This article originally appeared in the June 2002 issue of GammOnLine. Thank you to Kit Woolsey for his kind permission to reproduce it here. |
The Memorial Day tournament in Chicago was, as always an excellent and
enjoyable tournament. It was more enjoyable for me than usual, as I was
fortunate enough to win the Masters. My matches weren't particularly
interesting -- mostly just good dice. Here was a position from my
quarter-final match against David Wells.
For the most part, complex match equity calculations aren't necessary in tournament play. Understanding general concepts, using common sense, and following obvious principles such as being more conservative with the cube when ahead in the match and more aggressive when behind will usually see you to the correct decision. Occasionally, however, it is necessary to buckle down and do the dirty work. I had held a moderate lead throughout much of the race, but clearly not enough to consider redoubling. I didn't need to do any calculations to know that at this match score David wouldn't need very much to take, and that I would need a big advantage before I could justify redoubling and giving him a chance to put the cube on 8. Now, however, I had a big advantage. Is it big enough? I will try to reproduce my thinking process. First of all, I forget about the actual position and concentrate just on the match equities involved. Once I have worked out what his odds on the take are and what my odds on the double are, then I can examine the position and see what I think is going on. I don't want to get bogged down by worrying about the equity in the position before I know what to do with that equity. My first question is: what does he need to take. If he takes, he will of course immediately send the cube back to 8. From his point of view:
He passes: Behind 10-2 (1 away, 9 away) Crawford, about 5% equity.
David would be risking 5% (5%-0%) to gain 70% (75%-5%), so he would be getting 14 to 1 odds on his take. He can take if he can win more than 1 time in 15. Next question: What odds am I getting on my double. This is a bit more complex. The possible scenarios are:
I don't double and win: Ahead 10-2 (1 away, 9 away) Crawford, 95% equity.
By doubling I would be risking 51% (76% - 25%) to gain 5% (100% - 95%), so I am getting roughly 10 to 1 odds on my double. Since a decision is likely to be reached after the next exchange, I can probably treat this as a last roll decision, so I need to win 10/11 of the time to justify doubling. Perhaps a bit more, since in some variations I may be able to make efficient use of the cube next turn. Now that we know the relevant odds on the double and take, it is time to turn our attention to the actual position. What does David need to win this game? The easiest way for him to win is to roll boxes on his first roll. That will happen 1/36 of the time. Of course I might not give him that chance if I roll 3-3 or higher, so his chances of winning with those boxes are a bit less -- let's estimate about 1/40 or 2.5%. If he doesn't roll those boxes, he will need a parlay. First, I will have to miss. Second, he will have to get off in two rolls. How often will all this happen? For starters, I have to miss. That means I will have to roll two aces, not including 1-1 (yes, I could roll 1-1 followed by 2-1 or vice versa, but that is a very small additional chance). My probabality of rolling an ace on one roll is 10/36, so the probability of that happing twice is 100/1296 or roughly 1/13. Given that I have missed, he still has to get off in two rolls. His initial good rolls are 6-1, 6-3, 6-4, 6-5, 3-3, 4-4, 5-5, and 6-6 -- that comes to 12 good rolls which are gin or near gin. On his 24 bad rolls, he still has the chance of recovering with doubles next turn. It looks like he will get off in two rolls a little less than half of the time. Multiplying this by the chance that I will miss, we get roughly 3 1/2% (it would be about 4% if he got off half the time). Adding this to the original 2 1/2% for his immediate boxes, his winning chances appear to be about 6%. So now we have the answer. I have a big double, and it looks like he has a close drop. I will admit that this wasn't my instinctive first guess -- initially I guessed he had a clear take and the double would be close. I did double, and David did in fact pass. Here is an interesting position from my first round match in the open against Harry Cohn.
Harry is on roll, holding a 2-cube, leading 6-3 in a 13-point match. What's going on? Too good to double? Double/pass? Double/take? Not good enough to double? Totally confused? As a bonus question, what should the proper cube action be in a money game? Harry did double. This caught me by surprise. Usually I am prepared when my opponent doubles. I will have anticipated that he is likely to double, and I will have been thinking about my cube decision while making my previous play. In fact, if I am on the bar, I will have been thinking about what my likely cube decision will be during my opponent's play and while I am shaking my dice. Preparing the cube decision in advance has two advantages. First of all, I can coordinate my checker play to possible cube actions if that is relevant. Secondly, if I have planned my action in advance, I am able to make the actual pass/take decision instantly if I choose to do so (which I usually do). This way my opponent doesn't get a read on my thinking. If I take or pass quickly he may think that I thought it was a clear-cut decision, when in fact I felt it was quite close but I had been making my decision in advance so I had time to think about it while the previous couple of moves had been made. If my opponent believes I thought I had an easy decision, he may misjudge his future cube actions based on a misconception of what I am thinking. For example, if it seems from previous play that my opponent tends to be a slow doubler, I will generally take quickly and pass slowly almost regardless of the position (unless my decision is so obvious that he would know I was trying to deceive him). This will make him think that I tend to take more than I actually do, so he will be likely to be even slower with his doubles. Conversely, if my opponent appears to be a fast doubler, I will generally pass quickly and take slowly. This may tend to make him double even earlier, which is what I want. In the actual position, I was not prepared for the double. I hadn't thought that Harry would ever be redoubling to 4 in a gammonish position such as this one at the match score. Now that he did double, I had to stop and think about it. What is going on here? I have two checkers on the bar against a four-point board, and my blot will almost certainly be picked up before the smoke has cleared, so I can look at the position almost as though I have three checkers on the bar. That much is bad. On the other hand, there are plenty of favorable aspects about my position. Harry is very short on builders, so if I do enter quickly he won't be in position to carry out an attack. My board is very strong. This is important for several reasons. First of all if he does hit loose and I hit back, he will be in immediate trouble. Secondly, I don't have to waste time putting my offense together once I get in -- my offense is already in place. Thirdly, he does have two checkers stuck behind my blockade. If he fails to escape these checkers before I get my men in, he will be in trouble. My anchor on his bar point will disrupt his smooth flow of builders for the blitz, and if I survive the blitz this point will be very helpful for getting a late shot in the outfield. He has a buried checker on his two point. To top everything else off, the match score argues for aggressive takes on my part in this sort of position. A lot of my losses will be gammons when he carries out the blitz successfully. He can't use the full 8 points, and if I redouble to 8 (which I will as soon as I see daylight, of course), then gammons for him become totally meaningless while I can use the full 8 points and some of the 16 potential points if I win a gammon with the cube on 8. Yes, the more I looked at the position, the more attractive the take appeared. So I finally snatched it up. The game went as might be expected. He had no trouble picking up the third checker, but in a couple of rolls I entered two of my men with a 5-3. He had to hit loose. Now I recubed with two men on the bar. I might not have even been the favorite, but at the match score and with plenty of gammon potential for both sided the recube was mandatory since if I hit the shot I would lose my market by a fair amount. Things went my way, and eventually I won the game to take an 11-6 lead, after which I won the match. So, what is really going on in the position. The truth of the matter is that his position isn't nearly as strong as it looks. Yes, he wins a lot of gammons, but I will win almost half the games. It is too easy for me to get an anchor and play from there, and he has too much to do and not enough time or checkers to do it all. I have a totally trivial take, and his redouble is premature. In fact, he doesn't even have a money redouble! The match score makes his redouble a monstrous blunder, and he was appropriately punished. An early round match between Jeremy Bagai and Neil Kazaross resulted in a very exciting sequence which had a major effect on the tournament since Jeremy wound up winning the open. We start from here:
Neil played 4/2, 4/1. I agree. Obviously 8/6, 4/1 will be easier to clean up if the shot is missed. However, there is a big difference between 11 shot numbers and 24 shot numbers. This leaves:
What's going on? Obviously it is a trivial take -- Neil will easily win 25% of the time, and he has more gammon potential than Jeremy has. What about the double? If Jeremy hits it is a monstrous market loss. In fact, Jeremy is going to win several gammons after he hits due to Neil's completely stripped position. If Jeremy misses, he is likely to get at least a double shot -- Neil would have to roll perfectly to clear safely. In fact, some of Neil's rolls are really awful -- take a look at 6-5 for example. Of course, if Jeremy misses that double shot he will be very sorry he let go of the cube. My guess is that he is supposed to wait and risk losing his market, but I'm far from sure on this one. Jeremy did double, and Neil took of course. Jeremy rolled 3-2 and correctly in my opinion played 7/2. 20/15 is possible to hold the five-prime, but hanging onto the anchor on the 20 point could prove to be very important. If Jeremy had to play awkwardly on the offensive front then leaving the 20 point would be better, but since the roll plays reasonable comfortably, 7/2 looks right. Neil obviously doesn't have anything close to a double. He might not even be the favorite. He rolled, and rolled a perfect 4-2, playing 8/6, 8/4. Jeremy rolled another 3-2, and played 7/2. This left us here:
Now what's going on? This one is very tricky. For money, there would be no question -- double and pass. Even though Neil is likely to leave at least one shot, Jeremy still has to hit that shot. Neil is a clear favorite to win the game, and a big chunk of his wins will be gammons. At the match score it is another story. The gammon doesn't hurt Jeremy much with the cube on 8, and if he redoubles to 16 (which he may very well do after he gets or hits a shot) it doesn't hurt him at all. For simplicity, let's suppose that Jeremy will always redouble to 16 if he gets a shot, so he will never lose his market (probably not his proper strategy, but since a 10-2 lead is so large anyway this assumption won't affect things too much). Let's also assume that 70% of Neil's wins will be 16 pointers, either because he wins a gammon or because Jeremy redoubles and subsequently loses the game (I'm guessing, but once again since there isn't much equity difference between a 10-2 lead and winning the match all I need is any moderately close guess). This gives us the following possibilities:
Jeremy passes: Behind 6-2 (7 away, 11 away), 30% equity.
Therefore, Jeremy is risking 27% (30% - 3%) to gain 70% (100% - 30%), so he is getting 70 to 27 odds on his take -- which comes out to about 28%. If Jeremy can win this game 28% of the time, he has a take. Can Jeremy win 28% of the time? It sure looks like he can. Neil's position is immediately very dangerous. Any two large numbers leave a shot immediately. If Neil survives this roll he still has to clear the six point, and may leave a double shot attempting to do so. Neil won't have very many men off when he leaves his shot, so if Jeremy hits he will be a big favorite. I'm pretty confident that Jeremy has a take. What about Neil's double in the first place? How is he fixed for market losers? Oddly enough, except for the 2-2 perfecta his biggest market losers involve leaving a shot first -- something like 6-5 by him and a miss by Jeremy. If Neil rolls 6-1, say, he would be able to double next turn and Jeremy would still have a take or at worst a very close pass. Also, there is a fair amount of overage involved for Neil if he redoubles to 8. He can only use a small piece of the gammon. If Neil rolls boxes and Jeremy misses, for example, Neil can just play on for the gammon, and assuming he gets it he will wind up ahead 10-2 anyway. For these reasons I believe it is correct for Neil to hang onto the cube, even though the position is volatile and Jeremy doesn't have a huge take. It is odd that for money this would be an easy redouble/pass, but at an even match score with both players having 11 points to go it isn't even good enough to redouble. Match equity analysis can lead to interesting and unintuitive results. Neil is one of the best match equity theorists in the world. He properly worked all this out, and held onto the cube. He rolled 3-1, and played 4/0. And Jeremy rolled 6-2, playing 20/12. This leaves us with the following:
Now what's going on? It might seem odd, but Neil's position has improved compared to last turn despite the fact that he has lost his four point and has a blot. The key is that except for the nightmare 6-5, the rest of his rolls play safely this turn. Also Jeremy no longer own's Neil's five point, so Neil's 1-1 roll becomes a joker. Clearing the six point will still be a problem, but it now looks like Jeremy will get only one bite at the apple. Is it a take? Good question. The previously computed parameters are still the same -- Jeremy needs to win about 28% of the time to justify taking? I guess he still has that. Neil will have problems clearing the six point, and may have to leave a double-shot. However, this one may be a very close decision. Is it a double? Of course it is. Simply the fact that it might well be a pass (or that Jeremy might think it is a pass) is sufficient justification for turning the cube. Also, this time Neil has some big market-losers. His doubles clean house. Neil did double, and Jeremy did take. Excellent judgment by both of these great players in my opinion. Neil rolled 5-4, playing 6/1, 4/0. Jeremy rolled 5-2, playing 12/7, 4/2. Neil now rolled 5-1 in this position:
Let's examine the three candidates in terms of men off and shot numbers, since there are basically the factors which matter. There are other minor considerations such as gammon potential and ease in clearing next turn, but the big thing is to win the game. We can figure these minor factors in later.
6/1, 1/0: 3 men off, 20 shot numbers (all aces and threes).
If Neil wants to take a checker off, I think he should hit. Both plays leave 20 shot numbers. Both are pretty safe if Jeremy misses, -- Neil would have to throw 3-3, 1-1, or 3-1 to be in trouble. At least the hit may prevent Jeremy from keeping his back men split. More important, the hit improves Neil's gammon chances considerably. Granted winning a gammon isn't a big thing, but since the hitting play might be the better play to win the game anyway Neil might as well make the play with the better gammon chances. My guess is that Neil completely forgot about the relevance of the gammon at the table. How important is that extra checker? If this were a money game, it wouldn't matter at all. If Jeremy hits and Neil fails to roll a 1-6, the game will end with the cube regardless of which play Neil has made. Therefore, for money it would be clear to play 6/1, 6/5*. As is often the case, the match situation can make a difference. Let's suppose Jeremy doubles at some point. How will things look to Neil? This one is simple. If Neil passes, he is behind 10-2 (3 away, 11 away), with 10% equity. If he takes it is a 16-cube, so it is for the match. Since Neil needs only 10% winning chances, he will have a take after he is hit when he takes a checker off. Suppose he doesn't take a checker off. Then he will have only two men off. That will probably still be a take until Jeremy completes his prime -- then it I think it is a pass. With three checkers off Neil's winning chances are above 10% even if he is closed out, so he won't be facing the cube immediately. How can we analyze this? For simplicity, let's number Jeremy's rolls 1 thru 36 (without saying what they are). Let's also assume that the same rolls will hit with either play, but there will be one more roll which hits after Neil's play. So, we can say rolls 1-19 hit with either play, roll 20 hits with Neil's play but not after 6/5*, 6/1, rolls 21-36 miss with either play. If Jeremy rolls 21-36, there will be very little difference which play Neil has made. Therefore, Neil is slightly happy with his play 19 times, but he is very unhappy with his play 1 time. The question is: Is the very 19 times bigger than the slightly. I think that it is. Neil only gains a tiny amount by being permitted to play on when he is closed out. Jeremy won't be able to double him out immediately, but if Jeremy clears his six point and Neil doesn't enter immediately, Jeremy will then be able to claim with the cube. Thus, only about 1/3 of the time will Neil even have a chance to make use of his second life, and when that chance comes he will still be a substantial underdog. Therefore, I believe that the cost of being hit that 1 extra time more than compensates for the slight gain of the checker off the other 19 times. In our rough calculations, we had assumed that Jeremy would be redoubling to 16 as soon as he got a shot. Of course he can do better. As we have seen, even if he hits and Neil flunks, Jeremy won't lose his market. Therefore, redoubling on the come is definitely wrong. Was there justice? You will have to judge that for yourself, depending on whether you prefer my analysis or Neil's. Anyway, Jeremy rolled a 6-3 (which would have missed had Neil played 6/5*, 6/1). He played 22/19*, 19/13. Neil flunked. This leaves us with:
Should Jeremy double: As we have seen, Neil's take point is about 10%. What about Jeremy's doubling point. That is easy to calculate:
Jeremy doesn't double and wins: Ahead 10-2, 90% equity.
Therefore Jeremy is risking 10% to gain 10%, so if this were the last roll of the game he could double any time his is the favorite. This should come as no surprise -- in fact, that is always the case at an even match score as it is for money. So Jeremy has a very wide doubling window. It opens at 50%, and closes at 90%. Obviously he is in the window -- in fact, he is quite near the top of it. Should he double? Now it comes to a question of market losers. Being in the window merely means that doubling might be correct, and would be correct if it were the last roll of the game. However, this is not the last roll of the game. As we have discussed, Neil will have a take even if he is closed out. Therefore, Jeremy cannot possibly lose his market on the next exchange. Doubling is never right. Jeremy will always be able to double later, and Neil will still have a take. If things go badly, however, Jeremy will be glad that he didn't double. The game played out as might be expected. Jeremy closed Neil out, successfully cleared his six point, and Neil flunked. Now Jeremy doubled, and Neil passed. Jeremy lost his market, but when losing your market gives you a 10-2 lead you aren't crying about it. Jeremy went on to win the match, and in the double elimination tournament continued his winning ways and won the winners bracket. In the losers bracket, Steve Sax struggled through many rounds to meet Jeremy in the finals. Steve would have to win two straight matches to win the tournament. He breezed through the first match, winning almost every game. The second match was much tighter. I thought this was the most interesting and certainly the pivotal game of the match. Click here to see the game. |
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