Appendix A: Mobility Maps

Appendix A: Mobility Maps

Unrestricted mobility (positional pressure) maps for the king knight in the starting position:

1 move map (black knight represents potential move):

2 move map (moving a piece out of the way or doing nothing takes a move):

3 move map:

┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │ . │   │ . │   │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │   │ . │   │ . │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │ o │ . │ o │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │ o │ . │ o │ . │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ . │ . │   │ ▒ │ . │ ▒ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │ o │ ▒ │ o │ . │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │ . │   │ o │   │ N │   │
└───┴───┴───┴───┴───┴───┴───┴───┘
influence diagram, w knight on g1

If a piece is on our 3-move map for the knight, then it is possible to attack it or defend it in 3 moves, which include waiting moves or moves which move a piece out of the way.

Keep in mind that we need to take into account the location of the other pieces on the chessboard when we generate our mobility maps for each piece. If we trace mobility through a friendly piece, we must consider whether or not we can move this piece out of the way before we can continue to trace mobility in that particular direction. In Example Number 1 in Appendix E we can see this process in more detail.

So looking at this 3-move map and a diagram of the starting position, We can determine that the White knight can attack 3 enemy pieces in 3 moves. We can defend 8 of our own pieces in 3 moves (the knight cannot defend itself).

Here are the mobility (positional pressure) maps for the king bishop in the starting position:

1 move map (Black bishop represents a potential move):

2 move map (need to count waiting moves or moving pieces out of the way):

3 move map:

┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ . │   │ . │   │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │   │ . │   │ . │   │ . │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ o │   │ . │   │ . │   │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ o │   │ . │   │ . │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │   │ o │   │ . │   │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │   │ . │   │ ▒ │   │ ▒ │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │ . │   │ B │   │ . │
└───┴───┴───┴───┴───┴───┴───┴───┘
influence diagram, w bishop on f1.

So (looking at a diagram of the starting position) in 3 moves we can *potentially* attack 4 enemy pieces and defend 6 of our own pieces.

Here are the mobility maps for white's queen and king rook in the starting position - keep in mind that we are taking in to account the presence of other pieces on the board and the fact that we must spend one move to move a friendly piece out of the way, if necessary:

1st order queen mobility map

2nd order queen mobility map

3rd order queen mobility map

┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │   │ . │   │ . │ . │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ . │   │ . │ . │ . │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ . │ . │ . │ . │ . │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ o │ . │ . │ o │ . │ . │ o │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ o │ . │ o │ . │ o │ . │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ ▒ │ ▒ │ ▒ │ . │ . │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │ . │ ▒ │ Q │ ▒ │ . │   │   │
└───┴───┴───┴───┴───┴───┴───┴───┘
influence diagram, w queen on d1

1st order rook mobility map

2nd order rook mobility map

3rd order rook mobility map
┌───┬───┬───┬───┬───┬───┬───┬───┐
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │   │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │ . │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │   │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│ . │ . │ . │ . │ . │ . │ . │ o │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │   │ . │ ▒ │
├───┼───┼───┼───┼───┼───┼───┼───┤
│   │   │   │   │   │ o │ ▒ │ R │
└───┴───┴───┴───┴───┴───┴───┴───┘
influence diagram, w rook on h1.

Now things get a little complicated when we generate these maps because we must consider capturing enemy pieces or moving friendly pieces out of the way. Our mobility tables must keep track of all this, in order to accurately determine the piece mobility (positional pressure).

The goal of this exercise is to demonstrate how a machine can be encouraged to form attacking and defensive plans by *accurately* rewarding the placement of a piece based on its attacking and defensive potential, in the form of positional pressure.

The generation of these "maps" is one step the machine could take. The assignment of points based on these maps is another issue, but even a rough guess would play a tolerable game of chess.