What is eXtreme Gammon

eXtreme Gammon is a a Neural Network based backgammon program. The Neural network is derived from GNU BG 0.00.  Modifications to the code (Neural Network) and the introduction of cube handling and match play has been made by GameSite 2000  Ltd.

eXtreme Gammon compares to  all existing backgammon programs and plays at a World-Class level.

Compared results against Snowie 2.0 3ply

Comments :

- Elo performance: +10 means that eXtreme Gammon has played at a level that would make it worth 10 points elo more than its opponent.

- In this case,  where the chances to win each game is very close to 50% , the part of luck can easily switch the result above or below 50%. This means that the results above show that there is a reasonable chance that eXtreme Gammon is  better than Snowie but it's not a certitude. In all cases the difference of level is very slight.

XG-Coach and XG-Professor

XG-Coach will comment after your move on any mistake it thinks you made, while XG-Professor can be asked advice and suggestions before your move (sending "H" in the kibitz). 
You will see that more information is given to you on cube decisions: the current chances to win the game and the chances to win the match if you double or not (or take a double or drop) are given. 

Here is an example of what you can see:

XG-Professor thinks you should not double
Chances to win this game:
You:46.7% (G:13.0% B: 0.8%)
Opp:53.3% (G:19.3% B: 0.8%)
Chances to win the match:
No double: 48.9%
Double/take: 44.4%
Double/Drop: 58.0%

This means that you have 46.7% to win the game from this position.
12.8% will be gammon, 0.8% will be backgammon and thus 33.7% will be single games.
The same way your opponent has 53.3% chances to win the game (100-46.7), 18.5% of winning gammon, 0.8% chances to win backgammon)

If you don't double you have 48.9% chance to win the whole match (this result comes form a 5 pt match at 0-0).
If you double and your opponent takes you have 44.4%
If you double and your opponent drops you have 58%

Of course your opponent will take (by dropping he would lose 13.6% chances to win the game). Thus, to know what to do, you have to compare the 2 first numbers. In this case you'd better not double because you'd lose 4.1% chance to win the game if you double.

Note about positions too good to double (where you'd better play for gammon).
in these positions you will see results like:

Chances to win the match:
No double: 65.9%
Double/take: 80.4%
Double/Drop: 70.0%

Here you can see that if you double, it's better for your opponent to drop. And if you don't double, you have more chance to win the match. This mean that you should play for the gammon.

What is a Neural Network program and how eXtreme Gammon works

A neural Network algorithm was first used in the 70's. The idea was to try to recreate how the human brain processes information to obtain an intelligent result. Even if later discoveries in biology showed that the neurons are not working the same way a computer neural network does, the results in some fields were so interesting that this kind of algorithm is still widely used in some fields such as image recognition.

Typically, a neural network works with 3 layers of neurons. The 1st row is the input and it receives data from the data to be analyzed.  The 2nd layer, called hidden, processes the data and feeds the 3rd and last layer, called the output.  Each neuron of each layer is connected to all the neurons of the next layer. This represents the axon.  An axon can carry more or less information depending on its weight. A hidden or output neuron receives the data from all the neurons in the previous level. It adds the input from all these neurons and processes the result with a simple mathematical function (1/(1-e^-x)) 

During the training process the neural network receives both the input and desired output. The neural network compares the output it computes to the desired one and modifies the weight of each axon to get a result closer to the desired one.

In the case of backgammon, the input layers contains 250 neurons. The input contains, for each player,  the position, information about prime, anchors, timing, etc. 

The output layer is composed of 5 neurons giving the following information: 

• Chances to win the game
• Chances to win a  gammon
• Chances to win a  backgammon
• Chances to lose  a gammon
• Chances to lose a backgammon

Note that the chance to lose the game is known from the 1st input.

The big interest in this algorithm is that it can compute results .
 eXtreme Gammon Neural Network commutes in less than 0.6 µs. (on an intel Core 2 Duo at 1.87Ghz)

The result can then be more accurate by having the program look ahead. This  means that when choosing to move it will check all the possible moves it can make. Then for each of the 36 rolls its opponent can get, it computes what would be the opponent's best move. The evaluation of the position is then the average of the value obtained for each roll.

The program can go deeper in the look ahead (sometime called ply) but in backgammon, the branching factor is huge. There are 36 possible rolls that can be reduced to 21 as 6-2 is the same as  2-6 and an average of 20 possible moves. This gives a branching factor of 400. It means that with no look ahead, it needs to evaluate 400 positions. With one level look ahead it will evaluate 400x400=160,000 positions, with  2 levels 64,000,000 positions !!  This explains why Backgammon programs can not be programmed using the same system as chess, where the branching factor is only about 20.

About the authors.

Xavier Dufaure de Citres has an extensive working knowledge with backgammon. He is the author of many popular programs related to backgammon: 

GammonSite server
NetGammon server
CyberGammon computer player
Master Gammon/Master Backgammon computer program (in cooperation with Oasya)

Credit is given to Gary Wong for his work on GnuBg.