Date: Thu, 15 Jun 2000
From: Jim Geary
To: cgp
Subject: Re: [cgp] Measuring luck
On Thu, 15 Jun 2000, James A. Cherry wrote:
Lawren Freebody writes:
>Blue Dragon wrote:
>> Maybe it's just my lack of education showing, but I'd be
>> more interested in seeing a comparison of the Standard Deviations
>> of the game spreads.
>
>That would be interesting, also (as opposed to instead of). Can you do that,
>James?
Oh, I can do it.
Player 1 Player 2 Spread
Avg SD Avg SD Avg SD CorrCoef
TWL 429.2 58.7 416.6 58.0 +12.6 97.2 -0.3876
DD 452.7 59.2 440.7 58.8 +12.0 97.5 -0.3644
As you can see, the SD of spread is slightly higher in SOWPODS than at
TWL. But this must be viewed in light of the correlation coefficients
(CCs) between players' scores.
Robert Parker has reported, and I have confirmed above, that in TWL
Scrabble, players' scores are negatively correlated with a CC of about
-0.4. CC is a number between -1.0 and +1.0 which tells you how much
one variable depends on another. You can get an intuitive idea of CC
by plotting points one variable against another on a pair of axes. CC
generally measures how closely clustered the points are around a
straight line; CC tends towards +1.0 if the line is sloped like a
forward slash "/", and tends towards -1.0 if the line is sloped like a
backslash "\". If you get a perfectly straight line going from the
lower left to the upper right, then the variables are "directly
correlated", and the CC is +1.0. If you get a straight line going
from the lower right to the upper left, then the variables are
"inversely correlated" and CC is -1.0.
For the 10,000 TWL games, here's a graph of player 2's score on the
vertical axis against player 1's score on the horizontal axis. The
number at each point tells how many games had a final result with
those scores. Thus, the "307" at the (400,400) coordinate means that
the final score for player 1 was between 400 and 424, _and_ the final
score for player 2 was between 400 and 424, in 307 games.
675| 1
650| 1 1
625| 1 1 2 3
600| 4 1 1 3 2 5
575| 3 6 9 5 10 5 1 1
550| 3 2 2 9 27 20 10 13 4 5 2 1
525| 2 6 17 29 29 46 27 26 19 1 8 2 1
500| 1 3 7 26 55 69 70 86 56 34 20 7 1 1
475| 15 39 65 109 145 128 108 67 48 12 9 4 1
450| 3 12 50 111 163 212 196 203 137 71 31 14 5 1
425| 1 13 46 98 158 236 267 268 186 124 74 31 6 4 2
400| 1 7 27 69 166 228 307 306 266 164 99 45 18 4 2 1 1
375| 1 16 43 125 201 257 267 264 180 106 61 20 14 5 1 1
350| 1 7 30 54 128 172 235 228 153 113 62 42 18 8 1
325| 1 2 27 45 110 145 132 107 77 57 27 12 5 1 1
300| 6 16 31 48 61 58 27 28 9 9 2 1 1
275| 3 12 7 25 24 16 11 12 3 2 1
250| 4 5 7 4 2
225| 1 1 1 1
200| 1
+------------------------------------------------------------------------
225 250 275 300 325 350 375 400 425 450 475 500 525 550 575 600 625 650
You can see the blob has a vaguely "lower right to upper left" shape,
hence the negative CC. The graph for the DD scores looks pretty
similar.
In TWL, the spread's SD is 97.2 with a CC of -0.388. In SOWPODS, the
spread's SD is 97.5 with a CC of -0.364. I believe the interpretation
of the CCs is: at TWL, you have slightly less chance to come back when
you're losing. In other words, when a stronger player starts losing
at TWL, s/he needs a bit more luck to come back and win than a
stronger player playing SOWPODS. This supports my previous conclusion
that SOWPODS marginally diminishes the luck factor compared to TWL.
no doy.
Last Modified 2/22/01
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