Playing greedy for the gammon

So we closed out one of Gary’s checkers and have started bearing off. We think we have a pretty good shot at a gammon, but we aren’t sure how hard we need to work to earn it. Should we be greedy and take off 2 checkers at all costs, at the risk of a bit of safety? Or should the safe play be paramount? There’s a method we can use to inform this decision, which we will describe below.

First, examine the position below:

This feels like a guaranteed gammon.¹ Does that mean we should play safe, since we have nothing to lose? If we do, we might let the gammon slip away!

First, let’s outline the candidate plays. We already learned how to identify the safe play in Bear off safely: avoid an odd number of checkers in the two highest points by playing 5/off, 4/2. By contrast, the greedy play ignores this advice and bears off two checkers immediately: 5/off, 2/off. It’s a small risk, increasing our losing chances from roughly 1% to over 2.5% by leaving a blot when rolling 66, 55, or 44. Is this risk worth it?

Here’s how to make a methodical assessment over the board. We start by counting crossovers on both sides of the board. A single crossover is the distance a checker needs to traverse a quadrant of the board, and it roughly scales with the number of rolls remaining.

How many crossovers do we have remaining? Before making our play, with 13 checkers on the board, we have 13 crossovers left to win the game. If we make the safe play, we will have 12 crossovers left. If we make the greedy play, we will have only 11.

How about Gary? On his end, we are only concerned with him getting 1 checker off the board to save the gammon. First, the 4 checkers on the 12-pt each need 2 crossovers to enter his homeboard (8 crossovers). The checker on the bar needs to enter and make its way around the board (4 crossovers). And finally, Gary needs to bear off a checker to get off the gammon (1 crossover). This totals 13 crossovers.

Seeing as how we are ahead of Gary in the race when making the safe play, we can be pretty confident that we don’t need to take any risks in this position.

Below, we make two small adjustments to this position and show how this affects our decision. First, we decrease Gary’s total crossover count, while keeping his pip count constant:

Now that he has only 11 crossovers, we become desperate to catch up to him, and so the greedy play (also with 11 crossovers) becomes the right choice.

In the second variation, we add a checker back to our home board:

This increases the crossover number for our safe play to 13, matching Gary’s original total. Now, the safe play becomes nearly a blunder, and we are encouraged to play greedily to beat Gary and earn our gammon.

This method is not foolproof, especially in positions where the counts are close. It doesn’t take into account the distribution of checkers within each quadrant, for example. However, when the counts are fairly lopsided (say, a difference of 3 crossovers or more), it’s fairly reliable at determining the right play.

1. The real gammon chances are above 75%! ↩︎