ates, rather than vot­ing for just one, is good. But your point sys­tem is sub­ject to strategy. Every­one will rate some can­did­ate they be­lieve can’t win in second place, to avoid giv­ing points to a ser­i­ous rival to their fa­vor­ite. So some­body could win pre­cisely be­cause nobody takes them ser­i­ously!”
Borda: “My method is made for hon­est men!”
Con­dorcet: “In­stead, you should use the rank­ings to see who would have a ma­jor­ity in every pos­sible pair­wise con­test. If some­body wins all such con­tests, ob­vi­ously they should be the over­all win­ner.”

In my view, Borda was the clear loser there. And most vot­ing the­or­ists today agree with me. The one ex­cep­tion is the math­em­atician Don­ald Saari, en­am­ored with the math­em­at­ical sym­metry of the Borda count. This is totally worth men­tion­ing be­cause his last name is a great source of puns.

But Con­dorcet soon real­ized there was a prob­lem with his pro­posal too: it’s pos­sible for A to beat B pair­wise, and B to beat C, while C still beats A. That is, pair­wise vic­tor­ies can be cyc­lical, not trans­it­ive. Nat­ur­ally speak­ing, this is rare; but if there’s a de­cision between A and B, the voters who fa­vor B might have the power to ar­ti­fi­cially cre­ate a “poison pill” amend­ment C which can beat A and then lose to B.

How would a Con­dorcet cycle oc­cur? Ima­gine the fol­low­ing elec­tion:

1: A>B>C

1: B>C>A

1: C>A>B

(This nota­tion means that there’s 1 voter of each of three types, and that the first voter prefers A over B over C.) In this elec­tion, A beats B by 2 to 1, and sim­il­arly B beats C and C beats A.

Fast-for­ward to 1950, when the­or­ists at the RAND cor­por­a­tion were in­vent­ing game the­ory in or­der to reason about the pos­sib­il­ity of nuc­lear war. One such sci­ent­ist, Ken­neth Ar­row, proved that the prob­lem that Con­dorcet (and Llull) had seen was in fact a fun­da­mental is­sue with any ranked vot­ing method. He posed 3 ba­sic “fair­ness cri­teria” and showed that no ranked method can meet all of them:

  • Ranked un­an­im­ity: if every voter prefers X to Y, then the out­come has X above Y.

  • Independ­ence of ir­rel­ev­ant al­tern­at­ives: If every voter’s pref­er­ences between some sub­set of can­did­ates re­main the same, the or­der of those can­did­ates in the out­come will re­main the same, even if other can­did­ates out­side the set are ad­ded, dropped, or changed.

  • Non-dic­tat­orial: the out­come de­pends on more than one bal­lot.

Ar­row’s res­ult was im­port­ant in and of it­self; in­tu­it­ively, most people might have guessed that a ranked vot­ing method could be fair in all those ways. But even more im­port­ant than the spe­cific res­ult was the idea of an im­possib­il­ity proof for vot­ing.

Using this idea, it wasn’t long un­til Gib­bard and Sat­ter­th­waite in­de­pend­ently came up with a fol­low-up the­orem, show­ing that no vot­ing sys­tem (ranked or oth­er­wise) could pos­sibly avoid cre­at­ing stra­tegic in­cent­ives for some voters in some situ­ations. That is to say, there is no non-dic­tat­orial vot­ing sys­tem for more than two pos­sible out­comes and more than two voters in which every voter has a single “hon­est” bal­lot that de­pends only on their own feel­ings about the can­did­ates, such that they can’t some­times get a bet­ter res­ult by cast­ing a bal­lot that isn’t their “hon­est” one.

There’s an­other way that Ar­row’s the­orem was an im­port­ant found­a­tion, par­tic­u­larly for ra­tion­al­ists. He was ex­pli­citly think­ing about vot­ing meth­ods not just as real-world ways of elect­ing politi­cians, but as the­or­et­ical pos­sib­il­it­ies for re­con­cil­ing val­ues. In this more philo­soph­ical sense, Ar­row’s the­orem says some­thing de­press­ing about mor­al­ity: if mor­al­ity is to be based on (po­ten­tially re­vealed) pref­er­ences rather than in­ter­per­sonal com­par­ison of (sub­ject­ive) util­it­ies, it can­not simply be a demo­cratic mat­ter; “the greatest good for the greatest num­ber” doesn’t work without in­her­ently-sub­ject­ive com­par­is­ons of good­ness. Amartya Sen con­tin­ued ex­plor­ing the philo­soph­ical im­plic­a­tions of vot­ing the­ory, show­ing that the idea of “private autonomy” is in­com­pat­ible with Pareto ef­fi­ciency.

Now, in dis­cuss­ing Ar­row’s the­orem, I’ve said sev­eral times that it only ap­plies to “ranked” vot­ing sys­tems. What does that mean? “Ranked” (also some­times termed “or­dinal” or “pref­er­en­tial”) sys­tems are those where valid bal­lots con­sist of noth­ing be­sides a trans­it­ive pref­er­en­tial or­der­ing of the can­did­ates (par­tial or com­plete). That is, you can say that you prefer A over B or B over A (or in some cases, that you like both of them equally), but you can­not say how strong each pref­er­ence is, or provide other in­form­a­tion that’s used to choose a win­ner. In Ar­row’s view, the vot­ing method is then re­spons­ible for or­der­ing the can­did­ates, pick­ing not just a win­ner but a second place etc. Since neut­ral­ity wasn’t one of Ar­row’s cri­teria, ties can be broken ar­bit­rar­ily.

This ex­cludes an im­port­ant class of vot­ing meth­ods from con­sid­er­a­tion: those I’d call rated (or graded or eval­u­ational), where you as a voter can give in­form­a­tion about strength of pref­er­ence. Ar­row con­sciously ex­cluded those meth­ods be­cause he be­lieved (as Gib­bard and Sat­ter­th­waite later con­firmed) that they’d in­ev­it­ably be sub­ject to stra­tegic vot­ing. But since ranked vot­ing sys­tems are also in­ev­it­ably sub­ject to strategy, that isn’t ne­ces­sar­ily a good reason. In any case, Ar­row’s choice to ig­nore such sys­tems set a trend; it wasn’t un­til ap­proval vot­ing was re­in­ven­ted around 1980 and score vot­ing around 2000 that rated meth­ods came into their own. Per­son­ally, for reas­ons I’ll ex­plain fur­ther be­low, I tend to prefer rated sys­tems over purely ranked ones, so I think that Ar­row’s ini­tial neg­lect of ranked meth­ods got the field off on a bit of a wrong foot.

And there’s an­other way I feel that Ar­row set us off in the wrong dir­ec­tion. His idea of reas­on­ing ax­io­mat­ic­ally about vot­ing meth­ods was bril­liant, but ul­ti­mately, I think the field has been too fo­cused on this ax­io­matic “Ar­rovian” paradigm, where the en­tire goal is to prove cer­tain cri­teria can be met by some spe­cific vot­ing method, or can­not be met by any method. Since it’s im­possible to meet all de­sir­able cri­teria in all cases, I’d rather look at things in a more prob­ab­il­istic and quant­it­at­ive way: how of­ten and how badly does a given sys­tem fail de­sir­able cri­teria.

The per­son I con­sider to be the founder of this lat­ter, “stat­ist­ical” paradigm for eval­u­at­ing vot­ing meth­ods is War­ren Smith. Now, where Ken­neth Ar­row won the No­bel Pr­ize, War­ren Smith has to my know­ledge never man­aged to pub­lish a pa­per in a peer-re­viewed journal. He’s a smart and cre­at­ive math­em­atician, but… let’s just say, not ex­em­plary for his so­cial graces. In par­tic­u­lar, he’s not re­luct­ant to opine in var­ied fields of polit­ics where he lacks ob­vi­ous cre­den­tials. So there’s plenty in the aca­demic world who’d just dis­miss him as a crack­pot, if they are even aware of his ex­ist­ence. This is un­for­tu­nate, be­cause his work on vot­ing the­ory is ground­break­ing.

In his 2000 pa­per on “Range Vot­ing” (what we’d now call Score Vot­ing), he per­formed sys­tem­atic util­it­arian Monte-Carlo eval­u­ation of a wide range of vot­ing sys­tems un­der a wide range of as­sump­tions about how voters vote. In other words, in each of his sim­u­la­tions, he as­sumed cer­tain num­bers of can­did­ates and of voters, as well as a stat­ist­ical model for voter util­it­ies and a strategy model for voters. Using the stat­ist­ical model, he as­signed each vir­tual voter a util­ity for each can­did­ate; us­ing the strategy model, he turned those util­it­ies into a bal­lot in each vot­ing method; and then he meas­ured the total util­ity of the win­ning can­did­ate, as com­pared to that of the highest-total-util­ity can­did­ate in the race. Nowadays the name for the dif­fer­ence between these num­bers, scaled so that the lat­ter would be 100% and the av­er­age ran­domly-se­lec­ted can­did­ate would be 0%, is “Voter Sat­is­fac­tion Ef­fi­ciency” (VSE).

Smith wasn’t the first to do some­thing like this. But he was cer­tainly the first to do it so sys­tem­at­ic­ally, across vari­ous vot­ing meth­ods, util­ity mod­els, and stra­tegic mod­els. Be­cause he did such a sens­it­iv­ity ana­lysis across util­ity and stra­tegic mod­els, he was able to see which vot­ing meth­ods con­sist­ently out­per­formed oth­ers, al­most re­gard­less of the spe­cif­ics of the mod­els he used. In par­tic­u­lar, score vot­ing, in which each voter gives each can­did­ate a nu­mer­ical score from a cer­tain range (say, 0 to 100) and the highest total score wins, was al­most al­ways on top, while FPTP was al­most al­ways near the bot­tom.

More re­cently, I’ve done fur­ther work on VSE, us­ing more-real­istic voter and strategy mod­els than what Smith had, and adding a vari­ety of “me­dia” mod­els to al­low vary­ing the in­form­a­tion on which the vir­tual voters base their strategiz­ing. While this work con­firmed many of Smith’s res­ults — for in­stance, I still con­sist­ently find that FPTP is lower than IRV is lower than ap­proval is lower than score — it has un­seated score vot­ing as the un­dis­puted highest-VSE method. Other meth­ods with bet­ter strategy res­ist­ance can end up do­ing bet­ter than score.

Of course, some­thing else happened in the year 2000 that was im­port­ant to the field of single-win­ner vot­ing the­ory: the Bush-Gore elec­tion, in which Bush won the state of Flor­ida and thus the pres­id­ency of the USA by a mi­cro­scopic mar­gin of about 500 votes. Along with the many “elect­oral sys­tem” ir­reg­u­lar­it­ies in the Flor­ida elec­tion (a mass purge of the voter rolls of those with the same name as known felons, a con­fus­ing bal­lot design in Palm Beach, an­ti­quated punch-card bal­lots with dif­fi­cult-to-in­ter­pret “hanging chads”, etc.) was one im­port­ant “vot­ing method” ir­reg­u­lar­ity: the fact that Ralph Nader, a can­did­ate whom most con­sidered to be ideo­lo­gic­ally closer to Gore than to Bush, got far more votes than the mar­gin between the two, lead­ing many to ar­gue that un­der al­most any al­tern­at­ive vot­ing method, Gore would have won. This, un­der­stand­ably, in­creased many people’s in­terest in vot­ing the­ory and vot­ing re­form. Like Smith, many other am­a­teurs began to make worth­while pro­gress in vari­ous ways, pro­gress which was of­ten not well covered in the aca­demic lit­er­at­ure.

In the years since, sub­stan­tial pro­gress has been made. But we act­iv­ists for vot­ing re­form still haven’t man­aged to use our com­mon hatred for FPTP to unite be­hind a com­mon pro­posal. (The irony that our ex­pert­ise in meth­ods for re­con­cil­ing dif­fer­ent pri­or­it­ies into a com­mon pur­pose hasn’t let us do so in our own field is not lost on us.)

In my opin­ion, aside from the util­it­arian per­spect­ive offered by VSE, the key to eval­u­at­ing vot­ing meth­ods is an un­der­stand­ing of stra­tegic vot­ing; this is what I’d call the “mech­an­ism design” per­spect­ive. I’d say that there are 5 com­mon “anti-pat­terns” that vot­ing meth­ods can fall into; either where vot­ing strategy can lead to patho­lo­gical res­ults, or vice versa. I’d pose them as a series of 5 in­creas­ingly-dif­fi­cult hurdles for a vot­ing method to pass. Be­cause the earlier hurdles deal with situ­ations that are more com­mon or more ser­i­ous, I’d say that if a method trips on an earlier hurdle, it doesn’t much mat­ter that it could have passed a later hurdle. Here they are:

(0. Dark Horse. As in Con­dorcet’s take­down of Borda above, this is where a can­did­ate wins pre­cisely be­cause nobody ex­pects them to. Very bad, but not a ser­i­ous prob­lem in most vot­ing meth­ods, ex­cept for the Borda Count.)
1. Vote-split­ting /​ “spoiled” elec­tions. Ad­ding a minor can­did­ate causes a sim­ilar ma­jor can­did­ate to lose. Very bad be­cause it leads to rampant stra­tegic dis­hon­esty and in ex­treme cases 2-party dom­in­ance, as in Duver­ger’s Law. Prob­lem­atic in FPTP, re­solved by most other vot­ing meth­ods.
2. Center squeeze. A cent­rist can­did­ate is elim­in­ated be­cause they have lost first-choice sup­port to rivals on both sides, so that one of the rivals wins, even though the cent­rist could have beaten either one of them in a one-on-one (pair­wise) elec­tion. Though the dir­ect con­sequences of this patho­logy are much less severe than those of vote-split­ting, the in­dir­ect con­sequences of voters strategiz­ing to avoid the prob­lem would be ex­actly the same: self-per­petu­at­ing two-party dom­in­ance. This prob­lem is re­lated to fail­ures of the “fa­vor­ite be­trayal cri­terion” (FBC). Prob­lem­atic in IRV, re­solved by most other meth­ods.
3. Chicken di­lemma (aka Burr di­lemma, for Hamilton fans). Two sim­ilar can­did­ates must com­bine strength in or­der to beat a third rival. But whichever of the two co­oper­ates less will be the win­ner, lead­ing to a game of “chicken” where both can end up los­ing to the rival. This prob­lem is re­lated to fail­ures of the “later-no-harm” (LNH) cri­terion. Be­cause LNH is in­com­pat­ible with FBC, it is im­possible to com­pletely avoid the chicken di­lemma without cre­at­ing a cen­ter squeeze vul­ner­ab­il­ity, but sys­tems like STAR vot­ing or 3-2-1 min­im­ize it.
4. Con­dorcet cycle. As above, a situ­ation where, with hon­est votes, A beats B beats C beats A. There is no “cor­rect” win­ner in this case, and so no vot­ing method can really do any­thing to avoid get­ting a “wrong” win­ner. Luck­ily, in nat­ural elec­tions (that is, where bad act­ors are not able to cre­ate ar­ti­fi­cial Con­dorcet cycles by stra­tegic­ally en­gin­eer­ing “poison pills”), this prob­ably hap­pens less than 5% of the time.

Note that there’s a gen­eral pat­tern in the patho­lo­gies above: the out­come of hon­est vot­ing and that of stra­tegic vot­ing are in some sense po­lar op­pos­ites. For in­stance, un­der hon­est vot­ing, vote-split­ting destabil­izes ma­jor parties; but un­der stra­tegic vot­ing, it makes their status un­as­sail­able. This is a com­mon oc­cur­rence in vot­ing the­ory. And it’s a reason that na­ive at­tempts to “fix” a prob­lem in a vot­ing sys­tem by adding rules can ac­tu­ally make the ori­ginal prob­lem worse.

(I wrote a sep­ar­ate art­icle with fur­ther dis­cus­sion of these patho­lo­gies)

Here are a few of the vari­ous single-win­ner vot­ing sys­tems people fa­vor, and a few (biased) words about the groups that fa­vor them:

FPTP (aka plur­al­ity vot­ing, or choose-one single-win­ner): Univer­sally re­viled by vot­ing the­or­ists, this is still favored by vari­ous groups who like the status quo in coun­tries like the US, Canada, and the UK. In par­tic­u­lar, in­cum­bent politi­cians and lob­by­ists tend to be at best skep­tical and at worst out­right re­ac­tion­ary in re­sponse to re­formers.

IRV (In­stant run­off vot­ing), aka Al­tern­at­ive Vote or RCV (Ranked Choice Vot­ing… I hate that name, which de­lib­er­ately ap­pro­pri­ates the en­tire “ranked” cat­egory for this one spe­cific method): This is a ranked sys­tem where to start out with, only first-choice votes are tal­lied. To find the win­ner, you suc­cess­ively elim­in­ate the last-place can­did­ate, trans­fer­ring those votes to their next sur­viv­ing pref­er­ence (if any), un­til some can­did­ate has a ma­jor­ity of the votes re­main­ing. It’s sup­por­ted by FairVote, the largest elect­oral re­form non­profit in the US, which grew out of the move­ment for STV pro­por­tional rep­res­ent­a­tion (see the multi-win­ner sec­tion be­low for more de­tails). IRV sup­port­ers tend to think that dis­cuss­ing its the­or­et­ical char­ac­ter­ist­ics is a waste of time, since it’s so ob­vi­ous that FPTP is bad and since IRV is the re­form pro­posal with by far the longest track re­cord and most well-de­veloped move­ment be­hind it. In­so­far as they do con­sider the­ory, they fa­vor the “later-no-harm” cri­terion, and prefer to ig­nore things like the fa­vor­ite be­trayal cri­terion, sum­mab­il­ity, or spoiled bal­lots. They also don’t talk about the failed Al­tern­at­ive Vote ref­er­en­dum in the UK.

Ap­proval vot­ing: This is the sys­tem where voters can ap­prove (or not) each can­did­ate, and the can­did­ate ap­proved by the most voters wins. Be­cause of its sim­pli­city, it’s some­thing of a “Schelling point” for re­formers of vari­ous stripes; that is, a nat­ural point of agree­ment as an ini­tial re­form for those who don’t agree on which method would be an ideal end state. This method was used in Greek elec­tions from about 1860-1920, but was not “in­ven­ted” as a sub­ject of vot­ing the­ory un­til the late 70s by Brams and Fish­burn. It can be seen as a simplistic spe­cial case of many other vot­ing meth­ods, in par­tic­u­lar score vot­ing, so it does well on War­ren Smith’s util­it­arian meas­ures, and fans of his work tend to sup­port it. This is the sys­tem pro­moted by the Center for Elec­tion Science (elect­o­logy.org), a vot­ing re­form non­profit that was foun­ded in 2012 by people frus­trated with FairVote’s anti-vot­ing-the­ory tend­en­cies. (Full dis­clos­ure: I’m on the board of the CES, which is grow­ing sub­stan­tially this year due to a sig­ni­fic­ant grant by the Open Phil­an­thropy Pro­ject. Thanks!)

Con­dorcet meth­ods: These are meth­ods that are guar­an­teed to elect a pair­wise beats-all win­ner (Con­dorcet win­ner) if one ex­ists. Sup­por­ted by people like Erik Maskin (a No­bel prize win­ner in eco­nom­ics here at Har­vard; bril­liant, but seem­ingly out of touch with the non-aca­demic work on vot­ing meth­ods), and Markus Schulze (a cap­able self-pro­moter who in­ven­ted a spe­cific Con­dorcet method and has got­ten groups like De­bian to use it in their in­ternal vot­ing). In my view, these meth­ods give good out­comes, but the com­plic­a­tions of resolv­ing spoil their the­or­et­ical clean­ness, while the dif­fi­culty of read­ing a mat­rix makes present­ing res­ults in an easy-to-grasp form ba­sic­ally im­possible. So I per­son­ally wouldn’t re­com­mend these meth­ods for real-world ad­op­tion in most cases. Re­cent work in “im­proved” Con­dorcet meth­ods has showed that these meth­ods can be made good at avoid­ing the chicken di­lemma, but I would hate to try to ex­plain that work to a layper­son.

Buck­lin meth­ods (aka me­dian-based meth­ods; es­pe­cially, Ma­jor­ity Judg­ment): Based on choos­ing a win­ner with the highest me­dian rat­ing, just as score vot­ing is based on choos­ing one with the highest av­er­age rat­ing. Be­cause me­di­ans are more ro­bust to out­liers than av­er­ages, me­dian meth­ods are more ro­bust to strategy than score. Sup­por­ted by French re­search­ers Bal­in­ski and Laraki, these meth­ods have an in­ter­est­ing his­tory in the pro­gress­ive-era USA. Their VSE is not out­stand­ing though; bet­ter than IRV, plur­al­ity, and Borda, but not as good as most other meth­ods.

Deleg­a­tion-based meth­ods, es­pe­cially SODA (simple op­tion­ally-del­eg­ated ap­proval): It turns out that this kind of method can ac­tu­ally do the im­possible and “avoid the Gib­bard-Sat­ter­th­waite the­orem in prac­tice”. The key words there are “in prac­tice” — the proof re­lies on a do­main re­stric­tion, in which voters hon­est pref­er­ences all agree with their fa­vor­ite can­did­ate, and these pref­er­ence or­ders are non-cyc­lical, and voters mu­tu­ally know each other to be ra­tional. Still, this is the only vot­ing sys­tem I know of that’s 100% strategy free (in­clud­ing chicken di­lemma) in even such a lim­ited do­main. (The proof of this is based on com­plic­ated ar­gu­ments about con­vex­ity in high-di­men­sional space, so Saari, it doesn’t fit here.) Due to its com­plex­ity, this is prob­ably not a prac­tical pro­posal, though.

Rated run­off meth­ods (in par­tic­u­lar STAR and 3-2-1): These are meth­ods where rated bal­lots are used to re­duce the field to two can­did­ates, who are then com­pared pair­wise us­ing those same bal­lots. They com­bine the VSE ad­vant­ages of score or ap­proval with ex­tra res­ist­ance to the chicken di­lemma. These are cur­rently my own fa­vor­ites as ul­ti­mate goals for prac­tical re­form, though I still sup­port ap­proval as the first step.

Quad­ratic vot­ing: Un­like all the meth­ods above, this is based on the uni­ver­sal solvent of mech­an­ism design: money (or other fi­nite trans­fer­rable re­sources). Voters can buy votes, and the cost for n votes is pro­por­tional to n². This has some ex­cel­lent char­ac­ter­ist­ics with hon­est voters, and so I’ve seen that vari­ous ra­tion­al­ists think it’s a good idea; but in my opin­ion, it’s got ir­resolv­able prob­lems with co­ordin­ated strategies. I real­ize that there are re­sponses to these ob­jec­tions, but as far as I can tell every prob­lem you fix with this idea leads to two more.

TL; DR?

  • Plur­al­ity vot­ing is really bad. (Borda count is too.)

  • Ar­row’s the­orem shows no ranked vot­ing method is per­fect.

  • Gib­bard-Sat­ter­th­waite the­orem shows that no vot­ing method, ranked or not, is strategy-free in all cases.

  • Rated vot­ing meth­ods such as ap­proval or score can get around Ar­row, but not Gib­bard-Sat­ter­th­waite.

  • Util­it­arian meas­ures, known as VSE, are one use­ful way to eval­u­ate vot­ing meth­ods.

  • Another way is mech­an­ism design. There are (1+)4 vot­ing patho­lo­gies to worry about. Start­ing from the most im­port­ant and go­ing down: (Dark horse rules out Borda;) vote-split­ting rules out plur­al­ity; cen­ter squeeze would rule out IRV; chicken di­lemma ar­gues against ap­proval or score and in fa­vor of rated run­off meth­ods; and Con­dorcet cycles mean that even the best vot­ing meth­ods will “fail” in a few per­cent of cases.

What are the ba­sics of multi-win­ner vot­ing the­ory?

Multi-win­ner vot­ing the­ory ori­gin­ated un­der par­lia­ment­ary sys­tems, where the­or­ists wanted a sys­tem to guar­an­tee that seats in a le­gis­lature would be awar­ded in pro­por­tion to votes. This is known as pro­por­tional rep­res­ent­a­tion (PR, prop-rep, or #PropRep). Early the­or­ists in­clude Henry Droop and Charles Dodg­son (Lewis Car­roll). We should also re­cog­nize Tho­mas Jef­fer­son and Daniel Web­ster’s work on the re­lated prob­lem of ap­por­tion­ing con­gres­sional seats across states.

Be­cause there are a num­ber of seats to al­loc­ate, it’s gen­er­ally easier to get a good an­swer to this prob­lem than in the case of single-win­ner vot­ing. It’s es­pe­cially easy in the case where we’re al­lowed to give win­ners dif­fer­ent vot­ing weights; in that case, a simple chain of del­eg­ated vot­ing weight guar­an­tees per­fect pro­por­tion­al­ity. (This idea has been known by many names: Dodg­son’s method, as­set vot­ing, del­eg­ated proxy, li­quid demo­cracy, etc. There are still some de­tails to work out if there is to be a lower bound on fi­nal vot­ing weight, but gen­er­ally it’s not hard to find ways to re­solve those.)

When seats are con­strained to be equally-weighted, there is in­ev­it­ably an ele­ment of round­ing er­ror in pro­por­tion­al­ity. Gen­er­ally, for each kind of method, there are two main ver­sions: those that tend to round to­wards smal­ler parties (Sainte-Laguë, Web­ster, Hare, etc.) and those that tend to round to­wards lar­ger ones (D’Hondt, Jef­fer­son, Droop, etc.).

Most ab­stract pro­por­tional vot­ing meth­ods can be con­sidered as greedy meth­ods to op­tim­ize some out­come meas­ure. Non-greedy meth­ods ex­ist, but al­gorithms for find­ing non-greedy op­tima are of­ten con­sidered too com­plex for use in pub­lic elec­tions. (I be­lieve that these prob­lems are NP-com­plete in many cases, but fast al­gorithms to find prov­ably-op­timal out­comes in all prac­tical cases usu­ally ex­ist. But most people don’t want to trust vot­ing to al­gorithms that nobody they know ac­tu­ally un­der­stands.)

Basic­ally, the out­come meas­ures be­ing im­pli­citly op­tim­ized are either “least re­mainder” (as in STV, single trans­fer­able vote), or “least squares” (not used by any real-world sys­tem, but pro­posed in Sweden in the 1890s by Thiele and Phrag­men). STV’s greedy al­gorithm is based on elim­in­a­tion, which can lead to prob­lems, as with IRV’s cen­ter-squeeze. A bet­ter solu­tion, akin to Buck­lin/​me­dian meth­ods in the single-win­ner case, is BTV (Buck­lin trans­fer­able vote). But the dif­fer­ence is prob­ably not a big enough deal to over­come STV’s ad­vant­age in terms of real-world track re­cord.

Both STV and BTV are meth­ods that rely on re­weight­ing bal­lots when they help elect a win­ner. There are vari­ous re­weight­ing for­mu­las that each lead to pro­por­tion­al­ity in the case of pure par­tisan vot­ing. This leads to an ex­plo­sion of pos­sible vot­ing meth­ods, all the­or­et­ic­ally reas­on­able.

Be­cause the the­or­et­ical pros and cons of vari­ous multi-win­ner meth­ods are much smal­ler than those of single-win­ner ones, the de­bate tends to fo­cus on prac­tical as­pects that are im­port­ant polit­ic­ally but that a math­em­atician would con­sider trivial or ad hoc. Among these are:

  • The role of parties. For in­stance, STV makes par­tisan la­bels form­ally ir­rel­ev­ant, while list pro­por­tional meth­ods (widely used, but the best ex­ample sys­tem is Bav­aria’s MMP/​mixed mem­ber pro­por­tional method) put parties at the cen­ter of the de­cision. STV’s non-par­tisan nature helped it get some trac­tion in the US in the 1920s-1960s, but the only rem­nant of that is Cam­bridge, MA (which hap­pens to be where I’m sit­ting). (The other rem­nant is that former STV ad­voc­ates were key in found­ing FairVote in the 1990s and push­ing for IRV after the 2000 elec­tion.) Polit­ical sci­ent­ist @jacksan­tucci is the ex­pert on this his­tory.

  • Bal­lot sim­pli­city and pre­cinct sum­mab­il­ity. STV re­quires voters to rank can­did­ates, and then re­quires keep­ing track of how many bal­lots of each type there are, with the num­ber of pos­sible types ex­ceed­ing the factorial of the num­ber of can­did­ates. In prac­tice, that means that vote-count­ing must be cent­ral­ized, rather than be­ing per­formed at the pre­cinct level and then summed. That cre­ates lo­gist­ical hurdles and fraud vul­ner­ab­il­it­ies. Tra­di­tion­ally, the way to re­solve this has been list meth­ods, in­clud­ing mixed meth­ods with lists in one part. Re­cent pro­pos­als for del­eg­ated meth­ods such as my PLACE vot­ing (pro­por­tional, loc­ally-ac­count­able, can­did­ate en­dorse­ment; here’s an ex­ample) provide an­other way out of the bind.

  • Local­ity. Voters who are used to FPTP (plur­al­ity in single-mem­ber dis­tricts) are used to hav­ing “their local rep­res­ent­at­ive”, while pure pro­por­tional meth­ods ig­nore geo­graphy. If you want both loc­al­ity and pro­por­tion­al­ity, you can either use hy­brid meth­ods like MMP, or bi­pro­por­tional meth­ods like LPR, DMP, or PLACE.

  • Breadth of choice. Ideally, voters should be able to choose from as many vi­able op­tions as pos­sible, without over­whelm­ing them with bal­lot com­plex­ity. My pro­posal of PLACE is de­signed to meet that ideal.

Prop-rep meth­ods would solve the prob­lem of ger­ry­man­der­ing in the US. I be­lieve that PLACE is the most vi­able pro­posal in that re­gard: main­tains the loc­al­ity and bal­lot sim­pli­city of the cur­rent sys­tem, is re­l­at­ively non-dis­rupt­ive to in­cum­bents, and max­im­izes breadth of voter choice to help in­crease turnout.

Oh, I should also prob­ably men­tion that I was the main de­signer, in col­lab­or­a­tion with dozens of com­menters on the web­site Mak­ing Light, of the pro­por­tional vot­ing method E Pluribus Hugo, which is now used by the Hugo Awards to min­im­ize the im­pact and in­cent­ives of bloc vot­ing in the nom­in­a­tions phase.

Anticli­mactic sign-off

OK, that’s a long art­icle, but it does a bet­ter job of brain-dump­ing my >20 years of in­terest in this topic than any­thing I’ve ever writ­ten. On the sub­ject of single-win­ner meth­ods, I’ll be put­ting out a play­able ex­plor­a­tion ver­sion of all of this some­time this sum­mer, based off the work of the in­valu­able nicky case (as well as other col­lab­or­at­ors).

I’ve now ad­ded a third art­icle on this topic, in which I in­cluded a para­graph at the end ask­ing people to con­tact me if they’re in­ter­ested in act­iv­ism on this. I be­lieve this is a vi­able tar­get for ef­fect­ive al­tru­ism.