Imperfect Voting Systems

Stalin once (sup­pos­edly) said that “He who casts the votes de­ter­mines noth­ing; he who counts the votes de­ter­mines ev­ery­thing “ But he was be­ing in­suffi­ciently cyn­i­cal. He who chooses the vot­ing sys­tem may de­ter­mine just as much as the other two play­ers.

The Art of Strat­egy gives some good ex­am­ples of this prin­ci­ple: here’s an adap­ta­tion of one of them. Three man­agers are de­bat­ing whether to give a Dist­in­guished Em­ployee Award to a cer­tain worker. If the worker gets the award, she must re­ceive one of two prizes: a $50 gift cer­tifi­cate, or a $10,000 bonus.

One man­ager loves the em­ployee and wants her to get the $10,000; if she can’t get the $10,000, she should at least get a gift cer­tifi­cate. A sec­ond man­ager ac­knowl­edges her con­tri­bu­tion but is mostly driven by cost-cut­ting; she’d be hap­piest giv­ing her the gift cer­tifi­cate, but would rather re­fuse to rec­og­nize her en­tirely than lose $10,000. And the third man­ager dis­likes her and doesn’t want to rec­og­nize her at all—but she also doesn’t want the com­pany to gain a rep­u­ta­tion for sting­i­ness, so if she gets rec­og­nized she’d rather give her the $10,000 than be so pa­thetic as to give her the cheap cer­tifi­cate.

The man­agers ar­range a meet­ing to de­ter­mine the em­ployee’s fate. If the agenda tells them to vote for or against giv­ing her an award, and then pro­ceed to de­ter­mine the prize af­ter­wards if she wins, then things will not go well for the em­ployee. Why not? Be­cause the man­agers rea­son as fol­lows: if she gets the award, Man­ager 1 and Man­ager 3 will vote for the $10,000 prize, and Man­ager 2 will vote for the cer­tifi­cate. There­fore, vot­ing for her to get the award is prac­ti­cally the same as vot­ing for her to get the $10,000 prize. That means Man­ager 1, who wants her to get the prize, will vote yes on the award, but Man­agers 2 and 3, who both pre­fer no award to the $10,000, will strate­gi­cally vote not to give her the award. Re­sult: she doesn’t get rec­og­nized for her dis­t­in­guished ser­vice.

But sup­pose the em­ployee in­volved hap­pens to be the sec­re­tary ar­rang­ing the meet­ing where the vote will take place. She makes a seem­ingly triv­ial change to the agenda: the man­agers will vote for what the prize should be first, and then vote on whether to give it to her.

If the man­agers de­cide the ap­pro­pri­ate prize is $10,000, then the mo­tion to give the award will fail for ex­actly the same rea­sons it did above. But if the man­agers de­cide the cer­tifi­cate is ap­pro­pri­ate, then Man­ager 1 and 2, who both pre­fer the cer­tifi­cate to noth­ing, will vote in fa­vor of giv­ing the award. So the three man­agers, think­ing strate­gi­cally, re­al­ize that the de­ci­sion be­fore them, which looks like “$10 grand or cer­tifi­cate”, is re­ally “No award or cer­tifi­cate”. Since 1 and 2 both pre­fer the cer­tifi­cate to noth­ing, they vote that the cer­tifi­cate is the ap­pro­pri­ate prize (even though Man­ager 1 doesn’t re­ally be­lieve this) and the em­ployee ends out with the gift cer­tifi­cate.

But if the sec­re­tary is re­ally smart, she may set the agenda as fol­lows: The man­agers first vote whether or not to give $10,000, and if that fails, they next vote whether or not to give the cer­tifi­cate; if both votes fail the em­ployee gets noth­ing. Here the man­agers re­al­ize that if the first vote (for $10,000) fails, the next vote (cer­tifi­cate or noth­ing) will pass, since two man­agers pre­fer cer­tifi­cate to noth­ing as men­tioned be­fore. So the true choice in the first vote is “$10,000 ver­sus cer­tifi­cate”. Since two man­agers (1 and 3) pre­fer the $10,000 to the cer­tifi­cate, those two start by vot­ing to give the full $10,000, and this is what the em­ployee gets.

So we see that all three op­tions are pos­si­ble out­comes, and that the true power rests not in the hands of any in­di­vi­d­ual man­ager, but in the sec­re­tary who de­ter­mines how the vot­ing takes place.

Amer­i­cans have a head start in un­der­stand­ing the pit­falls of vot­ing sys­tems thanks to the so-called two party sys­tem. Every four years, they face quan­daries like “If leftists like me vote for Nader in­stead of Gore just be­cause we like him bet­ter, are we go­ing to end up elect­ing Bush be­cause we’ve split the leftist vote?”

Em­piri­cally, yes. The 60,000 Florida cit­i­zens who voted Green in 2000 didn’t elect Nader. How­ever, they did make Gore lose to Bush by a mere 500 votes. The last post dis­cussed a Vick­rey auc­tion, a style of auc­tion in which you have have no in­cen­tive to bid any­thing ex­cept your true value. Wouldn’t it be nice if we had an elec­toral sys­tem with the same prop­erty: one where you should always vote for the can­di­date you ac­tu­ally sup­port? If such a sys­tem ex­isted, we would have am­ple rea­son to in­sti­tute it and could rest as­sured that no mod­ern-day Stalin was ma­nipu­lat­ing us via the choice of vot­ing sys­tem we used.

Some coun­tries do claim to have bet­ter sys­tems than the sim­ple win­ner-takes-all ap­proach of the United States. My own adopted home­land of Ire­land uses a sys­tem called “sin­gle trans­fer­able vote” (also called in­stant-runoff vote), in which vot­ers rank the X can­di­dates from 1 to X. If a can­di­date has the ma­jor­ity of first prefer­ence votes (or a num­ber of first prefer­ence votes greater than the num­ber of po­si­tions to fill di­vided by the num­ber of can­di­dates, in elec­tions with mul­ti­ple po­ten­tial win­ners like leg­is­la­tive elec­tions), then that can­di­date wins and any sur­plus votes go to their vot­ers’ next prefer­ence. If no one meets the quota, then the least pop­u­lar can­di­date is elimi­nated and their sec­ond prefer­ence votes be­come first prefer­ences. The sys­tem con­tinues un­til all available seats are full.

For ex­am­ple, sup­pose I voted (1: Nader), (2: Gore), (3: Bush). The elec­tion offi­cials tally all the votes and find that Gore has 49 mil­lion first prefer­ences, Bush has 50 mil­lion, and Nader has 5 mil­lion. There’s only one pres­i­dency, so a can­di­date would have to have a ma­jor­ity of votes (greater than 52 mil­lion out of 104 mil­lion) to win. Since no one meets that quota, the low­est ranked can­di­date gets elimi­nated—in this case, Nader. My vote now goes to my sec­ond prefer­ence, Gore. If 4 mil­lion Nader vot­ers put Gore sec­ond ver­sus 1 mil­lion who put Bush sec­ond, the tally’s now at 53 mil­lion Gore, 51 mil­lion Bush. Gore has greater than 52 mil­lion and wins the elec­tion—the op­po­site re­sult from if we’d elected a pres­i­dent the tra­di­tional way.

Another sys­tem called Con­dorcet vot­ing also uses a list of all can­di­dates ranked in or­der, but uses the in­for­ma­tion to run mock runoffs be­tween each of them. So a Con­dorcet sys­tem would use the bal­lots to run a Gore/​Nader match (which Gore would win), a Gore/​Bush match (which Gore would win), and a Bush/​Nader match (which Bush would win). Since Gore won all of his matches, he be­comes Pres­i­dent. This be­comes com­pli­cated when no can­di­date wins all of his matches (imag­ine Gore beat­ing Nader, Bush beat­ing Gore, but Nader beat­ing Bush in a sort of Pres­i­den­tial rock-pa­per-scis­sors.) Con­dorcet vot­ing has var­i­ous op­tions to re­solve this; some sys­tems give vic­tory to the can­di­date whose great­est loss was by the small­est mar­gin, and oth­ers to can­di­dates who defeated the great­est num­ber of other can­di­dates.

Do these sys­tems avoid the strate­gic vot­ing that plagues Amer­i­can elec­tions? No. For ex­am­ple, both Sin­gle Trans­fer­able Vote and Con­dorcet vot­ing some­times provide in­cen­tives to rank a can­di­date with a greater chance of win­ning higher than a can­di­date you pre­fer—that is, the same “vote Gore in­stead of Nader” dilemma you get in tra­di­tional first-past-the-post.

There are many other elec­toral sys­tems in use around the world, in­clud­ing sev­eral more with rank­ing of can­di­dates, a few that do differ­ent sorts of runoffs, and even some that ask you to give a nu­mer­i­cal rat­ing to each can­di­date (for ex­am­ple “Nader 10, Gore 6, Bush −100000”). Some of them even man­age to elimi­nate the temp­ta­tion to rank a non-preferred can­di­date first. But these work only at the ex­pense of in­cen­tiviz­ing other strate­gic manuev­ers, like defin­ing “ap­proved can­di­date” differ­ently or ex­ag­ger­at­ing the differ­ence be­tween two can­di­dates.

So is there any vot­ing sys­tem that au­to­mat­i­cally re­flects the will of the pop­u­lace in ev­ery way with­out en­courag­ing tac­ti­cal vot­ing? No. Var­i­ous proofs, in­clud­ing the Gib­bard-Sat­terth­waite The­o­rem and the bet­ter-known Ar­row Im­pos­si­bil­ity The­o­rem show that many of the crite­ria by which we would nat­u­rally judge vot­ing sys­tems are mu­tu­ally in­com­pat­i­ble and that all rea­son­able sys­tems must con­tain at least some small el­e­ment of tac­tics (one ex­am­ple of an un­rea­son­able sys­tem that elimi­nates tac­ti­cal vot­ing is pick­ing one bal­lot at ran­dom and de­ter­min­ing the re­sults based solely on its prefer­ences; the pre­cise text of the the­o­rem rules out “non­de­ter­minis­tic or dic­ta­to­rial” meth­ods).

This means that each vot­ing sys­tem has its own benefits and draw­backs, and that which one peo­ple use is largely a mat­ter of prefer­ence. Some of these prefer­ences re­flect gen­uine con­cern about the differ­ences be­tween vot­ing sys­tems: for ex­am­ple, is it bet­ter to make sure your sys­tem always elects the Con­dorcet win­ner, even if that means the sys­tem pe­nal­izes can­di­dates who are too similar to other can­di­dates? Is it bet­ter to have a sys­tem where you can guaran­tee that par­ti­ci­pat­ing in the elec­tion always makes your can­di­date more likely to win, or one where you can be sure that ev­ery­one vot­ing ex­actly the op­po­site will never elect the same can­di­date?

But in prac­tice, these prefer­ences tend to be poli­ti­cal and self-in­ter­ested. This was re­cently ap­par­ent in Bri­tain, which voted last year on a refer­en­dum to change the vot­ing sys­tem. The Liberal Democrats, who were per­pet­u­ally stuck in the same third-place situ­a­tion as Nader in the States, sup­ported a change to a form of in­stant runoff vot­ing which would have made vot­ing Lib Dem a much more palat­able op­tion; the two ma­jor par­ties op­posed it prob­a­bly for ex­actly that rea­son.

Although no sin­gle vot­ing sys­tem is math­e­mat­i­cally perfect, sev­eral do seem to do bet­ter on the crite­ria that real peo­ple care about; look over Wikipe­dia’s sec­tion on the strengths and weak­nesses of differ­ent vot­ing sys­tems to see which one looks best.