Contact and the Race

Contact and the Race
by Walter Trice – 8 August 2006

One of the most frequently cited principles of backgammon strategy is to try to play a racing game if you are ahead in the race, but to try to “mix it up” and hit a blot somehow if you are behind in the race.

This is an extremely useful principle, and it is a factor in many checker-play decisions, even in complex situations where it seems unlikely that the game can turn into a pure race for many rolls.

But like all generalizations about the game it needs to be qualified. It tends to point to the right play with “other things being equal.” Usually, though, other things are not equal. Just because a play that ends contact makes you a 55% favorite to win the game, there is no guarantee that some other play that maintains contact won’t make you a 60% favorite.

Problem 1: Red to play 6-5.

In Problem 1 Red is winning the race, so he should just break contact and race, right? It happens that Red is winning the race. He is down two pips before the roll, and after he plays his 11-pip 6-5 he will be ahead by nine pips. (Note: generally you are winning the race when your pip count is at least five lower than your opponent’s after the roll. If your count is four pips lower then the race is effectively even.)

It also happens to be the case that Red should play 16/10, 16/11, breaking contact. The Snowie rollout (at the bottom of this article) shows that any other play is a big blunder. But the logical connection between the correct play and the reason that I have suggested is not so clear!

Problem 2: Red to play 6-5.

In Problem 2 Red is losing the race, but he should still break contact with 16/10, 16/11. Playing a pure race only gives him a 42.7% chance to win the game, but this is the best he can do.

Why so? The essence of Red’s difficulty is that he has a “timing” problem due to the fact that his outfield point is further from home than White’s. Red’s two men on the 16 point represent six pips more than White’s two men on the midpoint.

If Red plays 11/5, 6/1 he will still have two spare checkers in the outfield in addition to the point that he owns. White will also have two spares. If both players adhere strictly to the policy of “saving sixes” (that is, if they avoid moving spares home unless forced to) then White, being on roll, will be likelier to leave the first outfield shot. Thus one of the typical problems that result from a timing deficit – the greater likelihood of leaving a shot – does not apply to Red here.

But suppose, after Red’s 11/5, 6/1, that several rolls go by with neither player leaving a shot or clearing the outfield point. Then White rates to benefit. White’s timing advantage is manifested, in part, by his spare checkers on the high points in his home board. As a result he will be able to make constructive plays in his home board for longer than Red. Red’s home board is likely to crash before White’s. This will hurt Red in the race, since he will have to waste more of his pip count than White. In addition, if White gets a shot at Red it is likely to be a higher quality shot than a corresponding shot for Red, because he rates to have the stronger home board, the better to contain a hit blot.

Finally, suppose that play goes on for a while with both boards crashing, until one player or the other has to break for home, and nobody gets hit. Then Red starts the race for home with a six pip deficit.

So even though Red will be effectively down three pips, he does best to break contact and just race.

Problem 3: Red to play 6-5.

Of course if you put Red further back in the race, eventually it becomes right for him to stay put and pray for the shot. Position 3 seems to be close to the break-even point for this sort of configuration. Note that it is not purely a question of the racing chances, because raising Red’s pip count also improves his timing.

Problem 4: Red to play 6-5.

Taking away White’s outfield spares makes a huge difference. In Position 4 Red starts out one pip down and on roll, so he is ahead in the race even before rolling the dice. On top of that we have given him a 6-5 to play, which is almost three pips better than average. After 16/10, 16/11 he would be a 3-to-2 favorite in the race. But holding the point and closing his board with 10/5, 7/1 is just about as good, and possibly better if you factor in the stray gammons that result from the favorable contact.

White leaves an immediate shot with 6-1, 6-2, 6-3, and 6-4. Red hits on his next roll 2*(11+12+14+15)/1296 = 8% of the time. No roll forces Red to leave a shot on his next turn, so he gets two full rolls worth of first-strike potential, hitting perhaps 12% more often than White altogether. (That’s a pure guess, by the way.) When the game is not decided by hitting, Red still has his lead in the race.

The decision to maintain contact will become clearer if Position 4 is modified by sliding Red’s outfield blots back farther from his home board. We can add up to six pips to his count leaving him ahead, and it will become increasingly preferable for him to maintain contact. These modifications will improve his timing as well as reducing his chances in the racing variations, and both factors increase his incentive to maintain contact.

 

Here are the rollouts for the positions above:

Problem 1: Red to play 6-5.

Problem 2: Red to play 6-5.

Problem 3: Red to play 6-5.

Problem 4: Red to play 6-5.