Problem of the Week #91: Solution


Tournament match to 7 points. Black leads 4-0 and owns a 2-cube.





(a) Assume you are Black and playing a very strong player. Should Black double? If Black doubles, should White take or drop?

(b) Same question, but now you are Black playing a weak player.


First, let’s note that I made a small oversight in the original statement of the problem. I should have said that in each case you are Black and you are a strong player. That will clarify the decision-making later, especially in part (a). Sorry about that.

We’ll start by figuring out the correct cube action assuming you and your opponent will make correct decisions in the future. Then we can see if the theoretically correct action requires any adjustment depending on the strength of your opponent.

To figure out the right cube action, we need to estimate three numbers:

(1) Black’s actual chance of winning the game from this point, disregarding the cube.

(2) The value of a 6-0 lead in a 7-point match (which happens if Black doubles to 4 and White drops, or Black doesn’t double and simply wins the game with the cube on 2.)

(3) The value of a 4-2 lead in a 7-point match (which happens if Black doesn’t double and White pulls the game out.)

Part (1) is pretty easy. White only wins if Black first doesn’t bear off his three checkers, which happens 31/36 of the time (note that 1-1 is not a winning double for Black), and White then rolls a double, which happens 1/6 of the time.

31/36 * 1/6 = 31/216 = 14.4%

[You might ask, “How do players actually do these calculations over the board?” I use a lot of numerical tricks, as do most other players. After you’ve played a lot of tournament backgammon, you realize that some tricks are particularly useful and some numbers recur constantly. In this calculation, I’d note that 31/36 is only a little larger than 5/6, so what’s 5/6 times 1/6? Well, 1/6 is about 16.7%, so 5/6 of that will be a little less than 14%. We want a number a little bigger than that, so our answer is 14%+. That’s plenty good enough for our purposes. Trying to get exact answers in your head is pretty hard, but close approximations are much easier and almost always good enough.]

What about part (2), the value of a 6-0 lead in a 7-point match? Since the Crawford Rule is in effect, White must win the next game, taking him to a 6-1 deficit. (It doesn’t matter if he wins a gammon in the Crawford Game or not.) He’ll then double to 2 at the start of every game. If he then never wins a gammon, he’ll need to win three more games to win the match, for a total of four straight wins overall. The probability of four straight wins between two equal players is

½ * ½ * ½ * ½ = 6.25%

But if White wins a gammon in either the second or third game, he wins 4 points and saves a game, so he only needs to win three straight. The probability of that is 12.5%.

The probability of winning the match is therefore between 6.25% and 12.5%, and a little closer to the lower number, since you’re not favored to win a gammon in a two-game sequence. A good approximation is 9% for White’s chances, and therefore 91% for Black’s chances.

Finally, for Part (3) we need the value of leading 4-2 in a 7-point match. Different match equity tables give slightly different numbers here, but the range is roughly 64% to 66%. I’ll use 65% as a good average value.

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Now we’re ready to figure out the optimal doubling and taking decisions at this score. Let’s start with White’s take/drop decision if he gets doubled.

> If White takes and redoubles to 8 when he can, he wins the match 14.4% of the time (from Part (1)).

> If White drops, he trails 0-6 to 7 and wins the match 9% of the time (Part (2)).

So if White gets doubled, he should take and reship when he can.

Now let’s look at Black’s doubling decision.

> If Black doubles, he must assume White takes and reships (we’re postulating correct play on both sides) so he’ll win the match 85.6% of the time.

> If Black doesn’t double, he’ll win this game 85.6%, getting to 6-0, and will lose 14.4%, getting to 4-2. His total winning chances then look like this:

85.6% of the time he’s leading 6-0, and wins 91% of those.
Wins in this variation = 85.6% * 91% = 77.9%.

14.4% of the time he’s leading 4-2, and wins 65% of those.
Wins in this variation = 14.4% * 65% = 9.4%.

Total wins from not doubling = 77.9% + 9.4% = 87.3%.

So he wins 87.3% if he doesn’t double, 85.6% if he does. So Black shouldn’t double, even in this two-roll position.



Now we’re ready to tackle the two main questions. What happens when we face real opponents of varying strengths?

(a) Against a very strong player (and assuming Black is a strong player himself) Black should just make the theoretically correct play and not double. If Black errs and doubles, White should take and redouble to 8 if Black misses.

(b) What about if Black is a strong player and is facing a weak opponent? This is a really interesting question and in fact is the whole point of this problem.

The theory of how to play against weak players originated in the 1970s with the publication of Barclay Cooke’s The Cruelest Game. Cooke expanded on his notions in two later books, Paradoxes and Probabilities and Championship Backgammon. Cooke’s idea was that you should be very conservative with the cube against weak players, doubling only when you were pretty sure you’d get a pass, and aiming to grind them down in a long series of 1-point and 2-point games, and giving your huge skill advantage in checker play the maximum chance to work. The worst possible disaster was to give your weak opponent the cube in a volatile position, allowing him to rewhip to 4, win an 8-point gammon, and turn the match around.

Cooke was a well-liked fellow, a real gentleman of the old school. In addition, he was an absolutely superb writer, who was able to convey better than anyone else the glamour and excitement of high-level backgammon. The Cruelest Game, published in 1975, was one of the big influences driving the backgammon explosion of the mid-1970s. To this day, it remains the one book I would recommend if a friend who didn’t know the game came to me and wanted to understand why backgammon was popular and what the fuss was all about. Cooke captured the drama of the game better than anyone else, before or since.

Cooke’s ultra-conservative approach to playing weak players soon became accepted wisdom and was echoed in other books in the 1970s and 1980s. But it is, I think, completely the wrong approach. Let’s see why.

In Greek mythology, the gryphon was a majestic creature combining some disparate parts: the body of a lion with the wings of an eagle. Cooke’s “weak player” is a little like that: a hybrid containing components not likely to be found in nature. He plays the checkers so poorly that you’re a huge favorite to grind him down one or two points at a time. But he handles doubling decisions superbly; he scoops up cubes in volatile but takeable positions that might have a strong player scratching his head, then he whips it back when the game starts to turn in his favor, applying maximum pressure. What a tiger!

Do real “weak players” actually play like this? In my experience, almost none do. In the real world, bad players handle the checkers poorly, but they handle the cube even worse. Their cube action is mostly tentative; they know they don’t play well, so they try to postpone decisions that might make them look foolish. They double late, or not at all, because they’re waiting for positions that are so strong that doubling can’t be a mistake. When doubled, they’d rather drop than take, because dropping only loses a point, while taking might lose four points. Besides, they know you’re a better player, and they’re picking up cues from you. If you’re doubling, then you must believe you have a big advantage. Who are they to argue? Better to drop.

The best way to play against weak players is to be very aggressive. Double a little early, especially in volatile positions. They’ll probably drop, but if they take, you’re better off in a number of ways:

> Their checker errors will now be occurring with the cube on 2 rather than 1.

> They may redouble prematurely, giving you an extra edge.

> They may redouble late or not at all, giving you a huge edge.

All of these edges add up to huge vigorish over time.

Interestingly, poker players tend to handle weak opponents better than backgammon players do. Poker players understand the value of relentless aggression, and they apply it ruthlessly, raising their limps, 3-betting their raises, and pushing the action after the flop. In part, I think, this is because poker players don’t really attach much importance to individual hands; there are many more hands in a poker session than games in a backgammon session, so poker players find it easier to just make what their experience tells them is the best move against this particular opponent, and let the chips fall where they may (pun intended).

Now, after this long intermezzo, back to Part (b). What do we do if we’re a strong player and our opponent is a weak player?

We double, and we do so quickly, without giving any hint that there might be something to think about. Our opponent will drop, because it’s a two-roll position, and everybody knows that’s a pass. And we pocket our two points and our 91-9 edge without risking losing the game.


Solution:
(a) No double and take.
(b) Double.

I'm really not buying this at all. Obviously doubling early against people who make awful drops gives you an advantage, and depending on how often they drop, may turn the assessment of this position, but your three points seem completely irrelevant to me in a match situation.

1) They'll make the same checker errors in the next game too, as long as the match goes on. You're not gaining anything. To make this obvious, imagine you're at some match score (like up 4-0 to 13) and can play a position twice with a 1-cube that you can't turn first, or once with a 2-cube that he owns. Clearly you, as the better player, prefer the former, and it's not even close, even though the game equity is identical.

2 and 3 are the same logic in reverse. Would you rather be playing the position where you should get recubed once where they own a 2-cube or twice where they "own" a 1-cube? Again, you want the smaller cube AINEC.

And neither one are factoring in the value of actually having access to the cube later.

In a cash game, the bad drop advantage is enough by itself (plus time value of money), but in a match, the other factors are actively working against you. If he drops way too much, it can still be +MWC, but your defense beyond that- roughly "double bad players because they'll be bad for 2x as much" is just wrong in matches IMO.

We had a different definition of weak and strong players. I assumed in my answer that a weak player could do a match equity calculation and that the strong player was stronger than I am, which is apparently wrong for this problem

I read "weak" to mean "not as good" as opposed to "weak" like "weak-tight." But after reading your explanation, it seems like you think of it as the latter.

I'm not really convinced that your argument holds for match play in general. It seems that if you're playing against a player who takes too much, you should be more reserved with the cube (to maximize his cube errors by having him take when he should drop), but if you're playing against a player who drops to much, you should be aggressive with the cube (for the reasons you stated).

I think part of the problem for this position is that your opponent won't have the chance to make the types of mistakes you put forth. In other words, you can't really "collect" on his errors in the same way that you can with an early position cube.

All the weak players I play with(and some of them are REALLY weak) takes way too much and not the other way around. The not-so-weak but not really skilled either is in the other cubicle. Passes too much. So I think from my experience with the really weak players, at least in Norway, is that they take WAY too much. And as such this solution is just wrong.

To give Cooke his due, he almost always conditioned his descriptions of the leverage the weak player holds against an expert with the caveat, "if only he knew!"

Bill, how does your thinking compare with the recommendations of Danny Kleinman against his character Colonel Whiteflag? Kleinman emphasizes time and again that you will maximize your equity by doubling early, at the point of first pass, against the Colonel or any other timid player who tends to drop takable cubes. You seem to echo Kleinman in presuming the novice will too often pass your early double, but then contradict him by suggesting that your equity will rise when the novice takes an early double. Will your strategy really maximize equity? Is Keinman wrong?

Perhaps this is a moot comparison. Colonel Whiteflag is an otherwise competent player who systematically drops early. Your novice may be an inconsistent player who takes and drops erratically.

Any thoughts?

Thanks, Bill, for this kind of practical information. RolledUpTrips used this type of over-the-board thinking in his solution to Problem #91. That's why I thought his was the best response of all (based on the assumptions he made regarding player strengths):

Source: http://forumserver.twoplustwo.com/13...ry-9-a-957887/

I perhaps should have been clearer about what "weak" meant. In a tournament situation, I'm not talking about a strong open player matched against an average open player; I mean a strong open player matched against an average intermediate who's decided to play up for a change. These kinds of pairings are common in the side events, but they happen in main events as well. In chouettes, I mean a player who's clearly a steady loser in the game, whether he's weak-tight or loose.

In my view, chess, backgammon, and poker all share a common theme: you best exploit weak players by relentless aggression, not by playing close to the vest. Weak players make plenty of mistakes, and when you attack, they make even more mistakes. In chess, aggressive players win the most games, and in backgammon and poker they win the most money. This phenomenon is most apparent for backgamon in chouettes, where good tight players will make a little money from the weak players, but good loose players make a ton of money. But it applies to tournament play almost as well.

A good example from the world of chess involves the careers of Tal and Petrosian. Both came up at about the same time and both were phenomenally talented. Tal played a relentless attacking style, especially against players he thought were weak. Over the course of his career, he won dozens of tournaments. Petrosian played a tight defensive style, and he lost hardly any games. But over his career, he won only a handful of tournaments.

By the way, Part (b) of problem 91 isn't really an example of this theme. You're not doubling because it's aggressive (there's no more play in the game), but just to collect your points, because weak players don't do match equity calculations; they just drop. (And if they do take, they will often forget to redouble.)

I think, Mr. Robertie, your opinion about Barcley Cooke hasn't changed since "Reno 86". In "Can a fish taste twice as good" from Trice/Jacobs there is an approach to use different MET for doubling in an unequal match. Is this approach valid und would you buy it?

Meh I barely play backgammon at all. I take slight issue with part B tho.

You say just give him the cube and take your 91%. Surely your equity is higher vs this villian in the 6-0 match. He will be making checker mistakes. It will be even moreso in the 4-2 match variation, when the cube and gammons come back into play for you.

You are going to need him to not take the cube a good bit more than the 91% vs 87.3% vs 85.6% equities you have calculated vs a strong player would suggest.

Now maybe you know this to be the case, but if you further amend the description of the villian to "putz that won't take the cube", then it is not really a problem at all. Sure just give it to him.

I agree with most of what you say, but I have to still disagree on the backgammon solution. I have rarely seen anyone forget an automatic redouble in this type of situation, and it seems crazy to me to (essentially) push negative equity vs an inferior player. I've actually found a lot of weak players that love the idea of taking and reshipping when behind in a match, and do so too often and to their detriment. They simply can't pass up the opportunity to quickly win the match when they were fairly far behind.

On a side note, I do think the problem is fairly interesting from a match equity/optimal play standpoint...

I never read the Trice/Jacobs book. Too many tables.

Usually, you can tell if a weak player is going to take or drop just by looking at him. Their body language will usually tell you all you need to know.

This is largely meaningless. A backgammon tournament is a series of matches where winning the match is the only relevant concern. You play to maximize the moneyline of the match. A chess tournament isn't about maximizing the moneyline of every game (where 10% win, 90% draw, 0% loss is a guaranteed win in a match, but almost impossible to win a tournament with), so it's not a valid comparison. And Petrosian won more world championship matches than Tal (2 to 1). Cash games, chouettes (as I understand them), and poker cash games are a different animal.

Let's make a hypothetical player for a match, call him Johnny Nocube. He never turns the cube. Shockingly, you have a pretty good edge over him. Let's say, for any individual game, you are: Win 1 30%, win 2 20%, win 4 10%, lose 1 20%, lose 2 15%, lose 4 5%. Numbers are clearly pulled out of my ass, but they will make a point. You have an expected +.4 equity per game. In a match to 11 (ignoring matchpoint considerations, all games are the same), you win just over 75% of the time.

Now, in game 1, you have a double situation. You can play on and score a single 75% of the time or lose a single 25% of the time (game equity +.5). Or you can double and win 2 65% and lose 2 35%, (game equity +.6) (I have you winning fewer because you can't double-drop to cash in later). Obviously it's a cash game double. This isn't even an "early" double that's -game equity. As far as the match goes

No double:
You from 1-0: 79.2%
You from 0-1: 70.0%
MWC = .75*79.2 + .25*70 = 76.9

Double:
You from 2-0: 82.8%
You from 0-2: 64.1%
MWC = .65*82.8 + .35*64.1 = 76.3

Doubling is a blunder here where it would be clearly correct against an equal player.


Now let's look at a position where 4 is in play. If you play on, you win 1 50%, win 2 20%, lose 1 25%, lose 2 5% (game equity +.55). If you double, you win 2 45%, win 4 20%, lose 2 30%, lose 4 5% (game equity +.9)

Adding in
You from 4-0: 89.2%
You from 0-4: 49.9%

No double: .5*79.2+.2*82.8+.25*70.0+.05*64.1= 76.87%
Double: .45*82.8+.2*89.2+.3*64.1+.05*49.9= 76.83%

Even with a big double and a relatively narrow take (on game equity), this still comes out to be a blunder. God help you on an early double where 4 is in play.

The reason for this behavior is obvious- when you go up 2-0, you've picked up 7.6% MWC. When you go down 0-2, you've lost 11.1%. When you go up 4-0, you've picked up 14%. When you go down 0-4, you've lost 25.3%. Your doubles need to be MUCH stronger than normal, either by virtue of the position, or because you know he's going to drop a hell of a lot. Doubling against a donkey in a marginal spot and having him take is a disaster. And these results were against somebody who never even recubed and put 4+ singles and 8+ gammons in play.

Alternatively, imagine a no-gammon game where you always start with the 1-cube. Match to 11. You win 60% of the games if you never cube and win the match 82.5% of the time. You have a 75-25 spot in the first game.. If you play on, you have .75*87.3+.25*75.5% = 84.4%. If he takes and never reships, you have .75*91.1+ .25*66.7 = 85%. So doubling looks like it might be good..

But if he announces recube in the dark, then you don't cube again, from 4-0, you win 96.5%. From 0-4, you win 44.8%. For a MWC of 83.6%. Variance is so bad for you that you lose equity by turning the cube from 1 to 4 in a 75% position from 11-away/11-away. And obviously another cube/recube for the match is even worse for you (you only win 75%).