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StrategyChecker play
Estimating in volatile situations

Steve Peterson wrote:
> 1 2 3 4 5 6 7 8 9 10 11 12
> ++ O: opponent  score: 0
>  O O O   O O O 
>  O O   O O O 
>  O O   
>    
>    
>  BAR v 5point match
>    
>    
>    
>    
>  O   O X 
> ++ X: pip  score: 0
> 24 23 22 21 20 19 18 17 16 15 14 13
>
> BAR: O0 X0 OFF: O0 X14 Cube: 2 (owned by opponent) turn: pip
> You roll 1 and 5.
> Please move 2 pieces.
>
> What's the right play here?
> What's the correct play for money?
While getting an exact answer for a problem like this is virtually
impossible at the table unless you have brought your computer along (and
it's pretty difficult even with pencil, paper, and much time), one can
get pretty close. The key is to use simplifying assumptions and take
comparative differences between plays. This cuts down on the mental
arithmetic necessary.
In the above position, if X hits O has 21 return shots. If X doesn't
hit, O has 11 shots. That part is easy. So, what do we do with the
information.
It is clear that if X doesn't hit he doesn't get a backgammon. When X
hits, O has 15 missing rolls. On two of these (55 and 66), he gets off
the backgammon anyway. If he rolls one of the remaining 13, X still
might come back with 21 (a 17 to 1 possibility). So, we make the
simplifying assumption that on 12 of O's rolls he will be backgammoned.
Note that this isn't exact, but it is close. The idea is to avoid
fractions and a lot of multiplication.
We know that if X plays safe O hits 11 times, while if X goes for the
gusto O hits 21 times. Thus, O hits 10 more times when X goes for the
gusto. What happens when O hits? Not clear. Most of the time X will
win a single game. However O may fail to complete the prime, and X may
slither around and win a gammon anyway. Also, O might get lucky and
actually win the game. Note that O gains twice as much from winning the
game (from 2 to +2) as he loses when he fails to contain the checker
(from 2 to 4). I'm going to make the simplifying assumption that the
scenario where X gets away after being hit occurs about twice as often as
the scenario where O wins the game. Under this assumption, we can say
that if O hits he always loses a single game.
Now we have the numbers we need. X wins a backgammon instead of a gammon
12 times, and wins a single game instead of a gammon 10 times, so he is
getting 12 to 10 odds. For money going for the gusto is clear.
Now, look at the match situation. If X wins a gammon, he has 85%
equity. If he wins a backgammon he has 100% equity. If he wins a single
game, he has 66% equity. So, he is risking 19% to gain 15%. This is
more than the 12 to 10 odds he is getting, so he is slightly better off
playing safe.
My assumptions may not be valid. If you think X will get away more than
twice as often as he will lose the game (which I do), then you may revise
the match decision. Also, the assumption that X wins 12 backgammons was
slightly low. Thus, at the match situation I would make it too close to
call.
Note that I never had to do any great mental arithmetic or remember a
bunch of numbers. The above calculations can be done at the table quite
easily. We may not get the right answer exactly, but that is not too
important. What is important is to make sure we are in the ball park and
don't make a big blunder. Estimating backgammon positions is usually
guesswork anyway, so exact calculations are impossible. In this
position, we conclude that it is very clear to go for the gusto for
money, while at the match score it is a pretty close decision. That's
the best we can expect to do.
Kit




StrategyChecker play
 Avoiding major oversights (Chuck Bower+, Mar 2008)
 Bearing off with contact (Walter Trice, Dec 1999)
 Bearing off with contact (Daniel Murphy, Mar 1998)
 Blitzing strategy (Michael J. Zehr, July 1997)
 Blitzing strategy (Fredrik Dahl, July 1997)
 Blitzing technique (Albert Silver+, July 2003)
 Breaking anchor (abc, Mar 2004)
 Breaking contact (Alan Webb+, Oct 1999)
 Coming under the gun (Kit Woolsey, July 1996)
 Common errors (David Levy, Oct 2009)
 Containment positions (Brian Sheppard, July 1998)
 Coup Classique (Paul Epstein+, Dec 2006)
 Cube ownership considerations (Kit Woolsey, Apr 1996)
 Cubeinfluenced checker play (Rew Francis+, Apr 2003)
 Defending against a blitz (Michael J. Zehr, Jan 1995)
 Estimating in volatile situations (Kit Woolsey, Mar 1997)
 Gammonish positions (Michael Manolios, Nov 1999)
 Golden point (Henry Logan+, Nov 2002)
 Hitting loose in your home board (Douglas Zare, June 2000)
 Holding games (Casual_Observer, Jan 1999)
 How to trap an anchor (Timothy Chow+, Apr 2010)
 Jacoby rule consideration (Ron Karr, Nov 1996)
 Kamikaze plays (christian munkchristensen+, Nov 2010)
 Kleinman Count for bringing checkers home (Øystein Johansen, Feb 2001)
 Late loose hits (Douglas Zare+, Aug 2007)
 Mutual holding game (Ron Karr, Dec 1996)
 Pay now or pay later? (Stuart Katz, MD, Nov 1997)
 Pay now or pay later? (Stephen Turner, Mar 1997)
 Pay now or play later? (Hank Youngerman+, Sept 1998)
 Play versus a novice (Courtney S Foster+, Apr 2004)
 Playing doublets (Grunty, Jan 2008)
 Playing when opponent has one man back (Kit Woolsey, May 1995)
 Prime versus prime (Albert Silver+, Aug 2006)
 Prime versus prime (Michael J. Zehr, Mar 1996)
 Saving gammon (Bill Riles, Oct 2009)
 Saving gammon (Ron Karr, Dec 1997)
 Splitting your back men (KL Gerber+, Nov 2002)
 Splitting your back men (David Montgomery, June 1995)
 Trap play problem (Brian Sheppard, Feb 1997)
 When in doubt (Stick+, Apr 2011)
 When to run the last checker (Stick Rice+, Jan 2009)
 When you can't decide (John O'Hagan, Oct 2009)
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