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In backgammon, people always speak about 'equity', but never about 'winning probability'. To me, 'winning probability' seems much more simple and much more intuitive than 'equity'. So why is 'equity' much more commonly used than 'winning probability'?

And is there a mathematical formula that links 'equity' and 'winning probability'?

5

The reason that equity is used instead of winning probability is because it is possible to win a single game, a double game (gammon) or triple game (backgammon).

Let's say that the value of the game, or bet, is $1. (That would occur if the cube is in the middle. If it has been turned, you multiply by 2, 4, or whatever the number is on the cube.)

Let's further say that White is Playing red, they are both "bearing off," but White is ahead. White has a 55% chance of winning, and Red has a 45% chance of winning. Then White's equity in that $1 bet is .55*$1 or 55 cents, while Red's equity is .45*$1 or 45 cents.

Assume a second situation. White has a good "back game" against Red, who has started bearing off. White's chances of winning are still 55%, and Red's is 45%, but Red will probably score a gammon if White doesn't hit him before Red starts bearing off (that is 45%). For now, let's ignore the possibilities that Red will score a backgammon (a triple game) and that Red will win a single game, and assume that he gets a double game, if he doesn't get hit.

White's equity is .55*$1, or 55 cents. But because Red can win a double game, his equity is .45*$2, or 90 cents because of the gammon possibility. In this example, Red has more equity, even thought White has the higher win probability.

For a mathematica formula, let's say your chances of winning a single, double, and triple game are a, b and c, respectively. Your win probability is just a+b+c. But your equity is a*$1+b*$2+c*$3. The gammon and backgammon possibilities double and triple the value of your probabilities, b and c.

To address a comment below, in the examples above, I quoted "gross" equity. It is normally calculated as "net" equity. So if White's equity from winning a simple game were 55 cents, and Red's chances of winning a simple game were 45 cents, White's "net" equity would be 55 cents-50 cents (his share of the $1 bet), or 5 cents, and Red's "net" equity would be 45 cents- 50 cents or negative 5 cents.

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You can "translate" the word "equity" as a value of the particular position. Lets imagine we are one roll away from ending the match and we only have two checkers on deuce point.

We will win with the probabilty of 26/36 and we will lose with the probability of 10/36.

Lets also imagine that there is a friend who offers us some money and asks us to abandon the game for that price. Now what would the fare price be?

That "fare price" is what equity in fact is.

Equity = probability of winning - probability of losing.

Which means that Equity can be a negative number too.

And to return to our previous example, if our game stake was $100, how much would have an honest buyer offer us?

[(26/36)-(10/36)]*$100 = $44.44

and to complete the example, our opponent in the same match, should "sell" his position, or bail himself out, paying $44.44

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Equity is especially useful because it pre-calculates some of the analysis used in utilizing the Doubling Cube.

Consider the situation described by @Skytten: We have two chequers on the 2-point to roll; opponent has two chequers on the 1- and 2-points (a guaranteed win if he/she gets to roll). Our equity is $44.44. If we offer a double to opponent our equity becomes either $100.00 (if opponent declines) or $88.88 (if opponent accepts.

Unintuitively, opponent should accept the offered double because his/her choices are to have an equity of -$100.00 (declining) or of -$88.88 (accepting); the latter is a lesser evil.

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