Safe or Greedy by John O'Hagan

In the position below, the question for Red is whether to take checkers off and leave both checkers on the 6 point, or clear the 6 point.  Of course, clearing the 6 point is safer in terms of game wins, but you win more gammons by taking more off.  First, decide what you would do, and then scroll down and see how John O'Hagan views this problem.



Safe or greedy?  The gammon value is the normal 1/2 in this money game which means peeling needs to create a little over twice as many extra gammons as it does extra losses to be correct.  Why "a little over twice as many"?  Because we need to account for the fact that the opponent owns the cube.

 

How often do we lose after peeling?  All non-aces clear the 6-point, 4 numbers leave a shot, and the other 7 do neither.  This averages out to about 5 shots in 36 games.  The opponent hits about 30% of these and then (taking the cube into account) is probably a small favorite in the game.  That's around 2.3% cubeless chances of losing after peeling which translates to around 2.7% or so cubeful.  Peeling therefore needs to win over 5.4% extra gammons (compared to clearing) to make it the correct play.  Can it do this?

 

How many gammons do you win after clearing the 6-point?  You'll have 10 checkers left with a pip count of 27 so it's a 5+ roll position.  A pure 5-roll position (all 10 checkers on the ace and deuce points) would have an Effective Pip Count (a.k.a. Trice Count) of 36 so I'll guesstimate this position has a TC of 39.5.  The opponent, on the other hand, has a gammon count of 47 outside pips plus 2 to bear a checker off the deuce point for a total of 49.  At 7 pips per roll (to account for wastage), that's about a 7-roll position to get off the gammon.  After White's next roll, he'll have a 6-roll position with a TC of 43.  We will then be on roll with a TC of 39.5 vs. White's 43.  What kind of chances does that give us?  If White had 35.5, our gammon chances would be about pick-em and if White had 42, we'd be around 78%.  White has 43 so that should put us a little over 82% gammons after clearing the 6-point.

 

Can peeling give us over 87.4% gammons?  I will assume that Red never wins a gammon on the 1.5/36 games where he's hit which means he'll need 31.7 gammons in the 34.5  games (about 92%) where he's not to reach 87.4% gammons overall.  Peeling will leave Red with 8 checkers left and a raw pip count of 27.  The TC is difficult to estimate here since Red has future peel or safe choices with 43,42,32, and 22 but I'll guess it's around 36.  Is a TC of 36 on roll over 92% vs. a TC of 43?  32 would be 50% while 38 would make Red about a 78/22 favorite so each pip in that range is worth about 4.67%.  It's not linear beyond that however so I'll made an educated guess that the progression per pip over 38 is 4%,3%,2%,1%, .5% which totals 88.5% gammons, a bit short of the required 92%. 

 

Clearing the 6-point therefore looks correct. And XG agrees, as the evaluation below shows.





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