Game-Playing

Read Chapter 6

Adversarial Search

Game Types

• Two-person games vs multi-person– chess vs monopoly

• Perfect Information vs Imperfect– checkers vs card games

• Deterministic vs Non-deterministic– go vs backgammon

Two-Person Games: Perfect Information

• BF = branching factor (average)

• Chess: BF ~36– expert level

• Checkers: BF ~ 8, world champion

• Othello: BF ~10, better world champion

• Go: BF ~200– $2 million prize

MiniMax Algorithm (perfect information, 2 person game)

• Assume: evaluation of terminal position• Win = +1, Loss = -1, Draw = 0.• Descendants of max node is min node, etc.• Algorithm: recursive

– Value Max Node = max(descendants of node)– Value Min Node = min(descendants of node)– Value of terminal node: by evaluation function

• Applies to any tree with values assigned to leaves.• Needed if full tree too large.

MiniMax Example

4

3

2

4

47

63 8

4 2 6

Optimal Play

• Make move that yields highest minimax score.

• Computation: search: depth-first

• Time = b^d

• Memory= b*d

Applied to Chess• Average game is 40+ moves

• Tree to large to reach terminal positions

• Static board evaluation of worthiness

• Uses Partial Tree

• MiniMax yields optimal value for restricted tree, with values assigned by evaluation.

• No theorems connecting valuation on partial tree to estimates for complete tree.

Alpha-Beta Algorithm

• Yields exactly same value as minimax

• Knuth analyzed: time or nodes = O(b^d/2)

• Doubles depth of search with same time.

• Constant depends on ordering of nodes

• Iterative deepening alpha/beta achieves better ordering. (reorder after depth)

Alpha-beta Algorithm• Each node is assigned a range of values:

[alpha,beta]. The real value will lie between.• The root is assigned [-inf,+inf]. • For any max node N with values [A,B]

– if a son has value >=C, then N has new range [C,B].– If interval is empty, all nodes below cut.

• For any min node N with values [A,B]– if son has value <=D, then N updated to [A,D].

• Formal code in text.• http://www.cs.mcgill.ca/~cs251/OldCourses/

1997/topic11/

Alpha-Beta Example

alphabeta

alphabeta

alphabeta

alphabeta

47

63 8 4 2 6

Alpha-Beta Example

4

<=3no help

<=4no help

4f>=4

4 76f<=6

3f<=3

8cut

4f<=4

2cut

6cut

Max level

Min Level

(1,2,2) Nim

Multi-player Games

• Extension of minimax– assign a vector of values to each position– vector has value relative to each player– Each player maximizes choice– Equals minimax for 2 person game

• No variations like alpha-beta

Games with Uncertainty

• Card games like hearts or bridge

• Backgammon (roles of dice)

• Expectimax– Does it work?– Theoretically nice, but where’s the meat – for

what games was it successful?

Certainty from Uncertainty

• Simulation– Replace unknown world by specific world– simulate (or use alpha-beta)– Each simulation yields a play– Vote

• Works for hearts and bridge play– bridge high level card play can’t make

information gathering plans

What about War

• Games are games – restricted uncertainty

• What are the operators in war?– unknown effects– unknown number

• What is the state?– unknown