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BGonline.org Forums
'variance reduction' versus 'MWC error totals'
Posted By: Johan Segers
Date: Friday, 9 September 2011, at 7:21 a.m.
In another thread on this forum, the distinction between 'variance reduction' (also called 'luck adjusted result') and 'difference in MWC error total' has been discussed (MWC = match winning chances). However, I wonder whether these shouldn't yield the same thing?
Intuitively, a player's gain or loss in MWC is entirely explained from a combination of luck and errors.
Formally, consider the following quantities:
- gain_A = +50% if A wins the match, -50% if A loses the match
- gain_B = +50% if B wins the match, -50% if B loses the match
- luck_A = the total amount of luck (= MWC after the roll - MWC before the roll) throughout A's rolls (positive or negative)
- luck_B = the total amount of luck throughout B's rolls (positive or negative)
- error_A = the total amount of MWC wasted by A through errors (zero or positive)
- error_B = the total amount of MWC wasted by B through errors (zero or positive)
In the style of an old-style physicist, allow me to postulate the law of conservation of MWC:
gain_A = luck_A - luck_B - error_A + error_BAs a consequence:
gain_A - luck_A + luck_B = error_B - error_AThe left-hand side, gain_A - luck_A + luck_B, is, to my understanding, known as the 'luck adjusted result'. The right-hand side, error_B - error_A, is just the difference in MWC error totals.
Despite its intuitive appeal, the law fails a numerical test: in GNU Backgammon, the above equalities do not hold, not even for 1-point matches. The difference in error totals (MWC) is in general quite different from the luck adjusted result.
Questions:
- Is the law correct?
- If yes, then where do the deviations from the law in GNU come from? And which number is the most reliable?
- Does other software, e.g. XG, give similar deviations?
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