This article originally appeared in the October 1999 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
We are all used to playing for money or at an early stage in a match, where
the value of a gammon is not affected by the score. Under these circumstances
(assuming the cube is at 1) winning a gammon gains 1 point while losing the
game costs two points. Therefore you must win twice as many gammons as you
lose games in order to justify a risky play which goes after a gammon.
The match score can change things considerably. Obviously at double match point all that matters is winning the game -- gammons don't count at all. This is also essentially true at the Crawford game if the trailer has an odd number of points to go. The only gain he gets from winning a gammon is a free drop, which is of very small value. Therefore, assuming a possible backgammon isn't in the picture (backgammons are of course very valuable for the trailer at this score), it is correct for both players to play as though it were double match point.
Under other circumstances, gammons may have greater value to the trailer than they do for money. Consider the score 1 away, 2 away Crawford. If the trailer wins a single game the match is tied, while if he wins a gammon he wins the whole match. Of course if he loses the game he loses the match. Therefore he can afford to play more aggressively than usual for the gammon. If a play increases his gammon chances more than it increases his losing chances, it is the correct play, since he stands to gain 50% match equity from winning a gammon (vs. winning a single game) while he loses 50% match equity from losing the game (vs. winning a single game). Let's look at a simple example:
Play Cubeless equity % wins % gammons 3/0, 3/2 1.247 100.0 24.5 3/0, 1/0 1.165 89.7 36.1Blue can guarantee the win by playing safe. As seen by the equity figure, this is the correct play for money. Blue does not win twice as many extra gammons as he loses games.
At the 2 away, 1 away Crawford score, it is another story. Blue does win more extra gammons than he loses games, so the risky 3/0, 1/0 is the correct play. This may seem logical, but it is very difficult for most players to suddenly shift gears and make an abnormal play such as this due to the match score.
At more lopsided match scores, such as 10 away 1 away Crawford, the numbers change. Gammons are still valuable for the trailer, of course, but they are not as valuable as at the 2 away, 1 away Crawford score. The reason is that the trailer will have opportunities to win a gammon later, so the gain from getting the gammon now is not as great. In order to simplify the discussion, for the rest of the article I will be assuming only the scores of double match point and 2 away, 1 away Crawford in any analysis.
At double match point, what should we be playing for? Gammons don't help us, but they don't hurt us either. Blitzes are clearly of less value than under normal circumstances. On the other side of the coin, things aren't as bad as we are used to if we are being blitzed. This means that we should be more willing than usual to leave our anchor, if the alternative is staying back and having our game crunch. We might save the gammon by holding the anchor, but we won't win the game. If we make a bold run for it we may get blitzed, but we also might survive and win.
Suppose we have the strong position. At double match point, we should be playing more of a priming game than an attacking game. If we can force our opponent to crunch his board, he will be very unlikely to win. Under normal conditions it might be right to attack and risk letting him get some timing in exchange for the additional gammon possibilities.
What about back games? As we all know, a really well-timed back game will win more often than not. For money it isn't a favorite due to the gammon dangers, but at double match point it is fine. Consequently it is good to get into a well-timed backgame. Of course making sure the timing is adequate is always tricky. At double match point, the timing issue is even more important than usual. The person playing the back game can go out of his way to avoid crunching, leaving blots strewn all over the place if necessary, while his opponent must be very careful to avoid letting the timing for the back game become decent.
Now let's look at the 2 away, 1 away score, where one player needs gammons very badly but the other player has no use for them. The player who needs gammons should be playing for gammonish positions, such as blitzes and priming battles. It is vital for him to build up his inner board fast. Also he doesn't mind having checkers sent back as much as he might under other circumstances. True this hurts him in the race, but the gammon danger of having checkers sent back is irrelevant. For example, consider how to play an opening 2-1. For money, the plays of 24/23, 13/11 and 13/11, 6/5 are about equal. Not true at this match score. The player who is behind must play 13/11, 6/5. If he gets away with it he will make his five point, which will improve his gammon chances. If he is hit he will get gammoned more often, but that doesn't matter.
How about for the leader? Just as the trailer wants gammonish positions, the leader wants to avoid gammons. No blitzes for him. Just nice simple play. Most important is for him to form an advanced anchor. It is very hard for you opponent to win a gammon if you have an advanced anchor. With that same opening 2-1, it is vital for the leader to play 24/23, 13/11. The split greatly improves the chances of making an advanced anchor, as well as making it more difficult for the opponent to make inner board points. There is some danger of being blasted away with 4-4 or 5-5, but you have to live with that.
Once the leader has made that advanced anchor, he should be reluctant to give it up unless he sees a clear route home. The game plan is to sit on that anchor until he rolls big doubles, and then run away for the win without taking any risk of being gammoned. It should be noted that the leader doesn't mind complicated games where both sides have a lot of checkers back. These usually reduce to mutual holding games, which is fine. What he wants to avoid are games where he is being attacked.
I thought it would be useful to look at a few positions from the latest match of the month, which went to the Crawford game and a couple of games beyond. I will list Jellyfish rollouts of the plays in question, giving the percentage of wins and gammons for each side as well as the cubeless equity. In this way we can see how certain plays might differ depending on the match score.
It must be noted that the Jellyfish rollouts for this sort of problem have a reliability problem other than the usual problems associated with rollouts. The difficulty is that if we are analyzing a double match point problem, the program doesn't realize that is what we are doing. Thus in the rollout it plays aggressively for the gammon if it thinks that is correct, not knowing that it shouldn't be doing that. Consequently, we may get some misleading results for that reason. However we can still look at the rollouts and get an idea of what is going on.
Play Equity Win gammon Win game Lose Game Lose gammon 24/18, 8/4 -.551 10.0% 35.8% 64.2% 35.9% 10/6, 10/4 -.562 7.8% 30.3% 69.7% 23.8%This is a classic position which is affected by the score. White can run one back checker and risk getting blitzed, or hang back and die a slow death. The equity figures show that for money this is a very close decision. At double match point, it is another story. The running play wins more than 5% more often than staying, so it is clearly correct. Since Blue has an odd number of points to go at the Crawford game, 24/18, 8/4 is correct since the play should be the same as at double match point.
Suppose instead Blue had an even number of points to go, so gammons counted extra for Blue. The running play loses far more gammons than staying. It also wins more gammons, which is useless to White. Thus 10/6, 10/4 would be very correct. Of course if it were Blue who were ahead in the match then running would be even more correct since that play generates both more wins and more gammons, and losing a gammon wouldn't matter.
Play Equity Win gammon Win game Lose Game Lose gammon 23/20, 16/10 +.637 20.6% 73.5% 26.5% 4.7% 20/11 +.625 24.3% 72.2% 27.8% 7.3% 23/14 +.610 26.2% 71.3% 28.7% 9.5%Blue wins more games by locking up the anchor, largely because he can't be attacked so there isn't much White can do. However Blue both wins and loses more gammons by making one of the other plays. These plays mix the checkers up more. When gammons don't matter Blue should play 23/20, 16/10, but if Blue can use gammons while White can't then he should play either 20/11 or 23/14.
Play Equity Win gammon Win game Lose Game Lose gammon 13/9*, 9/7 -.731 4.5% 24.3% 75.7% 24.7% 6/2, 4/2 -.743 3.1% 22.0% 78.0% 20.5%White can hit and pray, or he can make the two point and hope something good happens later. Hitting is surely the best shot at winning the game. Blue might stay on the bar for a while giving White a chance to win frontwards, or Blue might hit something after which White might be able to time an ace-point game. At the match score where gammons almost don't matter, White should clearly hit. However if the score were 2 away, 1 away, then the gammons would cost. Since White loses more extra gammons after hitting than he wins games, it would be correct to make the two point. Note that the plays are pretty close in money equity, but at the different match scores it is quite clear which play is superior.
Play Equity Win gammon Win game Lose Game Lose gammon 10/8, 10/4 -.759 2.0% 17.8% 82.2% 13.2% 24/16 -.835 5.0% 21.0% 79.0% 29.6%If gammons mattered it would clearly be folly for White to play 24/16. The rollouts demonstrate what we already knew -- that running considerably increases the gammon danger. However if White's only care is winning the game, then 24/16 is the best shot. If White doesn't run now he will almost certainly be saddled with a poorly timed ace-point game, and those just don't win very often. White has a strong enough board to be a contender if he hits a shot, so if he runs and survives the first salvo he is right in the game. This play looks intuitively so awful that it is difficult to find at the table, but when winning is everything it is the right play.
Play Equity Win gammon Win game Lose Game Lose gammon 6/3, 2/0 +1.017 15.0% 93.2% 6.8% 0.0% 6/4, 6/3 +1.010 13.0% 93.9% 6.1% 0.0%The safest play is 6/4, 6/3. That play gives Blue very good distribution, with spares on the four and three points. This is the play to make at double match point, since if Blue doesn't get hit he wins. However if gammons are pulling extra weight then Blue should play 6/3, 2/0. The extra checker off can swing the gammon, and keeping the spare on the six point may enable Blue to hold the point for an extra roll, which may be enough to contain the White checkers so Blue can win a gammon. Blue will get hit more often and lose more games, but the extra gammons are worth it.
Play Equity Win gammon Win game Lose Game Lose gammon 4/0 +1.040 14.7% 94.5% 5.5% 0.0% 6/5, 6/3 + .996 8.9% 95.3% 4.7% 0.0%This is very similar to the previous position. If gammons counted, 4/0 is clearly the best. The extra danger is minimal, and the increase in gammon potential from the extra checker off and the held six point is considerable. However if only winning matters than clearing the six point is best.
Play Equity Win gammon Win game Lose Game Lose gammon 20/10(2) +.458 14.0% 68.3% 31.7% 5.1% 20/15(2), 6/1(2)* +.411 15.6% 65.8% 34.2% 6.4% 8/3(2), 6/1(2)*. +.376 19.7% 61.9% 38.1% 6.3%20/10(2) is clearly the best overall play. This brings the checkers around together and properly takes advantage of Blue's big racing lead. The plays which attack on the ace point leave Blue with a divided game, and if White enters quickly Blue will have problems. Even with the gain from winning a gammon equal to the cost of losing the game it turns out that 20/10(2) is best, but 8/3(2), 6/1(2)* comes in a very close second -- 5.7% more gammons, and 6.4% more losses. It is only the relatively low general gammon threat (because of White's advanced anchor) which makes the blitzing play lose out when gammons pull extra weight.
Play Equity Win gammon Win game Lose Game Lose gammon B/21, 20/16, 8/4(2) +.317 21.4% 59.5% 40.5% 9.5% B/21, 20/16, 11/7(2) +.304 18.2% 60.1% 39.9% 8.6%This is a very typical result for an early position. The purer play of making the bar point both produces more wins and loses fewer gammons, so for the player ahead in the match it is clearly correct. However if the player behind in the match had this play to make, making the four point would be the right idea. It is inner board points which produce gammons, since they lead to future blitzes. Primes are nice to lock up games, but they won't get the gammons which blitzes get.
Play Equity Win gammon Win game Lose Game Lose gammon B/20 +.211 15.7% 57.2% 42.8% 9.4% B/23, 6/3* +.211 18.0% 56.6% 43.4% 10.5%It should be no surprise by now that hitting on the three point stirs the checkers up and produces more gammons for both sides (and fewer wins for White). While for money the plays are close, at this match score it appears clear for White to play B/20.
Play Equity Win gammon Win game Lose Game Lose gammon 13/10, 8/5, 4/1(2)* +.945 49.3% 73.8% 26.2% 5.9% 13/7(2) +.876 36.9% 76.6% 23.4% 4.7%This is an excellent prototype position illustrating the difference between various plays depending upon the match score. For money, 13/7(2) would be a serious error. The plays which involve shifting to the ace point are clearly superior. White has excellent chances to blitz Blue off the board with the shifting play, and this play will produce a lot of extra gammons. However when gammons don't help White, then the quiet 13/7(2) is the big winner. The solid five prime makes it very difficult for Blue to win the game. He doesn't figure to have timing to play a satisfactory ace-point game, yet winning frontwards will be very difficult. It is to be noted that the pure play also loses fewer gammons than the blitzing play, which is no surprise. If the positions were reversed and the player behind in the match had the double-threes to play, then failing to blitz would be a monstrous error.
Play Equity Win gammon Win game Lose Game Lose gammon 21/16*, 16/14 +1.100 45.1% 81.8% 18.2% 2.4% 10/3 +1.063 39.5% 83.5% 16.5% 2.4%When your opponent is playing a deep anchor or back game, every checker you send back increases your gammon chances. In this position, hitting the blot is the money play or the play if White is behind in the match and can use the gammon. However for just winning the game, the simple 10/3 completing the prime is the winner.
Play Equity Win gammon Win game Lose Game Lose gammon 14/11, 13/12* +1.504 59.2% 92.7% 7.3% 0.1% 14/10 +1.200 32.8% 92.9% 7.0% 0.1%If gammons mattered, it would be a huge blunder to not hit the blot. However if White is just playing for the win, it appears that not hitting is better. By hitting White gives Blue a chance to time his ace-point game if Blue can stay on the bar for a while. If Blue is forced to play, his timing is almost certain to be shot.
The above positions are some illustrations of the big differences in checker play which can occur at double match point or when one side can use a gammon but the other can't. When you reach such a match score it is important to be aware of your goal and play accordingly. Seemingly obvious plays might be very wrong due to the match score.