Forum Archive : Puzzles

Pruce's paradox

From:   Alan Pruce
Address:   prucea@gmail.com
Date:   21 December 2012
Subject:   Crawford Rule Brainteaser
Forum:   rec.games.backgammon

You are playing an equal opponent in a match for $100, and are trailing 3-
away, 2-away. How much would you be willing to pay to turn off the Crawford
Rule for the duration of the match?

Sam Pottle  writes:

Nothing. I'm better off with it on.

Joe Russell  writes:

The trailer should pay up to $5.40 to keep the Crawford rule. The reason
the Crawford rule favors the trailer is the amount of increase in winning
chances after winning 2-points, his most common result. His winning chances
are 16.46% greater with the Crawford rule in effect.

Daniel Murphy  writes:

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


Freq    Result
1/6     I double, pass. Score is -2-2. Crawford rule doesn't matter.
2/6     He doubles, I take. The game decides the match (ignore last roll
        doubles with no redouble possible). Crawford rule doesn't matter.
1/6     He 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
2/6     I 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.

Alan Pruce  writes:

Wow a lot of great responses already, so I guess I will post my analysis

Clearly, the Crawford Rule helps the Leader when he reaches the Crawford
Game, so it seemed intuitive to me that the leader should always benefit
from having the Crawford Rule in effect. However, like many of the posters
have figured out, it seems the trailer actually benefits from having the
Crawford Rule in effect at the {-2,-3} score, so he is actually willing to
pay to keep the Crawford Rule on. Based on my calculations, the trailer is
willing to pay up to ~$3.60 to keep the rule in effect.

Also, I want to point out that I did not just find this effect at {-2,-3}.
The Crawford Rule at {-4,-5} hurts the leader even more than at {-2,-3},
and there were several other scores where the Crawford Rule decreased the
leader's MWC.

Now if I could only get myself to spend 10% of the time studying actual bg
as opposed to doing this type of mental buffoonery, maybe I could finally
get my PR in the single digits ;)
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