"Equity"
From: Chuck Bower
Date: 1 October 1996
Subject: Equity
Forum: rec.games.backgammon
> Can someone please explain equity to me and how it is calculated? I'll try to answer this question in steps. Hopefully I will neither insult the reader (by explaining something already known) nor lose you (by making things too complicated and using "jargon"). Here goes: 1) in general, "equity" is "value". The word is common in financial circles, especially in accounting. 2) in backgammon, "equity" is equivalent to "expectation"--a term used by mathematicians, especially statisticians. 3) There are two kinds of equity in backgammon: "Cubeless", and "Cubeful". Cubeless equity is the "value" of a position if THERE IS NO DOUBLING CUBE around, so that the game will be played to completion, and six outcomes are possible: either side can win a simple game, a gammon, or a backgammon. So, if we define: W = player on roll's chances of winning a simple game, G = player on roll's chances of winning a gammon, B = player on roll's chances of winning a backgammon, w = player NOT on roll's chances of winning a simple game, g = player NOT on roll's chances of winning a gammon, b = player NOT on roll's chances of winning a backgammon, then we can define player on roll's equity as: W + 2*G + 3*B - w - 2*g - 3*b. (* is multiplication sign). Some things worth noting: a) W + G + B + w + g + b = 1. That is becuase SOMETHING MUST HAPPEN! b) The equity for the player NOT on roll is just (-1) times the equity of the player on roll. Another way of saying this is that if we add the equity for the player on roll to that of his/her opponent, the sum is ZERO. c) cubeless equity will be somewhere in the range -3 to 3. Now, what about "cubeful" equity? This is a slightly more complicated case. Two things must now be included. If we asume the same (cubeless) definitions above, then a) multiply the current value of the doubling cube times the above "cubeless" numbers to get equity. (Cubeless equity can be bigger than 3 or less than -3). b) ownership of the doubling cube now plays a role, so some adjustment must be made for that. Typically +5-10% is added to the equity of the player on roll (and equivalently subtracted from the player NOT on roll) to account for cube ownership in money games. There are more complicated ways to figure it, but this message is ALREADY GETTING LONG! At matches, there may or may not be cube ownership equity, depending on where you are in the match. That, too, can be quantified. WARNING #1: With the doubling cube in play, the cubeless quantities (W, G, etc.) are NOT each players true winning/losing chances. That is because the cube can (and often will) be used to end games prematurely AND convert some losses into wins. That is why the adjustment [b) above] was made. It is also why you need to find out if rollout results (human or robot) were performed with a doubling cube in play. WARNING #2: For anyone familiar with Jellyfish evaluation and rollouts, note that the numbers in the JF results windows are NOT the values I defined above. Here is how JF defines things: Wj = player 1's chances of winning a this game (any flavor), Gj = player 1's chances of winning a gammon or backgammon, Bj = player 1's chances of winning a backgammon, wj = player 2's chances of winning a this game, gj = player 2's chances of winning a gammon or backgammon, bj = player 2's chances of winning a backgammon, NOW, equity for player 1 is: Wj + Gj + Bj - wj - gj - bj. Also, Wj + wj = 1, but the sum of all six values is not, in general. Method is equivalent, just using "oranges" instead of "apples". Be careful not to interchange the fruit, though... Chuck [email protected]