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Game Tree

Chapter-3

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Game Tree

A finite game is a game in which every possible

combination of moves leads to an end of the game.

That is, it is impossible for the game to go on forever!

A game tree is a directed graph whose nodes are

positions in a game and whose edges are moves.

In a game tree, every node corresponds to a board

position. The children of each node N are thedifferent positions that result from a single move in

N, made by the player whose turn it is at N.

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Game Trees

Problem spaces for typical games represented as trees.

Root node represents the board configuration at which

a decision must be made as to what is the best single

move to make next. (not necessarily the initial

configuration)

Evaluator function rates a board position. f(board) (a

real number).

Arcs represent the possible legal moves for a player (no

costs associates to arcs

Terminal nodes represent end-game configurations (the

result must be one of win, lose, and draw, possibly

with numerical payoff)

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Game Tree

The root of the tree is the starting position, or the

position from which we want to find a move.

The leaves (nodes which have no children) in the

game tree correspond to terminating positions;

positions in which the game has ended.

A portion of the game tree for "Tic-tac-toe" is shown

in Figure. Note that each level in the tree pertains to the moves

made by one particular player.

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Importance

Game trees are important in artificial intelligence

because one way to pick the best move in a game is to

search the game tree using the minimax algorithm or

its variants.

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An example (partial) game tree for Tic-Tac-Toe

-

f(n) = +1 if the position is a

win for X.

f(n) = -1 if the position is a

win for O.

f(n) = 0 if the position is a

draw.

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Some Chess

Positions

and theirEvaluations

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Minimax Rule

Goal of game tree search: to determine one move for Max

player that maximizes the guaranteed payofffor a given

game tree for MAX

Regardless of the moves the MIN will take

The value of each node (Max and MIN) is determined by

(back up from) the values of its children

MAX plays the worst case scenario:

Always assume MIN to take moves to maximize his pay-

off (i.e., to minimize the pay-off of MAX)

For a MAX node, the backed up value is the maximum of

the values associated with its children

For a MIN node, the backed up value is the minimum of

the values associated with its children

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Game Tree

Each node represents a board position, and the children of eachnode are the legal moves from that position.

To score each position, we will give each position which isfavorable for player 1 a positive number (the more positive,the more favorable).

Similarly, we will give each position which is favorable forplayer 2 a negative number (the more negative, the morefavorable).

In our tic tac toe example, player 1 is 'X', player 2 is 'O', andthe only three scores we will have are +1 for a win by 'X', -1for a win by 'O', and 0 for a draw.

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A player can play perfect tic-tac-toe if they choose the move with thehighest priority in the following table[3].

1) Win: If you have two in a row, play the third to get three in a row.

2) Block: If the opponent has two in a row, play the third to blockthem.

3) Fork: Create an opportunity where you can win in two ways.

4) Block Opponent's Fork:

Option 1: Create two in a row to force the opponent into defending,as long as it doesn't result in them creating a fork or winning. Forexample, if "X" has a corner, "O" has the center, and "X" has theopposite corner as well, "O" must not play a corner in order to win.(Playing a corner in this scenario creates a fork for "X" to win.)

Option 2: If there is a configuration where the opponent can fork,block that fork.

5) Center: Play the center. 6) Opposite Corner: If the opponent is in the corner, play the

opposite corner.

7) Empty Corner: Play an empty corner.

8) Empty Side: Play an empty side.