Min-Max Trees

1
Min-Max Trees
Yishay Mansour

• Based on slides by
• Rob Powers
• Ian Gent

2
Two Players Games

• One Search Tree for both Players
• Even layers Max Player move
• Odd Layers Min Player move
• The state evaluated according to heuristic
  function.

3
MinMax search strategy

• Generate the whole game tree. (Or up to a
  constant depth)
• Evaluate Terminal states (Leafs)
• propagate Min-Max values up from leafs
• Search for MAX best next move, so that no matter
  what MIN does MAX will be better off
• For branching factor b and depth search d the
  complexity is O(bd)

4
MinMax first Example
5
1
1
-2
-1
6
Cuting Off Search

• We want to prune the tree stop exploring
  subtrees with values that will not influence the
  final MinMax root decision
• In the worst case, no pruning.
• The complexity is O(bd).
• In practice, O(bd/2), with branching factor of
  b1/2 instead of b.

7
Alpha Beta First Example
8
Alpha and Beta values

• At a Max node we will store an alpha value
• the alpha value is lower bound on the exact
  minimax score
• the true value might be ? ?
• if we know Min can choose moves with score lt ?
• then Min will never choose to let Max go to a
  node where the score will be ? or more
• At a Min node, ? value is similar but opposite
• Alpha-Beta search uses these values to cut search

9
Alpha Beta in Action

• Why can we cut off search?
• Beta 1 lt alpha 2 where the alpha value is at
  an ancestor node
• At the ancestor node, Max had a choice to get a
  score of at least 2 (maybe more)
• Max is not going to move right to let Min
  guarantee a score of 1 (maybe less)

10
Alpha and Beta values

• Max node has ? value
• the alpha value is lower bound on the exact
  minimax score
• with best play Max can guarantee scoring at least
  ?
• Min node has ? value
• the beta value is upper bound on the exact
  minimax score
• with best play Min can guarantee scoring no more
  than ?
• At Max node, if an ancestor Min node has ? lt ?
• Mins best play must never let Max move to this
  node
• therefore this node is irrelevant
• if ? ?, Min can do as well without letting Max
  get here
• so again we need not continue

11
Alpha-Beta Pruning Rule

• Two key points
• alpha values can never decrease
• beta values can never increase
• Search can be discontinued at a node if
• Max node
• the alpha value is ? the beta of any Min ancestor
• this is beta cutoff
• Min node
• the beta value is ? the alpha of any Max
  ancestor
• this is alpha cutoff

12

2b Left-gtRight
? -?, ? ?
13
2b Left-gtRight
? 7, ? ?
? -?, ? 4
? 6, ? ?
? 4, ? 6
? -?, ? ?
? -?, ? 4
? 8, ? ?
? 6, ? 8
(Alpha pruning)
? 4, ? ?
? 8, ? 6
? 4, ? 3
? 4, ? 8
? 5, ? 6
(Alpha pruning)
? 8, ? ?
? 4, ? 8
?5, ?8
?5, ?6
14
Beta Pruning
? 4, ? ?
? -?, ? 4
? 4, ? 8
? -?, ? ?
? -?, ? 4
? 8, ? ?
? 4, ? 8
(Alpha pruning)
? 4, ? ?
? 8, ? 6
? 4, ? 3
? 4, ? 8
(Alpha pruning)
9
? 8, ? ?
? 4, ? 8
15
Beta Pruning
? 4, ? ?
? -?, ? 4
? 4, ? 8
? -?, ? ?
? -?, ? 4
? 8, ? ?
? 9, ? 8
(Beta pruning)
(Alpha pruning)
? 4, ? ?
? 8, ? 6
? 4, ? 3
? 4, ? 8
(Alpha pruning)
9
? 8, ? ?
? 4, ? 8
16
2b Right-gtLeft
? -?, ? ?
17
2b Right-gtLeft
? 7, ? ?
? -?, ? ?
? 7, ? 6
? 7, ? 7
(Alpha pruning)
(Alpha pruning)
? 7, ? ?
? 7, ? ?
(Alpha pruning)
? 7, ? 6
? 7, ? ?
? 7, ? -1
(Alpha pruning)
?7, ??
?7, ??
?7, ??