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The position where both players have one checker remaining on their 6 pts.,
with the player on roll having access to the cube, is a very special
position: it is a double and an optional take/pass, even though the second
player only has a 3/16 = 18.75% cubeless winning probability. In fact for
gammon-less positions, this represents the theoretical smallest cubeless
winning percentage in which a player may still have a take. The reason for
this is that player 2 wins exactly 25% of the time with the cube (if player
1 rolls less than 6, than player 2 can double player 1 out of the game...
actually this pass is optional, although the equity remains the same
whether the redouble is accepted or dropped - see below), and his redouble
if player 1 rolls less than 6 is also perfect in the sense that it is made
with the lowest possible winning percentage (player 2 now has a 75% winning
chance) that will force player 1 to pass (optionally player 1 can take the
redouble for the same equity).
Below is a proof that was given to me by bobk.
--
From: koca@orie.cornell.edu (Robert Koca)
Subject: .8125 proof
Preliminary lemmas
Lemma 1) Suppose playing continuous game (in sense of Zadeh)
and opponent is not allowed to double but you are. Then should
double when attain probability .75 of winning.
Proof. Easy. Equilibrate opponent take/drop equity.
Lemma 2) Suppose are playing backgammon and opponent is not allowed to
double but you are. Suppose Pr(winning cubeless)=p. If given choice it is
at least as good to start a continuous game starting with Prob(winning
cubeless)=p.
Proof. With condition that opponent can never double added, Zadeh's
arguments now make sense.
Proof of theorem. Suppose in a bg game, Pr(A wins cubeless)=.8125 and
there are no gammon or backgammon chances.
Suppose A offers a double, and pledges that he will never reredouble.
Then B would do at least as well in a continuous game starting with
.1875 cubeless chance. Since A is nice he offers to switch to such a
game also.
Then payoff to B to accepting after all these "favors" from A
equals +2 * Pr(B reaches .75 cubelss chance) - 2*Pr( B doesn't reach
.75 cubeless chance).
Pr(B reaches .75 cubeless)= .1875/(.5625+.1875)=.25.
Thus take equity = 2*(.25)-2*(.75)=-1.
If started with A's cubeless chances > .8125, then calculations give B's
take equity < -1 which implies not a take in original (not more favorable)
bg game.
It is interesting to note that 6 away 6 away attains this bound.
Pretty sure this is airtight, Bob Koca
--
I would like to go farther though. In gammon-less positions, the
theoretical lowest (cubeless) winning percentage that a player MAY still
have and still have a correct beaver is 37.5% by similar reasoning as
above. However I am unable to find such a position. Three questions:
1. Does such a position exist?
2. If no such position is known to exist, then what (currently) is known
as the "worst" possible gammon-less position that is still a correct
beaver.
Here "worst" refers to the cubeless winning position of the player
contemplating a beaver.
3. Same as 2, but now also considering positions involving gammons.
Here, "worst" refers to the equity of the player contemplating a beaver,
*IF* the cube were to be taken out of action after the beaver. In these
positions, for consistency, assume that one of the players has just been
doubled to 2, and is contemplating beavering to 4.
A few more things:
1. (Of course) I am referring to money play here.
2. By "gammons," I really mean "gammons AND backgammons"
3. Although the initial double (before the beaver) will in almost all
(ALL for #2) cases be incorrect, assume that after this, both players have
perfect checker and cube play.
4. In #2, if the position is simple enough, the exact cubeless winning
percentage may be determined exactly. However for #3, with positions in
which gammons are possible, it would seem unlikely that the equities could
be determined exactly - in this case a rollout program would probably have
to be used.
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Puzzles
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Alice, who is not on the bar, discovers that however she plays she ends up with 13 blots. What is her position and roll?
- All-time best roll (Kit Woolsey+, Dec 1997)
What position and roll give the greatest gain in equity?
- All-time worst roll (Tim Chow+, Feb 2009)
Find a position that goes from White being too good to double to Black being too good to double.
- All-time worst roll (Michael J. Zehr, Jan 1998)
What position and roll give the greatest loss in equity?
- Back to Nack (Zorba+, Oct 2005)
How can you go from the backgammon starting position to Nackgammon?
- Cube ownership determines correct play (Kit Woolsey, Jan 1995)
Find a position and roll where the correct play depends on who owns the cube.
- Highest possible gammon rate (Robert-Jan Veldhuizen+, May 2004)
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What is the highest possible gammon rate in an undecided game?
- Infinite loops (Timothy Chow, Mar 2013)
- Is this position reachable? (Timothy Chow+, Feb 2013)
- Janowski Paradox (Robert-Jan Veldhuizen+, Nov 2000)
Position that's a redouble but not a double?
- Least shots on a blot within direct range (Raymond Kershaw, Dec 1998)
Find a position with no men on bar that has the least number of shots out of 36 to hit a blot within direct range.
- Legal but not likely (David desJardins, July 2000)
Find a position that can be legally reached but never through optimum play.
- Lowest probability of winning (masque de Z+, Apr 2012)
What is the smallest win probability in backgammon, greater than zero.
- Mirror puzzle (Nack Ballard, Apr 2010)
Go from the starting position to the mirror position (colors reversed)
- Most checkers on the bar (Tommy K., May 1997)
What is the maximum total possible checkers on the bar?
- Most possible plays (Kees van den Doel+, May 2002)
Find the position and dice roll which have the most possible plays.
- Not-so-greedy bearoff (Kit Woolsey, Mar 1997)
Find a no-contact position where it is better to move a checker than bear one off.
- Not-so-greedy bearoff (Walter Trice, Dec 1994)
Find a no-contact position where it is better to move a checker than bear one off.
- Priming puzzle (Gregg Cattanach+, May 2005)
From the starting position, form a full 6-prime in three rolls.
- Pruce's paradox (Alan Pruce+, Dec 2012)
- Quiz (Martin Krainer, Oct 2003)
- Replace the missing checkers (Gary Wong+, Oct 1998)
- Returning to the start (Nack Ballard, May 2010)
What is the least number of rolls that can return a game to the starting position?
- Returning to the start (Tom Keith+, Nov 1996)
What is the least number of rolls that can return a game to the starting position?
- Shortest game (Stephen Turner+, Jan 1996)
What is the shortest (cubeless) game in which both players play reasonably?
- Small chance of ending in doubles (Walter Trice, Dec 1999)
Find a position where the probability of the game ending in doubles is less than 1/6.
- Three-cube position (Timothy Chow+, Sept 2011)
Find a position and roll for which three different checker plays are best, depending on the location of the cube.
- Trivia question (Walter Trice, Dec 1998)
What is the symmetric bearoff with the smallest pip count that is not an initial double?
- Worst possible checker play (Gregg Cattanach+, June 2004)
What position and roll have the largest difference between best and worst play?
- Worst possible opening move (Gregg Cattanach, June 2004)
What is the worst possible first move given any choice of dice?
- Worst symmetric bearoff of 8 checkers (Gregg Cattanach+, Jan 2004)
What symmetric arrangement of 8 checkers in each player's home board gives roller least chance to win?
- Worst takable position (Christopher Yep, Jan 1994)
What position has lowest chance of winning but is a correct take if doubled?
- Zero equity positions (Kit Woolsey, Apr 1995)
Find a position with exactly zero equity in (1) money play or (2) cubeless.
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