Chess, Mathematics, and Trees

Imagine chess mapped out as a tree of possibilities. White has 20 moves from which to choose on the first turn. For each of those, black can select one of 20 possible responses. And so on.

Obviously, some moves are considered worse than others, otherwise GM's would be simply rolling dice and flipping coins to determine their moves. How do you know what move is better?

Chess Axioms

1. It is possible to force an opponent to make certain moves. When a certain set of moves can force the winning of material, it is known as a tactic.

2. One's position is defined as the movement possibilities of one's pieces. A person has a better position when the person's pieces have more possibility for movement. (Note that I say movement, not necessarily space--although the two often coincide. Take the Hedgehog position as an example--although one side is cramped, both sides may be bursting equally with possibilities.) Returning to the tree analogy, the person with more branches on his or her turn has the better position.

In other words, if you take a certain position where one side is winning, then use a random number generator to pick the next move, the person with the better position will have a higher chance of winning, because there are more winning possibilities given that position. I've never tried a simulation, so you'll have to either do it yourself, or take my word for it.

3. Material is a positional advantage. Having one more piece on the board means that the piece has movement/attacking potential, too. If we go back to the definition of "position", it becomes clear why material helps the position: each piece represents an entire branch of the tree. More branches tends to mean a bigger tree. Of course, exceptions exist: when branches are thin and bare, it's not much of an addition to the tree. Likewise, the positions of two sides may be considered equal if both sides have equal possibilities, despite differences in material. (Take the King's Gambit for example, where white is down a pawn for more freedom of movement.)

There are likely other solid and universal ideas in chess that can be added to the list of axioms, but these are what I was able to come up with and back up with mathematical ideas.

Now that we have gone over the axioms, let's build off of those ideas.

In chess, one must always keep position in mind. To newbies like me, the idea seems so abstract, it is often difficult to determine how to improve one's position in the long term. Recalling our second axiom, it makes sense that limiting your opponent's moves is one way to improve your own position--take the prophylactic h3 in the Ruy Lopez, for example, or even the bishop pin that defines the Ruy on the third move. Thus, a good positional move is like a well-planned trimming of a limb. We are essentially pruning our chess tree, leaving branches with the most number of desirable fruit.

Just because a person is good at hacking off limbs from the tree doesn't make them a good chess player. They must be able to peer into the branches and cut only the ones that bear less fruit. In other words, if you play chess and try to only play forcing moves, the forced moves might actually be against your favor. As a drastic example:

24. Qxd7 absolutely forces black to reply Kxd7. However, white would obviously be losing as a result of the move (without the queen, white has significantly reduced the possibilities of his pieces' movement). The move is a very dull example of a tactical failure. Even though the player of white recognized that the move greatly limited black's movement possibilities on the next turn, the player of white failed to see the long-term effects of black's forced one-move tactic: Kxd7. When playing a tactically unsound move, you lose material, and your position (by definition) worsens. Thus, we can compare tactical play to being able to "see" into the branches: the further a player can calculate, the better his or her ability to work out tactics, i.e. spot the actual fruit within those branches.

This ends my analysis of chess for the night, but hopefully gives players of all levels a different way to look at chess, and the relationship between "tactics" and "position".

P.S. Please stop making those threads about tactics vs. position debates, now that you know how they are truly intertwined!

I can't believe no one's commented on this - I think it's great! I'm sure it could use some improvement so I'll start looking into this in detail.

"Imagine chess mapped out as a tree of possibilities."

Some have indeed imagined this. I think GM Kotov uses the idea of candidate moves in his books. And then computer chess in general uses this type of branching structure to assign a value to all possible strings of moves.

Now...there are only so many moves in chess. For example K x Q d7 is one of a finite number of moves possible in chess, but it all depends on context.