22 January 2015

Voting on our next RPG

(This is only incidentally about RPGs and mostly about voting methods.)

I took the opportunity of choosing the next RPG my group would play to try out an instant-runoff election. Which has also led me to learning more in-depth about voting methods.

The first thing to say is that this election is a bit weird because there are more candidates than voters. The second thing to say is that any election method used with three or more candidates has to make compromises. There is no perfect solution.

The actual voting was conducted via OpaVote. Each voter had to rank the 11 choices from most favorite to least favorite.

Instant-runoff voting (IRV) was probably a poor choice for this particular application, but it worked for picking the first-place winner. One of the criticisms of IRV is that it can fail to elect the Condorcet winner when there is a Condorcet winner. A Condorcet winner is a candidate who would beat all other candidates in one-on-one contests. In this case, there was a Condorcet winner, and the IRV method choose it.

I then tried to rank the other candidates by removing the winner and running the election again. This seemed to work OK for the first-, second-, and third-place. Then I started to run into one of the limitations of IRV.

Say you have three candidates: A, B, and C. Four of the voters choose A as their first choice, B as their second choice, and C as their third choice: A>B>C. Three of the voters vote: B>C>A. The last three vote: C>B>A.

A wins the first IRV round. One of the first-round losers needs to be eliminated for the second round (the runoff). The losers, however, are tied. If you eliminate B, then C wins. If you eliminate C, then B wins. If you eliminate both B and C, A wins. But B is the Condorcet winner who would beat both A and C in one-on-one races. The OpaVote IRV method chooses to break these sorts of ties randomly, which seems insane to me.

In practice, the larger the number of voters, the less likely you are to run into this. With a small number of voters, however, it becomes more likely.

The problem with just choosing the Condorcet winner is that there isn’t always a Condorcet winner. There are various methods that use Condorcet and then use some sort of “completion rule” if there is no Condorcet winner. Two of the best seem to be ranked pairs (Tideman) and beatpath (Schultz). Both of these also have the ability to rank all the candidates rather than just declaring the first-place winner. Due to my small number of voters, though, even these had to declare a five-way tie for third place.

So, I came up with my own way to break the tie. (Although, no doubt, this is an already known method. Perhaps Condorcet-IRV?) I ranked them by largest number of votes in the most preferred rank. e.g. Two voters picked D&D3e as their first choice, but no voters had chosen any of the others in the tie as their first choice. So, I gave third-place to 3e. (Which was where it placed in my IRV-based ranking attempt too.)

So, the final ranking I came up with is...

  1. Toon
  2. Shadowrun
  3. D&D3e
  4. Forbidden
  5. Dragon Age
  6. Star Frontiers
  7. Dragon Warriors
  8. Dungeon Crawl Classics
  9. Lejendary Adventures
  10. Lords of Creation
  11. SLASH!

I should also mention the Borda method. I’ve been aware of it, but I’ve never been particularly fond of it. Although I think it might have potential if the scores were calculated differently. It favors “broadly acceptable” winners rather than majority winners. Though there is an argument that that might be appropriate for this case. Here are the Borda results... (There were a couple of ties.)

  1. Shadowrun
  2. Toon
  3. D&D3e and Star Frontiers
  4. Dragon Age
  5. Dungeon Crawl Classics and Forbidden
  6. Dragon Warriors
  7. Lejendary Adventures
  8. SLASH!
  9. Lords of Creation

I’m now much less of a fan of IRV than I was. I think I agree with those who say a Condorcet winner, when there is one, ought to be the overall winner despite the drawbacks of the Condorcet methods. I am undecided on which of the completion rules I prefer.

The ironic thing that kept hitting me while reading about voting methods is that you could have a vote to choose a voting method.

Besides OpaVote, Eric Gorr’s Condorcet calculator was helpful in trying different counting methods.

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