democracy
1. A form of government in which political power resides in all the people and is exercised by them directly (pure democracy), or is given to elected representatives (representational democracy), with each citizen sharing equally in political privilege and duty, and with his rights to do so protected by free elections and other guarantees. 2. A state so governed. 3. The spirit or practice of political, legal, or social equality. 4. The common people. [ < French democratie < Medieval Latin democratia < Greek demokratia < demos people + krateein to rule ]

Funk & Wagnalls Standard College Dictionary

The Average Ranking Method


The average ranking method

To judge a voting method, first we need to know the goal of voting. What's the goal of voting, exactly? To select the best alternative? Certainly not. (Once in a programming class the teacher asked us to vote on what the correct syntax was for a function pointer in C. We all laughed. We knew our opinions had nothing to do with what the correct answer was.) Majority rules? Doesn't work when there's more than two alternatives. Make everyone happy? Only happens if everyone agrees.

I'll assume the goal of voting is to find the alternative that makes voters happiest. That is, find the alternative c that maximizes

sum(happiness(v,c)) over all voters v

The way you reach that goal follows immediately from its definition. First, you have to actually measure how happy each voter is with each alternative. Second, you have to sum all those measurements. Finally, the highest sum wins.

In practice, each voter would rank each alternative in the real-numbered interval of [0.0,1.0] (or 1..5 or 0..10 or -100..100, any closed interval of the real numbers will do), with the low end being the worst alternative imaginable and the high end being the best alternative imaginable. Measurements for each alternative are summed over the entire population. The alternative with the highest sum wins.

This method is already in use -- it's the way schools rank students. It's the way diving, gymnastics, and figure skating are judged in the Olympics, the way Consumer Reports ranks products, the way Amazon ranks books. It's pretty common.

In order for this to work, voters must rank every alternative for a given decision, or that vote for that issue must be discarded. Another possibility (really required if write-in candidates are allowed) is to have a "default" alternative per decision which applies to any alternative left unranked. Another possibility is to take an average of the votes that were cast, which is equivalent to setting the default for each alternative to the average of the rankings of all voters who ranked that alternative. I'll assume from here on that every decision offers a "default" alternative.

Another requirement of a voting scheme is that it be decisive, that is, there are no ties. I'll assume from here on that ties are resolved by an arbitrary rule: alphabetical order of the tied alternatives.

An example

Here's what my ballot would have looked like for the Gore-Bush-Nader-Buchanan election.

Candidate Bob's Vote
George Bush 2
Al Gore 6
Ralph Nader 5
Pat Buchanan 1
Default 2
Note that George Bush and Default are tied.

Arrow's theorem

Here are several, possibly desirable, criteria for judging a voting method, which I quote from the first site Google came up with when I typed in "Arrow's Theorem":

  1. Universality. The voting method should provide a complete ranking of all alternatives from any set of individual preference ballots.
  2. Monotonicity criterion. If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed in such a way that the only alternative that has a higher ranking on any preference ballots is X, then the method should still rank X above Y.
  3. Criterion of independence of irrelevant alternatives. If one set of preference ballots would lead to an an overall ranking of alternative X above alternative Y and if some preference ballots are changed without changing the relative rank of X and Y, then the method should still rank X above Y.
  4. Citizen Sovereignty. Every possible ranking of alternatives can be achieved from some set of individual preference ballots.
  5. Non-dictatorship. There should not be one specific voter whose preference ballot is always adopted.

Arrow's Impossibility Theorem is states that no voting method based on rankings of candidates can satisfy all five of these criteria. However, the average ranking method isn't necessarily covered by Arrow's theorem. Arrow assumed that votes consisted of just an ordering of the candidates, while the average ranking method uses votes that are real-numbered rankings in a range. These aren't equivalent. Any arrow ballots can be derived from an average ranking ballot, but not vice versa.

In fact, the average ranking method is NOT covered by Arrow's theorem. It satisfies all five of the criteria. It allows no ties (1), increasing the rank of an alternative on one ballot always increases the sum for that alternative (2), removing alternatives doesn't affect the rankings of other alternatives, and shouldn't change the rankings even if votes were recast because it can't change the best and worst alternatives imaginable (3), every voter can rank the alternatives in any order they want (4), and there is no voter whose ballot matches the outcome no matter how all other voters no matter how the other voters vote (5).

Further, deciding what alternative voters are happiest with has an implicit deterministic algorithm, and that algorithm is the average ranking method, so the average ranking method is unique in that it can always determine what voters are happiest with. It is the unique ideal voting method if your goal is to select the alternative that people are happiest with (if you assume that every voter's ranking of every alternative is equally important).

Vote inflation in the average ranking method

Consider my sample ballot again:

Candidate Bob's Vote
George Bush 2
Al Gore 6
Ralph Nader 5
Pat Buchanan 1
Default 2

Notice my actual opinions are between 1..6, not 0..10. I can make my vote count more simply by scaling my opinions to 0..10, like so:

Candidate Bob's Vote
George Bush 4
Al Gore 10
Ralph Nader 8
Pat Buchanan 0
Default 3

I can make my vote matter even more than that by giving each candidate a score of either 0 or 10, like so:

Candidate Bob's Vote
George Bush 0
Al Gore 10
Ralph Nader 10
Pat Buchanan 0
Default 0

Is this true? How do I decide where to put the 0-10 split? Mike Ossipoff first pointed this out to me. My goal is to maximize the chances of my vote deciding the election to my liking. For every pair of candidates, there is some probability that they will be the top two candidates. If a pair really is the top two candidates, my chance of deciding the election is proportional to the distance between my rankings of those two candidates. So, I want to:

   Maximize the sum over all pairs of
     ( probability of that pair being the top two candidates )*
     ( ranking of more preferred candidate - 
       ranking of less preferred candidate )
However, if I look at one candidate, the sum looks like
   ( ranking of that candidate )*
   ( sum of probabilities of pairs where this candidate is preferred -
     sum of probabilities of pairs where this candidate is not preferred )
Notice that the probabilities of pairs is independent of my rankings (ignoring the miniscule effect of my vote on the election outcome). If that sum of probabilities is positive, I should rank the candidate as high as possible, otherwise I should rank that candidate as low as possible. Notice that I'll prefer all candidates I rank 10 over all candidates I rank 0: for the two candidates closest to the 0-10 split, the only difference in their probability sums is for the pair containing those two candidates, and that will contribute a positive probability to the sum for the candidate I most favor in that pair and a negative to the one I don't.

If everyone follows this strategy and only votes 0 or 10 for every candidate, then average ranking reduces to the approval method. The approval method, unlike average ranking, is covered by Arrow's Theorem.

Comparing voting methods

Now that we have a well-defined goal of voting (find what voters are happiest with, with no ties) and a method that achieves that goal (the average ranking method without vote inflation), we can compare it to existing methods and see how biased they are.

Voting in the United States

The US voting method is equivalent to giving the first choice a 10 and everything else a 0 (FPTP, "First Past The Post"). For example, I voted for Gore. This isn't really a good representation of my views -- see, I much preferred Nader to Bush, but my vote weighted them equally. And my vote distinguished greatly between Nader and Gore, even though I actually considered them about the same.

US presidential elections have two other wrinkles:

  1. All the votes in a given state are given to the same candidate.
  2. Votes from different states have different worth (voters in less populated states are worth slightly more).
Go figure. The news always reports the vote tallies in real time both with and without these two biases, so these biases aren't necessary from any technical standpoint.

A special case of the US method is the yes/no vote. (That's really three choices: yes, no, or don't vote.) Consider how a vote with the system above would go. If you can't decide, you don't vote or you give both choices equal weight. Otherwise you make yourself more important by making your first choice a 10 and the other a 0. And, surprise, that's identical to each voter having one vote to cast, either yes or no.

For yes/no votes, if you're undecided, you shouldn't vote! If you wish to inflate your importance (or the importance of your coin flip) by voting, you can do so, but doing so is antisocial.

Voting in Australia

Australia has a system where candidates are ranked 1..n, with 1 for the first choice, 2 for the second choice, 3 for the third and so on. You tally votes by throwing out the candidate with the fewest first choices. Next you adjust the ballots for whoever voted for the loser so their second choice is their first choice, third is second and so on. Then you count all the ballots from scratch and repeat. Eventually you have as many candidates as open positions and you're done.

It's hard to quantify just what this computes. I did a few trials by hand and found this method makes it unusually likely for unknowns to win. Unknowns tend to get 3rd place for most people, so people's ballots fall wholeheartedly to them once their first and second choices get ruled out. Consider this average ranking judgement table:

Voter 1 Voter 2 Voter 3 Voter 4 Average
George Bush 10 0 10 5 6.25
Bill Clinton 5 10 10 5 7.50
Ross Perot 5 0 10 5 5.00
Chevy Chase 0 0 0 5 1.25
It translates into this Australian voting table:
Voter 1 Voter 2 Voter 3 Voter 4 Results
George Bush 1 2 2 3 2nd
Bill Clinton 3 1 3 2 3rd
Ross Perot 2 3 1 1 1st
Chevy Chase 4 4 4 4 4th
The Australian method gives about the opposite results as the average ranking method for these ballots. The US method would select Perot. (In most cases the three methods would choose the same winner.)

The Approval Method

The Approval Method allows you to select all the choices you approve of. It is the same as the average ranking method, with the restriction that only scores of 0 or 10 are allowed (nothing in between).

It is considered a good method. It is a good approximation for fanatics. It is a bad approximation for people whose opinions are many shades of gray. It has the advantage over the average ranking method that it is easy to conduct by show of hands. If the voters try to maximize the effect of their vote in the average ranking method, even if their true opinion is shades of gray, the average ranking method reduces to the approval method.

The Borda Point Method

The Borda Point Method orders candidates by preference, numbers them 1 through N, then adds up the numbers and the lowest total wins. It is the same as the average ranking method, with the restriction that if there are N choices you can only choose scores that are 10i/N for some i in 1..N, and once you use a score for one candidate you're not allowed to reuse it for any other candidate. This forces every i to be used for exactly one candidate.

It is considered a good method. It is a good approximation of the average ranking method when opinions are many shades of gray. It is a bad approximation for fanatics. It does not allow you to approve of just one candidate or disapprove of just one candidate.

The Condorcet Method

The Condorcet method can be viewed as carrying vote inflation in the average ranking method to its logical extreme. Rather than pretending that the best and worst alternatives imaginable are actually on the ballot, it pretends that the best and worst alternatives imaginable are any two alternatives on the ballot. That is, when considering any two alternatives, it pretends the other alternatives don't exist and scales one of them to 10 and the other to 0. Votes are cast by an ordering of the n alternatives, and votes are tabulated by an nxn matrix counting how many times the row beat the column. If any alternative beats all other alternatives, it wins. If any alternative loses to all other alternatives, it is eliminated. This allows for ties, and there are various ways to resolve the ties.

I haven't looked into this much yet.

Criteria for judging voting methods

I found several more criteria for judging voting methods here:

  1. Condorcet's Criterion:

    If there's a candidate who, when compared separately to each one of the other candidates, is voted above him/her by more voters than vice-versa, then that candidate should win. (That's the candidate that I've been calling a "BeatsAll winner"). (Of course if one ranks A, and doesn't rank B, that counts as ranking A over B).

    The average ranking method does not satisfy this. Consider this choice and these votes:

    X Y Z
    Voter A109 0
    Voter B109 0
    Voter C109 0
    Voter D 010 0

    The averages will be X=7.5, Y=9.75, Z=0, so Y wins, even though the majority of voters preferred X over Y. (I think this is a bad criteria. It ignores the possibility that the majority held only weak opinions.)

  2. Smith Criterion:

    If there's a set of candidates such that every candidate in the set beats every candidate outside the set, then the winner should come from that set. (A beats B if more voters prefer A to B than vice-versa).

    The average ranking method doesn't satisfy this, see above.

  3. Majority Criterion:

    This is the very familiar criterion that says that if a majority of all the voters vote X alone in 1st place (or vote for him in 1-vote Plurality), then X should win. It's met by every proposed method except for the notorious Borda point system.

    The average ranking method doesn't satisfy this, see above.

  4. Mutual Majority Criterion:

    This one sounds similar to Smith, but is different. If a specific group of voters consisting of a majority of all the voters vote every candidate in set S over every candidate outside that set, then the winner should come from S.

    The average ranking method doesn't satisfy this, see above.

  5. Condorcet Loser Criterion:

    If there's a candidate who, when compared separately to each one of the others, is ranked below that other candidate by more voters than vice-versa, then he/she shouldn't win.

    The average ranking method doesn't satisfy this, for example

    X Y Z
    Voter A 0 110
    Voter B 0 110
    Voter C 0 110
    Voter D10 010

  6. Majority Loser Criterion:

    If a candidate is ranked last by a majority of all the voters, then he shouldn't win.

    The average ranking method doesn't satisfy this, see above.

  7. Monotonicity Criterion:

    Changing your vote so as to vote someone higher should never make him lose when he'd have otherwise won. Changing your vote so as ton vote someone lower should never make him win when he'd have otherwise lost.

    The average ranking method satisfies this.

  8. Irrelevant Choice Criterion:

    Adding or removing nonwinners should not change who the winner is.

    The average ranking method satisfies this.

It seems to me, in every case where the average ranking method violates some criteria, it's the criteria that is flawed, not the average ranking method. The average ranking method (without vote inflation) is the ideal way to measure the will of the voters, by definition of what the will of the voters is.

Congresses

That's all well and good if there is only one choice to be made, or if all the choices are obviously independent. But when choosing members of a congress, the choices aren't obviously independent. (I'd rather have a congress with members with different temperments, so my vote for the last congress member would depend on who is already in congress, so the choices aren't independent.)

One way around this is to make each set of choices a choice and to judge each set of choices separately. That exactly maximizes people's happiness with the choice; we can use that as a standard to compare other methods against.

To elect a 100-person congress with 500 people running, that's (500 choose 100) sets of choices. This standard method isn't practical. Simplifying assumptions have to be made. Which simplifying assumptions you make have a large effect on what sort of congress you get.

I've seen two strategies. One is to allow everyone to vote for as many candidates as they want, then take those n candidates with the most votes. The other is to divide up the voters somehow and have each group of voters select their own representative. There are lots of ways to divide up voters, for example by age, location, political party, religion, race, alphabetically by last name ...

Actually, it's easy to form a congress that is approximately representative of the voters in all respects. Simply choose the appropriate number of registered voters by random lottery and make them the congress. Is this a good idea? No?

Well, how about having people's vote count in proportion to the taxes they pay, and select representatives at random by the same proportions? It would eliminate people who don't know how to manage money. Government's job is basically to spend their money, so they should have a vote in proportion to the amount of the government's money that was theirs. No? What if we be nice and give people who get more from the government than they pay a zero vote instead of a negative vote? Still no? What if we include a voting-stipend for people who have done military service, seeing how they're really contributing too? Hum. That might actually work. But I still think it's missing the point.

I think representatives aren't supposed to be a representative sampling. They're not just representing themselves, they shouldn't give to just themselves all the government's money and power. They're supposed to fight for the interests of the people they represent. This requires some very nonrepresentative qualities, such as being able to follow to multiple conversations at once in a crowded room, the ability think and argue well on your feet, the ability to read and write legalese, an interest in specifying and refining laws that people can actually live by, and a lack of a goal of making themselves rich and powerful at the expense of others. Given that representatives aren't just representing themselves, it helps to let them know just who they represent.


Orders of Magnitude
Gravitational orbits in Java
The Hash Function Sieve
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