Adversarial Search and Game Playing Examples

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Presentation transcript:

Adversarial Search and Game Playing Examples

Game Tree MAX nodes MAX’s play  MIN’s play  Terminal state (win for MAX)  MIN nodes Here, symmetries have been used to reduce the branching factor

Game Tree MAX’s play  MIN’s play  Terminal state (win for MAX)  In general, the branching factor and the depth of terminal states are large Chess: Number of states: ~1040 Branching factor: ~35 Number of total moves in a game: ~100

Example: Tic-tac-Toe e(s) = number of rows, columns, and diagonals open for MAX - number of rows, columns, and diagonals open for MIN 8-8 = 0 6-4 = 2 3-3 = 0

Backing up Values Tic-Tac-Toe tree at horizon = 2 1 -1 1 -2 Best move 6-5=1 5-6=-1 5-5=0 4-5=-1 6-4=2 5-4=1 6-6=0 4-6=-2

Continuation 1 1 1 3 2 1 2 1 2 3 1 1

Example

Example The beta value of a MIN node is an upper bound on the final backed-up value. It can never increase b = 2 2

Example 1 b = 1 2 The beta value of a MIN node is an upper bound on the final backed-up value. It can never increase

Example 1 b = 1 2 a = 1 The alpha value of a MAX node is a lower bound on the final backed-up value. It can never decrease

Example 1 b = 1 2 a = 1 -1 b = -1

Example 1 b = 1 2 a = 1 -1 b = -1 Search can be discontinued below any MIN node whose beta value is less than or equal to the alpha value of one of its MAX ancestors

An example of Alpha-beta pruning 5 -3 3 2 -2 max min -3 3 -3

Example of Alpha-beta pruning Final tree max min max min max 5 -3 3 3 -3 2 -2 3

An example of Alpha-beta pruning 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 -3 3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 -3 3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 -3 3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 -3 3 5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 2 -3 3 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 3 2 -3 3 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 -3 3 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 -3 3 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 -3 3 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 -3 3 2 5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 1 -3 3 2 1 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 1 -3 3 2 1 -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 2 3 2 1 -3 3 2 1 -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 1 2 1 3 2 1 -3 3 2 1 -3 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 1 2 1 3 2 1 -3 3 2 1 -3 -5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 1 2 1 3 2 1 -3 3 2 1 -3 -5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 1 2 1 -5 3 2 1 -5 -3 3 2 1 -3 -5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 2 1 2 1 -5 3 2 1 -5 -3 3 2 1 -3 -5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 1 2 1 2 1 -5 3 2 1 -5 -3 3 2 1 -3 -5 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 1 1 2 1 2 2 1 -5 2 3 2 1 -5 2 -3 3 2 1 -3 -5 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Example 1 1 2 1 2 2 1 -5 2 3 2 1 -5 2 -3 3 2 1 -3 -5 2 5 -3 3 3 -3 2 -2 3 5 2 5 -5 1 5 1 -3 -5 5 -3 3 2

Nondeterminstic games are the games with both an element of chance and Add chance nodes to tree Example with coin flip instead of dice 0.5 0.5 0.5 0.5 2 4 7 4 6 5 -2

Example with coin flip instead of dice (cont.) 3 3 -1 0.5 0.5 0.5 0.5 2 4 -2 2 4 7 4 6 5 -2

Alpha-Beta prunning in Tic-Tac-Toe