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.