Game-Playing

1
Game-Playing

• Read Chapter 6
• Adversarial Search

2
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

3
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

4
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.

5
MiniMax Example
6
Optimal Play

• Make move that yields highest minimax score.
• Computation search depth-first
• Time bd
• Memory bd

7
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.

8
Alpha-Beta Algorithm

• Yields exactly same value as minimax
• Knuth analyzed time or nodes O(bd/2)
• Doubles depth of search with same time.
• Constant depends on ordering of nodes
• Iterative deepening alpha/beta achieves better
  ordering. (reorder after depth)

9
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 gtC, 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 ltD, then N updated to A,D.
• Formal code in text.
• http//www.cs.mcgill.ca/cs251/OldCourses/1997/top
  ic11/

10
Alpha-Beta Example
11
Alpha-Beta Example
12
(1,2,2) Nim

13
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

14
Games with Uncertainty

• Card games like hearts or bridge
• Backgammon (roles of dice)
• Expectimax
• Does it work?
• Theoretically nice, but wheres the meat for
  what games was it successful?

15
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 cant make
  information gathering plans

16
What about War

• Games are games restricted uncertainty
• What are the operators in war?
• unknown effects
• unknown number
• What is the state?
• unknown