Equity
Sanjay Fernando
Can someone please explain equity to me and how it is calculated?
I have looked in a few places and what I have found has been a bit
confusing.
Thanks
Sanj
Chuck Bower
Chuck Bower
(Sorry for the truncated previous post. EMACS bit me again. CRB)
I'll try to answer this question in steps. Hopefully I will neither
insult the reader (by explaining something already known) nor lose you (by
making things too complicated and using "jargon"). Here goes:
1) in general, "equity" is "value". The word is common in financial
circles, especially in accounting.
2) in backgammon, "equity" is equivalent to "expectation"--a term used
by mathematicians, especially statisticians.
3) There are two kinds of equity in backgammon: "Cubeless", and "Cubeful".
Cubeless equity is the "value" of a position if THERE IS NO DOUBLING CUBE
around, so that the game will be played to completion, and six outcomes
are possible: either side can win a simple game, a gammon, or a backgammon.
So, if we define:
W = player on roll's chances of winning a simple game,
G = player on roll's chances of winning a gammon,
B = player on roll's chances of winning a backgammon,
w = player NOT on roll's chances of winning a simple game,
g = player NOT on roll's chances of winning a gammon,
b = player NOT on roll's chances of winning a backgammon,
then we can define player on roll's equity as:
W + 2*G + 3*B - w - 2*g - 3*b.
(* is multiplication sign).
Some things worth noting:
a) W + G + B + w + g + b = 1. That is becuase SOMETHING MUST HAPPEN!
b) The equity for the player NOT on roll is just (-1) times the equity of the
player on roll. Another way of saying this is that if we add the equity for
the player on roll to that of his/her opponent, the sum is ZERO.
c) cubeless equity will be somewhere in the range -3 to 3.
Now, what about "cubeful" equity? This is a slightly more complicated case.
Two things must now be included. If we asume the same (cubeless) definitions
above, then
a) multiply the current value of the doubling cube times the above "cubeless"
numbers to get equity. (Cubeless equity can be bigger than 3 or less than -3).
b) ownership of the doubling cube now plays a roll, so some adjustment must
be made for that. Typically +5-10% is added to the equity of the player on
roll (and equivalently subtracted from the player NOT on roll)
to account for cube ownership in money games. There are more complicated
ways to figure it, but this message is ALREADY GETTING LONG! At matches,
there may or may not be cube ownership equity, depending on where you are
in the match. That, too, can be quantified.
WARNING #1: With the doubling cube in play, the cubeless quantities (W, G,
etc.) are NOT each players true winning/losing chances. That is because the
cube can (and often will) be used to end games prematurely AND convert some
losses into wins. That is why the adjustment [b) above] was made. It is
also why you need to find out if rollout results (human or robot) were
performed with a doubling cube in play.
WARNING #2: For anyone familiar with Jellyfish evaluation and rollouts,
note that the numbers in the JF results windows are NOT the values I defined
above. Here is how JF defines things:
Wj = player 1's chances of winning a this game (any flavor),
Gj = player 1's chances of winning a gammon or backgammon,
Bj = player 1's chances of winning a backgammon,
wj = player 2's chances of winning a this game,
gj = player 2's chances of winning a gammon or backgammon,
bj = player 2's chances of winning a backgammon,
NOW, equity for player 1 is: Wj + Gj + Bj - wj - gj - bj.
Also, Wj + wj = 1, but the sum of all six values is not, in general.
Method is equivalent, just using "oranges" instead of "apples". Be
careful not to interchange the fruit, though...
James Eibisch
On Tue, 01 Oct 1996 18:53:40 +0100, Sanjay Fernando
<San...@ferndo.demon.co.uk> wrote:
>Can someone please explain equity to me and how it is calculated?
>
>I have looked in a few places and what I have found has been a bit
>confusing.
Disclaimer: I'm a writer, not a mathematician. I can't explain in-depth
technical aspects of bg because I don't, and might never, understand
them.
A prescript: Peter Bell (USRobots on FIBS) wrote a magnificent article
explaining equity and how to apply it to use of the doubling cube. This
article is, AIUI, currently being made into a commercial paper which
prevents me from posting it here. My inferior explanation below is not
taken from Peter's article but is informed by it - it taught me what
equity is, after all. May I take this opportunity to thank Peter for his
original posting of this article.
There are two types of equity: game equity and match equity. Match
equity describes, holistically, your relative position within an overall
bg match for any number of points. I'll talk only about game equity,
which describes your relative position within a single game, beit for
money (single games) or within a match.
Very briefly (and simplistically), equity is simply a number describing
the current value of a game to you and to your opponent. The equity
number is expressed as a proportion of the game's stake.
Another way of describing it: it is the proportion of the game's stake
that you would be willing your opponent to pay you (or vice versa) were
you and your opponent to stop the game right now and settle your
financial differences. Your opponent's equity is always the negative of
your equity.
For example, if your game equity in a single money game is 0.43
(therefore your opponent's equity is -0.43), and you're playing for $1 a
point, then you would accept $0.43 in direct payment were you both to
agree to stop playing at that point.
To convert equity to money value, you not only take the stake into
account but the cube value also. You multiply the equity by the stake
*and* the cube value.
So if your equity is -0.12 and you're playing for $3 a point, and the
cube is currently on 2, then you would pay your opponent $3 x 0.12 x 2 =
$0.72 were you agreed to stop playing at that point. (In this example.
your equity is negative, meaning that you are the underdog in the game,
and you would pay your opponent, not vice versa).
Yet another way to describe equity is the average of all the equities
that would apply for each position were you and your opponent to play
the game out to the end. Please note that I don't understand this
description! (I said I'm no boffin.)
At the start of a game, before both players have rolled to see who moves
first, your game equity is 0. I.e., you would pay your opponent and he
would pay you, nothing if you decided to stop playing at this point.
That should make sense, as you haven't started playing yet.
Imagine you have only one man left to bear off and your opponent has
already borne one man off, for example:
1 2 3 4 5 6 7 8 9 10 11 12
+------------------------------------------+ O: - score: 0
O | O O O O O O | | |
| O O O O O O | | |
| O O | | |
| | | |
| | | |
| |BAR| |v 1-point match
XX | | | |
XX | | | |
XX | | | |
XX | | | |
XXX | X | | |
XXX +------------------------------------------+ X: - score: 0
24 23 22 21 20 19 18 17 16 15 14 13
X's equity is 1 (so O's equity is -1). X cannot lose the game,
regardless of who's on roll or what either player rolls, and cannot win
a gammon. O would do as well to pay X 1 x stake x cube-value as he would
to play the game to completion. If the stake is $5 and the cube is on 4,
O could pay X $20 at this point.
1 2 3 4 5 6 7 8 9 10 11 12
+------------------------------------------+ O: - score: 0
| O O O O O O | | |
| O O O O O O | | |
| O O | | |
| | | |
| | | |
| |BAR| |v 1-point match
XX | | | |
XX | | | |
XX | | | |
XX | | | |
XX | X X | | O |
XXX +------------------------------------------+ X: - score: 0
24 23 22 21 20 19 18 17 16 15 14 13
In this position, *if* X is on roll, X's equity is 2. That's because he
will win a gammon whatever happens. So a stake of $2.50 and a cube of 8
means O would do as well to pay X $40 now as he would do to play the
game out (it's over next roll, anyway)
Those were black and white positions - the result of the game is already
decided and they're only to illustrate simple examples of equity.
Things are usually more complex. In mid-game positions of almost any
type, it is nigh-on impossible to work out your exact equity. I was
going to go on to describe more, and I can see how Peter's article might
have come about, considering this was going to be only a ten-line reply,
and as it's time for bed, I'll stop here before I end up trying to
replicate Peter's previous efforts!
Anyway, I hope that's of some use - there's a full-length book in this
subject somewhere...
--
_
James Eibisch ('v') N : E : T : A : D : E : L : I : C : A
Reading, U.K. (,_,) http://www.revolver.demon.co.uk/
=======
Sheldon Richter
James Eibisch (jeib...@revolver.demon.co.uk) wrote:
: On Tue, 01 Oct 1996 18:53:40 +0100, Sanjay Fernando
: <San...@ferndo.demon.co.uk> wrote:
: >Can someone please explain equity to me and how it is calculated?
**SNIP** **SNIP**
: A prescript: Peter Bell (USRobots on FIBS) wrote a magnificent article
: explaining equity and how to apply it to use of the doubling cube. This
: article is, AIUI, currently being made into a commercial paper which
: prevents me from posting it here. My inferior explanation below is not
: taken from Peter's article but is informed by it - it taught me what
: equity is, after all. May I take this opportunity to thank Peter for his
: original posting of this article.
**SNIP** **SNIP**
Peter Bell's article sounds interesting. What was the title and/or date of
the original posting?
--
,,,
(o o)
---------------------------oOO--(_)--OOo-----------------------------
I speak only for myself
Peter Bell
In article <52ubim$o...@pioneer.uspto.gov>, ric...@uspto.gov (Sheldon
Richter) wrote:
< James Eibisch (jeib...@revolver.demon.co.uk) wrote:
< : On Tue, 01 Oct 1996 18:53:40 +0100, Sanjay Fernando
< : <San...@ferndo.demon.co.uk> wrote:
<
< : >Can someone please explain equity to me and how it is calculated?
<
< **SNIP** **SNIP**
<
< : A prescript: Peter Bell (USRobots on FIBS) wrote a magnificent article
< : explaining equity and how to apply it to use of the doubling cube. This
< : article is, AIUI, currently being made into a commercial paper which
< : prevents me from posting it here. My inferior explanation below is not
< : taken from Peter's article but is informed by it - it taught me what
< : equity is, after all. May I take this opportunity to thank Peter for his
< : original posting of this article.
<
< **SNIP** **SNIP**
<
< Peter Bell's article sounds interesting. What was the title and/or date of
< the original posting?
< --
< ,,,
< (o o)
< ---------------------------oOO--(_)--OOo-----------------------------
< I speak only for myself
Hi James,
Well, it's encouraging to see that somebody remembers me!! Thanks for the
plug, and thanks for the thanks!
All,
For those who are interested ... back about a year ago, I started trying
to figure out take/drop, equity, recube vigorish, match equity, etc.,
etc. Just for fun, I started writing an article about it, which soon grew
to six articles! (James commented how hard it is to stop once you get
going.) I posted them to this newsgroup, and was amazed by the
overwhelming number of positive responses I received. Apparently, nobody
had ever written up a treatment of the doubling cube for "general
consumption".
With the encouragement of several people (most notably Bill Robertie of
the Gammon Press), I decided to turn the articles into a book. As usually
happens with this type of project, many roadblocks ensued. After some
discussion, Bill and I decided that my initial draft of the book was
really an agglomeration of TWO books, one for beginners/intermediates and
one for advanced/expert players. Then, Bill got busy with other things
... then, I got busy with other things ... and now, finally, I can return
to this project!
Anyway, I am on the verge of finishing the beginner/intermediate book.
Best-case ETA is January '97. It will be available through the Gammon
Press. I'm not sure whether I'll ever publish the advanced/expert book
... because when it comes to the cube, there are very few experts, so the
market is very small!
I cannot repost the original articles for various reasons that aren't
worth going into again, and I've asked others not to repost them ... and,
everyone's been very willing to respect my wishes, for which I'm very
grateful!
As for equity ... James' and Chuck Bower's posts covered the concept very well!
Glad to be back,
Peter Bell (USRobots)