Problem of the Week #24: Solution


Cash game. White owns cube. Black on roll.




Part (a): Black to play 5-4.




Part (b): Black to play 5-4.




Part (c): Black to play 4-3.


Problems 24a through 24c are all taken from the same game. In Problem 24a, Black played 13/4. In Problem 24b, he played 15/10 15/11. And in Problem 24c he played 5/2 4/off. Not only were all three plays incorrect, they were all incorrect for the same reason! Let’s take a little closer look and see what ties all these positions together.

A significant leak that bedevils players of all levels (even world-class players) is the inability to spot plays that offer small increases in gammon chances at little or no risk of losing the game. These three positions are very typical. In each case, there is an alternative play that wins a few more gammons but costs a small number of extra losses. (Or in two of the three cases, no extra losses at all!)

Before we look at the right plays, let’s review a little theory. Suppose we were looking at a position, and our first candidate play (Play A) featured the following probabilities:

80% chance of winning a single game
10% chance of winning a gammon
10% chance of losing a single game.

In other words, we’re a really big favorite, with small chances of losing but also small chances of winning a gammon.

Now suppose we notice a second candidate play, which we’ll call Play B. Play B wins more gammons, but also loses more single games. In essence, Play B requires that we trade some of our single games for a mix of gammon wins and single losses. In order for Play B to be a better option, the number of extra gammon wins has to be at least double the number of extra losses. (The reason is that winning a gammon gains us two points, assuming the doubling cube is at 2, because we win four points for a gammon instead of two points for a single game; but losing costs us four points, because we now lose two instead of winning two.)

Let’s go back to our estimates for Play A and now imagine that we have a second play, Play B, with this breakdown:

67% chance of winning a single game
20% chance of winning a gammon
13% chance of losing a single game.

Moving from Play A to Play B trades 13 single wins for a mix of 10 extra gammons and three extra losses. Since the number of new gammons is more than twice the number of new losses, it’s a good trade, and Play B is better.

So what are the characteristics of plays that win more gammons? In general, gammonish plays will do one of four things:

(1) Hit opposing blots
(2) Make inner board points
(3) Hit loose in the inner board
(4) Bear off checkers.

(Note also what’s not on this list. Things like escaping back checkers, running around through the outfield, avoiding hits, and filling in primes aren’t plays that win a lot of gammons. Instead, those are plays that are more likely to win single games.)

With the theory now out of the way, let’s go back to our original three problems and look for alternatives that win more gammons. They’re easy to find.

In Part (a), the hitting play 7/2* 13/4, putting White on the bar and preparing to make the 2-point, will win more gammons than 13/4.

In Part (b), 13/8 13/9, getting ready to fill in the annoying 3-point gap at the cost of some outfield shots, will win more gammons than the super-safe 15/10 15/11.

In Part (c), 4/off 3/off, bearing off two checkers and getting down to 10 men rather than 11, will win more gammons than 5/2 4/off.

While it’s clear these plays win more gammons, it’s not immediately clear that they’re the right plays. If they lose a lot more games, it might be the case that they couldn’t win enough gammons to be worthwhile. And that’s the key to analyzing these decisions: take a careful look at the losses these apparently aggressive plays incur. If the downside risk turns out to be more apparent than real, then the gammonish play is likely to be better. So let’s look carefully at the downside to each of these plays.

Part (a). By hitting, Black is trying to fill in his board quickly and prevent White from making his 2-point. The 2-point is much more likely to both save a gammon and win the game than the 1-point anchor, so the hit makes a lot of sense and will, in fact, win more gammons. What’s the downside? If White hits back with a deuce or 3-1 or 1-1 or 3-6, he breaks Black’s momentum. But that’s about all. Black will have shots at White’s blot on the 23-point, and Black still has three White checkers stuck behind a pretty good 4-prime. Being hit is annoying, but it’s not swinging the game around.

Part (b). The point of 13/8 13/9 is to quickly fill the gap on the 3-point. That’s good for winning a gammon (because it’s another inner-board point) and also good for winning the game (because a gap guarantees a lot more bad numbers when you start your bearoff.) What’s the downside? White can hit you with 2-5 or 2-6. How costly is that? Not very. You’ll be shooting at three White blots, with either 17 immediate return shots (if White hit with 2-5) or 18 (if White hit with 2-6). Even if you miss everything you’re still a huge favorite to win the game, since White’s board is choppy and you have two of his checkers still stuck behind a broken 5-prime.

Part (c). Playing 4/off 3/off bears off two men and reduces Black to just 10 men left, with the possibility of being off in five more rolls even without a double. That’s very good for the gammon. Obviously the play leaves Black seriously odd-ended, but how much does that increase his chance of being hit? After taking two off, Black leaves a shot next turn with 19 rolls whether White runs or stays. White hits about 1/3 of the time, so Black gets hit a total of about 6/36 of the time. After 5/2 4/off, it’s a little more complicated. If White runs, Black leaves a shot 7/36. If White stays, Black leaves a shot on the same 7 rolls, but 4 of those 7 are double shots. On average, Black will leave a shot about 8/36 times, and get hit 1/3 of those times, so now Black gets hit about 2.5/36. The difference in hits is about 3.5/36, or about 10% of the time. That might cost Black a gammon win when it happens, but his losing chances hardly increase at all because White has no board.

In each of the three positions, we can analyze the downside risk of the more “aggressive” play as more apparent than real. In each case, the weakness of White’s position allows Black to recover pretty easily from a good shot by his opponent. When that’s the case, the more gammonish play is likely to be correct, especially since the gammonish plays tend to improve Black’s game in the most direct manner.

The rollout results for these positions provide solid backup for the effectiveness of the gammon plays.

Part (a): 7/2* 13/9 wins 5% more gammons than 13/4, while losing only 1% more games.

Part (b): 13/8 13/9 wins 3% more gammons than 15/10 15/11, while showing a tiny decrease (!) in losses.

Part (c): 4/off 3/off wins 4% more gammons than 5/2 4/off, while losing the same percentage of games.

What makes these positions difficult, especially given the speed and tension of over-the-board play, is that there’s nothing obviously wrong with the “safe” plays. When safe plays involve creating awkward stacks or breaking key points, we immediately start looking for better choices. But the safe variations in (a) through (c) look perfectly fine. In (a), 13/4 brings a builder to a reasonable spot. In (b), 15/10 15/11 clears a point and brings builders to bear on the 8-point. In (c), 5/2 4/off takes a checker off and stays even on the rear two points.

But in each case, the right play isn’t about Black’s position: it’s about the weakness of White’s position, which should encourage Black to take small liberties to win in the most direct manner. Use the holes in your opponent’s game as a gauge for amping up your own aggression.

Solutions:

Part (a): 7/2* 13/9

Part (b): 13/8 13/9

Part (c): 4/off 3/off

Nice. I was virtually positive I'd go 0 for 3. I suppose I'm gun-shy because I've been bitch-slapped by the universe a few too many times.

Thanks for the problem and detailed solution Bill!

Quick question, what does making the slotting play in (b), 13/8, 7/3, do to the gammon and winning chances?

I'm guessing that it loses more games than either 13/8 13/9 or 15/10 15/11 and probably does not produce many more gammons than 13/8 13/9 so it is not worth the risk.

13/8 7/3 is a pretty big overplay. It allows 20 shots (all aces and deuces), four of which are double hits (2-1 and 2-5). That's too much downside, considering that 13/8 13/9 gives good building potential with hardly any risk.