Justifying my double in Las Vegas

zzy...@earthlink.net

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Apr 30, 1997, 3:00:00 AM4/30/97

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I just lost a match to some kid from NYC in the Limited Jackpot,
quarter finals. The match was to 17 and the winner got $4800 with the
finals then worth $11000+ to the winner. I got notta.

The kid's name was Abe Mosseri. He was good, but lacked tournament and
match experience. When I doubled, he told me that at the club he
played at in NYC the position was nearly a beaver. If you use Jelly's
evaluation function, you get 50.0 wins for each side, so he would
almost be right. If you roll the position out using level 5, things
look better for me. Cubeless I win 51.7% and get 6.4% gammons vs. 1.5%
gammons for Abe.

So by now you're dying to see the position, so here it is:

| | | X | | O | | | | X | X | X | X | X |
| | | | | O | | | | X | X | X | X | X |
| | | | | | | | | | | X | X | X |

home boards >

| | | X | | | O | | | O | O | O | O | O |
| | | | | | | | | O | O | O | O | O |
| | | | | | | | | | | | O | O | |2|

Here are the rest of the facts:

Abe is X and he leads 10-6 in the 17 point match. I am O and I have
the cube on 2. It is my roll. Abe has 61 pips to go and I have 74. My
doubling window opens at 45.6% (.456). His take point is .277. So I
passed the window test. But its not often you should double toward
the short end of the window. So let's look at what is likely to
happen.In a non contact race with these pips, I only win 31%. However,
there is contact. He has 2 blots and I have 16 shots. I let Jellyfish
play 1296 games on level 5, and left the settlement value at .550.

If I hold the cube I win 57.6% and we both get 1.3% gammons for a
positive equity of .153

If I double he's not going to get to redouble me even when I miss.His
redouble window to 8 doesn't open until he is an 83.6% favorite. So I
gave Jellyfish him the cube and rolled it out again using my take
point (9.8%) to figure the settlement value, I used .804 (his 90.2%-
my 9.8%). What I got was 50.3% wins for me with 6.9% gammons vs. 1.0%
gammons for him. My equity is .066. If you multiply that by 2 (cause
the cube is twice as high) you can compare .132 to the equity of not
doubling = .153. Looks like its not a double by 2%.

However, let's look at what happens in this position:
By not doubling, If I hit and don't get hit back, I lose my market by
a mile but not quite enough to play for the gammon, though in all
cases I have a single or double shot at the other blot. If I roll 6-6,
5-5 or 4-4, those 3 rolls even out to an even race. If I miss, he
can't double back unless he makes a big mistake, and I'm likely to get
at least 1 more indirect shot, maybe 2 and maybe another direct shot.
If I do hit a shot on my first roll, I have to give him a few returns
unless I roll 1-1 or 2-1. I'm almost certain to have a direct shot at
another checker if I don't get hit back. How would Jellyfish play
this? Would he minimize returns and double if missed? I tried hitting
with a 2 and playing a 6 from the 17 point to the 11 point leaving 4
return hits, but leaving me 22 shots at the 2nd checker if he fans.
Jellyfish thinks this is a double, pass, but I would roll on for the
gammon, of which Jellyfish thinks I get 14% if I play on.

So I think I can play this position better than Jellyfish. And the
complications give my opponent a chance to make a mistake. What if he
redoubles when he is only 75 or 80%?

And what if I pull off the gammon on 4 and take the lead in the match
14-10? That might have a devestating effect on the opponent,
especially since the dice had been all his the whole match.

So the way I saw it, the cube was nearly dead, and if I hit, I would
hate to own the cube still when I was shooting at the 2nd man.

Oh, you want to know what actually happened? I rolled 3-1, my worst,
and he rolled 6-6, his best. I rolled another crap number, he doubled
and I passed. (my equity in the game was down to 3%). So much for a
dead cube!

ZZyzx

"Me, indecisive? I'm not so sure about that."

Steve Mellen

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Apr 30, 1997, 3:00:00 AM4/30/97

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For anyone (like me) who found Harry's position impossible to read, here
is a hopefully more readable reconstruction:

+13-14-15-16-17-18-+---+19-20-21-22-23-24-+
| O X | | O O O O O |

| X | | O O O O O |
| | | O O O |
| | | |
| | | |
| | | |
| | | |
| | | |

| | | X X |
| | | X X X X X |

| O X | | X X X X X |[2]
+12-11-10--9--8--7-+---+-6--5--4--3--2--1-+

Kit Woolsey

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May 2, 1997, 3:00:00 AM5/2/97

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zzy...@earthlink.net wrote:

+13-14-15-16-17-18-+---+19-20-21-22-23-24-+
| O X | | O O O O O |
| X | | O O O O O |
| | | O O O |
| | | |
| | | |
| | | |
| | | |
| | | |
| | | X X |
| | | X X X X X |
| O X | | X X X X X |[2]
+12-11-10--9--8--7-+---+-6--5--4--3--2--1-+

: Here are the rest of the facts:

: Abe is X and he leads 10-6 in the 17 point match. I am O and I have
: the cube on 2. It is my roll. Abe has 61 pips to go and I have 74. My
: doubling window opens at 45.6% (.456). His take point is .277. So I
: passed the window test. But its not often you should double toward
: the short end of the window. So let's look at what is likely to
: happen.In a non contact race with these pips, I only win 31%. However,
: there is contact. He has 2 blots and I have 16 shots. I let Jellyfish
: play 1296 games on level 5, and left the settlement value at .550.

: If I hold the cube I win 57.6% and we both get 1.3% gammons for a
: positive equity of .153

This is a very complex problem, due to the potential of doubling later if
you hold the cube. Here is how I would work it out at the table, using
as much simplifying assumptions as possible.

First: Using Neil's numbers (which give a close approximation of my
match equity table):

6-14: 10% equity
6-12: 20% equity
8-10: 39% equity
10-10: 50% equity

So if it were a now or never situation you would be getting 11 to 10 odds
on the double. However it isn't a now or never situation -- if you hold
the cube, you may get to double later and win some games which might
have been lost if you had to play them to conclusion.

First simplifying assumption: I assume that O will never redouble to 8
(since he can't redouble except as a huge favorite, this isn't far off).

Let's suppose that X hits a shot. If he doesn't double, I assume he
always wins the game, since unless O rolls a joker X will be able to
claim with the cube next turn. This doesn't take O's jokers into
account, but they are somewhat counterbalanced by the potential of X
playing for, and getting, a gammon when O flunks.
If X does double and hits a shot, he has to play to conclusion. I
estimate that X will win 88% of the time when this happens (actually I
think it is a bit lower, but to compensate X will win a few gammons).

Let's suppose that X misses. He is a clear underdog, but has some racing
and some hitting chances. Also, he will clearly win more often if he
hangs onto the cube, since he may get an efficient double later and not
have to play to conclusion a game he might lose. I will guess that X
will win 35% of the time if he let's the cube go, and 40% of the time if
he hangs onto the cube.

So, what does all this mean? Roughly speaking, it looks like about 1/8
of the games X would win if he hangs onto the cube will turn into losses
if he doubles.

How does this figure into the match equities? If X loses a game he would
have won if he hadn't doubled, this costs him 29% equity (difference
between 39% and 10%). Thus, of the games X would have won if he hadn't
doubled, he gains 11% 7/8 of the time and costs 29% 1/8 of the time.
This comes to an average gain of about 6%. As we have seen, X loses 10%
if he doubles and is wrong. Thus, when we take the cube value into
account it looks like X is actually giving 10 to 6 odds with his
redouble. He clearly isn't close to being worth this in this position.
So, if you accept my estimates, the redouble is incorrect. Of course my
estimates may well be quite wrong, and different estimates might lead to
a different conclusion.

Kit

Chuck Bower

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May 9, 1997, 3:00:00 AM5/9/97

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> | | | |
> | | | |
> | | | X X |
> | | | X X X X X |
> | O X | | X X X X X |[2]
> +12-11-10--9--8--7-+---+-6--5--4--3--2--1-+
>

This is a very interesting, instructive, and potentially controversial
position. The match score weighs heavily, and market losers abound.

My news server rolled off the original post before I got a chance
to reply. But I seem to remember several things, not all of which are
obviously true. Something like:

1) O is a young, cocky New Yorker;
2) X is unlucky;
3) O says this is a money beaver;
4) X thinks s/he plays better than Jellyfish.

I think I'll stick to the position at hand (but will comment on
point 3, also). Kit did an "over the table" analysis and concluded that
X should hang onto the cube for at least one more shake. With pencil,
paper, and Jellyfish, I come to a different conclusion.

Here is the JF v2.01 level-6 cubeless rollout result:

total g+bg bg

X wins 52.0 6.5 0.2
O wins 48.0 1.6 0.0

1296 trials (equivalent to >32000 at level 5). X's cubeless money
equity is +0.090 and the standard deviation is 0.005 (in equity units).

According to Rick Janowski's semi-continuous money cube model, at these
gammon fractions O's beaver point is 45%, so this would be a money beaver,
but that isn't very relevant, since this position occurred in a match with
a skewed score and the trailer already owning a 2-cube.

Let's take a look at the the drop/take decision first:

decision and outcome O's resulting O's match
score winning chances

O passes -7, -9 0.61
-------------------------------------------------------
O takes and wins simple -3, -11 0.90

O takes and loses simple -7, -7 0.50 }
} = 0.47
O takes and loses gammon -7, -3 0.25 }

By taking, O risks 61 - 47 = 14% to gain 90 - 61 = 29%. So O's take
point is 14 / (14 + 29) = 33% ASSUMING NO CUBE OWNERSHIP EQUITY.

If O takes and turns the game around, where is X's subsequent drop/take
line? Since the corresponding 8-cube, if accepted, could put O over the
top, X would certainly reship to 16, so the game would then be for the
match. Dropping the 8-cube would leave X with a 10% chance in the match
(complement of 90% number above--score of 11-away, 3-away) so that would
be X's drop/take line, which is 90% from O's point of view. Thus WITH
A PERFECTLY EFFICIENT CUBE O's current take point is lowered by 10% from
33% to 30%. Applying the 70% rule (go 70% from 33% down to 30%) to
approximate REAL CUBE EFFICEINCY gives us O's take point--31%.

Clearly O has a take in this position. Does X have a double? We
first look at X's doubling window opening point (also known as the last
roll doubling point--assuming cube now or never).

X loses score X's MWC X wins score X's MWC

X never simple: -11,-5 19% } simple: -9,-7 39% }
doubles } 19% } 40%
gammon: -11,-3 10% } gammon: -7,-7 50% }

X doubles simple: -11,-3 10% } simple: -7,-7 50% }
now (O can't } 10% } 53%
redouble) gammon: LOSE 0% } gammon: -3,-7 75% }

(NOTE: final percentages are proration of simple and gammon wins, which
don't affect the numbers when O wins since O captures very few gammons.)

X risks falling from 19% to 10% by doubling, but stands to go from 40%
to 53% if the gamble works. So his/her doubling window opens at

(19 - 10) / [ (19 - 10) + (53 - 40) ] = 41%

Clearly X would have a "last roll double" (X is in the doubling window)
but, is the double correct? Are there market losers? Hit followed by wiff
(16/36 * 25/36 = 31%) are certainly BIG market losers (JF level-7 evaluation
says X will win 85-90% of these). This argues rather strongly for a double.

Can we find other evidence? Yes, by using JF level-5 CUBEFUL rollouts.
Here, though, we must CAREFULLY choose the settlement limit. (For those
unfamiliar, JF will allow a player to cash if above the settlement limit
and on roll, with minor "too good to double--play for gammon" situations
which shouldn't be particularly relevent here).

If X doesn't double and then misses the shots, we can assume that
there will be no gammons (not 100% true, but pretty close) in which case
O's drop point becomes about 26% (settlement limit is 1 - 2*0.26 = 0.48).
(The motivated reader should calculate this drop/take point.) Note that
if X hits and O fans, s/he will have a cash, so this settlement
limit will work for that case as well. If X DOES double then O will cash
a 4-cube at 90%, so the settlement limit in that case is 0.80. Here is
what JF v2.01 level-5 CUBEFUL rollouts give for these two situations:

Case 1: X owns cube and can cash at settlement limit = 0.48. (Cube remains
on 2 for all cases.)

outcome branching X's MWC product
fraction

X wins simple 0.570 0.39 0.222
X wins gammon 0.012 0.50 0.006
O wins simple 0.401 0.19 0.076
O wins gammon 0.017 0.10 0.002
------------------------------------------------
totals 1.000 0.306 (= X's MWC)

Case 2: O owns 4-cube and can cash at settlement limit = 0.80.

outcome branching X's MWC product
fraction
X wins simple 0.441 0.50 0.220
X wins gammon 0.059 0.74 0.044
X wins bg 0.002 1.00 0.002
O wins simple 0.489 0.10 0.049
O wins gammon 0.009 0 0
------------------------------------------------
totals 1.000 0.315 (= X's MWC)

(Note that even with these very conservative cube handling restrictions,
O is almost even at money play owning the cube. This is further evidence
that this is a money beaver.)

Thus level-5 cubeful rollouts favor doubling now, giving X about
1% more match winning chances than by holding. Did we make assumptions
that may have skewed the outcome? Jellyfish cubeful rollouts have a
couple weaknesses. 1) The cube owner is forced to lose his/her market.
2) The decision to cash or play on for gammon ways heavily toward cashing.
Both of these simplifications hurt the "hold" case, meaning that the
above 0.306 number should be a bit higher. In addition, forcing O to
hang onto the 4-cube until the game is a lock also means that the 0.315
number should be slightly smaller. All of this argues for an even closer
decision than was just indicated.

On the other hand, with the cube on 4, X should play more aggressively
for gammons than at money play. (risk::gain is 8::5 instead of 10::5 at
money play.) Thus JF, which makes it's decisions for money play, is
actually not playing optimum strategy. With the cube on 2, however, the
ratio is 20::11, and thus closer to money play. This fact says that the
0.315 MWC (cube on 4) should be a bit higher.

Finally, what if O misplays the cube? It is conceivable that some
O's will pass the 4-cube, suffering from gammon-phobia (2 blots). Also,
even if O takes, some may wait too long to redouble (although it's hard
to lose much here, since 90% is O's cash point). O may also misjudge
the position if X waits to cube (and possibly gives O a more pressurized
choice).

My conclusion is that X should double, but it is very close.

Chuck
bo...@bigbang.astro.indiana.edu
c_ray on FIBS