7 Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent)Time complexity? O(bm)Space complexity? O(bm) (depth-first exploration)For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible
13 Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruningWith "perfect ordering," time complexity = O(bm/2) doubles depth of searchA simple example of the value of reasoning about which computations are relevant (a form of metareasoning)
14 Why is it called α-β?α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for maxIf v is worse than α, max will avoid it prune that branchDefine β similarly for min
17 How much do we gain? Assume a game tree of uniform branching factor b Minimax examines O(bh) nodes, so does alpha-beta in the worst-caseThe gain for alpha-beta is maximum when:The MIN children of a MAX node are ordered in decreasing backed up valuesThe MAX children of a MIN node are ordered in increasing backed up valuesThen alpha-beta examines O(bh/2) nodes [Knuth and Moore, 1975]But this requires an oracle (if we knew how to order nodes perfectly, we would not need to search the game tree)If nodes are ordered at random, then the average number of nodes examined by alpha-beta is ~O(b3h/4)
18 Heuristic Ordering of Nodes Order the nodes below the root according to the values backed-up at the previous iterationOrder MAX (resp. MIN) nodes in decreasing (increasing) values of the evaluation function computed at these nodes
19 Games of imperfect information Minimax and alpha-beta pruning require too much leaf-node evaluations.May be impractical within a reasonable amount of time.SHANNON (1950):Cut off search earlier (replace TERMINAL-TEST by CUTOFF-TEST)Apply heuristic evaluation function EVAL (replacing utility function of alpha-beta)
20 Cutting off search Change: if TERMINAL-TEST(state) then return UTILITY(state)intoif CUTOFF-TEST(state,depth) then return EVAL(state)Introduces a fixed-depth limit depthIs selected so that the amount of time will not exceed what the rules of the game allow.When cuttoff occurs, the evaluation is performed.
21 Heuristic EVALIdea: produce an estimate of the expected utility of the game from a given position.Performance depends on quality of EVAL.Requirements:EVAL should order terminal-nodes in the same way as UTILITY.Computation may not take too long.For non-terminal states the EVAL should be strongly correlated with the actual chance of winning.Only useful for quiescent (no wild swings in value in near future) states
24 Heuristic difficulties Heuristic counts pieces won
25 Horizon effect Fixed depth search thinks it can avoid the queening move
26 Games that include chance Possible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)
27 Games that include chance chance nodesPossible moves (5-10,5-11), (5-11,19-24),(5-10,10-16) and (5-11,11-16)[1,1], [6,6] chance 1/36, all other chance 1/18
28 Games that include chance [1,1], [6,6] chance 1/36, all other chance 1/18Can not calculate definite minimax value, only expected value
29 Expected minimax value EXPECTED-MINIMAX-VALUE(n)=UTILITY(n) if n is a terminalmaxs successors(n) MINIMAX-VALUE(s) if n is a max nodemins successors(n) MINIMAX-VALUE(s) if n is a max nodes successors(n) P(s) . EXPECTEDMINIMAX(s) if n is a chance nodeThese equations can be backed-up recursively all the way to the root of the game tree.
30 Position evaluation with chance nodes Left, A1 winsRight A2 winsOutcome of evaluation function may not change when values are scaled differently.Behavior is preserved only by a positive linear transformation of EVAL.פונקציות הערכה למשחקים עם מזל מתנהגות שונה מאשר אלה של שחמט למשל – שבהן רק הסדר קובע.כאן הסקלה יכולה להכריע.
32 Checkers: Tinsley vs. Chinook Name: Marion TinsleyProfession: Teach mathematicsHobby: CheckersRecord: Over 42 years loses only 3 games of checkersWorld champion for over 40 yearsMr. Tinsley suffered his 4th and 5th losses against Chinook
33 Chinook First computer to become official world champion of Checkers! Has all endgame table for 10 pieces or less: over 39 trillion entries.
34 Chess: Kasparov vs. Deep Blue 5’10”176 lbs34 years50 billion neurons2 pos/secExtensiveElectrical/chemicalEnormousHeightWeightAgeComputersSpeedKnowledgePower SourceEgoDeep Blue6’ 5”2,400 lbs4 years32 RISC processors VLSI chess engines200,000,000 pos/secPrimitiveElectricalNone1997: Deep Blue wins by 3 wins, 1 loss, and 2 draws
35 Chess: Kasparov vs. Deep Junior 8 CPU, 8 GB RAM, Win 20002,000,000 pos/secAvailable at $100August 2, 2003: Match ends in a 3/3 tie!
36 Othello: Murakami vs. Logistello Takeshi MurakamiWorld Othello Champion1997: The Logistello software crushed Murakami by 6 games to 0
37 Go: Goemate vs. ?? Name: Chen Zhixing Profession: Retired Computer skills:self-taught programmerAuthor of Goemate (arguably the best Go program available today)Gave Goemate a 9 stonehandicap and still easilybeat the program,thereby winning $15,000
38 Go: Goemate vs. ??Name: Chen ZhixingProfession: RetiredComputer skills:self-taught programmerAuthor of Goemate (arguably the strongest Go programs)Go has too high a branching factor for existing search techniquesCurrent and future software must rely on huge databases and pattern-recognition techniquesGave Goemate a 9 stonehandicap and still easilybeat the program,thereby winning $15,000Jonathan Schaeffer
39 Backgammon1995 TD-Gammon by Gerald Thesauro won world championship on 1995BGBlitz won 2008 computer backgammon olympiad
40 SecretsMany game programs are based on alpha-beta + iterative deepening + extended/singular search + transposition tables + huge databases + ...For instance, Chinook searched all checkers configurations with 8 pieces or less and created an endgame database of 444 billion board configurationsThe methods are general, but their implementation is dramatically improved by many specifically tuned-up enhancements (e.g., the evaluation functions) like an F1 racing car
41 Perspective on Games: Con and Pro Chess is the Drosophila of artificial intelligence. However, computer chess has developed much as genetics might have if the geneticists had concentrated their efforts starting in 1910 on breeding racing Drosophila. We would have some science, but mainly we would have very fast fruit flies.John McCarthySaying Deep Blue doesn’t really think about chess is like saying an airplane doesn't really fly because it doesn't flap its wings.Drew McDermott
42 Other Types of Games Multi-player games, with alliances or not Games with randomness in successor function (e.g., rolling a dice) Expectminimax algorithmGames with partially observable states (e.g., card games) Search of belief state spacesSee R&N p