End Game Doubling Strategy
by Joe Sullivan
16 January 2000
Some of you may have noticed that the topic of my posts at the GammonVillage Forum on Strategy have been on end-game doubling situations. This is an aspect of the game that often gets overlooked and sometimes even miscalculated.
In close games, no one really wants to "take a chance" and double the stakes. I am going to re-evaluate (and probably over-analyze) a specific position from my first post based on my caculations. Hopefully by looking at this in close detail, we can see how easy it. But when we're playing do we catch it? Here's the board:
| || |
| Red to roll: Here is why Red should double|
Clearly it can be seen that Red can bear off in three rolls and perhaps two rolls if any doublet other than 1-1 is tossed. What isn't clear is white's position. Initially, you might say "white has 4 checkers left, white must be in the lead." That is not necessarily correct. There are only 11 rolls that can give white a good chance to bear off in two rolls or even one in the case of 5/5 or 6/6.
Those 11 rolls are 6/6, 5/5, 4/4, 3/3, 2/2, 6/5, 5/6, 6/2, 5/2, 2/5 and 2/6. There are only a total of 36 possible rolls and 25 of them could mean 3 rolls for white to win. My calculator tells me that's a 69.4% chance that white will take 3 rolls to bear off. I like these odds. I double.
Since this is from a real match, I'll tell you what happened next. I (as Red) doubled and white accepted. I bore two checkers off from the 1 and 2 points. White rolled a 5-2, one of those 11 rolls mentioned above. I rolled again and bore off two more from the 1 point and 2 point. Now, lets analyze white's position again, just in case you're thinking white better re-double, right? Not so fast! The 5 and 2 is an ugly combination to try to bear off in one roll.
Let's count the rolls that do it: 5/2, 2/5, 5/3, 3/5, 5/4, 4/5, 5/5, 5/6, 6/5, 6/2, 2/6, 6/3, 3/6, 6/4, 4/6, 6/6, 4/4, 3/3 and 2/2. That's 19 rolls = 52.8%. In my opinion, too close to call for a re-double, since Red's odds of winning are 100%. However, maybe you would act differently. That's a judgement call. White didn't double and rolled a 4/3. Red won.
Here's the main point. You can basically "steal" points away from your opponents if they're not paying attention and just see that they have 4 stones left and you have 6. I stole on that game and on a few others as well. By doubling, you throw your opponents off, because they may still think they're in the lead. I know I always feel pressure to answer a double quickly. In an earlier article I wrote here, I said "always trust the statistically correct answer", even if your gut instinct tells you otherwise.
One more example:
| Red is on a roll and should double here also|
White looks to be in good shape, but only 14 rolls can save white this time and that's only about 39% chance. Red should double, white may want to take a chance (and trust that gut instinct) but Red has the better chance of winning the game.
Look at your end-games a little closer. You may pick up a few points and you'll look pretty smart doing it too. As one person said in one of those Forum replies, "this is one of those situations that separates the men from the boys." This is one of the tools we average players need to use to "take our game to the next level".
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