Voting Methods Report for the Free State Project

by Jason Sorens in Consultation with Steve Cobb

The Failings of Cumulative Count

For months now, people have been arguing that cumulative count is not the ideal voting system for the Free State Project. The problem proved to be worse than we had thought: cumulative count turns out to be one of the worst possible voting systems for our purposes.

The reason for this is simple: it turns out that the best way to cast your cumulative count ballot (technically, "the utility-maximizing way") is to give all your points to your favorite candidate (of those that are perceived as having a chance at winning).

This is a problem because it means that people have an incentive to misrepresent their preferences. Even if you think there are several good candidates, you should give all your points to just one of them if you want to maximize your expected satisfaction from the result. As a result, it is difficult for good compromise candidates to do well in a cumulative count vote. Also, people can develop prejudices against states they don't think have a chance of winning. This is the familiar "wasted vote" problem that plagues third parties in the U.S.

For several months we ran a "cumulative count practice poll" on the State Data page. The results of this poll were enlightening. By pulling up the individual vote figures we saw that 42% of voters had given all 10 of their points to a single state. If you included people who gave 8 or 9 points to a single state, the figure was over 50%. Most of the people voting in this poll therefore were misrepresenting their preferences. The dangerous thing is that they may not have realized on a conscious level that they were voting "strategically," but they did it anyway. Moreover, we saw that these "strategic votes" went disproportionately to one state. Therefore, these strategic voters were able to take advantage of the honest voters and to manipulate the result in their favor.

Thus, we have empirical confirmation of theory: cumulative count promotes strategic voting on a massive scale. The mathematical proof of the proposition that one should give all points to one's favorite state in a cumulative count election follows. Those who are not interested in reading through the proof will want to skip to the next section of the report.

Mathematical Proof

Assume that there are 4 candidates on the ballot, A, B, C, and D. Our hypothetical voter has the following preferences over these candidates should each win election:
U(A)=10
U(B)=8
U(C)=5
U(D)=1

The voter has 10 points to distribute. If voting sincerely, the voter would give 0.0 points to D, 2.0 points to C, 3.5 points to B, and 4.5 points to A.

Assume that each candidate has an equal chance of winning, p=.25. Assume that each point 1.0 given to a candidate increases that candidate's chance of winning by x and decreases every other candidate's chance of winning by (1/3)x or 0.33x, where x>0. Then:

EU(A=10)=(.25+10x)10+(.25-3.33x)8+(.25-3.33x)5+(.25-3.33x)=
2.5+100x+2-26.67x+1.25-16.67x+.25-3.33x=6+53.33x

and:

EU(A=4.5,B=3.5,C=2.0)=
(.25+4.5x-5.5(x/3))10+(.25+3.5x-6.5(x/3))8+(.25+2.0x-8.0(x/3))5+(.25-3.33x)=
2.5+2+1.25+.25+(4.5x-1.83x)10+(3.5x-2.17x)8+(2.0x-2.67x)5-3.33x=
2.5+2+1.25+.25+26.7x+10.64x-3.35x-3.33x=6+30.67x

and:

EU(A=9,B=1)=(.25+8.67x)10+(.25-2x)8+(.25-3.33x)5+(.25-3.33x)=
2.5+86.7x+2-16x+1.25-16.67x+.25-3.33x=6+50.7x

Thus, EU(A=10)>EU(A=9,B=1)>EU(A=4.5,B=3.5,C=2.0). The expected utility of giving all his points to his favorite candidate is larger than the expected utility of the other two options presented (and indeed any possible option).

The Alternatives to Cumulative Count

In the search for an alternative to cumulative count, we wanted a voting system that would: 1) encourage voters to express their true preferences, 2) allow "compromise states" a good shot at winning, 3) not allow for paradoxes in which a state wins even though an absolute majority of people would prefer another state, 4) be easy to explain and understand, 5) allow easy tabulation of ballots and presentation of results, 6) be inexpensive to execute.

Simple plurality voting (each voter gets to vote for one state, the state with the most votes wins) was rejected on grounds 1, 2, and 3. All voting systems involving multiple rounds of voting were thrown out, chiefly on ground 6.

The alternatives seriously considered were: Rating, Approval Vote, Instant Runoff Voting, (Serial) Borda Count, and Condorcet's Method.

Rating works by allowing each voter to "rate" each candidate on a scale (0-10 or 0-100 or whatever). Then you just add all the points up, and the candidate with the most wins. This is the system used in the Olympics. When a judge gives a high score to one performer, that doesn't mean she can't also give a high score to the next performer. This system is appropriate for the Olympics because of the sequential, graded nature of performances. In a large vote like the one we will have, however, Rating breaks down into Approval Vote: you'll want to give the highest possible score to all states you like and the lowest possible score to all states you don't like. In essence, it's just like giving 1s and 0s to all the candidates.

Approval Vote is the system in which you just give a 1 (for approval) or 0 (for disapproval) to each candidate. You can approve of as many or as few candidates as you like. The "1" scores are added up, and the candidate with the highest score wins. This system is a good method for picking the "least bad" candidate, or a candidate that everyone thinks is adequate. However, it limits voters from expressing their full range of preferences. It doesn't adequately distinguish among excellent, very good, good, adequate, inadequate, and terrible choices.

Instant Runoff Voting allows voters to rank all the candidates, from favorite to least favorite. The candidate with the least number of first-preference votes is eliminated, and then those ballots listing the eliminated candidate as first preference are checked for their second preferences. Those second preferences are then distributed among the remaining candidates as if they were first preferences. Then you repeat the process, until just two candidates are left, and one wins a majority against the other. For example, if you have three candidates, Harry, Dick, and Moe, and Harry and Dick are considered the only ones with a chance of winning, you can still vote for Moe and put Dick in 2nd place. When Moe gets eliminated because he gets the fewest first-preference votes, those who voted for him have their second preferences counted. So by putting Dick in 2nd place you get to express your liking of Moe and still help tilt the election in favor of Dick against Harry. Instant Runoff Voting thus discourages the strategic voting we have in simple plurality rule. Instant Runoff Voting is just what it sounds like: a way of doing multiple rounds of voting all at once. It's a simple system to understand, and it encourages voters to express their full range of preferences, but it does allow for paradoxes and is slightly less easy to tabulate than other methods. IRV is used in the national elections of Australia and Ireland and is generally considered a "pretty good" voting system, but not the best possible.

Borda Count also allows voters to rank all the candidates. To aggregate the votes, it assigns points to each candidate based on its ranking. For our purposes, a first-preference vote would be worth 9 points, a second-preference vote would be worth 8 points, and so on, until a last-preference vote would be worth zero. You just add all the points up to get a winner. This system is used in college sports polls and to determine winners of sports awards (Most Valuable Player and so on). It is notoriously subject to strategic voting, however. You want to rank states that you don't think have a chance of winning over states that do have a chance of winning but are not your first choice, even if you really prefer these states that do have a chance of winning to the states that don't have a chance of winning. Doing this allows your first choice a better shot at winning, because you hurt your favorite's credible opposition. An example of this would be if you are an Astros fan and you rank your MVP ballot with Astros player Jeff Bagwell at the top, and you leave Barry Bonds of the Giants in last place, because you know a lot of other people will be voting for him. If enough people do this, Bagwell might beat Bonds, even though if people were voting honestly, they would have ranked Bonds higher, and Bonds would have won. To avoid this, well-known game theorist Dr. Donald Saari advised us to do "serial" or "sequential" Borda Count if we did not use Condorcet's Method (described next). He advised eliminating all candidates that did not receive the average number of points and then doing a new Borda Count poll, using the same ballots, with the eliminated states removed from the rankings. However, this method is difficult to tabulate: it probably would require going through and checking each individual ballot, unless we could devise a fancy program to do it for us. But then the aggregation system would not be transparent to everyone, and perhaps some people would be suspicious of the result. Also, this method does not guarantee that we will avoid paradoxes.

Condorcet's Method

Condorcet's Method is the voting method favored by almost all game theorists and mathematicians, and it is the method that Steve and I propose adopting. The Election Methods website explains Condorcet voting in some detail, but we explain it here as well.

The way it works is that you again allow voters to rank all the candidates. They can indicate "ties" in their rankings if they wish; it doesn't matter. Then you compare each candidate to every other candidate: candidates score wins over other candidates that are lower in the rankings and losses to other candidates that are higher in the rankings (if candidates are tied on a ballot, a tie is scored). If a majority of voters prefer one candidate above each other candidate in "pairwise" (one-on-one) comparisons, that candidate wins. Simple! This method guarantees that you won't have paradoxes of the kind described above, unless voters have cyclical preferences. It is also easy to tabulate; Steve has created a spreadsheet demonstrating how all the vote counters need to do is to put down each voter's ranking, and it spits out a table comparing each state to every other state.

Does Condorcet's Method encourage sincere voting? Yes. Let's say you have the following preferences among 10 candidates: 1. A 2. B 3. C 4-10. D, E, F, G, H, I, J. Under cumulative count, you would want to give all your points to A and zero to everything else. Under Condorcet's Method, you want to rank your preferences sincerely: A first, B second, C third, and D-J tied for last. Why? Because if you ranked C as tied with D-J for last, that does not in any way benefit A or B. A and B still beat C no matter whether you rank it tied for last or in third place ahead of the others. But if you rank C sincerely, in third place ahead of D-J, then C beats D-J. So if A and B don't win the election, then C has a chance of winning. There's no reason to misrepresent your preferences.

Sometimes voters' preferences are cyclical, and there is no clear winner. (Here is an example of cyclical voting: There are three voters and three candidates. Voter 1 ranks the candidates as follows: 1. A 2. B 3. C. Voter 2 ranks the candidates this way: 1. B 2. C. 3. A. Voter 3 ranks them: 1. C 2. A. 3. B. In this situation A beats B 2-1, B beats C 2-1, and C beats A 2-1. A beats B, which beats C, which beats A, which beats B, which beats C, which beats A... It's a cycle! There's no clear winner here. Of course, cyclical preferences become less of a problem with more candidates and more voters. I doubt it will happen in our state vote, but we need to be prepared for the possibility.) There are actually several different "Condorcet methods" differing in how they deal with cyclical preferences. The simplest method, the one I prefer, is simply to eliminate the smallest-magnitude defeat - that is, the defeat with the fewest total votes against - until one candidate is unbeaten. So let's say for example that state A wins over all 9 of the other states, except for one, state D. However, state D receives only a few votes against state A. This could mean either a small margin of victory of D over A, or it could mean that a lot of voters are indifferent between A and D and have given them ties. So if A has a greater number of votes in all its contests against other candidates than D has in its contest over A, D's victory will be eliminated, and A emerges the winner.

The results of a Condorcet election are presented in a table. I've used the table from ElectionMethods.org as an example.

This table may appear confusing at first glance, but it's quite simple if you follow this explanation very slowly and carefully while looking at the above table. A, B, C, and D are the candidates here. To see how many votes a candidate got against another candidate, you find the candidate's name in the vertical list and follow the row over to where it intersects with the column representing the name of the opposing candidate in the horizontal list. So to see how many votes A got against B, you follow the A row over to the B column and see that A got 63 votes against B. If you look diagonally down and to the left, you see how many votes B got against A: 87. So B wins that contest. By the same token, A gets 89 votes against C, while C gets 69 votes against A. A wins that one. A then loses to D: 67-57. B, however, beats C (78-72) and D (73-51). So B is the winner, because it beats every other candidate. What follows is a table in which no candidate beats every other candidate - there isn't a "Condorcet winner," so we have to use a method to figure out which candidate should win.

In this one, A beats B, 40-37, A loses to C, 30-22, and A loses to D, 20-13. B loses to A, beats C, 50-35, and loses to D, 60-50. C beats A, loses to B, and beats D, 25-20. D beats A and B but loses to C. So both C and D have two wins and one loss, while A and B have one win and two losses. How do we figure out a winner? According to the method I just suggested, we eliminate the smallest-magnitude defeat first. That is the victory of D over A: D gets just 20 votes over A. If that contest is eliminated, then A has one victory and one defeat, and so does D. C still has two victories and one defeat. We have to continue until one candidate is unbeaten, however. So the next weakest defeat is C's victory over D. If that is eliminated, then D has one victory and no defeats, and it wins.

Of course, there are other ways of doing this. One criterion might be that the candidate with the most wins and fewest losses should win, but if two candidates are tied for most wins and fewest losses, then the winner of the head-to-head contest between the two should prevail. If that criterion were used, then C would instead prevail in the example above. Another way to do it is to eliminate smallest margins of defeat first, but game theorists generally agree this is the wrong way to go about it, because it doesn't take into account the number of ties, and it also opens things up to strategic voting. If we eliminated smallest margins of defeat, we would eliminate A over B first, then C over D. D would be the winner. In our state vote, I doubt many people will be giving ties, so the "margin of defeat" and "magnitude of defeat" criteria should reach about the same result, as they have here. In fact, it is highly unlikely in the first place that there will be no Condorcet winner: in most elections, one candidate does beat all the others in pairwise contests, and our limited experimental data with FSP members suggests that this will also be the case in our state vote.

The most complex method of resolving ambiguities when there is no Condorcet winner is called "Schwartz Sequential Dropping." This is the method favored by game theorists for its overall fairness (both Election Methods and Rob LeGrand, a friend of the FSP whom we contacted during our investigation, support this method). I will not describe it here but encourage those interested to check out these websites. I would argue that we not adopt Schwartz and instead stick with simple Condorcet, simply because several people have argued that they want a vote-aggregation method that is very simple and transparent. Simple Condorcet is certainly that - and it has the advantages of being difficult to manipulate, easy to understand and explain, easy to calculate, and amenable to the whole range of possible voter preferences. For our purposes, simple Condorcet is as close to ideal as we'll get.