Top 300 ranking of games using Copeland method (Condorcet)
Mauricio Noda
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I was playing with downloaded data from BGG in the last weeks and was trying to generate a ranking using different systems, and I'm posting the results in this list.

The idea was to use an absolute ranking system, which catches the concept of majorities from voting theory.

What's the difference compared to the current ranking system?

The current system uses an averaging based ranking, which is closely related to the range voting system. The problem with this system is it doesn't treat voters in an equal manner. Ratings which are far away from the average of a game have higher weight than ratings which are closer to the average. The effect is, extreme ratings like "1" and "10" tends to distort the resulting ranking. Another problem is, different voters tends to give different meanings to the same rating, which also distorts the final ranking.

I tried to generate another ranking using Condorcet methods. Each ballot used ratings to create a preference order. Games with no rating were not considered in the preference instead of considering them less preferred than rated games (this is called the "no opinion" variant).

1st attempt - Ranked Pairs. Failed due to excessive computing time (it would take months literally).

2nd attempt - Schulze using all games. Failed due to excessive computing time. (it would take days)

3rd attempt - Schulze using only games with 30+ votes. The ranking was generated successfully, but a bunch of games with 30 to 100 votes dominated the top. All games with many votes clustered in the middle of the ranking. Another bunch of games with few votes stayed in the bottom. I considered this attempt a failure due to lack of information in the ballots. For the curious, "La Bataille de la Moscowa (third edition)" was first place. Also, it took about 4 hours to calculate the strength of all paths.

4th attempt - Schulze using only games with 30+ votes, and without the "no opinion" variant. The result was a ranking which tracked the num voters column very closely. Games with more votes almost always ranked higher than games with fewer votes. When it didn't happen, the game stayed displaced only 1 or 2 ranks.

5th attempt - Copeland using only games with 30+ votes. SUCCESS. The resulting ranking looked convincing and it is the ranking I'm posting here. Took only the time to load data from the database. After that the ranking was calculated in tens of a second.

6th attempt - Copeland without the "no opinion" variant. The result was the same as 4th attempt.

How does Copeland method works?

You count all votes and calculate preferences like all Condorcet methods (takes a few minutes). Then, for each pairing of 2 games, you count if there are more votes in favor of one game or another. The game with more votes earns a win and the other a loss. If both games have the same number of votes, it is a tie (happens mostly when there are zero votes for both sides). Then you count 2 points for each win, 1 point for each tie and 0 points for each loss. Actually, the original scoring is 1 point for wins and 0.5 for ties. But working with integers is easier and gives the same result. After calculating points for everyone, you sort the games. Higher score goes higher in the ranking. In case of ties in scores, and it is not so uncommon, I used second order Copeland method for tie-breaking.

How does second order Copeland works?

For each game, you sum the score of all games it wins against. Then, sum half the score of all games it ties against. The total is used in tie-breaking. The idea is, wins against high ranked games are worth slightly more.

Copeland and second order Copeland are widely used in chess tournaments, but under a different name.

I hope you like the results presented below.
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21. Board Game: Caylus [Average Rating:7.90 Overall Rank:20]
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22. Board Game: 7 Wonders [Average Rating:7.87 Overall Rank:19]
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23. Board Game: Dominant Species [Average Rating:7.90 Overall Rank:23]
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24. Board Game: Navegador [Average Rating:7.61 Overall Rank:98]
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25. Board Game: Imperial [Average Rating:7.66 Overall Rank:81]
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