This article originally appeared in the January 2001 issue of GammOnLine.
Thank you to Kit Woolsey for his kind permission to reproduce it here.

Snowie Equities

By Kit Woolsey
I was playing Ray Fogerlund a 7-point match when this mundane position arose:

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87 

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Fogerlund 2



7 point match




Woolsey 4

Ray turned the cube to 2, with me leading 4-2. Should I take?

I am 6 pips behind with the race in the 80's, plus he has the ace shot. If he hits I am pretty much dead. If he misses his average roll will be higher than normal, so in effect I am probably down 7 or 8 pips if it becomes a race. That would be a take on the race alone, but only by a couple of pips. Since 11/36 of the time I would have a very small chance of winning and the other 25/36 I would have a take which wouldn't be a bargain, it seemed likely that this was a pass. In addition I was ahead in the match, so I wouldn't get much recube vig. I chose to pass.

When I fed the match to Snowie later, Snowie said that my pass was a huge blunder! Did I misestimate the position, or was there something tricky about the match score which I didn't understand?

In the win-loss column, Snowie's rollout had me winning 23.7% of the games. That was about what I had estimated. Sure I had some small recube vig, but if I ever recubed to four Ray could take and send it back for the match with as little as 15% winning chances, since if he passed he would be behind 6-2 Crawford. Thus, my recube potential wouldn't be worth much. In addition, there is always the small chance that I could be hit and wind up getting gammoned. Not very likely (1.2% by the rollouts), but it could happen. It appeared that the gammon chances and the recube vig were both small and would probably about cancel each other out. My winning percentage was the important thing. I was ahead in the match, so surely I would need more than 25% winning chances. Or would I?

Let's check it out. Using my match equity table and ignoring gammons and recube potential, we get:

If I pass I am ahead 4-3 (4 away, 3 away), 59% equity.
If I take and win I am ahead 6-2 (1 away, 5 away Crawford), 85% equity.
If I take and lose we are tied at 4-4, 50% equity.

Thus, I would be risking 9% in order to gain 26%. A little worse than 3 to 1 odds, which is what seemed intuitively correct since I was ahead in the match. Thus, I needed better than 25% winning chances to take the double.

By my calculations and accepting the Snowie rollouts, my pass is quite clear. However, Snowie insists that the pass is a blunder. What is going on?

I was well aware that Snowie is using a different match equity table than mine. Snowie assumes a gammon rate of 26%, while for my table I assumed a gammon rate of about 21%. I agree that Snowie's gammon rate is more accurate in theory. In practice most backgammon players (even experts) do not play aggressively enough for a gammon, since they are more concerned about winning the game. Thus the empirical gammon rate is considerably lower than Snowie's theoretical rate.

It occurred to me that I could determine the exact equity table which Snowie is using. By setting up a gin position and modifying the match score, I could see what Snowie's equity estimates on a take and a pass of a double would be. Since the player on roll is 100% to win the game, the Snowie match equity estimates would be exactly what the Snowie table is. Snowie's equity table is below:

Snowie Equity Table

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 50.00 68.50 74.78 81.94 84.30 89.08 90.71 93.49 94.44 96.10 96.88 97.67 98.01 98.61 98.82
2 31.50 50.00 59.42 66.42 73.48 79.07 83.16 86.60 89.27 91.58 93.23 94.68 95.73 96.66 97.32
3 25.22 40.58 50.00 57.16 64.46 70.65 75.45 79.69 83.19 86.32 88.67 90.80 92.41 93.88 94.96
4 18.06 33.58 42.84 50.00 57.33 63.72 69.01 73.57 77.69 81.31 84.27 86.84 88.99 90.86 92.38
5 15.70 26.52 35.54 42.67 50.00 56.48 62.09 67.07 71.64 75.73 79.19 82.28 84.89 87.23 89.15
6 10.92 20.93 29.35 36.28 43.52 50.00 55.85 61.01 65.92 70.32 74.21 77.65 80.70 83.40 85.73
7 9.29 16.84 24.55 30.99 37.91 44.15 50.00 55.25 60.32 64.93 69.11 72.87 76.25 79.29 81.97
8 6.51 13.40 20.31 26.43 32.93 38.99 44.75 50.00 55.12 59.85 64.22 68.20 71.84 75.15 78.13
9 5.56 10.73 16.81 22.31 28.36 34.08 39.68 44.88 50.00 54.79 59.29 63.45 67.31 70.85 74.08
10 3.90 8.42 13.68 18.69 24.27 29.68 35.07 40.15 45.21 50.00 54.57 58.84 62.85 66.57 70.01
11 3.32 6.77 11.33 15.73 20.81 25.79 30.89 35.78 40.71 45.43 50.00 54.31 58.42 62.28 65.88
12 2.33 5.32 9.20 13.16 17.72 22.35 27.13 31.80 36.55 41.16 45.69 50.00 54.15 58.09 61.80
13 1.99 4.27 7.59 11.01 15.11 19.30 23.75 28.16 33.69 37.15 41.58 45.85 50.00 53.97 57.76
14 1.39 3.34 6.12 9.14 12.77 16.60 20.71 24.85 29.15 33.43 37.72 41.81 46.03 50.00 53.83
15 1.18 2.68 5.04 7.62 10.85 14.27 18.03 21.87 25.92 29.99 34.12 38.20 42.24 46.17 50.00

By comparison, let's look at my equity table.

My Equity Table

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 50 70 75 83 85 90 91 94 95 97 97 98 98 99 99
2 30 50 60 68 75 81 85 88 91 93 94 95 96 97 98
3 25 40 50 59 66 71 76 80 84 87 90 92 94 95 96
4 17 32 41 50 58 64 70 75 79 83 86 88 90 92 93
5 15 25 34 42 50 57 63 68 73 77 81 84 87 89 90
6 10 19 29 36 43 50 56 62 67 72 76 79 82 85 87
7 9 15 24 30 37 44 50 56 61 66 70 74 78 81 84
8 6 12 20 25 32 38 44 50 55 60 65 69 73 77 80
9 5 9 16 21 27 33 39 45 50 55 60 64 68 72 76
10 3 7 13 17 23 28 34 40 45 50 55 60 64 68 71
11 3 6 10 14 19 24 30 35 40 45 50 55 59 63 67
12 2 5 8 12 16 21 26 31 36 40 45 50 54 58 62
13 2 4 6 10 13 18 22 27 32 36 41 46 50 54 58
14 1 3 5 8 11 15 19 23 28 32 37 42 46 50 54
15 1 2 4 7 10 13 16 20 24 29 33 38 42 46 50

It can easily be seen that for just about every entry Snowie's table gives the trailer better winning chances than my table. This makes sense. If the gammon rate is higher, the chance for a comeback is obviously improved. In particular for match scores where one side has a good-sized lead, the difference can range up to 2%. For example, my table has the winning chances ahead 13 away, 5 away, as 87%, while Snowie's table has the winning chances as 84.89%.

For the most part, it shouldn't make a whole lot of difference which table one uses. The reason is that when one does a match equity calculation, if the leader is favored throughout he will be favored for each potential score you use in the calculation. Thus, the two tables are likely to produce the same result on a pass/take decison anyway. This might not be true if taking the cube will end the match, but otherwise any differences figure to cancel out.

So, what is going on with my actual position. Guess we better do the equity calculations along with Snowie, using Snowie's equity table:

If I pass: I am ahead 4-3 (4 away, 3 away), 57.16% equity.
If I take and win, I am ahead 6-2 (1 away, 5 away Crawford), 84.30% equity.
If I take and lose the score is 4-4, 50% equity.

Thus, I am risking 7.16% in order to gain 27.16%. According to Snowie I am getting way better than 3 to 1 odds on the take -- almost 4 to 1 in fact. This is why Snowie says my pass was a big blunder.

So, what is going on? Why is there such a big discrepancy in the match equity calculations for this specific situation, when for the most part they seem to come out similar. Is there something special about this particular score?

Let's take a look at the scores around the critical 4 away 3 away score and see if something unusual might be occurring:

Kit's Table:

3 away, 2 away: 60%
4 away, 3 away: 59%
5 away, 4 away: 58%
6 away, 5 away: 57%

Snowie's Table:

3 away, 2 away: 59.42%
4 away, 3 away: 57.16%
5 away, 4 away: 57.33%
6 away, 5 away: 56.48%

While my table follows a natural intuitive progression, Snowie's table is saying something very odd. Snowie thinks that in a 5-point match it is better to be ahead 1-0 than to be ahead 2-1! Obviously Snowie is putting some kind of special value on being exactly 4 points away from winning the match.

Can this possibly be right? Isn't it always true that the leader in the match is happy to have the match get closer to finishing while he maintains the same lead? If you carefully check Snowie's table you will see that there is no other case where this happens -- in the upper right part of the table (where the leader's winning chances are given), if you go down the diagonals you will find no other place where there is a bump like this. The percentages consistently decrease, as they do for my table. Only at this particular score does this aberration exist.

So, what should we think? There is some logic to it. Being four points away means that a doubled gammon or a redoubled win puts you out exactly -- no wastage at all. That must be worth something. Still, I find it hard to believe. Would you really prefer to be ahead 1-0 than 2-1 in a 5-point match? Neither would I.

I have no idea what algorithm was used to generate Snowie's match equity tables. For the most part the results seem quite reasonable, given the 26% gammon rate estimate. For this one score, however, there is a sharp break in the continuity. Did the program actually come up with this result? Or perhaps a number was mis-transcribed into the equity table (58.16% would look just about consistent with the other numbers). Whatever the reason, it is clear that any match equity calculations which involve the 4 away 3 away score are going to lead to unusual results if we use the Snowie equity tables.

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