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The maximum number of games in a match
Posted By: 6stones In Response To: Trailer invariably wins at the Crawford (6stones)
Date: Sunday, 25 September 2011, at 9:30 a.m.
Thank you for the responses.
The trailer invariably wins the Crawford-game (but never a gammon/backgammon win) in matches in which the largest number of games is reached (2N-1, where N is the match length and N>1, e.g. five games in a three-point match, final result 3-2). Clearly, the longer the match, the lower the probability of occurrence, and one is more likely to find them in three-point or two-point matches. Has anyone come by 2N-1 matches of at least five-points in length?
Some trivial observations, hypotheses and questions :
- If the trailer in the Crawford-game wins the match, he wins all the post-Crawford games: if the score is not 1-away 1-away in the first post-Crawford game, the trailer commits a cubing error when not automatically doubling;
- The last game is always an (undoubled) Double Match Point;
- There are invariably odd number of games;
- All the games are single wins, but there are various reasons for neither side winning a gammon/backgammon:
- as for the cube, either the cube was never turned or all the cubes were rejected;
- in matches of at least three points in length cube action errors are perhaps more likely than that there never arose a position in which it was correct to double and take;
- errors for the (potential) cuber would be ND/P, ND/T, MD/T, MD/P, TGTD/P, TGTD/T;
- when doubled and passed, Correct No Double/Take (ND/T) and (less frequent) Too good to Double/Take (TGTD/T) are errors for both players;
- there are no Wrong Takes, but correct Takes(D/T, TGTD/T) which were passed would be errors;
- whether to take or pass is a more decisive factor than the doubling decision itself when cube is turned;
- in checker play perhaps no undoubled gammon wins because of playing too timidly, or/and the opponent knows how to save gammon;
- skill level in playing GammonSave and GammonGo in Crawford when gammons count;
- an ideal player conducive to incidence of 2N-1 matches would perhaps be both a late doubler and a shallow taker, and plays checkers too timidly but knows how to save an undoubled gammon (plus what else?);
- as expert players are less likely to make cube action and checker play errors, it is plausible to assume that 2N-1 matches are more likely to arise in matches by novice players; or if played by more advanced players, cube actions are more likely to be correct double/pass (?).
- The luck is practically even (?) (at least in all the matches I have seen, viz. three matches, XG analysis, within 0.020). Would it be possible that the luck difference never exceeds a certain limit in 2N-1 matches, and if so, what would that be? If on the other hand there are matches where luck is 'clearly uneven' (yet to be specified), what are the compensatory factors: skill difference, others?;
- More likely when players are roughly of equal skill level (?);
- How frequently do 2N-1 matches occur (when varying different parameters: skill difference, others?) in a two-point, three-point match, etc.?;
- How large proportion of matches is attained with the best possible cube actions?
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