Annotated Game 
Malcolm Davis vs. Chris Peterson

From Backgammon Times, Volume 2, Number 1, Winter 1982. 
Over the weekend of October 30 to November 1, the New England Backgammon Club hosted its premier tournament, the Boston Symphony Orchestra Benefit, at the Hyatt Regency in Cambridge. Normally held in the spring, the tournament was moved to the autumn this year to avoid a possible scheduling conflict with the Turnberry Isle event.
The Championship Division was won by Malcolm Davis of Texas who made his first appearance in the New England area. Davis beat Chris Peterson, formerly of Boston but now residing in New York, in the finals. Gino Scalamandre and I were semifinalists.
The prize winners in the other sections were a collection of former champs. Ellen JacobyLee, runnerup in 1979, beat Leslie Stone, the winner in 1980, in the consolation. Al Hodis, last spring's winner, and Kent Goulding were the semifinalists. In the Last Chance, Mike Valentine edged Doug Mayfield (runnerup last spring).
On his way to the championship, Davis put together one of the most amazing runs of good dice ever seen in a major tournament. In his 17point match with Uli Koch in the round of eight, Davis trailed 16–4 entering the Crawford game. He then proceeded to:
Although Peterson played well in the finals, Davis's dice never gave him a chance to get in the match. For analysis, I've selected the penultimate game of the finals, which began with Davis sitting on a hefty 16–7 lead.
Davis (Black)  Peterson (White)  
1.  21: 13/11, 6/5  
2.  52: 13/8, 13/11 
 White to play 61. 
2.  . . .  61: 24/18, 6/5 
An excellent play. Trailing 16–7 in the match, Peterson wants to create complex, doubleedged positions with gammon possibilities. Moving to the bar point puts immediate pressure on Davis's position. If Davis hits on the bar, he won't be able to make an inner point and will be vulnerable to several return hits.
Compare the position after Peterson's play with the position that would have resulted if he had played 11/5, 24/23 instead:


Notice in this position that Davis's rolls of 31, 42, 53, and 61 are all much better for him than in the actual game position. He can use them to make inner points without worrying about his vulnerable blot on the 11point.
 Black to play 54. 
3.  54: 11/7*, 6/1* 
One of Davis's worst rolls. He makes the best play possible.
 Should White double? 
3.  . . .  Double to 2 
Another good play. Objectively, this is not a particularly strong double. In a money game, for instance, it would be quite premature. When trying to come from behind in a match situation, however, handling the cube requires a bit of creativity and psychological insight. The basic strategy is to offer an early double at the precise moment when your opponent most wants (incorrectly) to pass.
Here Peterson guages the position perfectly. Right now, Davis's position looks scary. He has no points, two blots, back men still trapped on the 24point, and his opponent already in possession of the crucial 20point. A player could easily talk himself into dropping this cube. Next turn, however, Davis's game might look much stronger. If Peterson enters without hitting and Davis makes an inner point, or covers the bar and splits his back runners, he would have reached a position that almost no one would pass. Peterson correctly doubles at his optimal bluffing opportunity.
4.  Take 
Davis is a very experienced player, and if anything he tends to err on the side of aggression rather than caution. He properly scoops up the cube.
4.  . . .  21: bar/24*, bar/23 
5.  43: bar/21, 24/21  43: 24/20, 23/20 
 Black to play 53. 
6.  53: 8/3, 6/3 
Picking up the blot with 8/3, 7/4 leaves Peterson free to maneuver in the outer boards without fear. Davis needs some teeth in his game and the 3point is a start.
6.  . . .  66: 20/14(2), 13/7(2) 
 Black to play 52. 
7.  52: 21/14* 
This is not quite as berserk a play as it might appear. Twos, threes, and fours are Peterson's only working numbers. Still, it doesn't appear to gain very much for the potential longterm risk of losing the anchor. I generally advocate making big plays to win the game even if gammon chances are increased. In this position, however, Davis remains the underdog even if his play works, which makes the extra risk hardly worth the trouble. I would move 13/8, 7/5.
7.  . . .  54: bar/20, 8/4* 
 Black to play 51. 
8.  51: bar/24, 13/8 
14/9 looks like a more natural play with the 5.
 White to play 31. 
8.  . . .  31: 8/4 
Another fine play. 14/11*, 14/13 is tempting, particularly since Peterson needs a gammon given the match score. However, sending the third checker back doesn't increase his gammon chances. In fact, the extra checker actually makes Davis more likely to form an advanced anchor. Peterson has the right idea: lock in the back checkers securely, make the win certain, then look around for a gammon.
 Black to play 63. 
9.  63: 14/5* 
Certainly no one can accuse Davis of faintheartedness. This play is tremendously optimistic: his chances of priming a checker of Peterson's are miniscule at best. Furthermore, if Peterson can somehow hit both loose blots, his gammon chances will be very real. Just 14/8, 7/4 looks simple and best.
9.  . . .  62: bar/23, 14/8 
10.  51: 6/5, 7/2*  64: bar/21, 14/8 
 Black to play 62. 
11.  62: 24/16 
Excellent. This move gives some real winning chances at the cost of a very slightly increased risk of a gammon. Only 15 numbers hit the blot, and Davis would then be a favorite to reanchor before being closed out. This play is much stronger than the passive 8/2, 8/6.
11.  . . .  51: 21/16, 8/7 
 Black to play 62. 
12.  62: 24/16 
12.  . . .  31: 16/13, 7/6 
 Black to play 33. 
13.  33: 16/10(2) 
And now Davis takes the lead in the race! The pip count right now is Davis 108, Peterson 111.
13.  . . .  41: 13/9, 7/6 
 Black to play 43. 
14.  43: 8/4, 8/5 
Good technique. Davis moves to fill the gap on his 4point rather than stack more checkers on the 6point.
14.  . . .  11: 8/6, 7/5 
15.  32: 8/6, 8/5  65: 13/7, 13/8 
16.  65: 10/4, 13/8  31: 8/5, 7/6 
17.  62: 10/4, 13/11 
 White to play 54. 
17.  . . .  54: 9/4, 8/4 
Both sides have played the bearin well, giving themselves a good distribution of builders across the 4, 5, and 6points.
18.  11: 8/6, 11/9  32: 5/off 
19.  42: 9/5, 2/off 
 White to play 21. 
19.  . . .  21: 6/3 
A slight inaccuracy. Since Peterson can't bear a man off anyway, he should even out his distribution on the higher points with 6/4, 6/5. This would give him plenty of fours and fives to play in the bearoff. His three will play efficiently from 6/3 in any case.
20.  53: 5/off, 3/off  54: 5/off, 4/off 
 Black to play 52. 
21.  52: 5/off, 6/4 
4/2 is equally good.
21.  . . .  42: 19/off 

Should Black redouble to 4? 
Davis has finally moved into the lead with a 52 to 55 lead in the pip count and the roll. Premature to double, of course, even for money.
22.  62: 6/off, 4/2  21: 3/off 

Should Black redouble to 4? 
Fortunes change quickly in the bearoff. Now, in a money game, Davis would redouble and Peterson would have to pass. At this score, it is still too soon to turn the cube back.
23.  61: 6/off, 4/3  53: 5/off, 6/3 
24.  53: 5/off, 3/off  54: 5/off, 4/off 
25.  Redouble to 4 
 Black redoubles to 4. Should White take? 
This position merits close examination. Although Peterson has caught up by a couple of pips over the last two rolls, he still trails in the pip count by 29 to 35. With each side having only seven checkers left, this is too much of a discrepency to take a money double.
Here, however, we have a match situation where the notion of overage comes into play. Peterson has the option of taking and immediately redoubling to the 8 level. All 8 points are useful to Peterson, but only 5 of the 8 points are useful to Davis. The other 3 points are wasted for him, since he already has 16. Theoreticians refer to this situation as overage. When overage happens, the odds change rapidly from normal money play conditions. Let's see just what the correct strategy for this position really is.
First, let's decide if Peterson should take. Note that if he does take, he should immediately redouble to 8, since his chance of winning the match from a 20–7 deficit is minute (255 to 1 against, not counting gammons). To decide if he should take, we need to know three probabilities:
To determine probability (1), I asked Kent Goulding to input the position to his bearoff computer program. After 11,000 trials, it revealed that Davis was an 84 to 16 favorite (all games played to a finish, no cube involved).
To determine probabilities (2) and (3), I referred to match probability charts. (The calculations to generate these charts are too tedious to explore here.) They revealed that an 18–7 lead is equivalent to a 97 to 3 likelihood of winning the match, while a 16–15 lead means a 58 to 42 likelihood of winning.
Knowing these figures, we can calculate Peterson's chances of winning the match in each case:
So Peterson would more than double his chances of winning the match by taking.
Given that Peterson's best strategy is to take and redouble, what about Davis's double? Was it correct or not?
We know that Davis's winning chances if he doubles (and Peterson takes and redoubles) are 93%. It's no trivial matter to calculate his exact winning chances if he holds the cube, since his future cube actions depend on the position. Suppose, however, we compute his winning chances if he never doubles, no matter how good his position gets. Obviously, this is less than his optimal strategy, so his real winning chances must be higher than this number.
If he never doubles, his chances of winning the match are:
for a total winning probability of 95.3%. His real winning chances, using the cube effectively, are slightly higher than this, let's say around 96%. So by doubling from 2 to 4, he gives Peterson about a 50% better chance to win the match. It pays to be conservative with the cube when overage is involved. (Actually, this still isn't the whole story. A complete analysis of Davis's double would have to take into account two additional considerations: (1) the possibility that Peterson would incorrectly drop the double, and (2) the possibility that Peterson will take but fail to redouble next turn. These two possibilities make Davis's double much less incorrect than the pure mathematics would indicate. In fact, watch what happens.)
25.  . . .  Take 
26.  51: 5/off, 2/1  21: 3/off 
27.  64: 6/off, 4/off  52: 6/1, 6/4 
28.  41: 4/off, 1/off  31: 1/off 6/3 
29.  31: 3/off, 5/4  41: 4/off, 6/5 
30.  63: Wins 
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