In close elections one vote can only serve to differentiate between two candidates. A third party "C" can become a spoiler such that "A" gets elected while the "C" voters would have preferred "B" over "A". This essay and simulation explores several improved voting systems.
A voter has some preference for every candidate. The preference ranges from total dislike to total like, represented as -1.0 to 1.0.
A voter's happiness with respect to the outcome of an election is equal to the voter's preference for the candidate that won the election.
In simulations so far, all preferences have been random and independent to the extent that the random() function is random and independent. No bias was introduced to organize a voter to have a coherent set of preferences, or to group the population into any kind of structure. The expected preference of any voter for any candidate is 0.0, no like or dislike. An average election thus simulated is closely contested, with no candidate having a major advantage over another.
To simulate a voter's imperfect self-knowledge of preferences, or equivalently, confusion about what a candidate stood for I introduced an error factor that ranged from 0.0 to 2.0. Before voting, a random value from 0.0 to the error factor with a random sign is added to each preference of each voter for each candidate. The errant preference is still constrained to -1.0 to 1.0. The systems are run on the errant voter preferences, and the happiness of the winner is determined from the original unmodified voter preferences.
This is the way things are. For each voter, cast a vote for most preferred candidate. This is of course oversimplified, it doesn't take into account a primary system or partial foreknowledge based on news polls and such.
This is the sum of the voter's internal model. The candidate with the greatest sum preference wins. It always maximizes the sum happiness.
It may however be impractical to implement in the real world, and is subject to human limitations to know and express internal desires.
In raw Fuzzy Vote, more opinionated voters get more say. While that maximizes the happiness measurement of this simulation, maybe it's not fail. So, counterweight all votes such that the sum of positive or negative votes cast (absolute value) by a voter is 1.0.
Conversely to counterweighting, scale up non-opinionated voters such that a voter's strongest opinion goes to -1.0 or 1.0, and scale other preferences similarly. Still not "fair" in that someone who votes all -1.0 or 1.0 gets more say that someone who expresses a 0.5 and a 1.0, but I thought it was an interesting option to try.
Behave as One Vote, counting only voter's first choice votes. Eliminate candidate with fewest votes and repeat until there is only one. (Voters with 1st choice eliminate vote for 2nd, or in general, highest non-eliminated choice.)
I saw this promoted on a couple web sites (http://www.fairvote.org/irv/ was most pointed at), and thought it was intuitively (to me) a non optimal idea that I could beat. So, I tried it.
Yes/No per candidate. In this simulation, a Yes is cast for every candidate a voter has a positive preference for. Most yesses wins.
Sort candidates by preference and for N candidates, give N points to the first choice, N-1 points to the second choice, and so on. (Optionally: No points to negative preference candidates.) Most points wins. (Also known as Borda count.)
This control case cheats and calculates the sum happiness score for each possible candidate win and selects the highest. This works out to the same thing as basic Fuzzy Vote, with a different loop nesting order. No strategy should ever get a higher happiness than Max Happiness.
Condorcet is based on comparing candidates pairwise. The candidate who transitively beats the other candidates by the greatest margin is the overall winner.
A population of voters is initialized with random preferences. Without changing that population, each voting system is run over the voters and the happiness data is collected. After all systems have been run over a set of voters, a new set of voters is created the process repeats.
I used various numbers of voters from 10 to 100,000. I also varied the number of candidates from 2 to 99. Each combination of numbers of voters and candidates was run on 10,000 sets of voters to collect mean happiness and standard deviation.
Data for determining the effect of error was gathered on simulations over the full range of error, 2 to 99 candidates, and 1,000 voters.
The result data is best portrayed by the graphs available on my web site.
http://bolson.org/voting/data/graph/index.htmlTraditional One Vote was reliably the worst performer, and it should be sacked.
IRV is only barely better than present methods. I believe that the flaw is in that it is still based initially on the present method of only counting one vote from every person. Although it can solve some fractured electorate problems it is still very limited and unexpressive.
Acceptance Voting, while collecting less data then IRV, processes all that data at once to better match the will of the people. It is also simpler to implement and explain to a real electorate.
Condorcet voting, while not the top performer, had a uniquely better response to error in the voters. For this it deserves further study.
Ranked Voting collects the same data as IRV but uses it to better effect because the overall will of the voters is considered at once. The variant where no points are awarded to disliked candidates does not actually aid the voter. That variant is just throwing away data, and if a voter doesn't get a desired candidate they have no influence to get a lesser evil.
The various Fuzzy Voting systems reliably got the highest happiness results. Unfortunately it would require relatively complicated or expensive, almost necessarily computerized, voting equipment. I'm imagining a GUI with a slider next to each candidate's name and image.
Based on these findings Acceptance Voting or Ranked Voting should be put into practice immediately. Acceptance Voting may be the easiest to implement with present equipment. The IRV advocacy sites I saw claimed that more and more voting equipment was capable of collecting IRV-type candidate rankings, but if such rankings can be collected, Ranked Voting should be applied to the data.
An interesting trend in Max Happiness: it increases with more candidates. In a field of otherwise undifferentiated candidates, when there are more candidates there is a better chance that there will be one candidate that more people will agree on preferring. This may be the most important reason to break away from the One Vote system to a system that doesn't punish the voters for the existence of many choices.
Suppose you wanted to do Fuzzy Voting without a complex GUI. Ok, so rate each candidate on an integer scale from 1 to 9, 1 is dislike, 5 is neutral, 9 is like. Or maybe on a scale of 1 to 5, which can be done on a simple and inexpensive Scantron machine that I took tests on in High School. How much does the result suffer from the quantization error?
Clear majorities aren't interesting. A clear majority wins the election and gets what they want and is happy. A happy majority is an easy average positive happiness.
It might be an interesting problem to explore systems that effect consensus building. Increase the weight of negative happiness in evaluating which systems make the fewest people the least unhappy. One real world problem with this is that it might be perceived as giving more power to people who chose to object.
Aside from mathematical perfection, a voting system must satisfy the voter's demand to be fairly represented. I'll play my own devil's advocate for a moment and try to make point and counterpoint on the above systems' fairness.
Our present one-vote system is the baseline of fairness. Everyone gets one vote, and thus equal representation. The exception is that duopoly is possible and third (and fourth, etc.) parties may go unrepresented. But, at least on a personal expression level we are all equal.
As I said above, Fuzzy Vote gives more opinionated voters more say, so I followed with counterweighted Fuzzy Vote which regains the ideal of everyone having equal expressive power. In favor of pure Fuzzy Vote, at present everyone has equal expressive power but a disgraceful percentage of eligible voters don't bother to express themselves, so the more opinionated people wind up having more say anyhow. Given the better mathematical results for pure Fuzzy Vote, it should be used because it doesn't really change the status quo.
Acceptance voting is bad if you don't want to accept any of the choices, I guess. But, that's a deeper problem than voting mechanics can solve.
IRV, Condorcet and Ranked voting operate from the same data, a ranking of the choices. Is it fair to require someone to assign "1,2,3,4,..."? What about "1,1,3,4,4,6,..."? Is the algorithm that handles that data fair? My abstract interpretation of the algorithms is that Condorcet and Ranked voting favor electing a consensus choice while IRV plays to more traditional factional politics. Which do we want? Perhaps the consensus building of Ranked and Condorcet explain their results, as more people are more happy.
Instant Runoff Voting is being enacted in a few places around the USA and the world. Because I find it to be so inadequate, I feel it important to address it directly.
Firstly, in all my tests, IRV degrades as choices are added beyond 7 choices. It doesn't just fail to give voters the benefit of a wide field of choices, it actually gets worse.
I've heard some IRV proponents claim it's better because it doesn't dilute one's vote for a most preferred choice (a Borda vote is a vote for the enemy). They assume (probably because it's true for themselves) that every voter has a strong single fist choice. When that is not the case, I have shown that any other method is superior. If they can show me a set of voters where IRV gets a better result than Ranked(Borda), Condorcet, and Acceptance, I'll be greatly amused. And then we can quibble about how likely such a set of voters are to occur in the wild.
Mathematically, IRV fails because it doesn't consider all the data at once in fitting the solution (election result) to the constraining data (voter preferences). IRV only considers a voter's first choice out of some (shrinking) field of candidates. This leads to suboptimal iterating through the solution space.