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Artificial Intelligence. Games 1: Game Tree Search. Ian Gent 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

Artificial Intelligence

Games 1: Game Tree Search

Ian Gent

Artificial intelligence1

Artificial Intelligence

Game Tree Search

Part I : Game Trees

Part II: MiniMax

Part III: A bit of Alpha-Beta

Perfect information games
Perfect Information Games

  • 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

Game trees
Game Trees

  • 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
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 = -

The problem with game trees
The problem with Game trees

  • Game trees are huge

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

    • Checkers/Draughts about 1040

    • Chess about 10 120

    • 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

Coping with impossibility
Coping with impossibility

  • 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

Static evaluation
Static evaluation

  • 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

O s and x s
O’s and X’s

  • 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

Minimax procedure
Minimax procedure

  • 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

What s wrong with minimax
What’s wrong with MiniMax

  • 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 search
Alpha-Beta search

  • 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

Alpha and beta values
Alpha and Beta values

  • 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

Alpha beta in action
Alpha Beta in Action

  • 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)

Summary and next lecture
Summary and Next Lecture

  • 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