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Crawford Rule Brainteaser

Posted By: Chuck Bower
Date: Friday, 21 December 2012, at 10:15 p.m.

In Response To: Crawford Rule Brainteaser (AP)

I think your OPPONENT should PAY YOU $2.20.

Assume you start the current (3a,2a) game at 50% cubeless total winning chances of which 1/4 (total 12.5%) of cubeless games are gammon wins for you. Your takepoint is 32.26% (your MWC at -3 post Crawford). Your opponent's takepoint can be calculated assuming the gammon fraction remains the same (1/4) as the game progresses. If he takes and wins the game he wins the match. If he takes and loses a gammon he loses the match. If he takes and loses simply he'll have 50% MWC. So the breakeven takepoint Pw is the solution of 1*Pw + (1 - Pw)*(0.25*0 + 0.75*0.5) = 0.5. That answer is Pw = 1/5 or 20% GWC.

Now use the Keeler&Spencer continuous model as illustrated by Kleinman's Football Field: The game begins at the 50 yardline. Match trailer must push his way to the opponent's 20 yardline to cash. Match leader must push his way to his opponent's 32.26 yardline to cash. The playable part of the field is 80 - 32.26 = 47.74 yards long. Match trailer must drive 30 of those yards while match leader only needs to drive 17.74 of those yards. Match leader accomplishes this 30/47.74 = 62.8% of the time and wins the game with a cash. The rest of the time (37.2%) Match trailer wins the game with a cash.

So the match equity (trailer's point of view) is 0.372*50%MWC + 0.628*32.26%MWC = 38.9%.

Value (at $100/match) = (0.389 - 0.611)*$100 = -$22.20. With Crawford intact it's (0.4 - 0.6)*$100 = -$20.00.

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