- Show (the logarithm of) the ratio of the number of opponents beaten to the number of games lost in the profile

Specifics:

- This could be instead of or in addition to the percentage of games won. It's similar to that percentage in that it's a simple statistic that reflects the proportion of games a player wins without trying to take into account different player strengths, but whereas the percentage depends strongly on how many players a player typically plays in a game, this ratio is independent of that and thus a more direct measure of a player's strength.

This will improve the following aspects of the site:

- relevance of statistics in profile for gauging player strength

This was originally a suggestion by Itrade in a General Discussion thread; I replied to it and someone liked the ideas and suggested to put the entire post in the suggestion box. For context, here's the URL for Itrade's original post:

Subject: Win Percentage Seems Cool, but is Actually Pretty Useless

Here are my comments on that thread:

It seems the discussion in this thread mixed a couple of objectives, and I want to try to disentangle it somewhat.

The objective of Itrade's original post, as I understood it, was to fix a specific problem with the win percentage, namely that it doesn't take into account the number of players per game. I found this post searching this forum because I wanted to make the same comment and a similar suggestion.

A lot of the ensuing discussion was about things like points per game and how best to reflect the skill levels of the opponents. This is an interesting discussion in itself, but it goes way beyond the original remark and proposal. The win percentage is a very simple stat whose attraction (it's clear from this and other threads that many people like it) lies in its simplicity and obviousness. A scheme for taking into account skill levels is inevitably more complicated and contains more free and somewhat arbitrary parameters, and no matter how nifty it is, there will inevitably be people who disagree with some detail of it.

So while I personally find the point system more interesting and would like to see a points per game stat (perhaps as a rolling average), this is a somewhat different topic from Itrade's proposal, which was about fixing a simple problem in a simple stat that people seem to value even though, or perhaps precisely because, it doesn't try to take into account player skills.

The solution Itrade proposed doesn't introduce any arbitrary parameters; it just ensures in a straightforward way that the stat tells you the most basic thing that one might want it to reflect, namely whether a player is doing better or worse than if the winners had been determined randomly.

So I second Itrade's proposal to introduce a comparison between # of players beaten and # of games lost. This could be instead of the win percentage, or if people really want to keep that, in addition to it.

One post that did comment on Itrade's original proposal was this:

lackattack wrote:Itrade wrote:

I personally like amount of players beaten divided by amount of games lost as a way to tell how good a player is. It doesn't take into account the skill level of the other players, though.

Not bad, but how about something simpler: instead of wins divided by games we use wins divided by # of opponents?

A subsequent post seconded this, and Itrade asked what the difference is.

In #wins/#opponents, the correction for the opponents is the wrong way around -- the more opponents you have, the less your wins count. So if you want to use the number of wins, the opponents would need to be in the numerator, with each factor normalized by the number of games: #wins/#games * #opponents/#games. That's not quite right yet, though, either: a player winning an average number of two-player games would get a score of 1/2 on this, whereas a player winning an average number of eight-player games would get a score of 7/8. What we need is actually #wins/#games * #players/#games. That would yield a score of 1 for an average player, no matter what sorts of games they play. It would have the advantage of looking like a slightly corrected version of the current stat #wins/#games. However, it still has a major disadvantage -- among good players, it favours those who tend to play more opponents. In the extreme case of a player who always wins, their score would be 2 if they only play two-player games, but 8 if they only play 8-player games.

Itrade's original proposal, #opponents beaten/#games lost, on the other hand, has none of these disadvantages: It's always 1 for an average player, it's always zero for a player who always loses, and it's always infinity for a player who always wins, irrespective of the number of opponents per game.

It also has the nice feature that it's similar to the point system, just without depending on the other players' skills. You earn points from players you beat and lose points to players you lose to, just instead of the points depending on your own and your opponents' current points, it's always exactly one point, and instead of starting out with 1000 points, you just take the ratio between the number of points you've earned and the number of points you've lost.

One more thing that I personally would like to see but that might perhaps seem too "mathematical" to others is to take the logarithm of the ratio -- that would remove the arbitrary asymmetry of putting one of the numbers in the numerator and the other in the denominator, and would transform the asymmetric range of 0 to 1 for below-average and 1 to infinity for above-average into the symmetric and more "natural" range of -infinity to 0 below average and 0 to infinity above average.