match winning chances & saving gammons
Morten Wang
I played a match today where I had a position where saving gammons and
computing match winning chances proved to give me quite a task.
during the game I only looked at the gammons, since I would lose the
match if I got gammoned. my opponent thought I made an error running
to save the gammons, and since I strongly disagreed I gave JellyFish
the task of sorting a few things out. here's the position:
O vs X, score is 2-2 in a 5-point match.
O owns 2 cube.
+24-23-22-21-20-19-+---+18-17-16-15-14-13-+
8| O O X O | | |
O| O O | | |
O| O O | | |
O| | | |
O| | | |
O| | | |
| | | |
| | | |
| X | | |
| X X X | | |
| X X X X X | | |
| X X X X X | | |
+-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
X rolls 5-1.
I played 22/16, thinking that gammons were a high possibility, a
shot coming my way was not very likely, and actually hitting the
not-so-likely-to-happen-blot even less so. my opponent said I
should've stayed. JellyFish' evaluation of the best moves on level 7
says 22/16 is clearly ahead, and all the top 5 moves have 22/15 as
a common denominator (is this correct English?).
the first move that stayed was making the ace-point (6/1, 2/1), number
6 on the list from JellyFish. I can fetch the numbers if anyone's
interested, since I only had two possible moves to think about (if I
hadn't run I'd make the ace point, period) I feel they'd only mess
things up a bit.
I ran the rollout, and the result was this:
Runs (22-16):
Wins G/BG BG
O: 100.0 10.1 0.0
X: 0.0 0.0 0.0
Equity O: 1.101
Sd: 0.002
Eq to: 22699
Stays and makes 1:
Wins G/BG BG
O: 96.2 25.4 0.2
X: 3.8 0.0 0.0
Equity O: 1.180
Sd: 0.003
Eq to: 32000
both rollouts had identical dice, were 1296 games in length, and
JellyFish played on level 6.
looking at a single game it's clear one has to run. this was in a
match so I tried to calculate the match winning chances for both moves
from the result of the rollouts. I used the Woolsey table for match
winning probabilities, and assumed that the players were equal in
strength (I was rated fairly higher than my opponent, but for
simplicity I feel no need to think about that untill after the
calculation).
according to the Woolsey table I get the following match-equities for
O (my opponent):
O loses, score 3-away, 1-away, 25% wins.
O wins single game, score 1-away, 3-away, 75% wins.
O wins gammon, wins game, 100% wins.
I calculated the match winning chance like this:
if I stay my opponent wins:
3.8% * 25% = 0.95%
96.2% * 75% = 72.15%
25.4% * 100% = 25.4%
(left column is percentage wins from JF rollout, mid column is
percentage wins from the Woolsey table, to the right is the calculated
total match wins)
adding the numbers I get that my opponent has a 98.5% of winning the match
if I stay.
if I run however, my opponent wins:
100.0% * 75% = 75%
10.1% * 100% = 10.1%
giving a total match winning percentage of 85.1%.
if I therefore have calculated it correctly it is incredibly clear
that I should run for all I'm worth.
as I noted earlier I was rated fairly higher than my opponent,
something I believe would make it even clearer that I should save the
gammons and play on since I would have higher possibilities of winning
later games.
comments to all my logic and calculations is of course welcome.
Morten!
--
"God does not deduct from our alloted life span
the time spent playing backgammon."
--> Morty on FIBS
--> Backgammon homepage: http://home.sn.no/~warnckew/gammon/
Hank Youngerman
Over the board, I think I would make the following assessment:
1. There is a big difference between my chances if I run now or if I
wait one roll.
2. It's not clear that I'll get a shot.
3. It's not clear that I'll hit the shot.
4. EVEN IF I GET AND HIT A SHOT it's not at all clear that I'll win.
Say my opponent rolls 5-4, 6-4, or 6-5, and I roll a 1. He is left
with 5 checkers on.
5. There is very little chance of picking up a second checker. Since
I pretty much have to make my ace-point if I don't run, my timing is
gone.
6. I'm still very much in the match if I lose. Suppose I were
trailing 6-0 to 15 and the cube at 8 (pretty extreme case, I know).
Now if I lose I'm down 14-0. I have to play this one for all the
marbles.
In other words, all the little factors that give me match winning
chances go against staying. I'm sure that all these are what factored
into the JellyFish rollouts.
Morten Daugbjerg Hansen
>I calculated the match winning chance like this:
>if I stay my opponent wins:
> 3.8% * 25% = 0.95%
>96.2% * 75% = 72.15%
>25.4% * 100% = 25.4%
>adding the numbers I get that my opponent has a 98.5% of winning the match
>if I stay.
98.5 is a lot !! and way too much. An easy way to see you made an error is
to count shots: 65, 64, 63, 54, 53 and 43 give a shot this roll. 12 shots
give approximately 4 hits, or about 10% of the games. IF you hit you
clearly have more than 15 %, and you also have fair chances of escaping
the gammon allthough you stay....
SO what is wrong ?
THE LEFT ROW ALWAYS HAVE TO ADD UP TO 100 % !!
In this case you counted your gammonwins as both a gammon and as a single win.
Here is how it should look:
3.8% * 25% = 0.95%
70.8% * 75% = 53.10%
25.4% * 100% = 25.4% a new total of 79.45%
>if I run however, my opponent wins:
>100.0% * 75% = 75%
> 10.1% * 100% = 10.1%
>giving a total match winning percentage of 85.1%.
Again you pretend you play more than 100% games...
89.9% * 75% = 67.43%
10.1% * 100% = 10.1% a new total of 77.53%
>if I therefore have calculated it correctly it is incredibly clear
>that I should run for all I'm worth.
Well, you didnt, but it is still correct to run by a fair margin,
almost 2%, but far from your original 13% !
> "God does not deduct from our alloted life span
> the time spent playing backgammon."
Gee, Im glad to know that...Monte Carlo 2167, here I come !!
Best regards,
Morten Daugbjerg aka md
Chuck Bower
In article <wk7m9bi...@ODIE.WANG.NO>,
Morten Wang <warn...@online.no> wrote:
>I played a match today where I had a position where saving gammons and
>computing match winning chances proved to give me quite a task.
>during the game I only looked at the gammons, since I would lose the
>match if I got gammoned. my opponent thought I made an error running
>to save the gammons, and since I strongly disagreed I gave JellyFish
>the task of sorting a few things out. here's the position:
>
>O vs X, score is 2-2 in a 5-point match.
>O owns 2 cube.
> +24-23-22-21-20-19-+---+18-17-16-15-14-13-+
> 8| O O X O | | |
> O| O O | | |
> O| O O | | |
> O| | | |
> O| | | |
> O| | | |
> | | | |
> | | | |
> | X | | |
> | X X X | | |
> | X X X X X | | |
> | X X X X X | | |
> +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
>X rolls 5-1.
(snip)
>I ran the rollout, and the result was this:
>
>Runs (22-16):
> Wins G/BG BG
>O: 100.0 10.1 0.0
>X: 0.0 0.0 0.0
>Equity O: 1.101
>Sd: 0.002
>Eq to: 22699
>
>Stays and makes 1:
> Wins G/BG BG
>O: 96.2 25.4 0.2
>X: 3.8 0.0 0.0
>Equity O: 1.180
>Sd: 0.003
>Eq to: 32000
>
>both rollouts had identical dice, were 1296 games in length, and
>JellyFish played on level 6.
(snip)
>according to the Woolsey table I get the following match-equities for
>O (my opponent):
>
>O loses, score 3-away, 1-away, 25% wins.
>O wins single game, score 1-away, 3-away, 75% wins.
>O wins gammon, wins game, 100% wins.
>
>I calculated the match winning chance like this:
>
>if I stay my opponent wins:
>
> 3.8% * 25% = 0.95%
>96.2% * 75% = 72.15%
>25.4% * 100% = 25.4%
(snip)
>adding the numbers I get that my opponent has a 98.5% of winning the match
>if I stay.
>
>if I run however, my opponent wins:
>100.0% * 75% = 75%
> 10.1% * 100% = 10.1%
>
>giving a total match winning percentage of 85.1%.
>
>if I therefore have calculated it correctly it is incredibly clear
>that I should run for all I'm worth.
Well, unfortunately you didn't calculate correctly, and although
it still looks clear that running is correct, it's not as big of a
difference as you come up with. In each case the left hand column
should add to 100%. You double counted gammons (that is, counted them
both as gammon wins and as simple wins). Using your numbers, your opp
wins 79.5% if you stick around one roll and 77.5% of the matches if you
bolt. Bottom line, though, is that it is correct to run here.
A good way to attack these problems is to compare risk vs. reward
where the "status quo" is a loss. If you go from losing simple to
losing gammon, you cost yourself 25% match winning chances (MWC). If
you go from a simple loss to a win, then you gain 50% MWC (25%-->75%).
So you stand to gain 50 by risking 25, or you receive 2::1. But the
JF rollout says your net is to trade 15.3% gammon losses for a paltry
3.8% wins. Even with net wins counting twice as much, you still end
up on the short end of the stick in the long run if you stay.
Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS
Stuart Katz, MD
On 11 Dec 1997 21:16:18 +0100, Morten Wang <warn...@online.no> wrote:
>I played a match today where I had a position where saving gammons and
>computing match winning chances proved to give me quite a task.
>during the game I only looked at the gammons, since I would lose the
>match if I got gammoned. my opponent thought I made an error running
>to save the gammons, and since I strongly disagreed I gave JellyFish
>the task of sorting a few things out. here's the position:
>
>O vs X, score is 2-2 in a 5-point match.
>O owns 2 cube.
> +24-23-22-21-20-19-+---+18-17-16-15-14-13-+
> 8| O O X O | | |
> O| O O | | |
> O| O O | | |
> O| | | |
> O| | | |
> O| | | |
> | | | |
> | | | |
> | X | | |
> | X X X | | |
> | X X X X X | | |
> | X X X X X | | |
> +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
>X rolls 5-1.
>
>I played 22/16, thinking that gammons were a high possibility, a
>shot coming my way was not very likely, and actually hitting the
>not-so-likely-to-happen-blot even less so. my opponent said I
>should've stayed....
[lots of math deleted]
My feeling is that you understand the theme here but have made the
calculations too complicated to be useful at the table. Previous
responses have been derived FROM the rollout estimates. Let me
offer a practical approach to making a reasonable real time guess.
Running gives up any chance to win but makes a gammon unlikely.
Staying back gives X some small winning chances but also raises
the likelihood of losing a gammon. Does the potential gain justify
the added risk?
If the expectation is a simple loss for 25% MWC, the added risk of
a gammon is 25-0 or 25%. The gain from converting a loss to a win
is 75-25 or 50%. Thus it would be correct to stay back if X can
generate 1 extra win for every 2 extra gammons given up, the same as
for money play.
After running with 22/16 X needs 10 pips to get in, 2 pips to get off.
With 3 crossovers to escape the gammon X is likely to be off in 2 more
shakes. O thus needs to roll two sets of doubles for a gammon, say
1/36. Of course X might roll small and require 3 shakes, so I
estimated the following outcomes:
wins gammons
O 100 5
X 0 0
Morten's JF rollout says Go would better be estimated at 10%, but
let's stick with my honest actual estimate.
>Runs (22-16):
> Wins G/BG BG
>O: 100.0 10.1 0.0
>X: 0.0 0.0 0.0
>Equity O: 1.101
>Sd: 0.002
>Eq to: 22699
If X stays with 6/1 5/1 I assumed X would be willing to stay for
only one shake. O leaves an immediate shot in 12/36 cases. X hits
11/36 of these so I estimated 1/3 x 1/3 or about 1/9 sequences
where X puts a man on the roof. X still has a lot of work to do. O
will have 9 or 10 off but will leave a blot on the ace-point. Moreover
O holds the cube. My guess was X would win half the time, so my table
estimate was 5% overall wins.
On the other hand X has less chance of escaping a gammon. X needs
16 pips to bear in and 1 to get off, a total of 4 crossovers. X
might get off in 2 but likely will need at least 3 shakes to avoid
the gammon. With O now on roll, my crude guess was 1/3 gammons in
8/9 cases, for an estimated 30% gammons.
Thus after staying with 6/1 2/1 I estimated:
wins gammons
O 95 30
X 5 0
Here is Morten's rollout data:
>Stays and makes 1:
> Wins G/BG BG
>O: 96.2 25.4 0.2
>X: 3.8 0.0 0.0
>Equity O: 1.180
>Sd: 0.003
>Eq to: 32000
In my model X generates 5% extra wins at a cost of 25% extra gammons,
so running is clearly correct. Using the JF rollouts the
corresponding ratio of extra wins to extra gammons is 3.8:(25.4-10.1)
or 1:4. I think this sort of analysis can be done quickly and
painlessly at the table.
Stuart
Ron Karr
Morten Wang wrote:
> >
> O vs X, score is 2-2 in a 5-point match.
> O owns 2 cube.
> +24-23-22-21-20-19-+---+18-17-16-15-14-13-+
> 8| O O X O | | |
> O| O O | | |
> O| O O | | |
> O| | | |
> O| | | |
> O| | | |
> | | | |
> | | | |
> | X | | |
> | X X X | | |
> | X X X X X | | |
> | X X X X X | | |
> +-1--2--3--4--5--6-+---+-7--8--9-10-11-12-+
> X rolls 5-1.
> >
>
> Wins G/BG BG
> O: 100.0 10.1 0.0
> X: 0.0 0.0 0.0
> Equity O: 1.101
> Sd: 0.002
> Eq to: 22699
>
> Wins G/BG BG
> O: 96.2 25.4 0.2
> X: 3.8 0.0 0.0
> Equity O: 1.180
> Sd: 0.003
> Eq to: 32000
>
>
> match so I tried to calculate the match winning chances for both moves
> from the result of the rollouts.
>
> O wins single game, score 1-away, 3-away, 75% wins.
> O wins gammon, wins game, 100% wins.
>
There are three parts to solving this problem (if you're a human being
and not Jellyfish):
1. What are your winning chances if you stay back?
2. What is the gammon price?
3. By how much do you reduce your gammon chances if you run?
None of these require Jellyfish to answer, although #3 isn't obvious
without some research, so it's useful to look at JF's results.
1. It's easy to figure out winning chances, since this is pretty much a
one-roll situation. ( If O fails to clear but doesn't leave a shot, X is
definitely running next time). So: O leaves a shot 1/3 of the time (I
find the easiest way to see this is that O plays SAFELY with all 1s and
2s (20 #s) + 4 doubles. = 24 numbers)
X hits with any 1 (30%). So 1/3 x 30% = 10%. How often does X then
win? Clearly he's an underdog since O has 9 or 10 off. I'd have
estimated slightly lower than JF's 3.8%, but somewhere around there.
2. The gammon price measures the relative value of a gammon vs. a
win-loss swing. It's useful in play decisions like this one; it's also
useful in cube decisions for determining how much gammons affect the
take point.
If you lose a gammon, you blow 2 points, while if you win the game, you
gain 4 points (compared with losing the game). Therefore the gammon
price for money is always 2/4 = 50%.
At different match scores, the gammon price can vary. At 3-away/3-away,
you gain 50% by winning the game and lose 25% by getting gammoned, so
again the gammon price is exactly 50%!. So there's really no difference
between this score and money as far as gammons are concerned.
The way to use this figure: in order for it to be correct to stay back,
my wins need to be at least 50% of my extra gammon losses. So if I'm
winning, say 4%, by staying back, am I losing an additional 8% gammons?
This requires some information.
3. If a gammon is very close, a single roll makes a huge difference..
around 30% gammon chances, I think. The number drops as the gammon gets
more likely or more unlikely. In this position, X is a favorite to get
off the gammon regardless (plus the roll is only 6 pips, slightly less
than average), so the extra gammon loss is only 15%, according to JF,
which makes sense. But this is still way more than twice as great as
the winning chances, so running is clearly right.
Ron