Match equities into take points
James Eibisch
(many thanks for these), I came up with this table of take points,
specifically for 5 point matches where you are being offered the
cube. Also, I've used match, rather than x-away scores - I realise
that this limits this table to 5 point matches, but I find it
easier to get a feeling for what's going on in the match overall
this way.
Another reason for this exercise was that I often see tables or
references to tables of match equities, but not take points. The
most I can do with equity tables is obtain take points from them,
although I'm sure they can be put to more sophisticated use. This
table, then, provides me with a lookup list for my basic needs
without going through another calculation process.
Finally, I'm not mathematically-minded - I just wanted a simple
list I could refer to without taxing my already-overloaded brain!
If anyone else is like me, I hope you find this useful!
From Patti's post, match equities for a 5-point match:
Opponent
1 2 3 4 5
1 50 70 75 83 85
Y 2 30 50 60 68 75
o 3 25 40 50 59 66
u 4 17 32 41 50 58
5 15 25 34 42 50
The big sums I did to convert this into take points:
100*((drop-take_lose)/((drop-take_lose)+(take_win-drop)))
I did these by hand, not programatically - that way they'll
sink in better. If there are any errors, please post them.
Take points when offered the cube to 2 in a 5-point match
---------------------------------------------------------
You Opponent Take point (%)
0 0 25
0 1 26.4
0 2 28
0 3 37
1 0 26.4
1 1 25
1 2 34
1 3 34
2 0 25.7
2 1 23
2 2 30
2 3 35
3 0 20
3 1 20
3 2 28
3 3 30
4 1 16.6
4 2 40
4 3 50
I left out obvious ones like opponent doubling when at 4. At some
scores, like, 3-3 (2-away, 2-away), special rules apply and the
numbers can be ignored. It's interesting to note that post-Crawford
strategy seems to be reversed. At 4 1, the take point is very low.
I thought it would be high: as your opponent has got an even number
of points to go, you've got a free drop, so your take point would be
high, giving you room to drop. Conversely, at 4 0, as your opponent
has an odd number of points to go, you haven't really got a free drop,
so your take point would be low. But it works out at 32% (I left it
off the table because your opponent can't double at 4 0). Maybe my
reasoning is wrong.
Lastly, taking a meaningful 4 cube, using the same forumla.
Take points when offered the cube to 4 in a 5-point match
---------------------------------------------------------
You Opponent Take point (%)
0 0 27
0 1 30 plus automatic redouble
0 2 20 plus automatic redouble
1 0 28
1 1 32
1 2 17
2 0 33
2 1 40
2 2 25
--
James Eibisch Liar. What's in your toolbox?
Reading, England
Christopher Yep
James Eibisch <jeib...@revolver.demon.co.uk> wrote:
>Following on from posts from Stephen and Patti on match equities
>(many thanks for these), I came up with this table of take points,
>specifically for 5 point matches where you are being offered the
>cube. Also, I've used match, rather than x-away scores - I realise
>that this limits this table to 5 point matches, but I find it
>easier to get a feeling for what's going on in the match overall
>this way.
The results can be used for other length matches too, when the scores are
within 5-away/5-away. This is why sometimes it's helpful to think about
match scores in terms of x-away/y-away.
>Another reason for this exercise was that I often see tables or
>references to tables of match equities, but not take points. The
>most I can do with equity tables is obtain take points from them,
>although I'm sure they can be put to more sophisticated use. This
>table, then, provides me with a lookup list for my basic needs
>without going through another calculation process.
>
>Finally, I'm not mathematically-minded - I just wanted a simple
>list I could refer to without taxing my already-overloaded brain!
>If anyone else is like me, I hope you find this useful!
>
>From Patti's post, match equities for a 5-point match:
>
> Opponent
> 1 2 3 4 5
> 1 50 70 75 83 85
> Y 2 30 50 60 68 75
> o 3 25 40 50 59 66
> u 4 17 32 41 50 58
> 5 15 25 34 42 50
Note that the equities, in the above table, when one player is at 1-away
(i.e. when one player has 4 pts.), refer to the crawford game, and not
post-crawford (i.e. the equities listed above assume that one player has
just reached 4 pts. (1-away), and that the next game to be played will be
the crawford game).
Post-crawford equities are approximately:
1-away/1-away, 2-away/1-away ~50%
3-away/1-away, 4-away/1-away ~30%
The free drop is worth a very small amount, in terms of equity.
But anyway, cube action post-crawford is not very complex compared to
pre-crawford cube action.
>The big sums I did to convert this into take points:
>100*((drop-take_lose)/((drop-take_lose)+(take_win-drop)))
i.e. 100 * (drop - take_lose)/ (take_win - take_lose)
>I did these by hand, not programatically - that way they'll
>sink in better. If there are any errors, please post them.
>Take points when offered the cube to 2 in a 5-point match
Correction
>You Opponent Take point (%)
>
>0 0 25
>0 1 26.4
>0 2 28
>0 3 37 21.4%
>1 0 26.4
>1 1 25
>1 2 34
>1 3 34 17%
>
>2 0 25.7
>2 1 23
>2 2 30
>2 3 35 25%
>
>3 0 20
>3 1 20
>3 2 28
>3 3 30
>
>4 1 16.6 ~50% (see below)
>4 2 40 very small, perhaps 2% (see below)
>4 3 50
>I left out obvious ones like opponent doubling when at 4. At some
>scores, like, 3-3 (2-away, 2-away), special rules apply and the
>numbers can be ignored.
But even at 2-away/2-away, if a player is doubled, he still should have
criteria for judging whether or not to take, i.e. the 30% criterion
as you indicated in the 3,3 entry of your chart above.
> It's interesting to note that post-Crawford strategy seems to be
> reversed. At 4 1, the take point is very low. I thought it would be high:
> as your opponent has got an even number of points to go, you've got a
> free drop, so your take point would be high, giving you room to drop.
> Conversely, at 4 0, as your opponent has an odd number of points to
> go, you haven't really got a free drop, so your take point would be
> low. But it works out at 32% (I left it off the table because your
> opponent can't double at 4 0). Maybe my reasoning is wrong.
Your pre-calculation intuition seems to be right. The reason why you
can't use Patti's numbers to calculate take points post-crawford is that
Patti's numbers for 1-away scores were for the crawford game, and not
post-crawford.
At 1-away/4-away post-crawford, you should probably use your free drop if
you have less than a 50% chance to win the game, assuming that your
opponent will win an average amount of gammons (i.e. ~21-23% or so of his
wins are gammons). If the position is more gammonish than this for your
opponent, then you should adjust your take point up slightly, and vice
versa if it's less gammonish for him. But it's unlikely that you'll have
to think much about this, because most of the time your opponent should
double at his first opportunity, so the position will be one of average
gammon rate.
It's possible that for positions of average "gammon rate," that your take
point is slightly less than 50% (i.e. if you are only a very slight
underdog, you may wish to save your free drop for the next game),
however, in general, I would advise against this, since there is a
significant chance that you will not have a chance to use the free drop
in the future.
At 1-away/2-away post-crawford, your take point is exactly 50%.
At 1-away/3-away post-crawford, you don't have a free drop (as you
indicated above). In a gammonless game, you would only want to drop if
you thought that gaining a free drop (ie in the next game at
1-away/2-away, you would have a free drop) was worth more than taking a
shot at winning the match with this game. As the free drop isn't worth
very much, I would estimate that your take point is ~2%. Note that
this is in a gammonless game.
None of the take points in the chart above (or below) take into account
the possibility of gammons. Also, the trailer often has significant
recube power, which often means that he can take with significantly less
winning chances than indicated in the charts. For example, when the
leader only has 2 points to go (2-away), if the trailer takes the
double at 2. then he should (almost always) instantly redouble to 4 on
his next turn. So I corrected the chart above to take this into
account. Also, for example, at 4-away/3-away, if the trailer is doubled,
then he can redouble out the leader (i.e. double to 4, and the leader
drops), by merely becoming > 60% favorite (compute this). I.e., his
redouble is very strong, because he can use all 4 points of a win,
whereas his opponent can't reap any gain from the overage of 1 pt. for
his wins. So in practice, the 4-away player can probably take with about
25% winning chances (if he could always be guaranteed of perfect efficiency
of his redoubles, then he could take with 60% * 34% = 20% winning chances; in
practice, I think 25% is a good take point). Note again that this isn't
considering gammons. If gammons are involved, then it's a different
calculation of the take point.
>Lastly, taking a meaningful 4 cube, using the same forumla.
>
>Take points when offered the cube to 4 in a 5-point match
>---------------------------------------------------------
Correction
>You Opponent Take point (%)
>
>0 0 27
>0 1 30 plus automatic redouble 25%
>0 2 20 plus automatic redouble 15%
>
>1 0 28
>1 1 32
>1 2 17
>
>2 0 33
>2 1 40
>2 2 25
>
>
>--
>James Eibisch Liar. What's in your toolbox?
>Reading, England
>
Chris