Game Playing

Introduction• One of the earliest areas in artificial

intelligence is game playing. • Two-person zero-sum game.• Games for which the state space is

small enough – generate the entire space.

• Games for which the entire space cannot be generated.

The Game NIM7

6-1 5-2 4-3

5-1-1 4-2-1 3-2-2 3-3-1

4-1-1-1 3-2-1-1 2-2-2-1

3-1-1-1 2-2-1-1

2-1-1-1-1-1

NIM- MAX Plays First

.

7

6-1 5-2 4-3

5-1-1 4-2-1 3-2-2 3-3-1

4-1-1-1 3-2-1-1 2-2-2-1

3-1-1-1 2-2-1-1

2-1-1-1-1-1

MAX

MIN

MAX

MAX

MIN

MIN

1

0

1

NIM- MIN Plays First

.

7

6-1 5-2 4-3

5-1-1 4-2-1 3-2-2 3-3-1

4-1-1-1 3-2-1-1 2-2-2-1

3-1-1-1 2-2-1-1

2-1-1-1-1-1

MIN

MAX

MIN

MIN

MAX

MAX

0

1

0

Minimax AlgorithmRepeat• If the limit of search has been reached, compute the

static value of the current position relative to the appropriate player. Report the result.

• Otherwise, if the level is a minimizing level, use the minimax on the children of the current position. Report the minimum value of the results.

• Otherwise, if the level is a maximizing level, use the minimax on the children of the current position. Report the maximum of the results.

Until the entire tree is traversed.

Minimax Applied to NIM

.

7

6-1 5-2 4-3

5-1-1 4-2-1 3-2-2 3-3-1

4-1-1-1 3-2-1-1 2-2-2-1

3-1-1-1 2-2-1-1

2-1-1-1-1-1

MIN

MAX

MIN

MIN

MAX 0

1

0

0

1

MAX

0

1 0

0 0 00

0 0

Generating the Game Tree to a Depth

• In some cases the game tree will be too large to generate.

• In this case the tree is generated to a certain depth or ply.

• Heuristic values are used to estimate how promising a node is.

• Horizon effect.

Example

3

3 0 2

3 9 0 7 2 6

2 3 5 9 0 7 4 2 1 5 6

MAX

MIN

MAX

MIN

Heuristic for Tic-Tac-Toe

• h(n) = x(n) - o(n) where –x(n) is the total of MAX’s possible

winning (we assume MAX is playing x) –o(n) is the total of the opponent’s,

i.e. MIN’s winning lines • h(n) is the total evaluation for a state

n.