srbgblitz.PositionDBu3FIaktNoLauthortLjava/lang/String;LcommenttLjava/lang/String;LcontacttLjava/lang/String;LintroductiontLjava/lang/String;LtitletLjava/lang/String;LvtLjava/util/Vector;LversiontLjava/lang/String;xpt
Stein KulsethtThis problem set was a side event of Oslo Open 1999. The time limit for the quiz
was several hours, but of course most people had a tournament to attend to, and
I don't think anyone used more than about half an hour, and at least half the field
rushed through the quiz in the last 10 minutes, which probably was too little time
for a quiz this hard.
12 persons/teams entered, and the contest was won by Eirik Milch Pedersen &
Fabio Cultrera (who BTW was among the last 10 minute rushers). They gave up
only 0.631 equity over all 18 problems.
The original problem set listed from 3 to 7 (more or less) plausible alternatives for
each problems, this BGB version gives open questions. If your preferred choice
isn't among the ones given in the answer it might be just as good as the lowest
ranked alternatives, but most likely it is even worse.
The numbers in the solutions give normalized equity, and score (equity difference -
on both sides for cube problems) for each alternative. The alternatives are numbered
as in the original problem set.
The correct answer is marked by ***.
Problems, rollouts and comments by Stein Kulseth.
Thanks to Fredrik Dahl for reviewing the problem set.
tstein.kulseth@fou.telenor.not1st Oslo Open Problems Tournament
May 1st 1999
Rules
1. Problems 1 through 18 are multiple answer. You are to choose the best checker
play or cube action from the given alternatives. You may assume that both sides
play optimally throughout the remainder of the game or match.
2. For match play problems you may assume that the Heinrich-Woolsey match
equity table (appendix A) gives the exact match winning probabilities for al
match scores.
3. Your score is the total normalized equity lost in all 18 problems.
A more accurate description of the term "normalized equity" is given in
appendix B, but please do not despair if you do not feel the urge to read this.
It all just boils down to your being punished more for making large errors than
for making small errors.
4. For cube problems you may lose equity on both sides, eg. if correct action for
black is double/drop and your answer is no double/take, you get both the equity
loss from not doubling as black and the equity loss from taking as white if black
had correctly doubled.
Beavers are used, but raccoons are not used.
5. Tied scores will be resolved by the tie-breaker problem 19. This is an open
question, you are to estimate black's cubeless winning probability, again
assuming optimal play from both sides. The one whose estimate is closest
to the rollout result will win.
For scores that are not tied problem 19 will not be considered.
6. The equity estimates are computed by JellyFish rollouts, and we have made
every possible effort to account for the value of the cube which cannot be
accurately represented in the JellyFish rollouts. Details on how the rollouts
are performed are found in appendix C, and again don't panic if you feel like
skipping this appendix.
Our guess is that you are better off using your time thinking about the problems
anyway.
7. No computers or calculators may be used!
8. Use of books or other printed material as reference is permitted.
9. Consulting with other participants is also permitted.
A group of participants may make a joint submission.
10. The participant must sign and submit his answer sheet to the tournament
director before the deadline given at Oslo Open.
Appendix A - Heinrich-Woolsey Match Equity Table
The probability (in %) of winning the match for all scores up to 9-away, 9-away.
-1 -2 -3 -4 -5 -6 -7 -8 -9
-1 70 75 83 85 90 91 94 95
-2 30 50 60 68 75 81 85 88 91
-3 25 40 50 59 66 71 76 80 84
-4 17 32 41 50 58 64 70 75 79
-5 15 25 34 42 50 57 63 68 73
-6 10 19 29 36 43 50 56 62 67
-7 9 15 24 30 37 44 50 56 61
-8 6 12 20 25 32 38 44 50 55
-9 5 9 16 21 27 33 39 45 50
Example: you are leading by 5 points to 3 in a 9 point match, that is you are at 4-away and your
opponent is at 6-away. Crossreference the -4 row with the -6 column to find the number 64.
Your chances of winning the match from this score is 64%.
Appendix B - Normalized Equity
The equity of a position is a measure of what the position is worth to the player. If a position has
equity +0.230 you can expect to win - on average - 0.23 points per game played from this position,
and if you played this position say 10,000 times your winnings would end up somewhere close to
2,300 points.
Normalized equity means that the current level of the cube is used as a unity measure, loosing a
single game at the current cube level is valued at exactly -1.0 while winning a single game is
valued at exactly +1.0
Formulas for equity
The basic formula for calculating the equity E from game winning chances is:
E = SW + 2*GW + 3*BW - (SL + 2*GL + 3*BL)
where
SW: Probability for a single point win
GW: Probability for a gammon win
BW: Probability for a backgammon win
SL: Probability for a single point loss
GL: Probability for a gammon loss
BL: Probability for a backgammon loss
Now this basic formula does not account for the fact that the cube may be turned to an arbitrarily
high level, but that won't matter here as the JellyFish level 5 rollouts used for calculating the
equity does not permit taken redoubles (see appendix C for details).
The formula above does not consider the cube value and gives normalized equity for money play.
For match play the calculation is a little trickier. According to the definition of normalized equity a
single win (as after a double/drop or an undoubled no gammon win) equals equity +1.00 and a
single loss equals equity -1.00. However gammon wins and losses may be valued at less or
more than 2.00 depending on the score, and similarily for backgammons and for doubled
wins/losses. Example:
Say you're trailing 4-away, 2-away with your opponent holding the 2-cube. Obviously losing a
single game loses the match (0% MWC), whereas winning a single game leaves you even
(50% MWC), and winning a gammon or backgammon wins the match (100% MWC). By the
normalizing requirement 0% MWC translates to equity -1.00 and 50% MWC translates to +1.00.
Linear extrapolation gives that 100% MWC translates to equity +3.00. Thus both your gammon
and backgammon wins are valued at +3.00 in this example, while your gammon and back-
gammon losses are only valued at -1.00 - same as single losses.
Appendix C - JellyFish Rollout Details
Both level 6 rollouts (7776 games) and level 5 rollouts (46656 games) have been used to
determine the equity of the positions. In the problems where the cube is dead only the level 6
rollouts were used. In the problems where the cube is live level 5 rollouts are used to calculate
all equities, but we checked that the level 5 cubeless equities are close to the level 6 equities -
indicating that JellyFish handles the position well playing on level 5.
Problems #4 and #5 are computed by exact calculation.
Level 5 settlement limits
All money game problems are rolled out using 0.550 as the settlement limit.
For match play problems the settlement limits are set by the following procedure:
· We find the take/drop point at the actual match score, given the gammon rates observed in the
rollouts.
· This take/drop point is converted to a take/drop money play equity, again adjusted for the
gammon rates observed in the rollouts.
· The settlement limit is set a little (typically 0.020) below this take/drop equity to account for equity
gained in double/takes.
t!Oslo Open 1999 Problem tournamentsrjava.util.Vectorٗ}[;IcapacityIncrementIelementCount[elementDatat[Ljava/lang/Object;xpur[Ljava.lang.Object;Xs)lxpsrbgblitz.Position~t oxp #Problem 1 - 9 point match - B:2 W:5Black to play - cube action?1. No double / take -0.727 -0.032
2. Double / take -0.695 0.000 ***
3. Double / drop -1.695
Black already risks losing a gammon and the match and she
should be prepared to double early, but how early is early?
If white had only one blot, he could get hit and closed
out and would still have a take, so then doubling would
be a serious error. Here he has two blots though, and
black's board is strong, which usually calls for a double,
yet black should note a couple of drawbacks to her position:
· She has just a pure 2-shot (20 numbers of 36 hit), no
indirects.
· The numbers that hit (1 and 4) are the same numbers that
cover the 2 point giving her few hit and cover numbers.
· If she doesn't hit with a 1, picking up the vital second
blot could prove to be very difficult.
These are strong arguments for "no double", yet the rollouts
show that black is strong enough to double, though only
barely.
Dropping is of course not on the list of things to do for
white.
Field result:
7 correct
5 small errors (no double)
0 blundersWhiteBlacksr
bgblitz.Board:'+[xpsq~sq~Problem 2 - Money GameBlack to play 6-2?51. 23/21 23/17 -0.308 -0.153
2. 23/15 -0.213 -0.058
3. 23/17 5/3* -0.155 0.000 ***
4. 13/5 -0.415 -0.260
5. 13/7 5/3* -0.265 -0.110
Were you tempted by the "safe play" 13/5, waiting for a
better opportunity?
Sorry, this is as good as it gets. After 13/5 white has
freedom to do anything, most likely he will make his 4
point.
So black must act now, and the most important thing is
to get the back men moving.
The hit looks scary, but white does miss nearly half the
time and then life is quite sweet.
Field result:
4 correct
6 13/5 blunders
2 other errorsWhiteBlacksq~sq~sq~Problem 3 - Money gameBlack to play 6-5?=1. 24/18 13/8 +0.025 -0.310
2. 24/18 10/5* +0.295 -0.040
3. 13/7 10/5* +0.335 0.000 ***
4. 10/4 9/4 +0.224 -0.111
5. 10/5* 9/3 +0.076 -0.259
If black had had less men up front a return hit on 5 would
be the end of the blitz and making the 4 point would probably
be best.
Here black has lots of ammunition ready and is prepared to
go for a full blitz. Thus hitting loose is the preferred
play - the main thing is to keep white from anchoring.
The six is best played 13/7 - again in accord with traditional
theory where the blitz has first priority. Yet the alternative
24/18 is not much weaker, getting the back men moving is
valuable. If white's blockade were just a little stronger
black should have used this opportunity to jar one man loose.
Field result:
9 correct
2 point-makers (10/4 9/4)
1 runner (24/18 10/5*)WhiteBlacksq~sq~sq~Problem 4 - Money gameBlack to play - cube action?[1. No double / beaver -0.086
2. No double / take +0.091 0.000 ***
3. Double / take +0.086 -0.005
4. Double / drop -0.929
Black is a tiny favourite with doubles to get off now and
nearly 50% to win later, and this is surely her last chance
to offer a takeable double, so doubling seems reasonable.
Yet by giving away the cube she will allow white to double
back next turn, and even if the gain from this redouble is
small it will happen often enough to offset black's gain
in the cases where she rolls doubles to win immediately.
Field result:
6 correct
4 small errors (double/take)
2 beavers
WhiteBlacksq~sq~Problem 5 - Money gameBlack to play - cube action?1. No double / beaver 0.000 ***
2. No double / take +0.091 -0.010
3. Double / take -0.010 -0.121
4. Double / drop -1.131
At a quick glance this may look like a two roll proposition
which is a clear double/drop. On closer inspection it is
easy to see that very often black will miss twice so that
white will surely have a take.
So does black even have a double?
No, and not only does she not, but white should even beaver.
Field result:
1 correct beaver
7 no-doublers
4 doublersWhiteBlacksq~sq~Problem 6 - Money gameBlack to play 2-1?<1. 24/21 -0.115 -0.118
2. 24/23 13/11 -0.145 -0.148
3. 13/10 -0.091 -0.094
4. 13/11 6/5 -0.001 -0.004
5. 7/5 6/5 +0.003 0.000
Black's main problem in the position is her lack of flexibility.
She cannot wait for points to make themselves, she must go for
them even if it means leaving some blots.
7/5 6/5 making the big 5-point and leaving the slot on the 7-point
is a tiny bit better than keeping the 7 and slotting the 5. All other plays are very much inferior.
Field result:
3 correct
3 almost-as-good 13/11 6/5
3 quiet plays 13/10
3 work-on-both-sides 24/23 13/11
WhiteBlacksq~sq~sq~#Problem 7 - 9 point match - B:7 W:6Black to play 5-3?1. 21/18 16/11 -0.913 -0.076
2. 21/18 11/6 -0.837 0.000 ***
3. 18/13 16/13 -0.917 -0.080
4. 18/15 16/11 -0.969 -0.132
5. 18/15 11/6 -0.887 -0.050
6. 16/11 5/2 -0.924 -0.087
7. 11/3* -0.841 -0.004
This is an exercise in catching the runner. Try to keep
your men back where they cover most of the field, or move
them up to attack.
Field result:
1 correct
5 almost-as-good hits
3 18/15 11/6
2 21/18 16/11
1 18/13 16/13
WhiteBlacksq~sq~sq~#Problem 8 - 9 point match - B:5 W:5Black to play - cube action?#1. No double / take +0.633 -0.367
2. Double / take +1.103 -0.103
3. Double / drop 0.000 ***
4. Gammon / drop -0.470
White has both forwards and backwards opportunities to win
and should be reluctant to pass. Yet at this score (4-away
4-away) doubled gammon losses are extra costly, and extra
caution is called for. If white takes he will have a strong
recube to 4 (black's take point on the 4-cube is 32%), but
this is offset somewhat by the fact that white cannot get
value from both the recube and the gammons he scores.
The rollouts show black winning only 57,3% after doubling,
but in nearly half of her wins (24,6% of all games) she
gammons white for the match.
This is enough to pass at this score. For money white
should take though.
Field result:
1 correct
6 takers
5 no-doublers
WhiteBlacksq~sq~Problem 9 - Money gameBlack to play 3-3?h1. 21/15(2) +0.458 -0.037
2. 21/18(2) 10/7(2) +0.411 -0.084
3. 21/18(2) 8/5(2) +0.495 0.000 ***
4. 21/18(2) 8/2 +0.374 -0.121
5. 10/7(2) 8/5(2) +0.405 -0.090
6. 10/1* 4/1 +0.134 -0.361
An exercise in choosing the best points. The result is not
surprising. First move the back men out so they don't get
primed. Then make the big 5 point. Don't bother with the
blot on 2, that is not important right now.
Field result:
3 correct
1 get-out-of-here 21/15(2)
5 solid primes 10/7(2) 8/5(2)
3 risk-averse 21/18(2) 8/2
I thought this one would not be very difficult, but
the result shows that opinions do differ.
WhiteBlacksq~sq~sq~Problem 10 - Money gameBlack to play 4-4?~1. 24/16(2) -0.358 0.000 ***
2. 20/16(2) 13/9(2) -0.408 -0.050
3. 20/16(2) 13/5 -0.471 -0.113
4. 20/16(2) 6/2(2) -0.570 -0.212
5. 13/5(2) -0.598 -0.240
Black cannot hold both her anchors with 13/5 - this would
release the direct pressure on the midpoint and leave her
with few good plays next turn. If she keeps the 24-point
and plays 20/16(2) she can afford to give up the midpoint
and should simply play 13/9(2) which doesn't give shots
and leaves her with two checkers to play.
Or she can play 24/16(2) which puts even more pressure on
the midpoint, but which gives up the possibility of hanging
back for a late shot.
Both of these are OK, but the play that puts the most
pressure on immediately is best. If black had had one more
checker to play with so she could hang back longer, she
probably should have.
Field result:
6 correct
5 next-best 20/16(2) 13/9(2)
1 20/16 13/5
WhiteBlacksq~sq~sq~Problem 11 - Money game1. No double / beaver 0.000 ***
2. No double / take +0.234 -0.160
3. Double / take -0.160 -0.714
4. Double / drop -1.874
5. Gammon / drop -1.320
A classic position, and if you don't know it by heart it is
time to learn it. 24 hitting numbers and a strong board and
prime seems tempting, but:
There is lots of work to do even after hitting. Black doesn't
even have the 5-prime yet, and the six-prime are quite a few
turns away. And every roll white will be threatening to escape.
Hit first, then double.
Field result:
2 correct
4 no double/takes
5 blundering doubles
1 super-blundering drop
This was the killer question - and split the field in two,
The ones who blundered here finished in the second half.
None of these would have made the top-three anyway though
even if they had got it right, but the double/dropper would
have come in 4th if he had.
Strangely all the top three finishers did not get this right
either, and lost quite a bit due to not beavering.
sq~sq~ $Problem 12 - 9 point match - B:8 W:8Black to play 5-5?;1. 23/8 18/13 +0.142 -0.004
2. 23/18 13/8(3) +0.034 -0.112
3. 23/18 13/3* 8/3 +0.146 0.000 ***
4. 23/18 8/3* 6/1*(2) +0.036 -0.110
5. 18/3* 8/3 +0.080 -0.066
6. 8/3*(2) 6/1*(2) +0.102 -0.044
Often the strong hitting plays that are right at money play
also prove to be the right play at DMP. Here moving the back
man plays so prettily that it just has to be right, and the
standard play 8/3*(2) 6/1*(2) is somewhat inferior to
23/8 18/13 and 23/18 13/3* 8/3.
Field result:
3 correct
3 almost-as-goods
5 double-hitting-is-more-fun-anyway
1 compromise (18/3* 8/3)
WhiteBlacksq~sq~sq~Problem 13 - Money gameBlack to play 5-2?1. 23/16* -0.406 0.000 ***
2. 13/6 -0.724 -0.318
3. 13/11 8/3 -0.700 -0.294
4. 13/8 5/3 -0.779 -0.373
5. 8/3 5/3 -0.638 -0.232
A no-brainer, right? You must hit, right?
Yes, it is right and by a reasonable margin, too. Yet, according
to the problem designer's experience, through excessive calculation
it is possible to convince oneself that the risk is too great and
go with some quiter play.
This is the time to just close your eyes and hit anything that moves.
Field result:
6 correct
6 do like I do, not like I say :-)
WhiteBlacksq~sq~sq~ $Problem 14 - 9 point match - B:0 W:2Black to play - cube action?1. No double / take +0.740 -0,307
2. Double / take +1.047 -0,047
3. Double / drop 0,000 ***
Basically, these position are money play double/takes when
black - as in this case - has 5 fewer checkers off than white.
Here both sides have important flaws:
· Black has already opened two points and will probably have
to open the 3rd this roll, making it easy for white to enter;
also she leaves a shot on 4-1, 5-1 and 6-1.
· White, on the other hand has two holes and may easily miss
in the bearoff, and the 1's needed to fill these holes are
poor racing rolls.
So does the rival flaws cancel out?
In fact, they just about do, and for money this is still a
comfortable take. In the match however, gammon losses are a
little bit more costly for white, and the redouble to 8 is
not as powerful as a money redouble. Together this shifts
the correct action to double/drop.
Field result:
4 correct
5 takers
3 no-doublers
WhiteBlacksq~sq~ $Problem 15 - 9 point match - B:5 W:7Black to play - cube action?1. No double / take +0.791 -0.270
2. Double / take +1.061 -0.061
3. Double / drop 0.000 ***
4. Gammon / drop -0.209
If you are not familiar with the score 4-away 2-away, you
are probably wondering why black is even considering doubling.
But at this score doubled gammon losses are extremely costly
for white, and even the faintest smell of gammon can turn into
a big stinking double for black.
Example: If 1/3 of black's wins are gammons, white's take
point would be a big 40%, compared to somewhere around 30%
for money.
Here black got her lucky break, she has a clear initiative,
and good things can happen. White will win more than 36%,
but black's gammon rate is close to 1/3 and white has to drop.
Field result:
0 correct!
6 takers
6 no-doublers
Hooray for the problem designer! But I am not surprised -
even if you do know your -4,-2 score it is very hard to
picture this as a drop.
WhiteBlacksq~sq~Problem 16 - Money gameBlack to play 6-1?I1. 24/17 -0.169 -0.030
2. 24/23 17/11 -0.534 -0.395
3. 17/11 8/7 -0.580 -0.441
4. 11/5 6/5 -0.728 -0.589
5. 8/2*/1* -0.139 0.000 ***
Sometimes a five-prime is just worthless. If black makes her
5-point white threatens to hit with 5's or 6's, and otherwise
he all but owns the timing battle. After making the 5-point
white can redouble immediately.
The real choice is between 24/17 and 8/2*/1*. The equities
are actually very close for to such differing alternatives,
but the double hit comes out a tiny bit better.
Field result:
7 correct
2 runners
3 other options - 1 of eachWhiteBlacksq~sq~sq~Problem 17 - Money gameBlack to play 2-1?`1. 24/23 24/22 -0.596 0.000 ***
2. 24/22 11/10 -0.668 -0.072
3. 24/23 11/9 -0.628 -0.032
4. 24/22 9/8 -0.645 -0.049
5. 9/7 8/7 -0.688 -0.092
Similar to the previous position, here too making the big
point just leaves black with the short stick in the priming
battle.
More interesting is that the plays that work on both sides
are not good enough. Now is the time when white does not
have ammunition to attack, and black must make every effort
to escape or make the advanced anchor now.
Field result:
3 correct
5 second best 24/23 11/9
1 with-too-many-blots 24/22 11/10
3 unsuccesful prime builders
WhiteBlacksq~sq~sq~ $Problem 18 - 9 point match - B:4 W:7Black to play - cube action?1. No double / take +0.913 -0.035
2. Double / take +0.948 0.000 ***
3. Double / drop -0.052
4. Gammon / drop -0.087
The pip count is 91-100 which is normally a money double,
and a comfortable take - usually white would take a 91-102
race but drop a 91-103.
Here things aren't that good for white: He has already put
4 men on the 1 and 2 points which is inefficient, and he
has 10 crossovers in the outfield to black's 5. Black's
only flaw - the 5-point hole - seems quite superficial.
She has many builders aimed at this point and will probably
be able to put a couple of men there before starting to
bear off.
Yet, pips are pips, and even with the poor distribution
white will win 21% from the position, a borderline money
drop.
Then one could think that white would be even more eager
to drop being ahead in the match. Strangely enough this
is not true. Being able to win the match exactly with two
points is extra valuable, and with no gammons he will never
risk more than the one extra point by taking. This means
that his take point in a no-gammon position is as low as
20% on this score, and he should take.
Field result:
5 correct
3 no-doublers
4 droppers
WhiteBlacksq~sq~Problem 19 - Tie-breaker9Black to play - what is black's cubeless win probability?Black wins 81,9%
WhiteBlacksq~pt1.0