This article originally appeared in the February 2002 issue of GammOnLine.|
Thank you to Kit Woolsey for his kind permission to reproduce it here.
When I first constructed a match equity table, I assumed a gammon rate
of about 21%. This was the accepted figure at the time, and empirical
evidence from analysis of many matches backed this figure up. There
were some who felt this was too low, but as a practical matter it seemed
to work well.
With the coming of the bots, things seemed to be quite different. When the bots played out full games against themselves, the gammon rate came to about 26 or 27%. This was quite a difference from what was happening with humans in the real world. What was going on?
At first I thought it had to do with the playing style and perhaps weaknesses of the bots. They did make more dangerous splits of the back checkers than humans did, so these plays may lead to getting gammoned more often. They did have a preference for attacking games rather than priming games. They didn't seem to handle back games and deep anchor games as well as humans did. These all appeared to be likely explanations for the high gammon rate from the bot rollouts.
Careful study of the plays of the bots revealed something else to me. The bots played very aggressively to get gammons. They would often take chances that human experts would never take, risking an almost sure victory in order to increase the gammon chances. Interesting. Upon looking at rollouts of these plays, it appeared that the bots were right! We humans had been playing far too conservatively in many positions. That was the main cause of the extra gammons for the bots. They got the gammons because they went after them. The bots loved pigging it for the gammon.
As we all know, in money play it is correct to make a risky play for a gammon if the increase in gammon chances is at least twice the increased losing danger. Assuming our opponent owns a 2-cube, we gain 2 points (from +2 to +4) if we get a gammon we wouldn't have gotten otherwise, while we cost ourselves 4 points (from +2 to -2) if we lose the game when we would otherwise have won it. Psychologically, it can be difficult for a player to throw away a sure thing. When the risky play backfires and you lose a game which was virtually locked up, that is very demoralizing. For this reason, most experts tend to play too conservatively, making sure of the win but not collecting enough gammons. The bots have no such psychological hangups. They crunch the numbers, and if they conclude that their expectancy is higher by making the risky play then they go for the gusto.
Let's look at a typical bearoff situation.
Blue can play 6/4, 5/1, which is quite safe. Nothing leaves a shot on the next roll, and the spares on the four and three points will probably give Blue the necessary flexibility to avoid leaving any shots as he bears off. Blue has some fair gammon chances, and these gammon chances will still be there after he makes this safe play.
What about playing 6/0? This risks leaving an immediate shot if Blue rolls 5-5 or 6-6 next turn. In addition, if Blue rolls any two numbers greater than three he will wind up with a dangerous position of three checkers on the five point and two checkers on the four point, after which any large doubles or two large numbers leave a shot. White's board is perfect, so if Blue is hit he will lose the game. Can it possibly be right to pig it and take the checker off?
What does taking that checker off gain? Well, it is one checker off. Also, it may allow Blue to hold the closed board for an extra roll. If Blue's next roll is 5-1, 5-2, or 5-3, he can again take one (or two) checkers off and still retain the closed board. Every roll Blue keeps White on the bar is important. The gammon race looks to be close, so this could make the difference. What is the right tradeoff?
Oink, oink say the bots. Rip that checker off. Yes Blue will lose more often, but he still wins almost all of the time. The extra checker off give Blue around 7% extra gammons, while losing only an extra 2 1/2% of the time. Keep in mind that even if Blue does leave a shot White still has to hit it. The pig play of 6/0 is the percentage play.
It should be noted that the gammon race being close is what makes taking this risk worthwhile. If the the gammon is very likely to be won assuming a shot isn't hit, then the safe play is better.
Here Blue is a big favorite to win a gammon, even if White enters at his first opportunity. The pig play of 6/0 will win more gammons than the safer 6/4, 5/1, but the increase in gammons is so small that it doesn't compensate for the extra losses when White hits a shot.
Simlarly, if Blue is very unlikely to win a gammon anyway, the safer play may be better.
Here Blue isn't likely to win a gammon whatever he does. The pig play will again get more gammons than the safer play, but since the probability of winning a gammon is so small to begin with the increase in the number of gammons isn't sufficient to make up for the extra losing potential.
So far we have been looking at positions where White's board is near-perfect. If White's board is crunched, it is another story. Now getting hit is not necessarily a loss for Blue, so he can afford to make the pig play even when a gammon is unlikely.
Blue still gets the improved gammon chances from 6/0, but he doesn't cost himself in the win department. Even if Blue leaves a shot and is hit, he is a favorite to enter and scamper around the board to win the game. In addition, and very important, White's crunched position means that there is another kind of race involved here. It is the race to win the game. If White barrels out 6-6 at his first opportunity, he could make it very close. It won't be so close if Blue has already taken a few checkers off. In fact, rollouts show that the safer play doesn't win any more often than the risky play.
Let's examine a pig play from later in the bearoff:
If Blue plays the safe 5/1, 4/3, it would take a major earthquake for him to lose the game. He will be able to play his next roll with 100% safety, and quite possibly the roll after that. In the meantime, White may be entering and be out of Blue's hair. Also if Blue does leave a shot he will have taken a lot of checkers off, so getting hit will not necessarily be fatal.
If Blue plays 5/0, it isn't so safe. 4-4, 5-5, and 6-6 all leave an immediate shot next turn, and if Blue rolls two big numbers he will be in a very dangerous position with three checkers on the four point and two on the three point. The danger of losing the game is definitely greater if Blue makes this play.
Despite the risks, Blue should make the pig play. The gammon race is close, and the extra gammons from the play are sufficient to justify being greedy. Offhand it might not seem to matter, since Blue will have 12 checkers left after either play, and with at least two checkers on each point he wouldn't be likely to miss. But that is an illusion. It is true if White enters, but what if White stays on the bar? Now Blue will have to be sure and play safe next turn, and unless he rolls 6-5 or doubles he will only be able to take one checker off. If White now enters Blue will have 11 men left, while if Blue had made the pig play he would have 10 men left, and that is quite likely to translate into an extra roll.
Let's examine a different type of pig play when bearing outer board checkers home.
You can't take them off until you bring them in is an old saying. Blue has a great position, but the gammon race may be close. Blue can speed up the bearoff process by bringing two checkers home with 9/6, 8/4, but this leaves White the 5-3 joker. Alternatively, Blue can play 100% safe for this roll with 9/2, but then he will still have two checkers in the outer board which need to come home before he can start bearing off.
Oink, oink! Once again, the greedy play is right. Blue wins sufficiently more gammons by bringing both checkers home to make up for the extra losses from the 5-3 joker. It should be noted that the safe play of 9/2 isn't as safe as it looks. In addition to leaving an immediate blot with 5-5 or 6-6, Blue still has the problem of clearing the eight point. If Blue rolls an ace or a three next turn he will again be forced to leave a joker or crunch his board, and if White enters Blue will have to try and play safe. Thus, the safe play wins the game only slightly more often than the greedy play.
Whether to hit loose and risk retailiation or play safe and coast on home is often a question. This next position is a good example.
Blue can hit loose with 11/5*, or he can play safe with 11/6, 4/3. The safe play will win the game more often, of course, but the loose hit will collect the gammons if it works. Are the increased gammons sufficient to justify the risk?
Once again, it is correct to go for the gusto. The difference in gammon potential is huge. If Blue is hit back he may still survive, and if White makes the anchor on Blue's five point Blue's gammon goes down the drain and there are still problems winning the game. However, the decision is pretty close.
Strengthen Blue's board and it is no longer close:
Now the loose hit is far better than the safe play. The made two point increases the gammon chances if White doesn't hit back, and also increases Blue's chances to survive if White does roll that five. It is no longer a close decision.
We can't ignore what is going on over there on White's side of the board. Let's strengthen White's position by giving him the five point.
Greed can be a terrible thing, and this time it is. As expected, the loose hit still produces far more gammons than the safe play. Here, however, if White hits back Blue is in very serious trouble. The extra losses from getting stuck behind the five-prime more than compensate for the extra gammons when the loose hit works. Blue should play safe with 11/6, 4/3.
Even when the win is a virtual lock with the safe play and the loss danger is quite real with the loose hit, it still may be better to pig it.
Blue has an almost sure win by playing safe with 11/5, 6/4, while he can easily lose if he plays 11/3* and is hit back. Despite the risk, it is still clear to hit. The safe play is unlikely to produce a gammon, particularly since White may well just run on his next roll. The loose hit is quite likely to produce a gammon if White doesn't hit back, and if White does hit back Blue had a decent chance to survive anyway. Definitely worth the risk.
Here is another example of a loose hit when the opponent is threatening to make an anchor.
In the old days, most players would play safe and sound with 13/7, 8/7. A full prime! Why risk an accident when you have a sure thing? There may still be a gammon. Today, no self-respecting student of the game would hesitate about playing 8/1*. This increases the gammon chances considerably. Sure Blue could get into trouble if White hits back, but sometimes you have to take chances. In addition, the safe play isn't as safe as it looks. If White gets the anchor, Blue will be facing a well-timed ace-point game and could easily lose to that. In fact, it is not even clear that making the bar point is more likely to win than the loose hit. It isn't even a close call.
As these examples show, it is quite often correct to make a pig play and go after a gammon. The bots know it, and now we know it also. Sometimes it will backfire, but the gains when the gammon comes home make it worthwhile. Happy oinking to you.