opinions-organizations-votingMethod

Criteria for voting methods

I have never found an analysis and comparison of voting methods that uses the following principle:

In situations in which the vote in sufficiently close, I don't care what the result is, provided it is selected from the "reasonable set"

In other words, I don't care about all these pathologies/violations of properties if they only occur when the vote is really close, and as long as they don't select a "dark horse", i.e. someone who is not one of the popular candidates who should "reasonably" be the only ones who can win. If the vote is close, I consider the will of the group to be indeterminate, and I wouldn't mind flipping a coin.

This doesn't rule out all consideration of pathologies. I won't tolerate the spoiler effect, in which a compromise candidate loses (even if they are everyone's second choice), except when elections are close. And as stated above, I won't tolerate pathologies like <a href="http://www.rangevoting.org/DH3.html">http://www.rangevoting.org/DH3.html</a>, even when they are close. One might distinguish between minor and major pathologies, major ones (such as DH3) being intolerable even when elections are close.

The rest of this webpage won't apply the previous principle much because I haven't seen much research working along those lines, and I have not yet had time to research it myself.

In addition to the previous principle, there is another principle that I have seen used, but not universally:

The system must return good results when any proportion of the population is sincere/strategic (i.e. from 100% sincere to 100% strategic, and anywhere in the middle).

In other words, a system is no good if it requires sincerity or if it requires strategy. However, I am willing to bend this in the direction of allowing a system which requires some strategy, provided that strategy is not very complex. (note: according to http://rangevoting.org/DonSaari.html , this is opposite to the preference of Saari, who apparently prefers to cater to sincere voters first)

"Good performance" means the following:

Finally, as for strategies, I am especially sensitive to the situation where each voter faction is strongly polarized and has trichotomous preferences about the candidates (i.e. each candidate is either a "good guy" or a "bad guy" or "unknown").

Which single-winner voting system do I currently prefer?

For single-winner elections, i am trying to decide between range voting and Condorcet. Currently i prefer range.

Range vs. Condorcet

i am still reading about it, but so far here are two arguments that i find persuasive in favor of range over Condorcet:

(1) Bd the DH3 pathology

When there are a number of competitive frontrunners, the strategic choice in Condorcet is to pretend that you prefer generally unpopular candidates over the opponent frontrunners. this can lead to the unpopular candidates winning.

<a href="http://www.rangevoting.org/DH3.html">http://www.rangevoting.org/DH3.html</a>

(DH3 = "dark horse 3")

(2) spoiler effect/wasted votes/favorite betrayal)

if you like a 3rd party (call them "C"), but you think they are not competitive, and you have a preference between the popular contenders (call them "A" and "B"; let's say you like "A"), then under condorcet you can have more influence in the result if you insincerely rank A over C. so 3rd parties are punished in Condorcet, similar to how it works in plurality.

<a href="http://rangevoting.org/IncentToExagg.html">http://rangevoting.org/IncentToExagg.html</a>

(3) More Condorcet than Condorcet

I don't yet understand the proofs of the arguments in this section as well, so I'm not as sure about this part.

(note that I use range voting and approval voting interchangably, since in some situations the optimal strategy for range voting may look a lot like approval voting)

The argument is that, paradoxically, although Condorcet methods are guaranteed to pick the "nominal" Condorcet winner, due to strategic incentives, Condorcet methods will actually pick the "real" Condorcet winner less often than range voting.

The "nominal" Condorcet winner is a candidate who fulfills the Condorcet criterion with respect to the preferences stated by the voters when they vote.

The "real" Condorcet winner is a candidate who fulfills the Condorcet criterion with respect to the sincere preferences of the voters.

The reason these might be different is that strategic voters might choose to misstate their preferences during voting.

Condorcet can fail to due to strategy

Because of strategic voting, it is possible for voters who are using a mixture of strategies with Condorcet to elect someone other than "real" Condorcet winner, even when a real winner exists: <a href="http://www.rangevoting.org/CondStratProb.html">http://www.rangevoting.org/CondStratProb.html</a>.

Under certain condition, approval voting always elects the Condorcet winner

When certain conditions hold, and when there is a Condorcet winner, approval voting always elects that Condorcet winner: <a href="http://www.rangevoting.org/AppCW.html">http://www.rangevoting.org/AppCW.html</a>. I am not very comforted by the following assumption, however: "they order the candidates from best-to-worst, then select a "threshold" T, and they approve the candidates above T." -- Wikipedia says, "An optimal approval vote will always vote for the most preferred candidate and not vote for the least preferred candidate. However, an optimal vote can require voting for a candidate and not voting for a more preferred candidate."

Wikipedia gives a stronger theorem: "... if all voters are rational and cast a strategically optimal vote based on a common knowledge of how all the other voters vote except for small-probability, statistically independent errors in recording the votes, then the winner will be the Condorcet winner, if one exists.", citing Laslier, J.-F. (2006) "Strategic approval voting in a large electorate," IDEP Working Papers No. 405 (Marseille, France: Institut D'Economie Publique). See also When Voters Strategize, Approval Voting Elects Condorcet Winners but Condorcet Methods can Elect Condorcet Losers.

Under certain assumptions, range is more likely to elect the Condorcet winner than Condorcet methods are

Under the previous assumption, as well as some others including the assumption that all voters have a good idea of the positions of other voters (perhaps this could be accomplished via iteration? I don't know of a theorem that shows convergence, though), it has been shown that, because of strategic voting, approval or range might be more likely to elect the real Condorcet winner than Condorcet methods themselves: <a href="http://www.rangevoting.org/RangeYieldsCondSumm.html">http://www.rangevoting.org/RangeYieldsCondSumm.html</a>.

Footnote 12 in this paper argues the same: http://www.nyu.edu/gsas/dept/politics/faculty/brams/theory_to_practice.pdf

Cases based on foreknowledge

The following page argues that strategic range voting (under the same assumptions as before) will always elect the Condorcet winner if one exists and if they are thought to be one of the top two contenders beforehand: http://www.rangevoting.org/ReasAssump.html . Furthermore, the Condorcet winner MIGHT be elected if they are the third contender. This is not the case for IRV or plurality.

Finally, if knowedge of the other voter's dispositions is given by a pre-vote poll in which everyone votes sincerely and in which votes are tallied in the same way as the actual vote, then that page argues that there is more chance that range voting will end up with the Condorcet winner placing first or second than for IRV, plurality, or approval voting.

That page contrasts this with Condorcet. If a sincere Condorcet pre-poll identifies the Condorcet winner, then strategic voters could make the actual Condorcet vote come out so that someone else wins.

Links

here is rangevoting.org's summary page for range vs condorcet:

<a href="http://www.rangevoting.org/CondorcetExec.html">http://www.rangevoting.org/CondorcetExec.html</a>

Possible disadvantages of range voting

Lessening the impact of moderates

The strategic choice for range voting is to give the highest possible rating to your favorites, and the lowest to your most feared opponents. "Moderate", non-strategic voters who don't give the top rating to their favorite candidate, nor the bottom rating to their favorite's major competitors, dilute the power of their vote. In effect, giving any score other than the highest or the lowest is saying, "I voluntarily give up some of my voting power because I don't care very strongly; I am willing to let the other, more opinionated voters have a greater say in the decision".

This could lead to a situation where people who are careful and measured in their opinion, and who sincerely let their votes reflect that, are underweighted, and people who are overly sure of themselves are overweighted. This would be bad because (if you assume that moderates are more intelligent than extremists) it underweights intelligent people.

However, I suspect that instead, what will happen is this. Intelligent people will discuss with other intelligent people and the fact that the best way to "get your way" is often to vote high-low will become widely known, at least among intelligent people, and probably among everybody. During a vote, intelligent people will vote high-low on those candidates that they have an opinion about. If they vote in the middle on other candidates (in a way that lessens the impact of their vote), it will only be when they make a conscious decision to let other people decide amongst their "middle choices", because they don't know enough or don't care enough about them. If they think that much of the voting public consists of people who are too ignorant or headstrong to be granted that power, then the intelligent people will not choose to cede their power that way.

So, in other words, intelligent people will not be "taken advantage of", unintentionally lowering the impact of their vote because they don't understand what's going on. Sooner or later, all of the intelligent people will learn the actual effects of voting various ways in range voting, and they will use this knowledge to wield range voting to their best advantage.

If anyone is "fooled" by range voting's ability to let you declare "I don't care as much" by voting in the middle, it will be less intelligent people. So the intelligent people will be overweighted, not underweighted.

Problems with range voting

Condorcet loser

Under some conditions, approval voting can elect the Condorcet loser! Here is the example from the wikipedia page on approval voting:

The voters all live in one of four cities. Depending on which city they live in, they like each of the other cities a certain amount. They want to vote on a city (i.e. the candidates are the 4 cities). Here is a table that tells you how many people live in each city, and how much the residents of each city like the other cities:

City Faction of Voters Memphis Nashville Chattanooga Knoxville
Memphis 42%10015100
Nashville 26%01002015
Chattanooga 15%01510035
Knoxville 17%01540100

Each voter votes for ("approves of") any candidate with an above-average likeability. In this case, each cities' voters vote for only their own city, except for Knoxville, whose voters approve of both Knoxville and Chattanooga. Memphis wins. But Memphis is the Condorcet loser! (Nashville is the Condorcet winner).

Note that in this example, the voters weren't strategically taking into account any information about what other voters were going to do. If, however, the voters believe that Nashville is the likely winner, and Memphis is the likely runner-up, and if they act on that belief by approving of cities which are preferred over the Nashville (and they approve of Nashville if they like it over Memphis), then Nashville wins. (if the voters think Memphis is favored to will win, followed by Chattanooga (the result of the "no-information" case that we discussed first), then there will be a three-way tie).

Perhaps there is a theorem similar to the "no-conflict theorem" in http://www.rangevoting.org/AppCW.html that shows that strategic approval voters with foreknowledge of everyone's preferences never elect a Condorcet loser? I would suppose that if it is thought that a Condorcet loser may be elected, everyone is going to vote against (not approve) them.

Indeterminacy

Given a population of sincere voters with fixed preferences, there can be many election outcomes in approval voting, depending on how the voters decide to set their thresholds, i.e. how many candidates they approve.

See Saari, D.G. and Van Newenhizen, J. (2004) "Is approval voting an ‘unmitigated evil?’ A response to Brams, Fishburn, and Merrill" Public Choice 59(2) pp. 133-147.

An example is given (I've changed it somewhat here). Let's say there are 100 voters. 99 have the same preference: they love A, they are neutral about B, and they hate C. So, they all approve both A and B. The 100th voter loves B, is neutral about C, and hates A. So he approves B and C. So B will win. As the paper says, that is because approval voting only allows the voter to partition the candidates into two groups. The first 99 voters had the power to separate A and B from C, but they couldn't also express a separation between A and B.

Note that strategy can prevent this outcome. If only a few of the 99-strong faction vote for just A, then A wins. Coordination is not needed; each voter in the faction could, with probability 90% vote A and B, with probability 10% vote just A. I note that this sort of strategy is easy to accomplish, because it's the sort of thing that might happen naturally anyway. Of course, in more contentious elections, there would be benefit to tightly coordinating the number of people doing one thing or another. Hopefully, by the time this becomes an issue, the election will have become so close that, under my first principle, I don't care if there are minor pathologies. But I'm not at all sure.

In response to implications that use of strategy can prevent bad outcomes, the article states that "With AV the voter needs strategies just to be sincere." This may be true, but if the choice is between requiring strategy or requiring sincerity (as noted above, Condorcet seems to require sincerity in order to elect the Condorcet winner in some cases), I would choose to require strategy.

In section "A positive feature of AV", the paper opines that approval voting works well when voters have dichotomous preferences, and badly if their preferences are more complex. I think that is the sort of situation that inspires strategic voting, however, so a method that is partially specialized to that situation is not so bad.

In the section "Strategic manipulation and sensitivity", the paper says that the results of approval voting can often be changed by a small number of individuals.

In general, I found this paper lacking in that it constantly made broad analytical claims about how most profiles have instability or sensitivity, but I feel there is a prior probability distribution over profiles. Hence, I don't care if the raw number of all possible situations makes approval voting come out bad, I only want to know if it is likely to come out bad, given my prior distribution. For example, the paper notes that AV works better when voters have dichotomous preferences, and I think that situation will come up often.

Unfortunately, I'm not exactly sure what my prior distribution is. However, I'm hoping that given examples which are particularly likely or unlikely, I would be able to identify them. This state of ignorance makes the analytical method of the paper inapplicable. What is needed is a boatload of examples of all these times where approval voting fails. Then I could look at the examples and determine if they seem likely to me; if they do not, perhaps I will be able to pull out a set of general properties that they share that I think are unlikely to occur, and then a theorem could be proved about approval voting being good under restricted conditions. But this paper is strongly in favor of the analytical approach and doesn't provide many examples.

In other words, there is room to hope that most of the bad outcomes that are mentioned in the paper come from scenarios that I would consider unlikely.

multiple round idea

i'm sure this has been thought of, but how about having multiple rounds of range voting, adding the results at the end? following the wikipedia Tennessee example (http://en.wikipedia.org/wiki/Approval_voting). We assume that on the first round, voters assume that all candidates are equally likely, and approve of anyone over the average utility. On each subsequent round, voters assume that the winner and runner-up of the previous round (NOT the winner and runner-up of the total sum) are the likely winner and runner-up. In all subsequent round, each voter votes for anyone they like better than the presumed winner, and they also vote for the presumed winner if they like them better than the runner-up. When the winner is tied, people vote for the city they like best out of the ties. When the runner-up is tied, people vote for the winner only if they like them better than both runner-ups.

Round Winner Runner-up Memphis Nashville Chattanooga Knoxville
1Memphis Chattanooga 42263217
2Three-way tie 42585858
total after 2Chattanooga Memphis, Nashville 84849075
3Nashville Memphis 42681517
total after 3Nashville Memphis 12615210592

Nashville over Memphis is stable (a fixpoint), and is the Condorcet winner.

Now let's assume that, after the first round, voters think the presumed winner and runner-up is the winner and runner-up of the total sum.

Round Winner Runner-up Memphis Nashville Chattanooga Knoxville
1Memphis Chattanooga 42263217
2Three-way tie 42585858
total after 2Chattanooga Memphis, Nashville 84849075
3Nashville Memphis 42683217
total after 3Nashville Memphis 12615212292

Presumably people's actual estimate of the presumed winner and runner-up is some interpolation of the previous two cases, perhaps with some extrapolation to the "future" of these two cases, or some finding of fixpoints.

It remains to be proven that such an interpolation converges, and how fast it converges.

If Memphis, seeing that it was also a runner-up, withheld its votes from Nashville in round 3 of the second case, then Nashville still wins but it takes 3 more rounds:

3Nashville Memphis 42263217
total after 3Memphis Chattanooga 12611012292
4Three-way tie 42585858
total after 4Chattanooga Memphis 170168180150
5Nashville Memphis 42683217
total after 5Nashville Memphis, Chattanooga 212236212167
6Memphis Chattanooga, Knoxville 42263232
total after 6Nashville Memphis 254262244199

IRL we can expect faster convergence because people will be trying to accurately assess everyone's utilities with polls and stuff, and also because people who take advantage of the range voting will probably help things out.

Links

here is the range voting advocacy site (advocates range for single-winner elections):

<a href="http://www.rangevoting.org/">http://www.rangevoting.org/</a>

Which multiseat (multiple winner) voting method do I prefer?

Todo.