Forum Archive : Theory

Undefined equity

From:   Paul Tanenbaum
Date:   15 August 1997
Subject:   Extremely theoretical
Google:   5t2nr3$7bc$

The diagram below should pique the interest of backgammon theoreticians.
Its invention was motivated by the paper: "Optimal Doubling in Backgammon",
E. Keeler & J. Spencer, Operations Research, Vol. 23, No.6. It is
presented as a problem: What makes this position extraordinary?  Those
without access to the aforementioned paper will find this a real challenge,
though there are some contributors to this newsgroup who are evidently
perspicacious enough to suss it out on their own.
    I have since been informed that Bob Floyd published a position
essentially identical in an obscure, now defunct, backgammon magazine
several years ago.  He therefore deserves recognition of first discovery.
This posting is submitted for the edification of those unfamiliar with
that article.
    Do any other positions exist which possess the same property?
    It should be noted that the Jacoby rule regarding an initial double
may or may not be in effect; the conclusion is unaltered.
Also, realize that it is quite legal.

                            Money game
              X on roll - or O on roll - it doesn't matter!

          O  X  X  ^  ^  ^   |   |   X  O  O  O  O  O
                             |   |      O  O  O  O  O
                             |   |
                             | X |                        ---
                             |   |                       | 1 |
                             | O |                        ---
                             |   |
                             |   |      X  X  X  X  X
          X  ^  O  O  ^  ^   |   |   O  X  X  X  X  X
                                   <   X home board   >

Paul Tanenbaum

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Gary Wong  writes:

Hmmm... looks like the first player to roll a 6 gets a tremendous
advantage (either an anchor and 2 of the opponent on the bar against a
5 point board for 6-1 to 6-5, or 3 of the opponent on the bar for
6-6).  And I guess 11/36 probability of this happening makes the
situation a double/take for whoever's on roll, right?  So the cube
gets turned every roll until somebody rolls a 6?

In that case, the expected gain of the player on roll (from this roll only)
is 11/36 of 2 points (after doubling, ignoring gammons and backgammons)
for 0.61 points; the expected gain of the opponent on the turn after is
25/36 (the first player missing) x 11/36 (the second player hitting) x 4
(the cube is turned twice) for 0.85 points; thereafter the expected gains
continue increasing at a factor of 2 (for the cube) x 25/36 (for the
probability of the game lasting that long) = 50/36 = 1.39.  So the total
expected gain is the difference of the odd or the even terms (depending
which player you mean) of the geometric series a = 22/36, r = 50/36.
But since r > 1, the series never converges, and so the expected gain
for both players is infinite, right?  (and so is the expected loss!)  That
sure counts as "extraordinary" in my book :-)

Gary.  (GaryW on FIBS)
  Gary Wong, Computer Science Department, University of Auckland, New

Brian Sheppard  writes:

Expected gain is infinite? Not at all! The expected gain in a backgammon
game is never more than triple the cube. I don't know where you made your
error, but you can rest assured that there are some.

The way to analyze this type of position is with a "recurrent equation."
In this type of equation the quantity you wish to evaluate comes up as a
term on both sides of the equation after you analyze a few rolls. In this
case, owing to the symmetry of the position, we obtain a "recurrence" of
the quantity after just one roll.

Let's analyze the cubeless case first. Call the equity of the side to move
X, and let the equity of the side to move given that he rolls a 6 be E.
Then we have the following equation:

    In 36 rolls, the side to move will have 11 sixes, winning E each time,
    and 25 misses, in which case he loses X, since the situation is exactly
    reversed. Therefore,

        36*X = 11*E - 25*X

    So X = E * 11/61. In words: multiply the value of the position
    after a 6 by 11/61 to obtain the equity for the side to move.

When we take the cube into account that changes things slightly. Let Y be
the equity after the opponent takes. In 36 rolls the side to move will win
E in 11 rolls, and will lose 2Y in 25 rolls (because the opponent will
redouble). Now the equation is 36*Y = 11*E - 25 * 2Y, so Y = E * 11/86.

To sum up: the side to roll should double, to raise his equity from 11/61 *
E to 2 * 11/86 * E. The other side should take because 11/61 * E is
definitely less than 1.


Bob Koca  writes:

Gary's only error, and it is a small one, was calling the expected
value infinite. Undefined is more appropriate.

    Brian's recursion method normally works,  however it depends on
the expected values actually existing. Here they don't.  Many people
find examples of this sort confusing and I believe some of that stems
from not knowing exactly what expected value means.

   Not getting overly technical, expected value can be thought of as
the long term average.  For example let's look at a very simple case.
You have two checkers on the 3 point, opponent has 2 on ace point and
the cube is centered.  The expected gain to the player on roll
is   - 1/18.  What this means is that as more and more games are
played from this starting position the average gain to the player on
roll will eventually get closer and closer to -1/18. (The exact
mathematical statement is basically this but rigorously deals with
"eventually" and "gets closer and closer")

    Suppose we played Gary's position many many times. The average
winnings will NEVER settle down to a limit.  The reason is that there
will be eventually huger and huger amounts of points lost on a single
game. Enough to knock off whatever progress was made towards the
average gain approaching a limit.

      As an aside, i've seen several posts in the past stating that
equities exist for backgammon and describing a recursive method
to calculate them. This only works IF positions such as Gary's
will never occur.

    The concept of undefined expected values is a little tricky and
a little counterintuitive. If you are interested in exploring it
further I would suggest looking at definitions of expected value,
the weak and strong laws of large numbers, and the cauchy distribution
(an example in which expected value does not exist,  even though the
distribution is symmetric).  All of these will probably be in a good
undergraduate level probability text.

Bob Koca
bobk on FIBS

Brian Sheppard  writes:

First I want to tender my apologies to Gary, who it seems has grokked
the real issue.

I can get my head around the concept of "undefined expected values"
for only moments at a time, and then it slips away. Perhaps if you
could answer a few questions it might help me to understand what is
going on.

First: am I correct to say that the situation arises because the payoff
is increasing at a factor of 2 per turn (because of the cube-turn) but
the probability of that payoff is decreasing at the rate 25/36 (which
is greater than 1/2)? The point is that the chance of a huge payoff is
decreasing, but the payoffs are increasing faster.

Second: suppose we take a sample of outcomes from this game, and
observe the average amount won as time goes on for an indefinite
number of games. Given an arbitrary positive number N, is it true
that the series of averages will eventually be larger than N? Is
it also true that the series of averages will eventually be less
than -N?

Third: is it correct to double? If it is not correct to double,
then the position does have a theoretical equity. (But, what is
the basis for deciding when to double if there is no equity to
reason about?)

Fourth: is it correct to take? If it is not correct to take, then
the position does have theoretical equity.

Fifth: If doubling and taking is correct, then this position has
undefined equity. That means that positions that lead to this have
undefined equity and so on. What process, if any, prevents backgammon
as a whole from having undefined equity?

Sixth: I am curious about computer evaluation of this situation. Does
JF double for the side to roll?

Thanks in advance to anyone who can answer any of these.

Did you find the information in this article useful?          

Do you have any comments you'd like to add?     



Derivation of drop points  (Michael J. Zehr, Apr 1998) 
Double/take/drop rates  (Gary Wong, June 1999) 
Drop rate on initial doubles  (Gary Wong, July 1998) 
Error rate--Why count forced moves?  (Ian Shaw+, Apr 2009) 
Error rates--Repeated ND errors  (Joe Russell+, July 2009) 
Inconsistencies in how EMG equity is calculated  (Jeremy Bagai, Nov 2007)  [GammOnLine forum]
Janowski's formulas  (Joern Thyssen+, Aug 2000) 
Janowski's formulas  (Stig Eide, Sept 1999) 
Jump Model for money game cube decisions  (Mark Higgins+, Mar 2012) 
Number of distinct positions  (Walter Trice, June 1997) 
Number of no-contact positions  (Darse Billings+, Mar 2004) 
Optimal strategy?  (Gary Wong, July 1998) 
Proof that backgammon terminates  (Robert Koca+, May 1994)  [Recommended reading]
Solvability of backgammon  (Gary Wong, June 1998) 
Undefined equity  (Paul Tanenbaum+, Aug 1997)  [Recommended reading]
Under-doubling dice  (Bill Taylor, Dec 1997)  [Recommended reading]
Variance reduction  (Oliver Riordan, July 2003)  [Long message]

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