return to indexFirst of all, let's review the Match Equity table for the five point match. It is as follows:
| 1 | 2 | 3 | 4 | 5 | |
| 1 | 50 | 70 | 75 | 83 | 85 |
| 2 | 30 | 50 | 60 | 68 | 75 |
| 3 | 25 | 40 | 50 | 59 | 66 |
| 4 | 17 | 32 | 41 | 50 | 58 |
| 5 | 15 | 25 | 34 | 42 | 50 |
The above figures give the winning chances for each match score, with the numbers on the top and to the left representing the number of points to go for each player. For example, suppose you are leading 2 to 0 in a five point match. You have three points to go, and your opponent has five points to go. Look to see where three and five intersect, and you will find the number 66. This means that the leader has 66% match winning chances, and the trailer has 34% match winning chances. When one player has one point to go, the equity is for the Crawford game at that score.
We will now look at each possible match score. If the leader has 1 point to go, we will examine both if it is the Crawford game and if the Crawford game has passed.
This is as simple as it gets. No cube, no gammons, just whoever wins the game wins the match. Double match point. Play can be quite different here than under normal circumstances. Since gammons won't help you, you should be less inclined to make blitzing oriented plays. These are plays which tend to win more gammons when they succeed, but leave you with anoverextended position which is more likely to lose when they fail. Similarly, you should be less concerned about being attacked than usual, since the cost isn't as great. The race becomes even more important than it is normally. Getting ahead in the race and bringing the position home is a very good approach for winning at double match point. However back games also have their value, since a well-timed back game will win more often than not. Since gammons don't matter, the player of the back game is the favorite. Howevera badly-timed back game is the kiss of death. Consequently timing and forcing your opponent to crunch are even more important than at a normal score.
Here are a couple of examples which illustrate the difference between play at double match point and at a normal score.
| 110 106 | White 4 5 point match Blue 4 |
Under normal circumstances, breaking the anchor would be too risky. Whitehas several builders ready to pounce, and Blue would be running too greata gammon risk. Blue is better off playing 7/1, 6/5 and hoping to get luckylater while minimizing the gammon risk. At double match point, it is anotherstory. Now getting gammoned is no longer a consideration. If Blue doesn'tescape now he will probably be forced to crunch his board shortly, afterwhich it will be difficult to win the game. His best winning chance is toget out into the outfield with 23/16. He may get blitzed, but a gammon isjust a loss. If White fails to carry out the attack, Blue has very realwinning chances by escaping the other back checker and controlling theoutfield.
| 153 151 | White 4 5 point match Blue 4 |
With gammons counting, it would be clear for Blue to blast away with 9/4(2)*,6/1(2)*. Blue has ammunition to carry out the blitz, and with two Whitecheckers on the bar against a three point board the blitz is a real possibity.At double match point, this isn't necessary. Blue should simply play 20/10(2)and coast home for the win. If he attacks and White survives, Blue may havedifficulty extracting the back checkers safely. Blue doesn't need a gammon,and 20/10(2) is the best play to win the game.
Now gammons come into play, since the trailer can use a gammon while theleader cannot. In fact, a gammon for the trailer (relative to simplywinning) is even more valuable than it is under normal circumstances.We can see this as follows:
Suppose you have a sure win available, but are considering making a riskyplay which increases your gammon chances but involves some danger oflosing the game. Early in the match (or for money), your odds are asfollows (assuming the cube is on 2):
If you win (no gammon) you are +2 points.
If you win a gammon, you are +4 points.
If you lose, you are -2 points.
So by making the dangerous play you are risking 4 points (from +2 to -2) inorder to gain 2 points (from +2 to +4), so you are giving 2 to 1 odds onyour play. Consequently, you need to have your play win twice as manyextra gammons as there are extra losses in order to justify the risk.
At the 1 away, 2 away score, it is another story. Let's suppose you arethe trailer, and again are considering the risky play (if you are theleader gammons have zero value, so obviously you would take no risksas far as winning the game goes in order to win a gammon). This timeinstead of points we will be looking at match equity:
If you win (no gammon), the score is 4-4 (1 away, 1 away), so your equityis 50%.
If you win a gammon you win the match, so your equity is 100%.
If you lost the game you lose the match, so your equity is 0%.
By making the dangerous play you are risking 50% equity (from 50% to 0%)in order to gain 50% equity (from 50% to 100%), so you are getting evenmoney odds on your play. If your play generates as many or more gammonsthan it generates losses, it is worth it. Quite a difference from normalcircumstances! An example:
| 114 80 | White 4* 5 point match Blue 3 |
Blue can virtually lock up the win by playing safe with 9/5, 9/3, but hethen has small chances of winning a gammon. The big play of 8/4*, 8/2increases his gammon chances considerably, but if he gets hit back he islikely to lose the game. Under normal circumstances it wouldn't be worththe risk. Blue will not win twice as many extra gammons as he will incurextra losses from the hitting play. At this match score it is anotherstory. If Blue survives after the hitting play his gammon chances arequite reasonable, and since he only needs to win more extra gammons thanhe loses extra games it appears worth the risk to play 8/4*, 8/2.
Early checker play should take the match score into account. The leadershould stive for gammon-free positions, while the trailer looks for positionswhich may lead to gammons. The leader likes races and holding games. Thetrailer likes blitzes, priming battles, and back games (for either side!).Thus, both players try to steer toward the type of game which suits themconsidering the match score. Most important for the leader is to establishan advanced anchor quickly. If he can do so the game is likely to turn intoa race or a holding game with relatively few gammons. The trailer wants toprevent the leader from doing this.
Opening play can be quite important. Consider how to play an opening 4-3.At a normal match score one could either make the building play of 13/10, 13/9,or one of the splitting plays 24/20, 13/10 or 24/21, 13/9. There is littledifference between these plays, and your choice should be what suits yourstyle and personal preference. Not so at the 2 away, 1 away match score!The leader is going after an advanced anchor and a race or holding game,so it is imperative that he make one of the splitting plays. Buildinga strong inner board is not as important to him as grabbing an advancedanchor. The trailer, on the other hand, is going after gammonish positions.He doesn't want to let the leader get that advanced anchor. He must play13/9, 13/10, steering for a priming battle.
Here is a simple example of an early play decision which is sharply affectedby the match score.
| 164 162 | White 4* 5 point match Blue 3 |
Blue can choose whether to make the offensive or the defensive five point.At an even match score it would be a close choice, with the nod probablygoing to 24/20, 23/20. However if Blue is behind 4-3, he clearly shouldplay 9/5, 8/5. That is the play which leads to more gammons for both sides,since it strenghens the offensive board but leaves the back checkers indangers. If Blue were ahead 4-3 it wouldn't be close. 24/20, 23/20 wouldbe very clear. Locking up the defensive five point cuts down considerablyon the chances of being gammoned.
It should be noted that this theme applies at all match scores where oneplayer is ahead. In general, gammons favor the trailer more than the leader.The reason is that the trailer can use the extra points from the gammonmore than the leader can, particularly after the cube is turned. Extrapoints from a gammon may be more than the leader needs to win the game, whilefor the trailer they may be enough to vault him into the lead.
At first glance, this looks very simple. Obviously the trailer will doubleat his first legal opportunity, since he has everything to gain and nothingto lose. After the leader takes the winner of the game will win the match,so in essence it is double match point.
It is true that once the cube has been turned and accepted both playersshould play as though they are playing at double match point, but the leaderis not required to accept the double. Let's see how it looks from hispoint of view:
If he takes, he is playing the game for the match.
If he drops the score is 4-4 (1 away, 1 away), and he has 50% equity.
It is clear that the leader should accept the double only when he is thefavorite. If he is the underdog, he should refuse the double. This is whatis called the free drop for the leader, since he can drop the double withoutgiving up anything.
Does this affect the play of the opening roll? If the leader wins the openingit won't matter. The trailer will double however the leader plays, and theleader will take unless he feels he is the underdog (which he might be onsome of the weaker opening rolls -- that is not clear). However if thetrailer wins the opening roll his play can make a significant difference.The reason is that the trailer can't double immediately. He has to waitfor the leader to roll and play, and the leader can now examine the positionand see who is the favorite.
For example, let's consider the opening roll of 2-1. Normally both thesplitting play of 24/23, 13/11 and the slotting play of 13/11, 6/5 arereasonable, and the choice of play is largely a matter of personalpreference. At this score, however, it isn't close. It is mandatory toavoid the slotting play.
Suppose the trailer plays 13/11, 6/5. If the leader hits the blot (or rollsa good doubles such as 1-1, 3-3, or 6-6) he is the clear favorite and willhappily accept the double. Otherwise he is the underdog, and will declinethe double. Consequently slotting the five point won't work, since thetrailer will not have the opportunity to cover the blot -- we will beplaying a new game. Therefore he should play 24/23, 13/11. Now most ofthe leader's rolls do not lead to a decisive advantage for one player,which is what the trailer would prefer.
It may seem obvious that the trailer should always double at his firstopportunity, but that is not clear. Suppose we have the best possiblestart for the trailer -- he rolls 3-1 and the leader rolls 6-2 (playing24/18, 13/11 let us say). Obviously if the trailer doubles the leadershould drop. But perhaps the trailer should play on for a gammon! It isquite unlikely that the leader will roll well enough on the next exchangeto take the advantage. If in the future the leader threatens to gain theadvantage the trailer can always double and move on to the next game, butif things continue to go the trailer's way he can continue to play on forthe gammon. It should be noted that you should not do this unless you aresure that your opponent understands the free drop concept. If there issome chance that he will just take the initial double, it would be foolishnot to double.
This is considerably different from the 4-3 score. Now gammons have verylittle value for the trailer. If the trailer wins this game he will bepermitted to double immediately next game, therefore the next game will beplayed for the match regardless of whether or not the trailer wins a gammonnow. The only difference is that the leader will not have a free drop nextgame if the trailer wins a gammon. The free drop gives the leader a verysmall advantage, but not enough to take into consideration. The trailershould play exactly as though it were double match point. He should notrisk losing the game in order to increase his gammon chances. Similarlythe leader should always make the best play to win the game, withoutworrying about gammon danger.
Backgammons, however, are another story. If the trailer wins a backgammon,he wins the match. This is a big improvement over the 50% (orslightly less if the leader has a free drop) equity he would have if hewins a single game or a gammon. Consequently, it may be worthwhile forthe trailer to take considerable risks in order to win a backgammon. Similarly,the leader should make an extra effort to avoid getting backgammoned. Whilethe leader should not be afraid to play a back game since gammons cost himvirtually nothing, he must beware of getting backgammoned -- perhaps runninghis back checkers slightly sooner than he normally would. Here is oneexample illustrating the extremes one might to go in order to scorea backgammon at this score:
| 177 15 | White 4* 5 point match Blue 2 |
At a normal match score it would be folly not to play safe. The gains fromthe extra backgammons would not nearly compensate for the losses from notwinning a gammon or possibly losing the game when the shot is hit if Bluetakes two checkers off. At this match score, things are different. Firstof all the difference between winning a gammon and winning a single gameis negligible. Secondly, there is a very high premium on winning a backgammon.Taking the extra checker off will result in Blue bearing off one roll soonerif the shot is missed, and that extra roll will often be the differencebetween winning a backgammon and not winning a backgammon. Also Blue willhave enough checkers off so even if the shot is hit he will still win more oftenthan not. Putting it all together Blue will win more extra backgammons fromtaking two checkers off than he will lose games, so 3/0, 2/0 is thecorrect play.
It is important to see that the leader does not have a free drop availablewhen the trailer has an odd number of points to go. The reason is that thetrailer will be turning the cube immediately for the rest of the match,so it is the same if the trailer has X points to go as if he has X-1 pointsto go, provided X is an even number. Thus, it is correct for the leaderto always take the double even if the trailer jumps out to an advantage.The theoretical action should always be double-take.
There is a way to potentially gain an advantage against an unknowledgableor unsuspecting opponent if you are the trailer. Instead of doublingimmediately, intentionally forget to double. Instead, wait until you havea very strong (but not gammonish) position. Now turn the cube. As theprevious analysis shows your opponent should take, of course, almost nomatter how bad his position is. If your opponent makes an error and drops,you have essentially won a free point. This may seem like a silly ploy,but I have used it successfully many times, often against quite competentopposition. Note that if the position starts to get gammonish it isnow necessary to double. If your game becomes so strong that it is morelikely you will win a gammon than lose the game, it now becomes correct foryour opponent to drop. Until there is a danger of that happening it is safeto wait, giving your opponent an opportunity to make an error when youfinally double.
After the cube has been turned and taken, the situation is exactly equivalentto the 4-3 (Crawford) game. If the trailer wins a single game the match is tied 4-4; ifthe leader wins he wins the match, and if the trailer wins a gammon he winsthe match. Thus the playing strategy is the same, with gammons being ofextra value for the trailer.
This appears to be similar to the 4-3 Crawford score. Gammons are of extra valueto the trailer, since winning a gammon would make the score 4-3 with theCrawford game out of the way. Since the trailer can now double, he isvirtually even money (except for the slight edge from the free drop). Itturns out that there is quite a difference. The reason is that if the trailer wins a single game(making the score 4-2), he doesn't necessarily have to win two more games towin the match. He might win a gammon. To get a ballpark figure for whatthe trailer's chances are behind 4-2 post-Crawford, let's assume thatthe trailer will then win the game 50% of the time, and that 24% of hiswins will be gammons (which comes to 12% of the total games). This gives us:
Leader wins 50% (100% match equity for leader)
Trailer wins single game 38% (50% match equity for leader)
Trailer wins gammon 12% (0% match equity for leader)
Multiplying this out, leader's equity is 50 X 100 plus 38 X 50 = 69%.Now we can calculate the odds the trailer is getting when he makes a riskyplay to win a gammon at the 4-1 score:
Leader wins: trailer's equity is 0%
Trailer wins single game: trailer's equity is 31%
Trailer wins gammon: trailer's equity is 49% (giving the leader 1% for thefree drop)
This means that if the trailer is considering making a big play which riskswinning the game in order to increase his gammon chances, he is risking31% match equity in order to gain 18%. These odds are far from the evenmoney odds the trailer gets at the 4-3 match score; in fact, they arefairly close to the normal 2 to 1 money odds. Thus, while gammons are ofimportance to the trailer, they are not much more important than they wouldbe for money.
Another key consideration at this match score is the unimportance ofbackgammons. As we have seen, there is little difference between 4-4 and4-3 post-Crawford. Consequently the leader can ignore backgammon risks ifthere is some chance of saving the gammon or even winning the game. As anextreme example:
| 4 83 | White 1 5 point match Blue 4* |
Under normal circumstances it would be foolish to do anything but play 24/18.The danger of losing a backgammon far outweighs the gains from staying for apossible last ditch shot. At this match score, with backgammons virtuallymeaningless, it is correct for Blue to play 11/5. If White does roll anace and Blue hits the shot Blue definitely gets off the gammon, and on a verygood day he could even win the game. The gains from this are greater thanthe miniscule cost of losing the backgammon vs. losing the gammon, sinceall saving the backgammon would do for Blue is retain his free drop next game.Even though this play will get Blue backgammoned far more often than it willsave him the gammon or win the game, it is the correct play at this match score.
Since the trailer has an even number of points to go, the leader has a freedrop available. This will always be the case at any post-Crawford gamewhere the trailer has an even number of points to go. Winning one point doesn'treally put the trailer any closer to winning, since the cube will be ontwo every game. Thus, the cube strategy is about the same as if thescore were 4-3 (1 away, 2 away) post-Crawford.
The only difference is that if the leader accepts the double and loses thegame without getting gammoned, he retains his free drop for the next game.This means that if when the leader is doubled he is a very slightunderdog it might be correct for him to take the double. This should notbe overdone. In general you want to use your free drop any time you arean underdog.
Once the cube has been turned and accepted, playing strategy is about thesame as if the score were 4-3 (1 away, 2 away) Crawford. The reason is thatif the trailer wins a single game he is now behind 4-3 and can turn the cubeimmediately next game, so for all practical intents and purposes it will bedouble match point. Of course if the trailer wins a gammon, he wins thematch. Once again gammons are super-valuable for the trailer, and bothplayers should adjust their play accordingly.
Strategy for this score is pretty much the same as 4-2 (1 away, 3 away)Crawford. Gammons aren't particularly important to the trailer, as thereis little difference between a 4-1 and a 4-2 score post-Crawford. Backgammons,however, are very valuable. Thus the play for both sides is basically likedouble match point, except if a backgammon is a possibility.
Just like the 4-2 score, the leader does not have a free drop available. Heshould take any double, regardless of the position. In theory the trailershould double immediately, but he may choose to wait and double later, hopingto induce the leader to make an error and pass.
If the trailer wins a gammon the match is tied at double match point, andif the trailer wins a single game he will be behind 4-2. Therefore once thecube has been turned and accepted the situation is almost equivalent to the4-1 (1 away, 4 away) Crawford game. Gammons are valuable for the trailer,so the trailer should strive for gammonish positions while the leader triesto avoid them. The main difference is that here backgammons are of importanceto the trailer (they win the match for him), so if a backgammon becomes apossibility it has to be taken into account.
There has been much written about cube strategy at this score. The truthis that it is very simple. You should double if it is at all possible thatafter the next exchange (you roll, he rolls), your opponent will have apass. Even if you double when you can't lose your market, that is not anerror. The only error you can make is to lose your market.
Why is this so? Normally, there are two possible not to double(not counting playing for the gammon) when you have the advantage.
1) You believe that even if things improve you will be able to doublelater (and your opponent will still have a take) if you wait. If youdouble now, only your opponent will have the opportunity to double inthe future.
2) If things go badly, your loss will be less if you don't double. Inaddition if you own the cube initially you prevent your opponent fromdoubling.
At the 2 away, 2 away score, these considerations do not apply. It will notbe any advantage for your opponent to own the cube once you double, sincethe cube will then be dead. Furthermore you don't gain by waiting, sinceyou can't prevent your opponent from doubling any time he wants withoutcost.
It can be shown to be correct to double any time you have a potentialmarket losing sequence as follows: Suppose you and I are playing a two-pointmatch, and I declare to you that I will follow the strategy I suggest(namely doubling any time I have a potential market losing sequence). Therefore,I will never lose my market. If you take a more conservative approach, it willbe possible for you to lose your market. Therefore, the only games whichwon't be played to completion will be those games where you waited too longand lost your market. Since you would have done better (on average) if thesegames had been played to completion, you have lost equity. Consequently ifyour opponent is playing correctly, it is vital for you to double if it isat all possible to lose your market.
While the above is theoretically correct, it might not be the best practicalstrategy. If you believe that your opponent may not understand the conceptsinvolved in a two-point match, then it may be worth taking a slight risk oflosing your market. Your hope is that your opponent will potentially makea bigger mistake, fail to double when he should, and lose his market. Howeverthis strategy can only gain if your opponent errs. Playing against anopponent who understands these concepts, you can only lose by failing todouble when you should.
If the above discussion is not clear to you, or if you are playing an expertwhom you believe understands the concepts better than you, there is a verysimple solution. Double at your first legal opportunity, regardless ofwhat happens! This will automatically make it a one-point match, and thereis nothing your more knowledgable opponent can do about it. This may seemsilly, doubling when your opponent wins the opening 3-1 roll, but it can'tcost anything. The point is that unless you roll a very strong response yourknowledgable opponent will double on his next turn, since it may be possiblefor him to lose his market. Since you can't prevent this (and it will happen),you don't lose anything by doubling yourself.
Once the cube has been turned and accepted, as it always should be unlesssomeone makes an error, you are simply playing double match point. Sinceit is expected that at some point a double will be taken, play strategyfrom the beginning of the game should be as though this is the doublematch point game.
Now we start to examine some of the more interesting match scores. Firstof all, let's look at what is going on when the trailer has the advantageand is potentially turning the cube.
How do things look from the leader's point of view? Cube ownership is ofno value to him, since he only needs two points to win the match. Thereforeif he takes, the game must be played to a conclusion. For the moment let'sassume that there is zero gammon potential in the position. Then theleader's possibilities are:
1) If he passes the score is 3-3, for 50% equity.
2) If he takes and wins he wins the match, for 100% equity.
3) If he takes and loses he is behind 4-3 Crawford, for 30% equity.
This means that the leader risks 20% equity (the difference between 50% and30%) in order to gain 50% equity (the difference between 100% and 50%).Not nearly as good as the 3 to 1 odds one gets for money when consideringwhether or not to take a double. In addition he doesn't have any use ofthe recube, which makes things still worse than for money. He will have towin the game played to a conclusion close to 30% of the time in order tojustify taking the double.
The above analysis assumed gammons weren't in the picture. Let's examinea typical position where there is a moderate gammon threat.
| 172 155 | White 3 5 point match Blue 2 |
Blue has a clear advantage, but White has a solid position with plenty ofplay. For money it appears that if Blue were to double White would havea trivial take, and that a double by Blue would be quite premature. A Snowierollout confirmed this:
Blue wins gammon: 18.8%Blue wins single game: 44.0%White wins single game: 29.3%White wins gammon: 7.9%
The above comes out to a cubeless equity of .373 for Blue. As expected itis a trivial take, and the position is clearly not volatile enough to justifya double.
At the match score, it is another story. Look at White's equity if he takes:
Blue wins gammon: 18.8% X 0 (Blue wins match)
Blue wins single: 44.0% X 30 (Blue leads 4-3)
White wins: 37.2% X 100 (White wins match)
This totals to 50.4%. If White passes the double the score is tied, whichwould give White 50% equity. So if these rollout results are accurate,White has a bare take at the match score. Strengthen Blue's position justa bit more and White should pass. Clearly Blue has a monster double. Quitea difference from the evaluation for money, where White has a trivial takeand Blue has a ways to go before he has a double.
The above analysis indicates that the leader is much quicker to pass thanhe would be for money, and consequently the trailer is much quicker to double.This is not surprising; that is usually the case in a match when one sidehas the lead. It also illustrates the importance of gammons. A gammonthreat gives the trailer more cube leverage. As usual the leader shouldavoid gammonish positions, while the trailer strives for them.
The trailer is not the only one who has access to the cube. The leader mayalso turn the cube if the time is right. Obviously if he does so and thedouble is taken the trailer will immediately redouble, so the game will beplayed for the match. Let's see how things look from the trailer's pointof view if he is doubled:
Trailer passes: Behind 4-2, 25% equity.
Trailer takes and wins: Wins match, 100% equity.
Trailer takes and loses: Loses match, 0% equity.
Thus, the trailer is risking 25% to gain 75%. Exactly 3 to 1 odds; the sameas money decisions. However, there are two factors which make the situationquite different from money. First, we are just talking about winning thegame. Gammons will not matter. Second, the trailer doesn't get the recubevig he would have for money. He will recube immediately, of course, but thenthe game has to be played to conclusion. For money you can take positionswith less than 25% winning chances, because the potential recube means thatyou will win some games you would have lost had the game been played toconclusion. Not so at this score -- the game has to be played to conclusion.
The above analysis indicates that if the position doesn't have much gammonpotential the trailer's pass/take decisions are pretty much the same asthey would be for money -- in fact, he will be a bit quicker to pass.It follows that the leader's doubles are about the same as they would befor money. He will be a bit cautious on the marginal doubles since hedoesn't want to give the trailer an easy road to win the match. The soliddoubles he definitely should make, and if the trailer chooses to accept thenthey just play the game for the match. For example:
| 127 107 | White 3 5 point match Blue 2 |
If Blue doubles for money, it looks like White has a close decision. Hiscube access means that if he hits a shot he will probably be able to claimwith the recube, and that may be sufficient to get his winning chances over25%. At this match score White will have to prove himself after hittingthe shot, and those few games which slip away after the shot is hit areprobably sufficient to drop his win percentage below 25% which makes ita pass at the match score. At any rate, Blue clearly has a boomingdouble.
If a lot of the leader's advantage is a gammon threat, it is another story.The leader can win the match without turning the cube if he wins a gammon,so he shouldn't give his opponent the choice of either passing and livingto see another day or taking and redoubling which would make the gammonmeaningless. If the leader has some decent gammon chances, it isvirtually never correct to turn the cube at this match score. This isanother reason why the leader should strive to avoid gammonish positions.For example:
| 164 147 | White 2 5 point match Blue 3 |
This position arises after White rolls 5-2 and splits, Blue rolls 5-5, andWhite flunks. For money this is a huge double and a pretty clear pass dueto the big gammon threat. At the match score, it would be a serious errorfor Blue to double. Most of his edge is the gammon threat, and by doublinghe loses that advantage. White will easily win 25% of the time, so he shouldquickly take and send it back for the match. However even if White mistakenlypasses, he is probably better off than he would have been had Blue simplyplayed on.
This score more than any other score emphasizes the importance of doubledgammons for the trailer, since he needs exactly four points to go out. Onthe other hand, the equity difference between one point and two points isn'tso great for the trailer. This means that if the trailer doubles there arevery different standards for the leader depending on gammon potential.
First, let's suppose there is no gammon possibility. From the leader's pointof view, we have:
If he passes, he is ahead 3-2 with 60% equity.
If he takes and wins he wins the match, for 100% equity.
If he takes and loses the score is 3-3, for 50% equity.
The leader is getting 4 to 1 odds on his take, better than the 3 to 1 hewould be getting for money. True he won't have any recube potential, butif he can win the game more than one time in five that is sufficient to take.Consequently in straight races the trailer should actually be moreconservative with his doubles than normal.
When gammons come into play, it is another story. A moderate gammon threatcan turn what would normally be an easy take into a pass for the leader.For example, consider the following position:
| 139 143 | White 3 5 point match Blue 1 |
Leader wins 40%
Trailer wins single game 40%
Trailer wins gammon 20%
This would be a huge take for money. At the 3-1 match score, however, if theleader takes the double we have:
Leader wins: Leader wins match for 40% X 100
Trailer wins single game: Score is 3-3 for 40% X 50
Trailer wins gammon: Trailer wins match for 20% X 0
This totals to 60% equity for the leader if he takes the double. If he passeshe will be ahead 3-2, also 60% equity. So what would be a very easy moneytake becomes a borderline pass/take at the match score.
What about when the leader has the advantage? If he doubles and the trailerpasses, the score will be 4-1 Crawford, with 17% equity for the trailer.If the trailer takes, he will immediately redouble for the match. Thusthe trailer can take with better than 17% winning chances. Provided thereare no gammon threats, the leader can double with close to 83% winningchances. Even if the trailer has a close take, the leader should be willing toplay for the match rather than risk losing his market by a lot. If thereare any real gammon chances, however, the leader should never double at thisscore. Doubling simply makes his gammon threats worthless.
The above analyses show how important it is for the leader to avoid gammonishpositions, while the trailer strives for them. This may at first not appearto be the case, since the leader needs exactly two points to win the match,so he would like winning an undoubled gammon. If the cube stays at one, wehave:
Leader wins single game: 4-1, for 83% equity.
Leader wins gammon: Leader wins match, for 100% equity.
Trailer wins single game: 3-2, for 60% equity.
Trailer wins gammon: 3-3, for 50% equity.
According to the above analysis it would appear that the leader gains 17%from winning a gammon while losing 10% from getting gammoned, so gammonsfavor him. The catch is that the cube isn't staying at one. As soon asthere are gammons in the air and the trailer has any potential marketlosing sequences he will be quick to turn the cube. With the cube at two,the leader gains nothing from winning a gammon, while he loses 50%. Forexample:
| 138 138 | White 3 5 point match Blue 1 |
This score has many similarities to the 3-1 score. Gammons still favorthe trailer, of course, and the leader wants to avoid them while thetrailer encourages them. However the effect of gammons isn't as sharp asit is at the 3-1. The reason is that winning a gammon doesn't put thetrailer out, while at 3-1 it puts him out exactly. To see this, let'sexamine the hypothetical position we looked at before, where the leaderwon 40%, the trailer won a single game 40%, and the trailer won a gammon 20%.At the 3-0 score, the possible results are:
Leader wins: Leader wins the match, for 40% X 100
Trailer wins single game: Score of 3-2, for 40% X 60
Trailer wins gammon: Score of 3-4, for 20% X 30
Adding these up, we see that the leader will have equity of 70% if hetakes the double. If he passes the score will be 3-1, for 68% equity.Thus what was a borderline decision at the 3-1 score becomes a very cleartake at 3-0.
If there are no gammon possibilities, the leader can once again be quiteloose with his takes. The reason, as before, is that two points puts theleader out exactly. Here are the numbers:
Pass the double: Ahead 3-1, 68% equity.
Take and win: Win the match, 100% equity.
Take and lose: Ahead 3-2, 60% equity.
The leader is risking 8% to gain 32% on his take, so he is getting 4 to 1odds. Once again in a gammon-free position the leader can take moreliberally than for money, and the trailer must be more cautious.
If the leader has the advantage, naturally he should not be doubling ifgammons are possible. With gammons not in the question, the cube actionis fairly normal. Let's look at the numbers from the trailer's point of view:
Trailer passes: Behind 4-0 (Crawford) for 15% equity.
Trailer takes and wins: Ahead 4-3 (Crawford) for 70% equity.
Trailer takes and loses: Loses the match, for 0% equity.
The trailer is risking 15% to gain 55%. He is getting between 3 and 4 to1 odds, and needs to win about 21.5% of the time to justify a take. This isquite different from the 3-1 score, where 17% is all the trailer needs.Since the trailer won't have use of the recube (he redoubles immediately, ofcourse, but then the cube is dead), we are talking about playing the game toconclusion. Thus the situation is not too different from a money double.There the trailer needs to win 25% of the time, but that takes his recubeleverage into account, so in practice he usually needs to win around 22% ofthe time.
Last roll positions, where the roll will decide the game, can lead to trickydecisions. Now the trailer can't count on getting the cube to 4, since theopportunity won't arise. This drastically changes the odds. For example,consider the following simple position:
| 2 6 | White 0 5 point match Blue 3 |
If Blue doubles, should White take? Blue will fail to get off 13 rollsout of 36, which is about 36% of the time. Blue is ahead in the matchand 36% is way higher than 25%, so obviously White has an easy take, right?Wrong!
White takes and wins: Behind 3-2, for 36% X 40 (remember, he doesn't get the recube)
WhIte takes and loses: Loses the match.
So if White takes his equity is only 14.4%. If he passes he is behind4-1 Crawford, for 15% equity. Passing is the percentage action.
To complete the analysis of this score, let's examine the same position butnow suppose that White is ahead 3-0. What should the cube action be? Wehave already seen that the leader can be quite liberal with his takes inraces, so obviously if Blue doubles White has an easy take. Should Blue double?Let's look at the numbers:
Blue doesn't double and wins: Behind 3-1, 32% equity.
Blue doubles and wins: Behind 3-2, 40% equity.
Blue doesn't double and loses: Behind 4-1 (Crawford), 15% equity.
Blue doubles and loses: Lose the match, 0% equity.
Thus Blue gains only 8% by doubling if he wins, while he costs himself15% if he doubles and loses. He needs to win over 65% of the time tojustify doubling. As we have seen he only wins 64% of the time, thereforehe should not double. It is remarkable that at this score it is doubleand pass if the leader doubles from the side which has checkers on the fourand two points, but it isn't even a double if the trailer has that position.This is completely contrary to our intuition which say that the leadershould be more conservative and the trailer more aggressive with thecube.
Play is close to normal backgammon at this score, but you should tend todrop quicker than usual. A look at the numbers illustrates why. Let'sassume gammons aren't an issue, and you are being doubled.
If you pass: You are behind 3-2, 60% equity.
If you take and win: You are ahead 4-2 (Crawford), 75% equity.
If you take and lose: You are behind 4-2, 25% equity.
As can be seen the taker is risking 15% to gain 35%. This is worse than thenormal money 3 to 1 odds -- you need 30% winning chances to justify the take.If there were anything special about the recube value it wouldn't be sobad, but the recube is normal. If your opponent passes, he will be behind4-2 (Crawford) for 25% equity. If he takes, it is for the match. Thereforeyour opponent will have the normal 3 to 1 odds on his take; his onlydisadvantage will be that the cube will now be dead so he will have to playthe game to conclusion.
Since the drops come quicker at this score, the doubles come quicker alsosince the Doubler doesn't want to lose his market. Otherwise, play isfairly normal.
This can be one of the trickiest scores to deal with, and even veteran playerscan become confused. The difficulties come when trying to figure in thepotential value of the recube. So far we haven't had to worry much aboutthis. When the trailer doubles and the leader is 2 away, he simply doesn'tredouble. When the leader is 2 away and doubles, if the trailer takes hehas an automatic redouble. At the 2-1 score there is no automatic actionas far as the redouble goes, and that can confuse issues.
Let's suppose the leader doubles. Forgetting about gammons, what percentwinning chances does the trailer need to take? If we don't take the recubeinto account, we get the following:
Trailer passes: He is behind 3-1, 32% equity.
Trailer takes and wins: He is ahead 3-2, 60% equity.
Trailer takes and loses: He is behind 4-1, 17% equity.
The trailer is risking 15% in order to gain 28%. Thus, he has to winalomst 35% of the time in order to justify taking. Can that possibly beright? Our instincts tell us that if anything the trailer should be moreliberal with his takes than for money, but apparently this is not thecase.
The error we have made is that we haven't taken the potential recube intoaccount. Certainly the trailer will be quick to redouble. How quick wecan discuss later, but he will be redoubling a lot. How can we figurethis redoubling potential into the picture?
As a start, let's suppose the trailer's strategy is to redouble immediately,regardless of the position. We know he can do better than that, but thisassumption will give us some idea about what is going on. Now we can throwthe previous calculations out the window. If the trailer redoubles immediately,that is for the match. Since the trailer would be behind 3-1 if he passes,this means that if he can win the game 32% of the time he will do betterby taking and redoubling immediately. This 32% is considerably less thanthe 35% we calculated by assuming that the trailer never redoubles. Thatillustrates how important the potential recube is, and why it is vital tofigure that recube into the equation.
The trailer doesn't have to redouble immediately and risk losing the wholematch. He can wait until things start going his way and then redouble. Hewill be very aggressive, of course, but if things never go his way he won'thave to redouble. Let's assume that the trailer never risks losing hismarket (which means he will always have redoubled when he wins), but thaton half of his losses he never saw daylight so he didn't redouble. Bythis assumption he will always win the match when he wins the game, but onhalf of his losses he only loses two points (which puts him behind 4-1 for17% equity). Then his average equity when he loses will be about 9% (midwaybetween 17% and 0%). We can now figure the percentage of games he needsto win in a more sensible manner:
Trailer passes: Behind 3-1, 32% equity.
Trailer takes and wins: Wins match (since by our assumption he always redoubles in time).
Trailer takes and loses: Average equity of 9%.
Now he gains 68% when he takes and wins, while losing an average of 23% whenhe takes an loses. This is close the the 3 to 1 odds which we intuitivelyknow it should be, rather than the 35% we initially calculated when weignored the redouble.
Speaking of redoubles, when should the trailer be redoubling once he hastaken the double. Let's look at it from the leader's point of view andconsider his pass/take decision:
If he passes, he is behind 3-2, 40% equity.
If he takes and wins, he wins the match.
If he takes and loses, he loses the match.
So, the leader needs 40% winning chances just to take the redouble. Itfollows that the trailer should redouble any time his winning chances getto even money, or perhaps a bit worse if the position is volatile. Notethe from the trailer's point of view he stands to gain 40% from asuccessful redouble (the difference between being ahead 3-2 and winning thematch), while he risks only 17% (the difference between being behind 4-1and losing the match), so he has a lot of margin for error. In fact, ifit were the last roll of the game he should redouble as a 2 to 1 underdog!The only reason he doesn't do so earlier is that he doesn't have marketlosers, so there is no need to redouble.
Now let's look at things when the trailer makes the initial double. Assumingno gammon threat and forgetting about redouble potential, we have:
Leader passes: 2-2, for 50% equity.
Leader takes and wins: Ahead 4-1 (Crawford) for 83% equity.
Leader takes and loses: Behind 3-2, 40% equity.
He is risking 10% to gain 33%, slightly better than 3 to 1 odds. This issomewhat compensated for by limited recube use. Since the trailer willbe able to take a redouble with 17% winning chances (because his alternativewould be to play from behind 4-1 Crawford), the leader will only be redoublingwhen he almost has a claim.
It should not be surprising that gammons change the picture considerably,since the trailer needs exactly four points to go out. Let's look at ourprototype from before (leader wins 40%, trailer wins single game 40%,trailer wins gammon 20%). For now we will ignore the leader's recube, andalso ignore any gammon potential the leader may have.
Leader wins: Ahead 4-1 (Crawford), for 40% X 83.
Leader loses single game: Behind 3-2, for 40% X 40
Leader loses gammon: Loses the match, for 0%.
This totals to 49.2%, which is less than the 50% the leader would have ifhe passed the double. Of course the leader does have some recube value evenif small, and if the game is complex he may have some gammon value also.These extra pieces are probably enough to put him over the 50% mark andgive him a take. But it is close, once again illustrating the huge valueof gammons to the trailer when he has four points to go.
Checker play is what you would expect. The leader wishes to avoid positionswith gammon threats, while gammons suit the trailer fine. Also when thereare gammons floating around the leader will be slow to double, while thetrailer can double very quickly. For positions with no gammon danger,cube action is approximately the same as it would be for money.
Many of the same considerations for the 2-1 score apply here. The leaderdoesn't need the full four points, while the trailer can use them. Thismeans that the leader should be more cautious about actions which potentiallyinvolve doubled gammons or the cube reaching the four-level. Of course thingsaren't quite as serious as at the 2-1 score, since winning four pointsdoesn't put the trailer out. Therefore the leader can be a bit more liberalwith his initial takes or his takes of redoubles.
There is, however, one further consideration which we haven't seen before.At this score, the trailer can use more than four points. Thus if theleader makes a loose double in a dangerous positions where he could begammoned if things go badly, the trailer has the potential to take thecube, redouble to 4 at an appropriate moment, and win a gammon to win thematch. The same redouble hangs over the leader's head at the 2-1 score,but not the threat of a redoubled gammon. This means that the leader shouldbe more cautious than usual making an initial double when either side mightget gammoned. Often it is just best to roll on. If things go well he cantry for an undoubled gammon, while if things go badly he will be glad hedidn't double. The becomes even more true when the leader has three pointsto go and the trailer is farther behind. The most extreme case occursat the score 3 away, 8 away. Here the leader should virtually neverdouble if there is some chance he could get gammoned. It is too easyfor the trailer to take, whip it back to 4 on any excuse, and win a gammonto suddenly win the match.
| 129 133 | White 2 5 point match Blue 0 |
Blue has a strong position and will win a lot of gammons, but White has plentyof potential counterplay if he survives Blue's first punch. In addition itis not out of the question for White to win a gammon if things go his way.White certainly figures to win at least 30% of the games, which gives himan easy take. At the 2-1 score I would double as Blue. There is just toomuch potential not to do so. At 2-0, however, I believe it is correct towait a roll. If Blue rolls well he can just play on for the gammon mostof the time, while if things go badly he will be glad he didn't double. Itis very dangerous to let White get his hands on a 2-cube in this sort ofposition when Blue has a good lead in the match.
Finally, both players can use all four points. It might seem as though cubeaction is the same as it would be for money, but there are some subtledifferences. First off all, let's suppose gammons aren't an issue andsomebody doubles.
Pass: Behind 2-1, 41% equity.
Take and win: Ahead 3-1, 68% equity.
Take and lost: Behind 3-1, 32% equity.
The taker is risking 9% to gain 27%; exactly 3 to 1 odds. So far, just likefor money. However when we take the recube into consideration we see anotherpicture. If the redouble is passed, the player is behind 3-1 with 32% equity.If he takes it is for the match. Therefore, the initial doubler must win32% of the time cubeless in order to justify a take of a potential redouble.This means that the original taker has far more recube leverage than normal,so he can take more liberally than for money. It is interesting that itis correct to pass more quickly than for money at the 2-2 score, but take more quickly atthe 1-1 score.
It should come as no surprise by now that gammons change the picture. Withthe doubler needing exactly four points, a gammonish cube should be treatedwith extra caution. Since passing the double means being behind only 2-1for 41% equity, it isn't a good idea to risk the whole match on a seriousgammon threat even with the powerful recube vig.
| 160 145 | White 1 5 point match Blue 1 |
For money White has a pretty clear take, despite the gammon threat. Whiteshould win at least 35% of the time, and if he survives the immediate blitz hewill probably be the favorite. At the 1-1 match score, it is probably correctto let it go. Blue simply wins too many gammons and they end the match. IfWhite could count on winning four points every game he wins while losing onlytwo points when he avoids getting gammoned, he could justify taking, but he can't do that.
The fact that both sides need exactly four points to go can generate somepretty interesting cube positions, particularly in complex games. Forexample, look at the following back game.
| 183 71 | White 1 5 point match Blue 1 |
White's timing is good enough that he figures to be the favorite to win thegame. Blue has the advantage, however, because of the gammon potential -- ahigh percentage of Blue's wins will be gammons. If Blue made the mistakeof doubling, White would happily take, and even if White redoubled immediately White would bethe favorite in the match. By turning the cube, Blue would be giving Whitethe opportunity to redouble and negate Blue's gammon potential. White,however, can't afford to double even when he gets his shot. Blue cancomfortably take, since Blue will still win at least 25% of the time. Andsince some of Blue's wins will be gammons, this would put Blue out exactly.Obviously Blue would never redouble unless the game becomes a race where hehas a sufficient advantage. Even if White hits a shot, he may have to waita while before doubling. Thus, due to the match score, it is wrong for bothplayers to double for quite a while even if things go their way.
Once again both players can use the full four points, so initial cube andchecker play is pretty much normal. Since the trailer can use more thanfour points, the leader should be a bit wary about turning the cube in2-way gammon positions, but otherwise play pretty much like money. It isthe redoubles which are more interesting.
Let's suppose the leader is redoubling. If the trailer takes, he will ofcourse send it back to 8 for the match. If he passes, he will be behind3-0 with 25% equity, so he has to be able to win 25% cubeless to justifytaking. Since the leader goes out with four points he should refrain fromredoubling if he has reasonable gammon threats; otherwise his redoublingstrategy is normal.
What about the trailer's redoubles. First let's assume no gammons. Fromthe leader's point of view:
If he passes, he is behind 2-1 with 41% equity.
If he takes and wins, he wins the match.
If he takes and loses, he is behind 4-1 with 17% equity.
Thus, he is risking 24% in order to gain 59%, which means he must win almost29% if the time without use of the recube. If there is gammon danger hisodds are worse, for if he loses a gammon his equity is 0% instead of 17%.Since the leader needs to win a higher percentage than normal, it followsthat the trailer will be quicker to redouble than normal. This is a themewhich is consistent for longer matches. Initial cube action is pretty muchlike money play unless one player is close to going out, but when it comesto higher cubes the trailer should be more aggressive and the leadermore conservative.
Finally, the five-point match. Armed with the knowledge of proper cubestrategy and checker play for all possible future scores, you are nowprepared to play the match to the best of your ability. Initial actionsare pretty normal. There may be subsequent actions involving redoubleswhich are out of the ordinary, but your knowledge of the possible scoresand how to tackle the problems should see you through any difficulties.Good luck, and I hope to meet you in the finals of the next blitz or lastchance tournament.