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"Slot Rule" and "Simborg Rule"

Posted By: Nack Ballard
Date: Wednesday, 20 January 2010, at 7:55 a.m.

In Response To: Slot-gammon and "The Simborg Rule" (Matt Ryder)

Nack proposes a variant of BG he calls “slot-gammon”:

"The only rule change from backgammon is that you must give your opponent a direct shot with his back checkers on the opening roll."

Phil proposes a variant of BG utilising "The Simborg Rule":

"...which simply states that you cannot make a point on the opening roll, and you cannot run with 6-5. Just the first roll. Find another way to play 3-1 and 4-2 and 5-3 and 6-4 and 6-5."

Nack's variant seems a little more punitive to the opener (presumably necessitating whoppers like 31$ and the Bruce Beckerian 65D where Phil would be content with 31U and 65S). But it seems to me that both variants would swing the equity balance massively in favour of the unrestricted player who rolls second, especially if the 2nd throw is a doublet. Wouldn't it be appropriate therefore to add a further modification that the responder should: a) roll again if a double is thrown; b) also be prohibited from making a point?

Actually, no. Evidently, people don't seem to realize how big the current first player's advangage is. The Slot Rule only reduces that advantage; it doesn't (quite) negate it.

With our suggested rules, Phil and I have the following objectives:

(1) Make games more exciting/complex through increased chance of recirculation.
(2) Create interesting new decisions on the first two or three rolls of the game.
(3) Decrease the advantage (luck) of getting the opening roll.

With the "Slot Rule" (or "Slot-gammon," whichever one wants to call it), I am mostly interested in (1), though I also like (2). I don't really care about (3), though I can see how it might be viewed as a benefit.

With the "Simborg Rule," I think Phil is mostly interested in (3), probably likes (2) as well, and as far as I know doesn't care at all about (1).

Since you chose to focus on (3), I'll address that. I don't remember if I have a more accurate number, so I'm just grabbing the Snowie eval margin, which is close enough for our purposes: the opening player's (standard) cubeful equity is .102.

The equity decreases (mostly according to full rollout, some truncs) of the opening Slot Rule rolls are

31 -.237
42 -.220
61 -.185
53 -.114
64 -.070
65 -.129
32 -.048
41 -.016
43 -.082
51 -.011
52 -.056
54 -.082
62 -.044
63 -.060

Total = -.1354, divided by 15 = -.090

The equity decreases of the opening Simborg Rule rolls are

31 -.235
42 -.154
61 -.185
53 -.060
64 -.024
65 -.061

Total is -.719, divided by 15 = -.048

In summary, the Slot Rule decreases the first player's edge (from .102) to a mere .012. The Simborg Rule decreases the first player's edge to .054.

As I said, I don't care much about this first-roll equalizing factor (which is (3) listed above), but to the degree that it matters, the Slot Rule accomplishes this goal much better than the Simborg Rule. Perhaps knowledge of these figures will cause Phil to reevaluate the Slot Rule, as I am under the impression that reducing or negating the first player's advantage was his primary (or least a very strong) objective.

Recapping, the opening roll advantage is

Standard: .102
Sim Rule: .054
Slot Rule: .012


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