Equities and GNUBG


Trice’s introduction to equity gives a good basic overview of what equity is, but if you are like me you are not often settling short bearoffs in money games. The other place that most of us will encounter equities in our bot evaluations. I would have thought everyone interested enough in backgammon to be reading this blog would have at least learned a little about using a bot, but I am finding that is no the case. Below is example of a GNUBG move analysis, without all of the rollout details. The position and details of the rollout are not important for our purposes.

# Ply Move Equity
1 R 9/4 6/4 +1.029
0.774 0.068 0.002 0.226 0.023 0.001 +0.595 +1.029
0.003 0.002 0.000 0.003 0.001 0.000 0.007 0.021
  2 R 8/1 +0.723 ( -0.306)
0.699 0.073 0.002 0.301 0.041 0.001 +0.430 +0.723
0.003 0.002 0.000 0.003 0.002 0.000 0.007 0.017
  3 0 14/9 6/4 +0.338 ( -0.691)
  0.611 0.104 0.004 – 0.389 0.089 0.002  
  0-ply cubeful prune [expert]  

The first two moves are from a rollout, as the R in the ply indicates. You can get all the details of the rollout if you want, ie. number of games, playing levels, etc. I did the rollout because I wanted to be able to look at the cubeless equities – more on that in a moment.

The first row of numbers under the move are the winning and losing probablities and equities, as follows:

wins, win gammons, win backgammons, lose, lose gammons, lose backgammons, cubeless equity, and cubeful equity.

The numbers under those are the standard deviations for those values, you could use those to calculate confidence intervals and statistical significance if you wanted to.

The first thing to keep in mind is that the wins and losses includes gammons and backgammons, and the gammons include backgammons. So to calculate the cubeless equity you need to correct for those, so to get the cubeless equity:


If you run this calculation you will get 0.594, the differences between this and the 0.595 is likely due to rounding as GNUBG will carry more decimal places than shown in the calculation.

So if we compare the cubeless equity for the first and second play we see that the first play is better by 0.595-0.430 or 0.165. Another way to think about this is that if you played 100 games for $1 a game from this position you would win $16.50 more by playing the first move instead of the second. (Of course this is only the expected amount, even if you played the 100 games out perfectly you would not likely get that exact result due to all the dice variables, if you played it out a million times you would get closer because you would begin to take into account all the possible dice rolls).

The main point is the first play is significantly better than the second. The last number is cubeful equity, this is the number that really matters in most cases since you will likely be playing with the cube. Getting to the cubeful equity is a lot harder. If you want to know more about calculation of this value you the gnubg documentation discusses it (http://www.gnubg.org/documentation/doku.php?id=appendix#8).

It is enough to know, however, that the value of the cube and doubles in included in this number. 

Once word of caution when looking at the best play in match is that the best play may change based on the current score. For example, you will find situations where the play with the highest equity is not the play that wins the most games but wins enough more gammons to make up for losing a few more games. But, if you evaluate that play in a match context where the person making the play only needs one point to win the match then the play that wins the most games will come out best because winning a gammon has no value in that situation.


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