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Crawford Rule Brainteaser

Posted By: Daniel Murphy
Date: Friday, 21 December 2012, at 4:17 p.m.

In Response To: Crawford Rule Brainteaser (AP)

It seems to me that Joe and Sam are right, that the Crawford rule favors the trailer at 3-away 2-away -- by quite a lot -- so I would pay nothing to turn the rule off.

I make these simplifying assumptions:

  • 2/3 of initial doubles should be taken
  • Players are equally likely to make the initial double at this score
  • 2/3 of initial doubles should be taken at this score
  • The doubler wins 70% of initial doubles at this score.
  • Gammon rate = 20%
  • Ignore backgammons and undoubled gammons
Then:

Frequency
1/6I double, pass. Score is -2-2. Crawford rule doesn't matter.
2/6He doubles, I take. The game decides the match (ignore last roll doubles with no redouble possible). Crawford rule doesn't matter.
1/6He doubles, I drop. Trailing 3-away 2-away, with Crawford rule, I have about 25% MWC, without the Crawford rule I have about 30% MWC. 30%-25% = 5%. 1/6 * 5% = 0.83% disadvantage with Crawford rule.
2/6I double, he takes. If I win a gammon, or lose, Crawford rule doesn't matter. The only games that matter are my single wins which make the score 1-away 2-away (as Joe wrote, "The reason the Crawford rule favors the trailer is the amount of increase in winning chances after winning 2-points, his most common result"). Single wins = 2/6 * .7 * .8 = 18.7% (that's 2/6 times the percentage of games I win times the percentage of games I win that aren't gammons). Leading 1-away 2-away, with Crawford rule, my MWC is 70%, without Crawford rule, about 50%. 70% - 50% = 20%. 20% * 18.7% = 3.7% advantage with Crawford rule.

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