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In backgammon, there comes a time when all pieces have passed each other; there is no more interaction, you are simply rolling it out until you can bear off all of your pieces. If one player is well ahead of the other at getting their pieces into their home board and bearing them off, then it might seem pointless to keep rolling it out; it may well be impossible, or at least extremely improbably, for the other player to make up the difference.

May the player who is behind resign in such a situation, or do they need to play it out in case of a gammon?

And if they are allowed to resign, are there any good rules of thumb to decide when it's worthwhile to do so, and when it's worthwhile to keep playing? For instance, has anyone calculated out the possibilities such that you can say "if player 1 has started bearing off, and player 2 has x pieces outside of their home board, there is no possible way for player 2 to win." Or, has anyone calculated out the probabilities, such that you can determine based on the position that there's only a one in a million chance of winning, and it's probably not worth hoping for that, while in another situation, you might have 10 to 1 odds and it's worth seeing if you make it.

edit to add: According to a thread on Backgammon Galore, along with their glossary, you can offer to resign with any outcome (backgammon, gammon, or a simple win). If you offer one of the lower outcomes (gammon or simple win), your opponent can refuse in order to take the chance at getting the higher outcome. Of course, this only applies if you're playing in a tournament, a rating system, or for money where the value of the win matters.

So, I've answered the first half of my question, but I'm still interested in the second half; when should you resign? Are there any good rules of thumb, or has anyone done a statistical analysis, of what points in the game one player can be guaranteed to lose, or have such a low probability of winning that it's worth it to just resign? Are there any good ways to tell when you should offer to resign with a gammon?

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4 Answers

up vote 13 down vote accepted

Since the edit, there are two parts to the remaining question: "When have you definitely lost?" and "How can you tell when you've probably lost?". The first is easier to answer than the second, so I'll start with that.

The minimum you can roll on a go is a two and a one (note that a double one would give you four moves). The maximum is double six. So (assuming all pieces have passed each other) you can determine if you have definitely lost by assuming you will roll double sixes for the rest of the game, and assuming your opponent will roll two/ones.

It would be really nice to include a formula here that takes pip count, piece count and whose turn it is, and computes if you've definitely lost, but such a formula escapes me. There are some situations that depend on more than these three variables - for example:

|24  23  22    |24  23  22
+---+---+---   +---+---+---
| O   O        | O       O
| O   O        | O
|              | O

In both these finishing positions the pip count and piece count are the same. However the maximum number of rolls to finish is two in the left hand picture, but three in the right hand picture.

There are two main metrics of backgammon positions (other than pip count). These are the Thorp count and the Kleinman ratio.

The Thorp count is calculated for each player as:

  1. Start with the pip count.
  2. Add 2 for each checker still on the board.
  3. Add 1 for each checker on the 1 point.
  4. Subtract 1 for each occupied home board point.
  5. If the player on roll's count is greater than 30, multiply by 1.1.

There is an article here explaining how the Thorp count can be used to estimate the game winning probability.

The Kleinman ratio is calculated as:

(D * D) / S

where D is the pip count difference of the two players, plus 4 to compensate for the player on roll's additional pips; and S is the pip count sum, minus 4 to compensate for the player on roll.

There is an article here explaining how the Kleinman ratio can be used to estimate the game winning probability.

Finally there is an article here which overlaps the previous two, but which recommends using the Kleinman ratio over the Thorp count assuming you are able to learn a large table of logarithms!

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Thanks! That gives me some good starting points. –  Brian Campbell Oct 30 '10 at 15:32
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Two good rules of thumb are, never resign in a position with contact, and resign if the result is "obvious". If you need to stop and think to work it out then it's probably quicker to just play on to a position of certainty anyway - certainly so online where forced moves (and optionally, greedy bearoffs) can be played automatically.

It's generally not considered polite to resign a single game if a gammon is still possible, however unlikely - some players will interpret it as a way to trick them out of the gammon possibility.

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In professional play I can't say but if it were just you and I playing then I'd say sure you could resign just to finish the game and start a new game. Is it possible that the losing player could win ... yeah but if it's such a remote possibility then I'd call the game just for the sake of entertainment (I.E. lets play again, it's boring just to finish the game out).

As for a rule of thumb ... I'm not sure of the mathematical calculations, but from just playing alot I tend to know when I'm beat. I'll keep playing but in my head I know that I've been beat so psychologically I throw in the towel and start thinking of new strategies for the next game. It's probably not the ideal mindset, but then again I don't play professionally, in tournaments or for money.

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You resign when it is mathematically impossible for you to win, and also, if it is mathematically impossible for your opponent to get a gammon or backgammon. This occurs after contact has been broken off between the two teams (no possibility of further hits).

"Mathematically" impossible to win means, give yourself 6-6 the rest of the way, give your opponent 2-1, and you still lose. That happens only at the very end of the game.

A gammon (and backgammon) is ruled out when you have borne off one piece, or will do so with the worst rolls for you, and the best rolls for your opponent.

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