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# Artificial Intelligence - PowerPoint PPT Presentation

Artificial Intelligence. Games 1: Game Tree Search. Ian Gent ipg@cs.st-and.ac.uk. Artificial Intelligence. Game Tree Search. Part I : Game Trees Part II: MiniMax Part III: A bit of Alpha-Beta . Perfect Information Games. Unlike Bridge, we consider 2 player perfect information games

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### Artificial Intelligence

Games 1: Game Tree Search

Ian Gent

ipg@cs.st-and.ac.uk

### Artificial Intelligence

Game Tree Search

Part I : Game Trees

Part II: MiniMax

Part III: A bit of Alpha-Beta

• Unlike Bridge, we consider 2 player perfect information games

• Perfect Information: both players know everything there is to know about the game position

• no hidden information (e.g. opponents hands in bridge)

• no random events (e.g. draws in poker)

• two players need not have same set of moves available

• examples are Chess, Go, Checkers, O’s and X’s

• Ginsberg made Bridge 2 player perfect information

• by assuming specific random locations of cards

• two players were North-South and East-West

• A game tree is like a search tree

• nodes are search states, with full details about a position

• e.g. chessboard + castling/en passant information

• edges between nodes correspond to moves

• leaf nodes correspond to determined positions

• e.g. Win/Lose/Draw

• number of points for or against player

• at each node it is one or other player’s turn to move

Game Trees  Search Trees

• Strong similarities with 8s puzzle search trees

• there may be loops/infinite branches

• typically no equivalent of variable ordering heuristic

• “variable” is always what move to make next

• One major difference with 8s puzzle

• The key difference is that you have an opponent!

• Call the two players Max and Min

• Max wants leaf node with max possible score

• e.g. Win = +

• Min wants leaf node with min score,

• e.g. Lose = -

• Game trees are huge

• O’s and X’s not bad, just 9! = 362,880

• Go utterly ludicrous, e.g. 361! 10750

• Recall from Search1 Lecture,

• It is not good enough to find a route to a win

• Have to find a winning strategy

• Unlike 8s/SAT/TSP, can’t just look for one leaf node

• typically need lots of different winning leaf nodes

• Much more of the tree needs to be explored

• It is usually impossible to solve games completely

• Connect 4 has been solved

• Checkers has not been

• we’ll see a brave attempt later

• This means we cannot search entire game tree

• we have to cut off search at a certain depth

• like depth bounded depth first, lose completeness

• Instead we have to estimate cost of internal nodes

• Do so using a static evaluation function

• A static evaluation function should estimate the true value of a node

• true value = value of node if we performed exhaustive search

• need not just be /0/- even if those are only final scores

• can indicate degree of position

• e.g. nodes might evaluate to +1, 0, -10

• Children learn a simple evaluation function for chess

• P = 1, N = B = 3, R = 5, Q = 9, K = 1000

• Static evaluation is difference in sum of scores

• chess programs have much more complicated functions

• A simple evaluation function for O’s and X’s is:

• Count lines still open for maX,

• Subtract number of lines still open for min

• evaluation at start of game is 0

• after X moves in center, score is +4

• Evaluation functions are only heuristics

• e.g. might have score -2 but maX can win at next move

• O - X

• - O X

• - - -

• Use combination of evaluation function and search

• Assume that both players play perfectly

• Therefore we cannot optimistically assume player will miss winning response to our moves

• E.g. consider Min’s strategy

• wants lowest possible score, ideally - 

• but must account for Max aiming for + 

• Min’s best strategy is:

• choose the move that minimises the score that will result when Max chooses the maximising move

• hence the name MiniMax

• Max does the opposite

• Statically evaluate positions at depth d

• From then on work upwards

• Score of max nodes is the max of child nodes

• Score of min nodes is the min of child nodes

• Doing this from the bottom up eventually gives score of possible moves from root node

• hence best move to make

• Can still do this depth first, so space efficient

• Minimax is horrendously inefficient

• If we go to depth d, branching rate b,

• we must explore bd nodes

• but many nodes are wasted

• We needlessly calculate the exact score at every node

• but at many nodes we don’t need to know exact score

• e.g. outlined nodes are irrelevant

• Alpha-Beta = 

• Uses same insight as branch and bound

• When we cannot do better than the best so far

• we can cut off search in this part of the tree

• More complicated because of opposite score functions

• To implement this we will manipulate alpha and beta values, and store them on internal nodes in the search tree

• At a Mx 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 < 

• then Min will never choose to let Max go to a node where the score will be  or more

• At a Min node, we will store a beta value

• the beta value is upper bound on the exact minimax score

• the true value might be  

• Alpha-Beta search uses these values to cut search

• Why can we cut off search?

• Beta = 1 < 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)

• Game trees are similar to search trees

• but have opposing players

• Minimax characterises the value of nodes in the tree

• but is horribly inefficient

• Use static evaluation when tree too big

• Alpha-beta can cut off nodes that need not be searched

• Next Time: More details on Alpha-Beta