 ## How the computer trainers work

Given is the distribution of the players on the field and the strengths (with two factors, upper number that changes, and lower (hidden) number, that remains constant during the match). Now each pass and each shot has a given fixed probability of success. Let p(A,B) be the probability that a pass from A to B is successful, and p(A) the probability that a shot from A results in a goal.

Let A1, A2, A3, ... be the players of our team. Assume A1 has the ball. What should we do?

The probability for a goal in one step is p(A1).

What about in two steps? The probability for a goal in two steps by passing first to A2 is p(A1,A2)*p(A2). Since we may pass to A3, ... as well, for the two-step option we would choose that Ai where p(A1,Ai)*p(Ai) is maximized.

We proceed in this direction until we investigate the depth-step option. "depth" is the value you can adjust on the applet. Higher depths means higher probabilities, but a high depth trainer may recommend very circumstantial play. So at the end we look at the probabilities for the different options:
1-step: p(A1),
2-step: maxip(A1,Ai)*p(Ai),
3-step: maxi,jp(A1,Ai)*p(Ai,Aj)*p(Aj),
and so on, until the depth-step option.
If A has the ball, the largest of these numbers is the base of the trainer recommendation for a pass or shot.

Usually there is only one recommendation for every player. The lines from all players to the recommendated ball receivers or the goal forms a tree. Of course this tree may change, since the strengths of the players change.

These probabilities above are computed recursively. To compute the 2-step options of the players we use the 1-step options, to compute the 3-step options we use the 2-step options, and so on.

Besides depths, what are the differences between the trainers? Vogts only knows what we know, the hidden strengths of the players are not known to him until half time, therefore before he only considers the first factors. Voeller knows the hidden factors of his team from the very beginning and takes them into account, whereas Hiddink knows all hidden factors from the very beginning. Therefore he is the best computer trainer.

Erich Prisner, October 2002