Problem of the Week #10: Solution




Cash game, center cube. Black on roll.

(a) Should Black double?

(b) If doubled, should White take, drop, or beaver?

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Up until now our problems have all been checker plays. Problem 10 is our first cube problem, so before we discuss the specifics of this position, let’s make a few general comments about doubling and taking decisions.

We’ll start by considering positions where no gammons are possible, as for example in a straight race. The defender (the side being doubled) needs to win the game at least 25% of the time to take a cube. If this isn’t immediately clear, consider what happens when the doubler is 75% to win and turns the cube from 1 to 2. Imagine that the two players agree to play four games from this position. If the defender drops, he loses four games at one point per game, for a total loss of four points. If he takes instead, he loses three games at two points each (total loss six points) and wins one game for a two point victory. His loss is -6 for his three losses and +2 for his single win, for a net loss of four points, exactly as before. So in a game with no gammons, 25% winning chances are the break-even point for the defender. If he wins less than 25% of the time, he should drop, and if he wins more than 25%, he should take.

Now what about the doubler? If the defender is taking as long as his chances are greater than 25%, what kind of winning chances does the doubler need to be doubling?

This is a tougher question and actually depends on what we call the ‘volatility’ of the position. Volatility is just a measure of how rapidly the position can change from roll to roll. In a long race, where only a few big doubles can change the evaluation of the position by a lot, volatility is low. Here the doubler will wait until he’s close to 75% before doubling; 69% to 73% is a typical range for turning the cube.

When the race is very short and only a few rolls remain in the game, volatility gets higher and higher with each roll, and less of an advantage is needed to double. Position A shows an extreme example:



Position A: Black on roll.

Here the volatility couldn’t be higher – Black will either win or lose the game on the next roll! As it happens, 19 of Black’s rolls win for him and 17 lose, so he’s about a 52.5% favorite. With volatility off the scale, that’s good enough to double. (White will of course take.)


Adding Gammons

The explanation above described positions where no gammons were possible. In that case, 75% wins for the doubler and 25% wins for the defender represented a marginal take; with fewer chances than that, the defender should be passing. But in almost all early and middle game positions, gammons are possible. Let’s see how we account for them.

Imagine an early game position, and one player has the advantage and doubles. His opponent considers taking. He estimates that the doubler will win a gammon about 10% of the time in this position. What’s the minimum winning percentage he needs to take?

We can reason this out if we remember the relationship between gammons won and extra losses incurred. If we turn a simple win (with the cube on 2) into a gammon, we gain two points, going from +2 to +4. But if we turn a simple win into a simple loss, we lose four points, going from +2 to -2. Gammons won are only half as important as extra games lost.

So a position where 10% of our wins are gammons can be balanced if the defender can win just 5% more games. Therefore, if the defender can win a total of 30% of the time, he will have a marginal take in a situation where the doubler expects to win 10% gammons.

Let’s summarize what we know so far:

With 0% gammons:
Doubler wins 75% single games
Defender wins 25% single games
is a marginal take.

With 10% gammons:
Doubler wins 60% single games
Doubler wins 10% gammons
Defender wins 30% single games
is also a marginal take.

We can continue this reasoning along, of course. If the doubler expected to win 25% gammons, then the defender would need to win 37.5% of the time just to have a marginal take! (He needs his base 25%, plus an extra 12.5% to balance the gammons, for a total of 37.5%.)

With 25% gammons:
Doubler wins 37.5% single games
Doubler wins 25% gammons
Defender wins 37.5% single games
is a marginal take as well.

For Problem 10, this last chart most closely resembles our actual situation. Black has 15 numbers that enter from the bar and hit at least one of White’s blots (all fours plus 3-1 and 3-5.) With no anchor and vulnerable blots, White needs to anchor immediately if he gets hit. He might, of course, but when he doesn’t, his gammon chances skyrocket. Rollouts indicate that Black’s gammon chances are squarely in the 25% range, meaning White needs 37.5% wins to give him a marginal take. He’s actually doing a bit better than that, with winning chances around 40%, so his take is pretty easy. But the double is correct nonetheless.

In 501 Essential Backgammon Problems, I labeled these positions ‘Action Doubles’ and the name has stuck. Action doubles are positions where the side on roll has a number of immediate shots that swing the game heavily in his favor with good gammon chances. If he fails to roll one of these numbers, the game will be about even.

To be a good double, an action position usually needs three strategic characteristics:

> The doubler has a good home board, at least as strong as his opponent’s.

> The doubler is shooting at several blots, and hitting one of these blots can swing the game dramatically.

> The defender doesn’t have an anchor.

If all of these characteristics are present, the position probably warrants an action double.

Many players are uncomfortable with action doubles because the positions are often easy takes for the defending side. In Position 10, for instance, White would be making a huge mistake by dropping. These players like the security of knowing that they’ll still retain a big edge even if they roll poorly after doubling. Action doubles don’t provide that security. If Black fans in Position 10, White is already a small favorite.

But the power of action doubles is that if Black rolls well (hitting) and White rolls poorly (dancing or at least not anchoring) Black becomes a huge favorite. In backgammon lingo, he will miss his market by a mile, meaning any subsequent double will be a huge drop. Missing your market is costly; better to double a little too loosely than to risk become a big gammon favorite only to collect a measly point.


Solution: Black should double, and White should take.

Curious question: is snowie not accurate for this type of position? or is it really close between double/take and no double/take? just asking because snowie recommends no double/take for this position

edit: after reading the post more thoroughly i think i understand

Try doing a rollout and watch the answer change.

Snowie's evaluation function is very good. Rollouts are much more accurate.

What sort of rollout settings do you recommend? Do you vary them by position?

Is there a way to quantify (in rough terms) when you have "enough" market losers or that you've lost your market by "enough" in those sequences?

This is obviously a hugely important question. A few years ago (around the time Snowie 4.0 came out) I decided to set up an experiment to answer this question. I selected a suite of 50 positions from various types of games, ranging from opening to back games to races. In each position, the object was to calculate Black's winning chances as closely as possible, as though Black were on roll and thinking about doubling. I decided to accept the results of a Snowie 3-ply, 1296 game rollout as the "correct" answer for each position, and then did rollouts using different numbers of trials and different play levels to see which level gave me the best answer in the least time.

The whole experiment took about six months, with two computers running all night, every night to get all the positions solved for all the ply levels and trial possibilities. But the result yielded a pretty clear winner: 2-ply and 648 trials is "good enough". Running more trials and/or going to 3-ply was overkill, while dropping to fewer trials or going to 1-ply resulted in quickly increasing discrepancies with the "correct" answer.

As I say, this was done some years ago using Snowie 4.0 or 4.1. Snowie is now up to 4.7, but I'd wager a similar result would still hold true.

If I were doing a backgame, I'd crank the rollout up to 3-ply and keep the trial number at 648. That might be unnecessary because Snowie 4.7 is noticeably better at playing backgames than the early versions were, but it's good insurance. For simple positions like races and blitzes, 1-ply rollouts are fine -- actually unnecessary because the evaluations are extremely accurate as well for these positions.