Forum Archive :
Inconsistencies in how EMG equity is calculated
I've written a paper that explores what I consider to be a troubling
inconsistency within EMG (Equivalent to Money Game) numbers.
Here's the basic idea:
A. 3-away is doubled to 2 by 2-away.
B. 3-away is doubled to 4 by 3-away.
C. 3-away is doubled to 8 by 5-away.
In each case 3-away must choose between taking for the match or passing and
trailing 3-away 1-away for 25% MWC. Suppose in each case she makes a bad
take with only 20% CPW, giving up 5% MWC. That 5% error gets transformed
into very different EMG errors at the different scores:
Case A: Case B: Case C:
MWC and EMG lost 3-away, 2-away 3-away, 3-away 3-away, 5-away
by taking Position 1: Doubled to 2 Doubled to 4 Doubled to 8
--------------- --------------- ---------------
MWC EMG MWC EMG MWC EMG
------ ------ ------ ------ ------ ------
Snowie 4.5 -5.26% -.425 -5.26% -.212 -5.26% -.141
gnubg 0.14 -5.10% -.407 -5.10% -.203 -5.10% -.136
I find it distressing that the same error (taking a 20% CPW for the match
rather than passing to 3-away 1-away) is sometimes reported as a .425 error
and sometimes reported as a .141 error. I am having a crisis of faith
wondering if our "whoppers" and "double whoppers" mean anything at all.
The paper goes over this in detail, considers the history of the EMG
transformation, and considers some possible alternatives. I don't have a
solution, so the paper feels unfinished and unsatisfactory. I'd love your
thoughts and comments.
(Note that while case A involves an automatic redouble, neither case B nor
case C does and they have different EMG errors, so the effect is not based
on automatic double weirdness.)
- 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)
- 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)
- Solvability of backgammon (Gary Wong, June 1998)
- Undefined equity (Paul Tanenbaum+, Aug 1997)
- Under-doubling dice (Bill Taylor, Dec 1997)
- Variance reduction (Oliver Riordan, July 2003)