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An introduction to equity

If you have ever played backgammon, you are very likely to have encountered what we call beginner's luck. Your opponent plays a move which seems just terrible to you, leaving two vital blots (single, uncovered checkers) when he could have left only one. And of course, you roll a double of the very number which would have hit if your opponent had played the correct move, but which now misses the two blots and loses the game outright. Well, a bit frustrating maybe, but luck is a part of the game, you think, and if your opponent plays such blunders you are very likely to do better in the next game. However you can't resist the temptation to explain him the mistake he did and how lucky he has just been. Now here is the big mistake of yours !

Your opponent bitterly disagrees with you ! What is that advice that you are trying to give him ? That he should have played the move which would have allowed you to hit, and filling your last gaps in the same time ? He is not that dumb ! It is absolutely obvious that his move was much better, as a simple look at the current position tells. From now on, he would very much appreciate if you could keep your comments for yourself.

What can go wrong when discussing checker play can get even worse when discussing cube decisions. The doubling cube is sometimes seen as a gambler's device, with cube decisions as well motivated as by "I was feeling lucky" or "What could I do but reject ? I know it is not my day". There is certainly no evil in resorting to such guesses if you don't mind winning or losing, but assuming that you prefer to win, couldn't your decision making be backed up by something slightly more solid ?

There is only one concept we need for that, it is the concept of equity. Equity is the value of a position, it says what it is worth, hence what positions are better than others. Equity is the key concept of any luck-based game, and the good news is that understanding it does not require more advanced mathematics than the four arithmetic operations ! Let us now see how we would compute an equity in a very simple position :

In the above position, Black to play will win unless he rolls a 2-1. The probability of rolling a 2-1 is 2/36 (2-1 or 1-2 will do it). It means that black will win two points - because the cube shows 2 - in 34 cases out of 36 and lose two points in 2 cases out of 36.

Despite the very unbalanced chances, anything can still happen in this particular game. But let us now imagine that we will play a very large number of games starting from this position, say 36 million games. It is almost certain that the number of black wins will be very close to 34 millions, and the number of losses very close to two millions ; so the score will be very close to 68'000'000 - 4'000'000 in favour of Black, a 64 million points lead. This means that one single game from this position is worth 64'000'0000 / 36'000'000 = 1.78 points for Black (of course, you can see that the millions were there only for the sake of argument, and we will use smaller numbers from now on). Those 1.78 points are called Black's equity. It means that the given position is worth 1.78 points for Black in exactly the same sense that a certain win would be 2 points worth for him. Or, as finance people would put it, if it was a money game played with one euro per point, and Black wanted to "sell" his game to somebody else, the fair price for it would be exactly 1.78 euros.

Now we can define what is good play at backgammon : whether a cube decision or a checker play, the good play is the one which leads to the position with the highest equity. Working out the equity of a position is rarely as easy as in the previous case, and in a typical backgammon position such a computation would require the exploration of zillions of possibilities. The experienced player will use his principles and instinct instead, but his goal will always be to maximize his equity ; this is just what is called playing well !

Some may object that talking in terms of equity is just a disguise for talking in terms of winning probabilities. It is true that both are equivalent when considering single games without the cube ; but the reason why backgammon players are almost always talking about equities instead of probabilities is that the situation gets a little bit more complicated when the doubling cube is involved, because not all games will be worth the same number of points. Here is an example of a how to make a decision based on an equity calculation :

In this well-known position, Black to play is again a clear favourite to win the game. Should he offer a double ? Doubling just before the last relevant move of the game is indeed always correct when you are a favourite ; but let us compute the precise equities in order to be sure ! Red has 27 winning rolls out of 36 (all rolls except 1-1, 1-2, 2-1, 1-3, 3-1, 1-4, 4-1, 2-3 and 3-2) ; after the 9 bad rolls, Black is 100% sure to lose the game. Let us play 36 games : if he doesn't double, Black should win in average 27-9 = 18 points out of 36 games, so that his equity is 18/36 = half a point ; if he does double and Red accepts, he will win 54-18 = 36 points, so his equity will be exactly one point. It comes as no surprise that after the cube has been set to 2, Black has exactly doubled his equity.

The most interesting question is now whether Red should accept or reject the double offer, and this one is now very easy to compute : we just learned that accepting the double offer would lead to an equity of one point for Black, hence of minus one point for Red. But the equity of rejecting the double is even simpler to compute and does not involve any probability : losing a single game is also worth minus one point. Both equities are exactly the same, which means that accepting or rejecting the double would make no difference at all in the long run if this position was played out a number of times. This is a almost a unique case !

So what should you do if you were Red ? Well, I would advise you to go back to this essential question : Are you feeling lucky ?