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> That's also something that Elo handles just fine? If every game is a coin flip then everyone will end up with the same Elo. If player A has x more Elo points

by dvt 6y ago

> That's also something that Elo handles just fine? If every game is a coin flip then everyone will end up with the same Elo. If player A has x more Elo points than player B, then they win y% of their games. If your game has a skill ceiling where even a complete beginner always wins, say, 20% of their games, then that just means no-one will ever be able to rise above a corresponding Elo rating. That's not how it works. The distribution you end up with will not be uniform, it will look like this (just ran Elo with a coinflip; 11 players, 1000 matches): https://imgur.com/9O82pRj https://imgur.com/9O82pRj On the long term, I think this will tend to a geometric distribution with a low p value.

lmm 6y ago
Show your working? If you're matchmaking players against equal-ranked players, then each match is just +/- 50 points, you'll get a binomial distribution which tends to normal as n gets large (assuming a large player pool so each player's results are independent). If players play players with different ratings then that will tend to push their rating back towards neutral. You certainly don't get a geometric distribution because the rating algorithm is completely symmetric.
- dvt 6y ago
  > each match is just +/- 50 points This only happens in the rare cases where you're matching players against (exactly) equally-ranked players. You can mitigate this by always trying to match as "close as possible," but it's only a mitigation. Try simulating random matchmaking with Elo, and you'll get something like this: https://i.imgur.com/1Y08jUB.png https://i.imgur.com/1Y08jUB.png (1000 players, 100,000 games). In my simulation, I set k (the Elo constant) = 50. I think it's going to tend to a geometric distribution for reasons discussed here (which is another interesting and non-intuitive result): http://www.decisionsciencenews.com/2017/06/19/counterintuitive-problem-everyone-room-keeps-giving-dollars-random-others-youll-never-guess-happens-next/ http://www.decisionsciencenews.com/2017/06/19/counterintuiti...
  - LudwigNagasena 6y ago
    Elo can be negative, so this doesn’t apply.
  - lmm 6y ago
    > Try simulating random matchmaking with Elo I will. I was hoping you'd post the actual simulation details rather than more unlabelled graphs.
    - dvt 6y ago
      https://gist.github.com/dvx/e3311a984e14dca7c5eeb9214e0be3d5 https://gist.github.com/dvx/e3311a984e14dca7c5eeb9214e0be3d5 Using: https://github.com/HankSheehan/EloPy https://github.com/HankSheehan/EloPy (with a custom k-value of 50).
      - lmm 6y ago
        > with a custom k-value of 50 So you've patched this library somehow? Because when I run your code I get a result that's just full of 0 ratings. But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way. (Did you test your plotting methodology against some simpler rating system before using it to draw conclusions about Elo?). Plot a normal histogram, or a density plot if you're feeling fancy: https://towardsdatascience.com/histograms-and-density-plots-in-python-f6bda88f5ac0 https://towardsdatascience.com/histograms-and-density-plots-... . I'm betting the result is just the bell curve that we'd want and expect.
        dvt 6y ago
        > So you've patched this library somehow? Yes, as mentioned, I set the k-value to 50 on this line: https://github.com/HankSheehan/EloPy/blob/master/elopy.py#L80 https://github.com/HankSheehan/EloPy/blob/master/elopy.py#L8... Author decided to do something fancy which will only work when number of players is less than 1/2 * starting Elo rating. > But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way. As mentioned, you end up with a geometric distribution. I covered a similar phenomenon in a blog post I wrote last year[1]. See Theorem 3.3 in this paper: https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.p... But in short, the geometric distribution has maximal entropy over (0,∞) given a known mean (in our case, the mean will always be 1000). [1] https://dvt.name/2017/07/10/confusing-math-with-morality/ https://dvt.name/2017/07/10/confusing-math-with-morality/

by dvt 6y ago

lmm 6y ago

Show your working? If you're matchmaking players against equal-ranked players, then each match is just +/- 50 points, you'll get a binomial distribution which tends to normal as n gets large (assuming a large player pool so each player's results are independent). If players play players with different ratings then that will tend to push their rating back towards neutral. You certainly don't get a geometric distribution because the rating algorithm is completely symmetric.

dvt 6y ago
> each match is just +/- 50 points This only happens in the rare cases where you're matching players against (exactly) equally-ranked players. You can mitigate this by always trying to match as "close as possible," but it's only a mitigation. Try simulating random matchmaking with Elo, and you'll get something like this: https://i.imgur.com/1Y08jUB.png https://i.imgur.com/1Y08jUB.png (1000 players, 100,000 games). In my simulation, I set k (the Elo constant) = 50. I think it's going to tend to a geometric distribution for reasons discussed here (which is another interesting and non-intuitive result): http://www.decisionsciencenews.com/2017/06/19/counterintuitive-problem-everyone-room-keeps-giving-dollars-random-others-youll-never-guess-happens-next/ http://www.decisionsciencenews.com/2017/06/19/counterintuiti...
- LudwigNagasena 6y ago
  Elo can be negative, so this doesn’t apply.
- lmm 6y ago
  > Try simulating random matchmaking with Elo I will. I was hoping you'd post the actual simulation details rather than more unlabelled graphs.
  - dvt 6y ago
    https://gist.github.com/dvx/e3311a984e14dca7c5eeb9214e0be3d5 https://gist.github.com/dvx/e3311a984e14dca7c5eeb9214e0be3d5 Using: https://github.com/HankSheehan/EloPy https://github.com/HankSheehan/EloPy (with a custom k-value of 50).
    - lmm 6y ago
      > with a custom k-value of 50 So you've patched this library somehow? Because when I run your code I get a result that's just full of 0 ratings. But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way. (Did you test your plotting methodology against some simpler rating system before using it to draw conclusions about Elo?). Plot a normal histogram, or a density plot if you're feeling fancy: https://towardsdatascience.com/histograms-and-density-plots-in-python-f6bda88f5ac0 https://towardsdatascience.com/histograms-and-density-plots-... . I'm betting the result is just the bell curve that we'd want and expect.
      - dvt 6y ago
        > So you've patched this library somehow? Yes, as mentioned, I set the k-value to 50 on this line: https://github.com/HankSheehan/EloPy/blob/master/elopy.py#L80 https://github.com/HankSheehan/EloPy/blob/master/elopy.py#L8... Author decided to do something fancy which will only work when number of players is less than 1/2 * starting Elo rating. > But in any case I'm not at all convinced that your charts don't just show the normal distribution that we'd expect, just in some weird way. As mentioned, you end up with a geometric distribution. I covered a similar phenomenon in a blog post I wrote last year[1]. See Theorem 3.3 in this paper: https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.pdf https://kconrad.math.uconn.edu/blurbs/analysis/entropypost.p... But in short, the geometric distribution has maximal entropy over (0,∞) given a known mean (in our case, the mean will always be 1000). [1] https://dvt.name/2017/07/10/confusing-math-with-morality/ https://dvt.name/2017/07/10/confusing-math-with-morality/