# Mathematics vs. Democracy: A Clear Winner or a Tie Game?August 27, 2007 12:11 PM   Subscribe

The Marquis de Condorcet and Admiral Jean-Charles de Borda were two men of the French Enlightenment who struggled with how to design voting systems that accurately reflected voters' preferences. Condorcet favored a method that required the winner in a multiparty election to win a series of head-to-head contests, but he also discovered that his method easily led to a paradoxes that produced no clear winners. The Borda method avoids the Condorcet paradox by requiring voters to rank choices numerically in order of preference, but this method is flawed because the withdrawal of a last-place candidate can reverse the election results. Mathematicians in the 19th century attempted to design better voting systems, including Lewis Carroll, who favored an early form of proportional representation. Economist Kenneth Arrow argued that designing a perfect voting system was futile, because his "impossibility theorem" proved that it's impossible to design a non-dictatorial voting system that fulfills five basic criteria of fairness. (more inside)
posted by jonp72 (43 comments total) 38 users marked this as a favorite

So what does this have to do with current politics? The United States uses a system of plurality voting that often violates the principle of the independence of irrelevant alternatives. Namely, the addition or subtraction of a third-party candidate to a two-candidate election can completely change what candidate wins the election, also known as the spoiler effect. In addition, according to Duverger's law, as created by French political scientist Maurice Durverger, a plurality voting system (such as the United States system) will almost always lead to the death of third parties, a two-party duopoly, and a system that penalizes sincere non-strategic voting. Meanwhile, the mathematician Warren D. Smith argues that range voting is superior to all other voting methods, because it minimizes Bayesian regret, a method of measuring the happiness or unhappiness a voting system produces. For the rest of you policy wonks and electoral math nerds, you may also want to check out Citizens for Approval Voting, Social Choice and Beyond, the Electowiki, Condorcet.org, and Fairvote.
posted by jonp72 at 12:11 PM on August 27, 2007 [11 favorites]

Great post -- will point thoughtful Americans here when they express any enthusiasm for third parties. Sorry, guys, it really is throwing your vote away!
posted by gum at 12:29 PM on August 27, 2007

There are a couple of other ways to look at electoral systems. One of the most profound I ran into one time was this:

The true goal of an electoral system is to convince the losers to accept their defeat, and to not rise in armed rebellion in response.

My own take on electoral systems is rather strange. They're a form of signal processing, attempting to extract a signal from an extremely noisy communication medium.

A lot of people object to "winner take all" because, they claim, it politically disenfranchises minority points of view. Such people look with approval at the Parliamentary system, where minority parties get seats in the Parliament and can become part of a ruling coalition and as such have a chance to wring concessions out of the coalition maker.

I see that as a bug, not as a feature, for an interesting reason: A "winner take all" system can permit more freedom of speech. In Europe, where parliamentary systems do exist, there are also laws restricting certain kinds of political speech. (For instance, laws regarding Nazi symbols.) Those laws are seen to be necessary because without them, such despicable fringe points of view might well not be easy to marginalize politically.

In the winner-take-all system, we can tolerate such things because they cannot influence government policy as long as their numbers are relatively small. That, combined with Duverger's law, is the reason that there is a legal American Nazi Party. I think they're assholes but I'm glad that it's possible for them to organize. They're an example of a canary-in-the-coalmine for political speech; if they vanish or are suppressed, I will begin to worry about my own right of free speech.

So content-based censorship of political speech isn't needed here because the system itself structurally rejects extreme minority points of view. In parliamentary systems -- or in systems which use things like Condorcet voting -- that's not the case, and I think there's a natural tendency in such systems for the majority to rely on content-based censorship to prevent such minorities from influencing the system. Which is often the first step on the slippery slope.
posted by Steven C. Den Beste at 12:34 PM on August 27, 2007 [4 favorites]

You forgot Florida
posted by Sk4n at 12:44 PM on August 27, 2007

Great post jonp. I wonder how internet communities have fared with range voting. My experience is that if given a sense of the ongoing average, people tend to vote at the extremes - lots of "0"s and "10s" - to try and push the average closer to what they think it should be, with all they can. In a situation where voters aren't presented with the 'running total', I wonder if their votes would be any more honest.

Range voting, in other words, might reward jingoism and punish reflective consideration. If party X's supporters all vote "10" for their candidate and "0" for others, will that effect overpower honest citizens giving more thoughtful ratings?
posted by anthill at 12:45 PM on August 27, 2007

I think that in Smith's IRV vs Range example, the "good" winner that Range picks got No Opinion from 1/3 of the voters.
posted by a robot made out of meat at 12:46 PM on August 27, 2007

Super post.
posted by facetious at 12:49 PM on August 27, 2007

If you like this kind of stuff, Chaotic Elections: A Mathematician Looks at Voting by Donald Saari is a great introduction to the main ideas, written at a quite accessible level.

SCDB, your point about the upside of marginalizing minority views is a good one. But I also agree with you that it seems very weird to analogize elections to the process of extracting a signal from noise. Do you think there really is a signal? That is, do you think there's something called "the people's will" and that elections are a kind of instrument to detect it? Or do you think "the people's will" is the word we use for the result of an election?
posted by escabeche at 12:50 PM on August 27, 2007 [2 favorites]

n Europe, where parliamentary systems do exist, there are also laws restricting certain kinds of political speech. (For instance, laws regarding Nazi symbols.) Those laws are seen to be necessary because without them, such despicable fringe points of view might well not be easy to marginalize politically

A fine theory, except (a) there is an organized Neo-Nazi party in Germany (they just have a safe "official" name), and (b) these speech limits are always on Nazis, race or religion. In a parliamentary system, fringe groups can and do organize, obtain some political power.

But most importantly, and the reason the US will ultimately move to abandoning the electoral college in the mid to distant future, is that fringe parties can become kingmakers. The US electoral system does not abide kingmakers other than the wealthy interests that can control the outcome with money.

In a parliamentary system, where winner does not take all, a coalition needs to be formed to reach a majority. The fringe holdouts can extract concessions from the plurality party because the fringes handful of votes are needed to reach majority and deny the other bloc its majority.

Imagine the 2000 outcome if the Green party could throw its votes to one or the other party. You can't, because in such a system where losers can form coalitions, elections splinter and there'd be many more factions. More importantly, each of those factions would have readily identifiable views and objectives - Christian Conservatives, fiscal conservatives, etc. There's no hiding the ideology or the ideologues.

The US system holds paramount the orderly transfer of power. For a system to so reliably transfer power in such a consistently orderly manner requires that regardless of who wins, the vested interests remain largely unaffected.
posted by Pastabagel at 12:58 PM on August 27, 2007

posted by Steven C. Den Beste at 12:59 PM on August 27, 2007

Pastabagel, it turns out that the American system also has that coalition aspect to it. But it happens during the primary process, not during the general election.

Tell me in five word what the Democratic party stands for. Or the Republican party. Jokes and prejudice aside, you can't do it -- because they don't actually stand for anything.

Each of the two parties is actually a loose assembly of groups with widely divergent points of view. Selection of candidates requires creating exactly the kind of coalition you like among the various groups within the party, so as to create a majority in the primary process.

BUT...

But then the resulting candidate has to face the entire voter population in the general election, so if extreme political points of view influence the primary process in one of the parties to a dangerous degree, the voters still have the candidate from the other party to select as the lesser of evils.

posted by Steven C. Den Beste at 1:22 PM on August 27, 2007

America doesn't even deserve an improved system. There is often the "throwing your vote" away mantra, and this shows a lack of independence of thought. Such behavior should be rewarded with less democracy, not improved and effective democracy. Enough people use the "throwing away your vote" mantra to actually swing elections, which would force parties, a la FDR's socialist lean, to adopt the platform to sway voters. Doesn't matter, American voters lack a strong freethought movement, and our current system is perfect for it. The country was founded on revolution and a constitution without equal, and now we've come to "throwing our votes away."
posted by Gnostic Novelist at 1:32 PM on August 27, 2007

SCDB:

I'm not disagreeing with you, except in the signal that the channel is carrying.

But it happens during the primary process, not during the general election.

It happens before the candidate runs. We already know what views the 2008 Republican nominee will have on abortion, gun control, etc. Even if the individual has different personal views, they will adopt this position.

In other words, the platform is selected for both parties on most issues but the most important, and on those issues the position is always and intentionally ambiguous. Use whatever code word you want here - "nuanced", "complex", etc. it means the same thing: "I'm not going to tell you."

To clarify what I said above in less of a lunatic fringe tone, the purpose of the American system, the purpose of the system as it has evolved - direct election of senators, P and VP run together, high \$ campaign costs to access mass media, is to push parties to the center. The "center" is the basket of positions held by most Americans.

But the center is both predictable and malleable. If the political view of Americans are subject to influence through marketing, PR, etc., AND the two major parties are going to run candidates whose positions occupy that center position + some delta to the left or right as appropriate, then it's clear that the same powers that can influence the political views of Americans are able to influence the outcomes.

In other words, they don't pick a Dem or a Republican as the winner, but rather they ensure that the winner, whoever it is and from whatever party, has a platform and a position on issues that conforms to their own.
posted by Pastabagel at 1:36 PM on August 27, 2007 [1 favorite]

Another type of paradox: In A Mathematician Reads the Newspaper by John Allen Paulos there is a section called 'Tsongkerclintkinbro Wins'. It describes a hypothetical election contest among 5 candidates. Depending on which method is used to determine the winner, each candidate could claim victory.
posted by MtDewd at 1:39 PM on August 27, 2007

In addition, according to Duverger's law, as created by French political scientist Maurice Durverger, a plurality voting system (such as the United States system) will almost always lead to the death of third parties

...except in Canada and the UK. But it's rightish insofar as plurality elections tend to breed either two-party systems or strong regional parties.

Economist Kenneth Arrow argued that designing a perfect voting system was futile

It's actually stronger than that. He proved that designing a perfect system of preference aggregation is futile. You can prove the theorem without any actual voting, relying only on the Pareto principle (if everyone, without exception, likes A better than B, we shouldn't choose B) and the definition of decisiveness (if A is decisive over some decision, only A's preferences are relevant to that decision).

But what it means is that nonvoting methods of preference aggregation, such as the aggregation of individual free choices in a pure market, are also fubared.
posted by ROU_Xenophobe at 1:46 PM on August 27, 2007 [1 favorite]

SCDB:
In my observation, you have it backwards. In proportional systems, the wingnuts can have their free speech and don't ever get dangerous power because no-body gets dangerous power. Contrast that with winner-takes-all, and just look at the mess the USA is in, because the system not only allowed wingnuts to take power, but it happily gives them absolute power, beholden to no other party, the policies of an ideologically-blinkered minority rammed through without reflection or caution. Winner takes all is a disaster for rational policy making, and not just in the USA. That's why many countries consider it obsolete and are moving away from it.

I watched the before and after of politics in New Zealand, which reformed from a winner-takes-all to a proportional system, and there is just no comparison. Proportional is just better, hands down. (Even despite the rocky years while adjusting to the alien paradigm, and plenty of the old guard still want the old days of us-vs-them back-vs-white back again).

Your stretch regarding free speech as an argument for winner-takes-all sounds to me more like rationalizing a pre-conceived opinion, not serious reasoning.

Laws regarding freedom of speech did not change when NZ reformed its electoral system - there was no need, because they are not linked in the way you suggest. My own impression of the differing cultural views on freedom of speech is that it has little, if anything to do with the political system. For example, people simply aren't worried about neo-nazi skinheads winning seats in parliment, they're worried about being bashed by them while walking down the street, and that if neo-nazis were free to preach hate, they can grow their gangs and bash more people.
posted by -harlequin- at 1:48 PM on August 27, 2007 [3 favorites]

Pastabagel, "pushing the result towards the center" sounds like a good thing. Isn't that what we'd really want an electoral system to do?

The system as designed makes it so that the two parties will always be different, but not really a lot different. Both of them will orbit around the political center of the nation, but over time they can shift, mostly as the political center of the nation shifts.

That all sounds desirable to me.

As to the malleability of the center, I think it's less malleable that you apparently do. And I don't think it is malleable in all ways. I also think that people who have extreme points of view should be permitted to express them, and to try to influence the political center. That can have bad results, but it usually doesn't, and without that ability we would never have ratified the 19th Amendment (for instance).
posted by Steven C. Den Beste at 1:50 PM on August 27, 2007

Fantastic post. I follow all these systems with great interest, and have since the 2000 debacle. Seems to me that ANYTHING is better than plurality voting, even if there are occasional paradoxes. Approval voting gets my vote, but I'd vote for more than one if I had the chance. :-)
posted by rouftop at 1:53 PM on August 27, 2007

To add to the fun, there are more weird paradoxes in voting. I think my favorite is nonmonotonicity. If a voting system is nonmonotonic, gaining more support -- having more people like you better -- can make you lose. ISTR that any voting system that eliminates alternatives as it goes is nonmonotonic, but I'd have to look it up. STV and other common proportional representation schemes are nonmonotonic, at least in principle.
posted by ROU_Xenophobe at 1:56 PM on August 27, 2007 [1 favorite]

Voting systems also have to be easy to operate for the voter (and, to a lesser extent, for the counter). Borda Count makes for difficult decisions in the middle order, and is tedious above more than four or five candidates.

This matters in practical cases. In UK General Elections, many constituencies will have candidates from Conservative, Labour, Lib Dem, Respect, UKIP and perhaps one or two others. Perhaps that's not too bad, but in local government elections, where each ward (division) can have up to three members, you can get upwards of 20 candidates on a ballot paper.

That's why I broadly speaking like approval voting.
posted by athenian at 3:26 PM on August 27, 2007

Range voting looks great, but why use 99 points?

The 5-star rating system is an unofficial standard (Amazon, iTunes, Netflix), and allows about as much granularity as you really need. (Is there really a difference between a 37-point candidate and a 38-point candidate?)

We should rate candidates on a scale of 1-5 stars or No Opinion, and take the candidate with the highest average rating. If it's good enough for Jeff Bezos, it's good enough for me.
posted by designbot at 3:33 PM on August 27, 2007

Mathematician here.
Plurality with Elimination (instant run-off) is my favorite method, and Condorcet (Pairwise-Comparison) is a close second. Both seem to me to upset fewer voters and provide for a relatively popular winner, rather than Borda (and related) which can have nobody's favorite winning handily and seems to favor compromise. In some cases, compromise is good. In others, not so much.
posted by monkeymadness at 3:36 PM on August 27, 2007

Fascinating stuff. I wish I'd known beforehand the futility of trying to establish any kind of "fair" voting procedure, such that I could get 14 guys to agree on 5 things to do in Vegas. For such an endeavor, management by tyranny becomes a viable option and you simply mollify the complainers with extra booze. Thusly is the fringe minority placated by catering to their shared vice.

I believe in the principle of democracy, but find it incredible that it functions at all. We might as well give it a name and worship it as some kind of deity that has it's mood swings; its will can be predicted somewhat through math and probability, but there's so much human element to it that you get righteously pissed off sometimes when it's not showing you favor.
posted by krippledkonscious at 3:39 PM on August 27, 2007

posted by designbot at 3:47 PM on August 27, 2007

Small correction: Obviously, you wouldn't just take the highest average rating, or Joe Schmoe with 1 5-star vote would win. You'd need to eliminate anyone without a quorum.
posted by designbot at 3:55 PM on August 27, 2007

America doesn't even deserve an improved system. There is often the "throwing your vote" away mantra, and this shows a lack of independence of thought. Such behavior should be rewarded with less democracy, not improved and effective democracy.

This is arrogant to the point of ridiculousness. The point isn't to give people the government they deserve. The point is to give people a government better than they deserve. If our government was really determined by what we deserve, we would probably all be in chains. It's as arrogant as the Leninists who insist on "breaking a few eggs to make an omelette" and making societal conditions worse until "the masses" see the wisdom of overthrowing capitalism. It's as arrogant as Grover Norquist who sees his function as inflicting corruption, cronyism, and mismanagement on the American public until "the masses" see the wisdom of drowning government in a bathtub. The voters aren't some rats in a behavior modification experiment who should be rewarded and/or punished based on whether you think they show enough "independence of thought."
posted by jonp72 at 4:27 PM on August 27, 2007 [1 favorite]

the reason the US will ultimately move to abandoning the electoral college in the mid to distant future

My god you're optimistic. But I agree that plurality politics doesn't, or doesn't have to, restrict free speech more than winner-take politics. The example of Nazi symbols in Germany doesn't really hold water since we have similar laws here in the U.S. They're called 'hate speech' laws and they're not joined to one particular symbol, but if you open a deli and put a giant nazi flag outside, I think you'd find out where the laws apply.

Now, I'm not objecting to hate speech laws, that's a whole different discussion and I'm not sure where I stand anyway. I'm just pointing out that we have much the same laws here as in Germany. Plus, as has been pointed out, Germany is one country with a pluralistic election system. I think one would have to show a correlation between countries adopting that system and passing those laws to prove your point.

And as has been mentioned before, minority points of view get no traction at all in American politics. Which is not such a big deal as long as the majority is sensible. I think a little more minority influenced debate leading up to the Iraq invasion would have benefited us greatly.
posted by lumpenprole at 4:27 PM on August 27, 2007

P.S. I know I'm responding to both sides of the argument and I'm sorry for conflating them with the pronoun 'you'.
posted by lumpenprole at 4:29 PM on August 27, 2007

The true goal of an electoral system is to convince the losers to accept their defeat, and to not rise in armed rebellion in response.

SCDB, I think your theory is very interesting, but I do have one counterexample: the election of Salvador Allende. Allende was elected with 36.3% of the vote in a 3-candidate election with plurality voting (the same system as the U.S.). The subsequent coup by Pinochet suggests that the losers didn't accept their defeat too well.
posted by jonp72 at 4:33 PM on August 27, 2007

This is a fantastic post on a subject I have long been fascinated with. And interesting discussion from Steven C. Den Beste, whose self-links I will be reading with great interest, in spite of being one of those parliamentary-leaning, Condorcet-loving commies myself.

Unfortunately, gum, this post won't win you any headway against my third-party enthusiasms, such as they are. I will couch an example in frankly biased terms. In 2004, I cast my presidential vote for Ralph Nader. My state's electoral votes went, quite predictably, to a detestable putz named John Kerry, and my nation's presidency went, as you know, to a detestable putz named George W. Bush.

Neither of these outcomes would have been swayed by my voting for either man. Therefore my vote was, indeed, thrown away. But consider: Somewhat over 59 million Kerry voters found that the final occupancy of the Oval Office was not swayed by their votes, either. That is to say, their votes were also thrown away. And by whom? Not by themselves; by the 62-million-odd Bush voters who prevailed (along with the smaller number of state and federal legislators who have left the Electoral College in its present state of partial dysfunction).

(Not excluding, either, the possibility of electoral fraud; and omitting, for the moment, the separate though engaging question of how greatly those 62 million also had their (future) choices limited, i.e., votes thrown away, in repercussions of their (/our) electing a cryptofascist.)

Once you think about it, this is quite to be expected. It's the necessary cost of living in a democracy (/republic): Half the votes get thrown away, every time it matters. Partial or part-time disenfranchisement happens if you share the franchise. Have cake/eat it, yo? To me, thus far, the benefits of freedom outweigh that cost; and if they ceased to—(insert well-known concerns about media influence and popular ignorance here)—I'd consider moving to Switzerland or New Zealand or Antarctica or somewhere.

One thing I wouldn't consider doing is giving even one vote's worth of apparent support to a detestable putz in high office. That would be a thrown-away vote.

Full disclosure: My 4-greats grandmother's first cousin ran for President on a third-party ticket in 1844. This was not because he expected to win.
posted by eritain at 4:34 PM on August 27, 2007 [1 favorite]

# anthill, I know that the Debian Linux distribution uses a Condorcet system for project leader elections.
posted by ijoshua at 5:06 PM on August 27, 2007

The subsequent coup by Pinochet suggests that the losers didn't accept their defeat too well.

Which means that in that particular case the electoral system didn't succeed in convincing the losers to accept defeat gracefully.

I never said it was guaranteed to work every time. Nothing works every time.
posted by Steven C. Den Beste at 5:25 PM on August 27, 2007

We should rate candidates on a scale of 1-5 stars or No Opinion, and take the candidate with the highest average rating. If it's good enough for Jeff Bezos, it's good enough for me.

Under that particular system is that it's possible for the candidate with the highest average rating to not have been any voter's first choice.
posted by Steven C. Den Beste at 5:30 PM on August 27, 2007

Unfortunately, gum, this post won't win you any headway against my third-party enthusiasms, such as they are. I will couch an example in frankly biased terms. In 2004, I cast my presidential vote for Ralph Nader. My state's electoral votes went, quite predictably, to a detestable putz named John Kerry, and my nation's presidency went, as you know, to a detestable putz named George W. Bush.

Eritain, third party "enthusiasms" are fine as long as your motivations for voting don't include helping your side win representation in government.

You're certainly not alone -- I'll bet a plurality of American voters are more interested voting as individual expression than voting as collective action. More interested in identity ("I and mine are not detestable putzes, not one of us") than outcomes ("I rubbed up against some detestable putzes in my coalition, but we accomplished some good stuff together before I ran home and washed my hands").

Any serious third party activist owes it to herself to get intimately acquainted with Duverger's Law. A single-member district plurality voting system as pure as the one constitutionally hard-wired in the United States stabilizes a two-party system and restricts third parties to gadfly/spoiler roles. It's not sound-bite simple, but it's really worth the effort to think through. (I've heard Duverger's Law cited as the strongest evidence available of actual "science" in the discipline of political science.)

I think the American plurality voting system is just awful. But I think the massive constitutional amendment project that would be necessary to put it out of business is almost impossible to imagine, since it would have to be proposed and ratified by representatives of the two major parties in Congress and the state legislatures. Far more practical to organize the revolution.

And meanwhile, if you want redemption, see if you can't substitute the fifteen-minute ceremony of voting for a third-party non-putz -- refreshing! zero calories! -- with the much more challenging and rewarding experience of engaging the putzes in the major party on your side and sending the best possible putzes into government to steamroll the much worse putzes on the other side.

Or move to a mature, robust democracy somewhere else (god knows I've thought about it!).
posted by gum at 5:36 PM on August 27, 2007

We should rate candidates on a scale of 1-5 stars or No Opinion, and take the candidate with the highest average rating. If it's good enough for Jeff Bezos, it's good enough for me.

SCDB: Under that particular system is that it's possible for the candidate with the highest average rating to not have been any voter's first choice.

That's why I think "range voting" is probably the best we're going to get. It is possible that the winning candidate wasn't anybody's first choice, but if that happens, it is likely that the winning candidate was the second or third choice of a lot of people instead. On the other hand, it does get rid of the "spoiler effect" for third parties. Approval voting has some of the same virtues as range voting, but it suffers from the Burr dilemma, referring to the Electoral College tie that occurred between Burr and Jefferson in 1800. The Burr dilemma suggests that approval voting can fail when strategic (i.e., non-sincere) voters falsify their preferences in order to "put all of their eggs in one basket" and withhold approval votes from everyone except their favored candidate. Range voting "fails" occasionally, but it fails much less often according to computer simulations, has less damaging consequences when it does fail (e.g., people are stuck with their second choice instead of their least-favorite choice), and is less likely to fail under real-world conditions (i.e., where many hardened partisan voters will give high scores to their party's candidate and a zero to everybody else).
posted by jonp72 at 6:11 PM on August 27, 2007

Under that particular system is that it's possible for the candidate with the highest average rating to not have been any voter's first choice.

Yeah, but the result would still be the least disappointing result possible for the largest number of people. I'd rather see my second choice be president than some guy I can't stand.
posted by designbot at 6:36 PM on August 27, 2007

This is fascinating stuff; thanks for posting it.

One thing stood out to me in the "election results" link as seeming really off:
10. One-party domination: One interesting consequence of the Borda's "teaming" effect above (that a party, by running many clones, increases its chances of victory) is that Borda can lead over time to one-party domination. (Sort of like two-party domination, but even less fun.) Here's an observation of that by Tim Hull.

I'm [a student] at the University of Michigan, and we use a variant of the Borda count for our elections where you get as many votes as open seats. Slates of candidates typically contest elections as "parties", and most discussion of elections revolves around these parties.
Anyway, the system as-is works better than at-large plurality, but it still leaves much to be desired. The biggest problem with the current system is that the largest party slate always wins a disproportionately high number of seats – so large, in fact, that competition has generally withered away.
But, how is range voting any better at preventing this?

Scenario: Party A has twelve candidates, and 60% support. Party B has four candidates and 40% support. Six seats are available. Range voting is used.

Party Aers all give scores in the nineties to Party A candidates and scores of 0 to Party B candidates (they have preferences among their own candidates, but strongly prefer any of them to a Party Ber). Party Bers do the same, but in reverse. Each candidate gets an average score in the nineties within their own party, and 0 in the opposing party.

So every Party A candidate gets an average score between 54 (90*.6) and 60 (100*.6), and every Party B candidate gets an average score between 36 (90*.4) and 40 (100*.4). So, all eight seats go to Party A candidates, even though Party B has a pretty heft percentage.

Also, I don't see how the system they describe (which seems to consist of giving everyone n votes and letting them distribute them among the candidates) will have the results they describe. I thought that with that kind of setup, it's a better strategy to have fewer candidates (to a certain point) to avoid splitting your party's vote up too much.

For instance: Same scenario as above, but now everyone gets six votes to distribute, instead of doing the range thing. There are a hundred voters, for simplicity. Party A gets 360 votes altogether, and they're split more-or-less evenly among their twelve candidates, so each gets about thirty. Party B gets 240 votes altogether, and they're split more-or-less evenly among their four candidates, so each gets about sixty. Party B gets four seats out of six, despite being in the minority. (If they'd had six, each of their candidates would have gotten about forty, and they would have had all the seats, unless Party Aers really prefered at least one of their candidates to the others.)

So with that kind of system, having a whole bunch of similar candidates for your side is basically giving the other side permission to completely dominate the election, even if they're smaller.

Can anyone explain this?
posted by Many bubbles at 8:54 PM on August 27, 2007

We should rate candidates on a scale of 1-5 stars or No Opinion, and take the candidate with the highest average rating.

In terms of Arrow's axioms, this one fails. As linked above, Arrow's axiom #1 says:

"If a candidate receives a majority of first place votes, that candidate should be the only winner."

Let's have an election with two choices (A, B) and 9 voters.

5 voters say A=5, B=4. 4 voters say A=0, B=5.

A has a score of 25. B has a score of 40. So even though A was the top choice of 5 out of 9 voters, B still wins.

Variations on this idea (e.g. every candidate has to be rated in order) fails axiom #3 (independence from irrelevant alternatives).
posted by Steven C. Den Beste at 9:12 PM on August 27, 2007

In terms of Arrow's axioms, this one fails. As linked above, Arrow's axiom #1 says:

"If a candidate receives a majority of first place votes, that candidate should be the only winner."

I just looked at the link, and while it's labeled that way, those aren't Arrow's axioms or conditions. Those are just five ways to evaluate voting systems.

Arrow's conditions can be stated a few different ways, but for me at least the most useful way is:

Social transitivity -- if we choose A over B, and we choose B over C, we should choose A over C.

Universal admissibility -- nobody's preferences will be ruled out because of the content of their preferences; any and all preference orderings are admissible.

Pareto optimality -- if everyone prefers A over B, we should choose A.

Independence from irrelevant alternatives -- if people who like chocolate ice cream better than vanilla change their minds about that, this shouldn't affect our choice between mashed potatoes and stuffing, since nobody changed their mind about that.

nonDictatorship -- there is no person whose voice is decisive over each and every decision.

Which, if you allow the goofy nonDicatorship, spells out STUPID which is arguably a useful mnemonic. For political science dorks out there, usage of this mnemonic is an easy way to play Spot The Aldrich Student

There are also several ways to prove the theorem, but they boil down to first showing that decisiveness is contagious -- If I'm decisive over the choice between A and B, that can spread to the choice over A and Y, or the choice between X and Y -- and then showing that someone is decisive over some choice.
posted by ROU_Xenophobe at 9:31 PM on August 27, 2007

I think we should just run the election on Digg.
posted by empath at 2:50 PM on August 28, 2007

designbot: Sorry I couldn't get on again until tonight.
jonp and Steven pretty much summed up range voting. It has results similar to Borda but a candidate with a majority of votes has the potential to lose, which really is an important point. In a non-practical sense I suppose the individual ballots could show lots of insight into how voters feel about the candidates. I always figured it would devolve into approval voting, though, with voters basically giving 1s or 5s and very few in between. Also, I don't trust individuals to "grade" without a rubric. Two people may feel the exact same way about a candidate, with one expressing that with a 4 and the other with a 3. Quantifying stuff like this is hard.

Thanks for posting, jonp. I love this stuff.
posted by monkeymadness at 7:34 PM on August 28, 2007

Gum, you have accurately refuted a position I don't hold. Regrettably, that means that the untangling of it in this post is going to be somewhat ... lengthy.

To start with, I think you've assumed that I assign the status 'detestable putz' lightly and often. No. Not even in politics. Even the most disagreeable people in my world seldom aspire to such a condemnation. I just really, really, shuddered to imagine either Kerry or Bush in the big chair. Let me also draw your attention to the hedging phrase, "such as they are," that immediately follows the one you jumped on, "my third party enthusiasms."

Eritain, third party "enthusiasms" are fine as long as your motivations for voting don't include helping your side win representation in government. [...] I'll bet a plurality of American voters are more interested [in] voting as individual expression than voting as collective action.

Let me note again that my state's electoral preference was an overwhelming given, and consequently my vote was guaranteed to have no effect on the outcome at large—a fact I knew when I took my ballot in hand. Whichever "side" I might have taken, on that particular line of that particular ballot I could neither help nor hinder its winning any representation at all. And with action ruled out, I saw an opportunity for expression and took it, on the theory that that's still better than nothing.

It so happens that among the things I expressed was a thorough dissatisfaction with both major-party "sides" (candidates really, and it is especially true in America that we vote for candidates and not for parties). But that doesn't mean disgust was the sole factor determining my vote. Had there been an opportunity to vote more meaningfully, I would have done so. But there wasn't. So I see no reason to insist that this was "throwing a vote away" as people tried to insist. Note to people who use that line: Anyone who openly discusses voting for third parties has heard it before. Some are too stupid to understand it, and the rest have refuted it to their satisfaction. So go straight for an intelligent discussion about preference paradoxes and cloneproofness, or go away.

OK, here's where the details come on heavy. Stay with me. (I'll call myself a pedant now and spare everyone the trouble: Pedant.) On the same ballot, in 2004, I cast two other third-party votes. In one race, my most-favored candidate (from one of the majors) was a frank shoo-in (somewhere around 80% in opinion polls). My next favorite was a third-party candidate who had very different views, but picked up my agreement on all the points where the first had not. The other major-party candidate brought nothing to the fight for my vote: one or two issue positions, very scanty qualifications and credibility, and that was the end of it. (As you can guess from that poll percentage above, said candidate didn't bring much to the fight for anyone else's vote, either. Polled ahead of the third-party candidate, of course, but that never takes much doing.)

So I thought, 'How can I do some good with this franchise I've got?', and I voted for my second-favorite, to help that party retain its legal (nominal) major-party status in the state . (Didn't happen. Oh well. Worth a shot.) If the other major-party candidate had done as well by me as the third-party kid did—or even anywhere close—I'd have had more important things to worry about than creating a little third-party-friendliness. But they didn't, and so that's how I swung.

In another race, I favored a third-party candidate (different third party! wouldn't agree with the other one! such are the paradoxes of categorization) who didn't stand a chance, but the two major-party candidates both looked like they'd do all right by me. Not great, but not much worse than my first choice; unlike the presidential election, my conscience would have been all right if I'd supported a major here. It was clear one of them would win, and yet neither one had given me any reason to care which. In that situation, it is precisely appropriate to stand back and let the rest of the populace decide.

But I could let them decide as well by voting sincerely as by not voting in that race, and it had the neat little side-effect of raising a tiny little blip in the numbers. Those tiny little blips are important. They serve as reminders of dissent from the two party lines—reminders that the two major parties are not the immutable cosmic opposites that our public discourse paints them as, but coalitions of diverse people and factions that Duverger's Law created; that they reflect the pressures of their electoral situation, and not just a simple sum of unanimous, sincere views within their membership. The blips say, "There will always be two parties. They will not always be these two parties. They might inherit their names, but their interests will differ." There are certain people whose power depends on promoting the illusions of intraparty unity and interparty polarity, and, well, illusion-based power doesn't sit well with me. (And now we have come back around to Duverger's Law, which I thought about and understood before that election, between that election and this FPP, in the FPP, and in your (gum's) comment, thank you very much.)

Now, gum, lest you take your condescending tone with me again, let me assure you that on other lines of that ballot I did vote for 13 candidates from one of the major parties and 10 candidates from the other. In no case was this because I agreed 100% with the one I voted for and disagreed 100% with the other. They were all compromise votes, with the aim of getting the best-qualified person in the seat, and the side effect of pushing some of each party one vote closer to my county and state government—very approximately as much of each party as there was of my thinking in that party. That is, it's the compromise game again. There were third-party candidates in some of those races, for whom I did not vote because I didn't think they were any good. I also voted in 16 nonpartisan races and about five other ballot measures (bond issues, initiatives, referenda, and the like). So no fair painting me as a rebel without a cluecause—"well, I'm just not gonna vote for either of 'em then!"—because that's not how it went down.

And meanwhile, if you want redemption, see if you can't substitute the fifteen-minute ceremony of voting for a third-party non-putz -- refreshing! zero calories! -- with the much more challenging and rewarding experience of engaging the putzes in the major party

Who says I ain't? (Yes, I voted in the primaries, too.) Y'know, professing as you do to appreciate democracy, you could show a little more faith in a fellow citizen.

Yes, I realize that now I'm fightin' dirty. I've just invited a fallacy of division—if the citizenry is competent, Citizen 'Tain is competent— and thereby set our readers up to make uncharitable inferences against your fairmindedness. But here I've called myself out, so tell you what, don't you fight dirty, 'n I won't either.

I do agree with you about the "massive constitutional amendment project" being impossible in practice. But I value the gap between "impossible in practice" and "unthinkable"—and that's (one reason) why I (sometimes!) vote third party.

Now, if you want to direct anyone to this post who is voting for third parties merely in a wasted effort to stick it to the man, or who is deliberately ignoring the situations that motivate strategic voting—have at it.

* * *

On an entirely separate subject, range voting appears to have advantages I had not been aware of. Rare for a system that's easy to explain to be free of crippling weaknesses. Thanks, jonp.
posted by eritain at 6:37 PM on September 1, 2007

Arrow’s impossibility theorem of democratic social choice assumes ranked preferences. It’s a reminder that both rank and range methods of voting violate the rationale for the majority rule (where all votes are equal, whether divided or not). The logic of single-winner does not assume methods of ballot ordering, or any incremental range votes, both of which require complex ballots. However, American-style plurality voting punishes a majority position with multiple candidates during an election by dividing it, often losing to the minority position—the spoiler effect. Approval voting, selecting everyone we like (virtually against those we don’t like), generates accidental spoilers by inviting unknown candidates to win by avoiding negative vote patterns.

Having more than two candidates should not challenge the utility of voting, nor threaten to be a tally of regrets and second-guesses. The problem with modern voting methods is that attempts to solve such problems have not always assumed the reasons for voting. A sincere voter could use a row of candidates to increase their chances of representation. However, the sincere voter in a robust election with good choices is naturally undecided between two or more candidates, and for different reasons. This should no longer be considered the problem, but the solution, because the voting function is to decide the most agreeable candidate among many choices.

I propose the following voting sieve method, which assumes one-member-one-vote equally, and offers a demand-side rationing of multiple votes without any regret in the process. In a single-winner election, if we allow the division of one vote into any number of equal parts, and then add the entire candidate column of those whole or fraction values, then the candidates with the least points can be eliminated by rounds. The sieve effect occurs after a candidate is removed from the tally, when the remaining fractions in the row are adjusted upwards, always adding up to the sum of one. For example, in a field of ten candidates, if I vote for three, then two of my three preferences will surely lose, but as they do, my marks change from one-third to one-half, and then to a whole vote as my selections are narrowed. Ties between candidates would be decided by the one with the most points in the first round.

The rational voting sieve rejects the candidate who lacks a corresponding depth of voter commitment in relation to other candidates, nor is this depth decided by ordinal ranks or spoiled range votes. There is also a high enough probability for a clear majority winner so that it can be required. This method also stands alone as a proportional vote for a parliamentary-style election without any rounds at all. Perhaps its most promising aspect is the reliable information that is gleaned from sincere polling, which can be used as a consumer recommendation model when compared with the preferences of others.