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Topic: Winning percentage for gold for h v. h 2100+ (Read 626 times) |
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PhilomathBret
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Arimaa player #2167
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Winning percentage for gold for h v. h 2100+
« on: Dec 16th, 2009, 9:42pm » |
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Sorry if I led anybody to believe that I knew the statistic. I was just wondering if it is possible to get a stat like that from the archive of games. It could give us some insight for the question "Who is winning from move 1 in Arimaa: gold or silver?".
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woh
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Re: Winning percentage for gold for h v. h 2100+
« Reply #1 on: Dec 17th, 2009, 4:21am » |
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For all HvH games, it is:
Gold 2563 wins vs 2844 wins or
Gold 47.4% vs Silver 52.6%
The expected result according to the gameroom ratings is:
Gold 2591.7 vs Silver 2815.3
The expected result according to the WHR ratings is:
Gold 2595.6 vs Silver 2811.4
Silver has done marginaly better than expected
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woh
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Arimaa player #2128
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Re: Winning percentage for gold for h v. h 2100+
« Reply #2 on: Dec 17th, 2009, 8:08am » |
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For HvH games where both player have a gameroom rating of 2100 or greater:
Gold 145 wins vs Silver 140 wins
Gold 50.9% vs Silver 49.1%
The expected result according to the gameroom ratings is:
Gold 141.6 vs Silver 143.6
The expected result according to the WHR ratings is:
Gold 140.7 vs Silver 144.3
In this case Gold has done marginaly better than expected
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Fritzlein
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Arimaa player #706
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Re: Winning percentage for gold for h v. h 2100+
« Reply #3 on: Dec 17th, 2009, 9:38am » |
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Thanks, woh. Interesting statistics.
The variance in a single game result is P(1-P), where P is the Gold player's probability of winning that game. The variance is maximum when P = 1/2, and less when P is closer to 1 or 0. Let's take a guess at an "average" variance of 3/16, based on P = 1/4. (With the actual rating gaps, woh could calculate the variance of each game and sum them, but I'm just doing a ballpark estimate here.)
In the 285 games that met PhilomathBret's criteria, namely HvH over 2100 rating, we can estimate the variance to be 285 * 3/16 = 53.43. The standard deviation is the square root of the variance, i.e. 7.3 wins.
Statisticians usually give confidence intervals of plus or minus two standard deviations. Thus to add a confidence interval to the predictions of the game room ratings, we would say
Gold should have won 141.6 of the games, plus or minus 14.6 games.
Equivalently, we could say
Gold should have won between 127.0 and 156.2 games.
The fact that Gold actually won 145 of the games instead of 141.6 is clearly statistically meaningless. Indeed, it is only 0.23 standard deviations from the expected value, which is closer to evidence of tampered data than it is to evidence of an advantage for either side. Only 19% of the time would the measured value be so close to the expected value. If our test had been "let's test whether someone is fudging the data to make it appear that Gold and Silver have precisely equal chances," and we had been only 0.05 standard deviations from the expected value, then by most views of statistical significance, that result would be too good to be true.
To apply the same calculation to the larger data set of 5407 games, I expect that there would be more mismatches, so I would guess an average variance of 5/36, based on P=1/6. That gives us a total variance of 5407 * 5/36 = 751, and a standard deviation the square root of that, i.e. 27.4. The actual result of 2563 wins for Gold compared to the expected result of 2591.7 wins for Gold is off by 1.05 standard deviations, i.e. off by enough that it is not suspiciously close to expected, but not off by the 2 standard deviations usually required for statistical significance.
These calculations are rough guesstimates, but it should convey the general sentiment quite clearly: we have no basis to conclude that either player has an advantage in Arimaa.
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PhilomathBret
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Arimaa player #2167
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Re: Winning percentage for gold for h v. h 2100+
« Reply #4 on: Dec 17th, 2009, 2:05pm » |
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Thanks woh and thanks Fritz! Shame on me for thinking 52% for silver is worth much, even though I've had many classes dealing with the null hypothesis.
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