Range Voting

A little known or discussed fact of life in the US is that, while we are a younger nation than most European countries, our constitution and our voting system is older than any of theirs. Why? Because they have updated their constitutions and/or voting systems while ours have remained static since their inception over 200 years ago. While other countries have taken the pragmatic view that upgrading (taking into account advances in voting theory) might be a good idea, our voting system along with all other aspects of our constitution (not to mention our economic system) might as well have been set in stone. To even question their continued validity or possible improvement is considered by some to be unpatriotic.

Our voting system is called plurality voting. This means that only one candidate out of however many are running for office is chosen by each individual voter, and then the votes are tallied and the one with the most votes wins providing he or she has a sizable enough percentage of the total votes cast. Otherwise there's a runoff. The literature is full of the pitfalls of this method of voting. Suffice it to say it practically eliminates more than two parties. Another disadvantage of our political system is that the US is divided up into districts with each district returning one representative to the bicameral House while each state returns two Senators to the Senate. So each voter is represented in Congress by only three people: one member of the House and two Senators. That's a very small proportion of the total number of lawmakers when you consider that there are 435 Congressmen and women in the House and 100 Senators. 3 out of 535? 0.5%? How does that make you feel, voters, to know that less than 1% of the lawmakers in Washington represent you or that you had anything to do with electing them? In addition there are the problems of the Electoral College (the President is not elected by a direct vote of the public) and gerrymandering which is an artificial way of dividing up political districts in such a way that favors one party or another. Of course, the party in power gets to do this in order to guarantee itself as many seats in perpetuity as possible. A more sensible way to choose a President is by a majority of all the votes cast eliminating the Electoral College altogether, and a more sensible way of choosing legislators would be a system in which each voters gets a chance to vote for more than three legislators.  

Fortunately, voting theorists have not been inactive. There are at least three methods which promise to be an improvement over the current state of affairs vying for the hearts and minds of the public - at least some members of the public, the cognoscenti, who have bothered to consider such things: range voting (RV - not to be confused with those behemoth gas guzzlers), instant runoff voting (IRV) and approval voting (AV). Each method has its supporters, defenders, detractors, protagonists, antagonists and critics. Approval voting is simple enough. Instead of voting for one out of however many candidates are running, you vote for all the  candidates you approve of. Instead of marking the ballot once in each race, you mark it multiple times. Obviously, it wouldn't make sense to mark it for every candidate who is running although I'm sure there are many who would. IRV is more complicated, but you can Google it and find out more than you really wanted to know. That leaves range voting.

You are already familiar with it as this is the method used to score Olympic athletes. You would rate each candidate on a scale from 1 to 10 or 0 to 20 or -99 to +99. The actual limits to the range are somewhat arbitrary as long as each voter can assign ratings over the same range. Then you tote up the score for each candidate and the one with the most points wins. Or you can take the average and the one with the highest average wins. There is one refinement, however, with reference to the above linked website. You do not have to actually vote for each candidate. If there are 100 running and you actually rate only 50 filling in an X for the remaining fifty, then the average is computed by dividing the total (over all voters) votes cast for a candidate divided by the sum of all the voters who actually cast a vote for that candidate (Xs excluded). A candidate needs a quorum of votes to win where a quorum is defined as half the total points of the highest point getter. So, theoretically, a candidate with half the total points of the highest total point getter could win the election if sufficiently few voters actually voted for him or her while sufficiently many voters put down an X for him or her. This seems to introduce a certain amount of arbitrariness to the method, but, obviously, something would have to be done to prevent a candidate whom only a handful of voters voted for (all presumably giving him the highest score) from winning. An alternative way of handling this situation would be to just use total point scores forcing each voter to make a decision regarding each candidate. For candidates unfamiliar to any particular voter, that voter could use a proxy rating. A proxy rating would be provided by an expert of the voter's own political persuasion or someone trusted by the voter who is familiar with the candidate. It could be a rating provided by the voter's political party. Without having done an extensive analysis, this seems somewhat less arbitrary to me. 

Donald Saari is a proponent of the Borda count. Steven Brams is a proponent of Approval Voting and Warren D Smith is a proponent of Range Voting. I also independently came up with a version of range voting which I called the Lawrence Count. Due to the serendipity of the internet, someone perusing my website, Social Choice and Beyond, brought to my attention that, unbeknownst to me at the time, there exists a major website promoting and expounding on a similar voting method. Thus are like-minded people all over the world brought into contact with one another thanks to the world wide web! This is truly amazing! As I said on my webpage: "The Lawrence Count is a modified Borda Count which seems so obvious that maybe someone has already discovered it. If so, I relinquish the name and any claim to being its progenitor." So I will change my webpage to indicate that I'm no longer pretentious enough to name something after myself especially if someone else has already discovered it. By the way Warren D Smith doesn't claim to have discovered it either so  I guess it was too obvious for anyone to have discovered it, and Smith claims that actually the ants and honey bees discovered it.

The three gentlemen named above go round and round trying to prove that their method is the best and showing the pitfalls of all the others. My opinion is that approval voting is a step above plurality voting but is too simple to take into account the range of expression a voter may wish to demonstrate. It's as if Olympic ice skaters were either approved or disapproved by the judges and then their scores computed. Would you be satisfied with that? The Borda count is too rigid. It has the anomaly that, if one candidate drops out of the race, point values must be reassigned with the result that someone could win who previously was ranked last. Range Voting, in my opinion, is a modified Borda count in that the underlying grid remains constant whether or not candidates enter or drop out of the election. Point values remain the same except for some strategic considerations which, again in my opinion, the voter has a right to take into consideration. There is maximum expressivenesss in that the voter not only gets to rank the candidates but also gets to indicate to an extent how much he or she favors one over another. If each voter gives his or her most favored candidate the highest possible score and his least favored the lowest posssible, then he or she will be getting the most strategic value out of his or her vote. If there is one candidate who is so horrible compared to all the others in the mind of a particular voter, it would not only be honest but strategically advantageous to give the horrible candidate a zero and all others the highest possible rating. Likewise, if there is one candidate who is far and away the most superior compared to the other candidates, it would be honest and strategic to give that candidate the highest rating and all others a zero. Another point is that the point spread from lowest to highest score (say 1 to 99) need only be as great as the sensitivity of the most sensitive voter where sensitivity is defined as the most perceptive voter's ability to make a one point distinction between two candidates. Presumably some voter would be able to rate some candidate in a meaningful way as a 53 and another as a 54, for example.

One reason for the lack of agreement regarding voting methods is Arrow's Impossibility Theorem which states in general terms that there is no democratic voting system which obeys certain rational and ethical criteria. This has given rise to the field of social choice theory. Not everyone would agree with Arrow's choice of rational and ethical criteria but, be that as it may, in over 60 years no one has been able to show that Arrow didn't know what he was talking about. So even if in some sense democratic voting systems are impossible, elections are still held and some voting methods are definitely better than others, and, even though there is no general agreement, there is still hope that we will be able to do better in the future than we have done in the past. Hope springs eternal!

Comments

[CLAY SHENTRUP]

If each voter gives his or her most favored candidate the highest possible score and his least favored the lowest posssible, then he or she will be getting the most strategic value out of his or her vote.

This is incorrect. "Polarizing" your most and least preferred candidates is only the first step in maximizing benefit. The general best strategy is to find some utility threshold that you feel you can expect from the result of the election, and to maximize the scores of all candidates above that threshold, and minimize the scores of the others. The exception to this is that one should strategically minimize the lesser-preferred candidate between the two perceived front-runners, even if one likes both front-runners.

As for Arrow's theorem, there is no doubt that it is a significant problem for ordinal (ranked choice) voting methods. I would go so far as to say that it proves the invalidity of the very notion of ordinal voting methods in the first place. Range Voting is a cardinal, not ordinal, voting method, and so it is immune to Arrow's paradox. See:
http://rangevoting.org/ArrowThm.html

Also, Approval Voting is Range Voting, with 2 options instead of 10, or 100 for example.

Now, since we're looking for the "best voting method", let's go ahead and discuss which method has the greatest social utility efficency, that being the ultimate metric of average/expected voter satisfaction with the election outcome. Here are some sample figures from
http://RangeVoting.org/vsr.html

Utility measurements: Group A: 5 candidates, 20 voters, random utilities; Each entry averages the results from 4,000,000 simulated elections. Group B: 5 candidates, 50 voters, utilities based on 2 issues, each entry averages the results from 2,222,222 simulated elections.

** Voting system SUE A SUE B **

Magically elect optimum winner 100.00% 100.00%
Range (honest voters) 96.71% 94.66%
Borda (honest voters) 91.31% 89.97%
Approval (honest voters) 86.30% 83.53%
Condorcet-LR (honest voters) 85.19% 85.43%
Range & Approval (strategic exaggerating voters) 78.99% 77.01%
IRV (honest voters) 78.49% 76.32%
Plurality (honest voters) 67.63% 62.29%
Borda (strategic exaggerating voters) 53.26% 51.78%
Condorcet-LR (strategic exaggerating voters) 42.56% 41.31%
IRV (strategic exaggerating voters) 39.07% 39.21%
Plurality (strategic voters) 39.07% 39.21%
Elect random winner 0.00% 0.00%

These results are from 2 of 720 different models, with millions of elections simulated for each model. Note that range voting is approximately as great an improvement over plurality voting, as plurality is over random selection; range voting effectively doubles the benefit brought about by the invention of democracy. These experimental results also strongly suggest that range voting is the least susceptible to strategic voting, of these common methods.

[John Lawrence]

"If each voter gives his or her most favored candidate the highest possible score and his least favored the lowest posssible, then he or she will be getting the most strategic value out of his or her vote.

This is incorrect. "Polarizing" your most and least preferred candidates is only the first step in maximizing benefit. The general best strategy is to find some utility threshold that you feel you can expect from the result of the election, and to maximize the scores of all candidates above that threshold, and minimize the scores of the others. The exception to this is that one should strategically minimize the lesser-preferred candidate between the two perceived front-runners, even if one likes both front-runners."

How do you know what "utility threshold that you feel you can expect" from an election? It seems to me that in the absence of knowledge, an honest expression of preferences would be the best policy since a distortion of this might result in your "getting what you asked for."

"The exception to this is that one should strategically minimize the lesser-preferred candidate between the two perceived front-runners, even if one likes both front-runners."

If one is pretty certain that the election will come down to a contest between the two front runners, wouldn't you give your favorite of the two the highest score and all others including the other front runner the lowest score?

"As for Arrow's theorem, there is no doubt that it is a significant problem for ordinal (ranked choice) voting methods. I would go so far as to say that it proves the invalidity of the very notion of ordinal voting methods in the first place. Range Voting is a cardinal, not ordinal, voting method, and so it is immune to Arrow's paradox."

I may be wrong about this, but I seem to recall from the literature that even ordinal methods escape from Arrow's paradox if all you want to do is pick the top ranked winner and not give the entire social ranking.

[CLAY SHENTRUP]

> How do you know what "utility threshold that you feel you can expect" from an election?

Pre-election polls, such as this one:
http://zohopolls.com/us/pres

> It seems to me that in the absence of knowledge, an honest expression of preferences would be the best policy since a distortion of this might result in your "getting what you asked for."

Nope. See page 2, theorems 2 and 3 here:
http://rangevoting.org/RVstrat.pdf

> If one is pretty certain that the election will come down to a contest between the two front runners

One can be 100% certain that it will. The uncertainty is in guessing the front-runners, based on pre-election polls and such.

> wouldn't you give your favorite of the two the highest score and all others including the other front runner the lowest score?

What about people you prefer to your favorite between the front-runners? That is, say Obama and McCain are the clear front-runners, and I like Obama a bit more than McCain, so I give McCain a 0 and Obama a 10. But my favorite candidate is Ron Paul. Why on Earth wouldn't I want to give Paul a 10 as well? The worst that could happen is that he wins, which is good for me.

> I may be wrong about this, but I seem to recall from the literature that even ordinal methods escape from Arrow's paradox if all you want to do is pick the top ranked winner and not give the entire social ranking.

Incorrect. Here's an example using plurality voting.

Actual rankings:

#voters their vote
28 A>B>C>D
25 B>C>D>A
24 C>D>B>A
23 D>C>B>A

If we held a plurality election here, A would win, even though a HUGE majority of the voters prefers all three other candidates over A. That is Arrow's paradox in effect.

A detailed description is available at the link I provided: http://rangevoting.org/ArrowThm.html

[John Lawrence]

Thanks for enlightening me, Clay!

Upon further reflection, it seems to me that, in the absence of polling data, one would indicate honest preferences after giving the most favored candidate the maximum rating and the least favored candidate the minimum rating. After polling data has been received, we can attach a "probability of being elected" to each candidate. Using this information we would change or distort our original "honest" ballot based on the probabilities associated with each candidate, but I don't think that this would necessarily mean that we would automatically give our preference of the two front runners the maximum rating because our original most favored candidate (as you pointed out) still has some chance of winning, and, even though we would elevate our favorite between the two front runners, we might not want to maximize him since this would reduce somewhat the chances of our actual favorite. I'm sure there must be an optimal way to assign points based on the associated probabilities. Has this theory been worked out?

Likewise, we might not want to minimize all candidates less favored than the two front runners because we do distinguish among them in our "honest" preferences, and, since any of them have at least some chance of winning, we might want to distinguish among them even after polling data has been received just in case one of them actually won. For instance, assume A and B are originally rated below your original most favored candidate and what turns out to be the two front runners. Then, even though neither of them are likely to win, they still have a probabilistic chance of winning. Let's say you preferred A to B in your original rating, and they are below the 2 front runners. If you rated them both zero after the polling data was in and then B actually won by 1 point, it seems that you would regret not having distinguished A from B by 2 points.

[CLAY SHENTRUP]

> Upon further reflection, it seems to me that, in the absence of polling data, one would indicate honest preferences after giving the most favored candidate the maximum rating and the least favored candidate the minimum rating.

Nope. If you had to vote on something without first getting any data about how others would vote, you would want to min/max all of the candidates such that the sums of the pairwise utility differences from max to min was maximized. So say you had candidates A-D with utility values as

A:1 B:13 C:15 D:32

If you voted as follows

A:0 B:0 C:10 D:10

then your sum of differences would be

C-A + C-B + D-A + D-B =
15-1 + 15-13 + 32-1 + 32-13 =

16 + 31 + 19 = 66

If you can find a way to polarize the candidates in such a way as to get that number higher than 66, then you've got a better strategy.

> but I don't think that this would necessarily mean that we would automatically give our preference of the two front runners the maximum rating because our original most favored candidate (as you pointed out) still has some chance of winning

No. If he had a chance of winning, he would _be_ a front runner.

> and, even though we would elevate our favorite between the two front runners, we might not want to maximize him since this would reduce somewhat the chances of our actual favorite.

Again, if you think your true favorite has a real chance to win, then he is a "front runner", and you would want to strategically minimize the other front-runner anyway.

> I'm sure there must be an optimal way to assign points based on the associated probabilities. Has this theory been worked out?

Yes. See the theorems in the PDF that I linked you to above. They explicitly describe the optimal strategies (which nearly always involve min/maxing every candidate).

> Likewise, we might not want to minimize all candidates less favored than the two front runners

Most certainly you don't. Say the two front-runners are your least favorite 10 candidates. You would still want to give one of them a min and the other a max, to maximize your effect on the real election. You then might as well maximize the scores for everyone else if you like.

I would re-read that PDF, as I think it addresses most of what you're bringing up here.

[John Lawrence]

>They explicitly describe the optimal strategically which nearly always >involve[s] min/maxing every candidate).

I got it! You would strategically maximize all the candidates you rate higher than your least favored of the two front runners and minimize everyone else. Wouldn't this be the same as approval voting with a twist?

Instructions to the voters could be very simple: 1) Rank all the candidates. 2) Based on the last poll before election day, maximize all candidates you rank higher than your least favored of the two front runners. 3) Minimize all others.

If range voting is used, all ballots would consist of only +99s or 0s. Wouldn't this make range voting moot? All you would need is strategic approval voting.

[CLAY SHENTRUP]

> I got it! You would strategically maximize all the candidates you rate higher than your least favored of the two front runners and minimize everyone else.

Nooo. You maximize any candidates you prefer to your MOST favored among the front runners. That is, you should maximize the score for anyone you like as much or more than anyone else you maximize. Likewise you should minimize your score for anyone you like _less_ than anyone else you minimize.

> Wouldn't this be the same as approval voting with a twist?

Under strategy, Range Voting essentially degrades to Approval Voting. Hence the social utility figures for strategic Range and Approval are the same in the SUE listings above. But in practice a lot of people will honestly use intermediate options, which increases the utility efficiency, and gives better representation (by a LOT) to minor party candidates.

> Instructions to the voters could be very simple: 1) Rank all the candidates. 2) Based on the last poll before election day, maximize all candidates you rank higher than your least favored of the two front runners. 3) Minimize all others.

NOOOO! Instructions should be, "Mark the score you wish to assign to each candidate. The candidate with the highest average (who meets the quorum) is the winner."

That's IT. If anything, you could specify for them to give their most favored candidate a "10" (or 99, or whatever range you decide to use) and their least favorite a 0, and score the other options with respect to those two. I basically did this in my exit poll last November in Beaumont, TX. See: http://RangeVoting.org/Beaumont.html

> If range voting is used, all ballots would consist of only +99s or 0s.

I'd suggest 0-10, because I think 0-99 is overkill.

> Wouldn't this make range voting moot? All you would need is strategic approval voting.

I don't know what you mean by "need". We don't "need" democracy. We, for the most part, want it. And if we're going to have democracy, we might as well pick the best winners possible. Range Voting has a significantly higher social utility efficiency than Approval Voting. If every voter voted strategically, it would be a moot point, but a lot of voters vote honestly - ESPECIALLY about candidates they do not perceive to be a threat. This offers a HUGE benefit over Approval voting. Say people honestly want to give Nader a 5. With Range Voting, they'll generally do it. With Approval Voting, it's like they can only give him a 0 or a 10 - nothing in between. To be on the safe side, they'll choose 0, and we'll have almost no idea how much real support minor candidates will have.

So in a hypothetical world where voters were more strategic, yes, I wouldn't advocate the added "complexity" 0-10 Range Voting over 0-1 Range Voting (Approval Voting). But we might as well take advantage of all that honesty out there.

[CLAY SHENTRUP]

Let me further add to clarify: explaining to the voters how to vote strategically is EXTREMELY bad. It reduces the social utility efficiency severely. The last thing on Earth you want people to do is polarize the two front-runners. They should only do that if they happen to stumble upon the idea and want to be strategic. We should do everything possible to encourage honesty so that the results produce the greatest average utility for the electorate.

[John Lawrence]

OK. If you wanted every voter to vote strategically (which I understand you don't), instructions to voters would be as follows: based on the last poll before election, maximize your favorite of the two front runners and everyone you rank above him or her and minimize everyone else. If you gave these instructions to the voters and everyone voted strategically, you would have an "honest" election procedure since all the strategy would have been taken out of it. No one would have a strategic advantage. It seems to me that this would be very advantageous compared to keeping voters ignorant except for the few who"stumbled" on the correct strategic procedure. In fact it seems almost imperative for a fair election. And again it would seem to make range voting superfluous.

I agree that, if everyone voted honestly, there would be a better election outcome in terms of the increased utility of society as a whole, but stable and fair election procedures would seem to me to be more important. Also, as you pointed out in your comments, there is an optimum procedure for maximizing the effectiveness of one's ballot even in the absence of polling.

Perhaps voters would be persuaded to vote honestly if they knew that the election system itself would strategically maximize their ballot ie the voter's ballot would be the input to a black box that would strategize and maximize the effectivenesss of the voter's ballot. This would be under the control of election officials and not the voter himself. Then the voter would have an incentive to vote honestly since to not do so would result in his not using his vote to maximum advantage. Then you could have it both ways. Honest voting and strategic voting. The black box would become part of the processing of the votes and hence part of the vote counting process.

[CLAY S]

If you wanted everyone to vote strategically you would tell them to maximize their favorite among the front-runners, as well as any non-front-runners that they liked even more than him. Then minimize the less-liked front-runner, and then polarize the remaining candidates in such a way as to maximize the pairwise utility differences, as I previously explained. I think that's about as good as an evil tactical voter could do.

[CLAY S]

stable and fair election procedures would seem to me to be more important.

1) I'd argue that the most important thing is having the highest social utility efficiency, even if the system weren't "fair" (say it gave some voters more ballots, but still produced greater average happiness than a "fair" system).
2) Range Voting is fair, because all voters get one ballot, and have the same options. It's also arguably the most stable method, because it gravitates the most strongly toward the ideological center, where the best compromise candidates lie.

Also, as you pointed out in your comments, there is an optimum procedure for maximizing the effectiveness of one's ballot even in the absence of polling.

So what? Even if everyone uses that, Range Voting still performs better than (or approximately as well as) IRV with 100% honest voters.

Perhaps voters would be persuaded to vote honestly if they knew that the election system itself would strategically maximize their ballot ie the voter's ballot would be the input to a black box that would strategize and maximize the effectivenesss of the voter's ballot.

Ugh...NOOOO. That would defeat the point of voting honestly in the first place. And it would effectively never terminate, in the case of a Condorcet cycle.

I can't blame you for bringing this up actually...it was one of the first things I asked about when I was new to the concept.

CLAY

[John Lawrence]

The study of a combination of honest and strategic voters - http://www.rangevoting.org/StratHonMix.html - shows that social utility will be somewhere between that for totally honest voters (maximum) and totally strategic voters (minimum). However, it's randomized strategic voters. What happens when all the strategic voters vote in a bloc and all the honest voters, well, vote honestly? This isn't far-fetched because of the following scenario: All Republicans vote strategically in a bloc and all Democrats vote honestly. Over a number of elections, who would get their way more often? The Republicans, of course. And isn't that what they're all about anyway? Taking as much power as possible using any means at their disposal? I maintain that Republicans' utility would consistently be more than the Democrats' utility, and this, I think, would defeat the notion that, if some voted honestly and some strategically, society would be better off than if all voted strategically. Yes, maybe when you compute the utility of society as a whole. But it's not fair to Democrats, in the above example, if you compute the utilities separately.

[Clay Shentrup]

What happens when all the strategic voters vote in a bloc and all the honest voters, well, vote honestly?

Read further down that page, and you'll notice Warren covered that type of scenario. IRV voters are paragons of virtue - perfect little angels - while 50% of Range voters are strategic, and Range Voting still produces the greatest voter satisfaction.

All Republicans vote strategically in a bloc and all Democrats vote honestly. Over a number of elections, who would get their way more often?

1) Range Voting would still be better than other voting methods for the minority/losers.
2) What makes you think such factional disparity would occur anyway?
3) All of this is covered in the link you referenced.

I maintain that Republicans' utility would consistently be more than the Democrats' utility, and this, I think, would defeat the notion that, if some voted honestly and some strategically, society would be better off than if all voted strategically.

Well, it would be. More people would be more happy. I agree that it's not good for honest people to be "taken advantage of" by strategic ones; but that's going to happen with any voting method. Range Voting at least greatly mitigates the harm caused by such behavior.