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
MinMax evaluation function

• function MinMax-Decision(game) returns an
  operator
• for each op in Operatorgame do
• Valueop MinMax-Value(Apply(op,game),gam
  e)
• end
• return the op with the highest Valueop
• function MinMax-Value(state,game) returns an
  utility value
• if Terminal-Testgame(state) then
• return Utilitygame(state)
• else if MAXs turn
• return the highest MinMax-Value of
  Successors(state)
• else (MINs turn)
• return the lowest MinMax-Value of
  Successors(state)
• end

7
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.

8
Alpha Beta First Example
9
Alpha-Beta search cutoff rules

• Keep track and update two values so far
• alpha is the value of best choice in the MAX path
  
• beta is the value of best choice in the MIN path
• Rule do not expand node n when beta lt
  alpha
• for MAX node return beta
• for MIN node return alpha

10
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

11
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)

12
Alpha and Beta values

• Max node has ? value
• the alpha value is lower bound on the exact
  minimax score
• with best play M ?x 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

13
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
• It is a Max node and
• the alpha value is ? the beta of any Min ancestor
• this is beta cutoff
• Or it is a Min node and
• the beta value is ? the alpha of any Max
  ancestor
• this is alpha cutoff

14
Alpha-Beta prunning

• function Max-Value(state,game,alpha,beta) returns
  
• minmax value of state
• if Cutoff-Test(state) then return Eval(state)
• for each s in Successor(state) do
• alpha Max(alpha,Min-Value(s,
  game,alpha,beta))
• if beta lt alpha then return beta
• end return alpha
• function Min-Value(state,game,alpha,beta) returns
  
• minmax value of state
• if Cutoff-Test(state) then return Eval(state)
• for each s in Successor(state) do
• beta Min(beta,Max-Value(s, game,alpha,beta))
• if beta lt alpha then return alpha
• end return beta

15

2b Left-gtRight
? -?, ? ?
16
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
17
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
18
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
19
2b Right-gtLeft
? -?, ? ?
20
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, ??