Optimal strategy Perfect play for deterministic games Minimax Value for a node n This definition is used recursively Idea: minimax value is the best achievable payoff against best play
Minimax example Perfect play for deterministic games Minimax Decision at root: choose the action a that lead to a maximal minimax value MAX is guaranteed for a utility which is at least the minimax value – if he plays rationally.
Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time complexity? O(b m ) Space complexity? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible
Multiplayer games Each node must hold a vector of values For example, for three players A, B, C (v A, v B, v C ) The backed up vector at node n will always be the one that maximizes the payoff of the player choosing at n
Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruning With "perfect ordering," time complexity = O(b m/2 ) doubles depth of search A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)
Why is it called α-β? is the value of the best (i.e., highest-value) choice found so far for MAX at any choice point along the path to the root. If v is worse than , MAX will avoid it prune that branch is the value of the best (i.e., lowest-value) choice found so far for MIN at any choice point along the path for to the root.
How much do we gain? Assume a game tree of uniform branching factor b Minimax examines O(b h ) nodes, so does alpha-beta in the worst-case The gain for alpha-beta is maximum when: The MIN children of a MAX node are ordered in decreasing backed up values The MAX children of a MIN node are ordered in increasing backed up values Then alpha-beta examines O(b h/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(b 3h/4 )
Heuristic Ordering of Nodes Order the nodes below the root according to the values backed-up at the previous iteration Order MAX (resp. MIN) nodes in decreasing (increasing) values of the evaluation function computed at these nodes
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)
Cutting off search Change: –if TERMINAL-TEST(state) then return UTILITY(state) into –if CUTOFF-TEST(state,depth) then return EVAL(state) Introduces a fixed-depth limit depth –Is selected so that the amount of time will not exceed what the rules of the game allow. When cuttoff occurs, the evaluation is performed.
Heuristic EVAL Idea: 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
Heuristic EVAL example Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) + … + w n f n (s)
Heuristic EVAL example Eval(s) = w 1 f 1 (s) + w 2 f 2 (s) + … + w n f n (s) Addition assumes independence
Heuristic difficulties Heuristic counts pieces won
Horizon effect Fixed depth search thinks it can avoid the queening move
Games that include chance Possible moves (5-10,5-11), (5-11,19-24),(5- 10,10-16) and (5-11,11-16)
Games that include chance Possible 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 chance nodes
Games that include chance [1,1], [6,6] chance 1/36, all other chance 1/18 Can not calculate definite minimax value, only expected value
Expected minimax value EXPECTED-MINIMAX-VALUE(n)= UTILITY(n) if n is a terminal max s successors(n) MINIMAX-VALUE(s) if n is a max node min s successors(n) MINIMAX-VALUE(s) if n is a max node s successors(n) P(s). EXPECTEDMINIMAX(s) if n is a chance node These equations can be backed-up recursively all the way to the root of the game tree.
Position evaluation with chance nodes Left, A1 wins Right A2 wins Outcome of evaluation function may not change when values are scaled differently. Behavior is preserved only by a positive linear transformation of EVAL.
Checkers: Tinsley vs. Chinook Name: Marion Tinsley Profession:Teach mathematics Hobby: Checkers Record: Over 42 years loses only 3 games of checkers World champion for over 40 years Mr. Tinsley suffered his 4th and 5th losses against Chinook
Chinook First computer to become official world champion of Checkers! Has all endgame table for 10 pieces or less: over 39 trillion entries.
Chess: Kasparov vs. Deep Blue Kasparov 5’10” 176 lbs 34 years 50 billion neurons 2 pos/sec Extensive Electrical/chemical Enormous Height Weight Age Computers Speed Knowledge Power Source Ego Deep Blue 6’ 5” 2,400 lbs 4 years 32 RISC processors + 256 VLSI chess engines 200,000,000 pos/sec Primitive Electrical None 1997: Deep Blue wins by 3 wins, 1 loss, and 2 draws
Chess: Kasparov vs. Deep Junior August 2, 2003: Match ends in a 3/3 tie! Deep Junior 8 CPU, 8 GB RAM, Win 2000 2,000,000 pos/sec Available at $100
Othello: Murakami vs. Logistello Takeshi Murakami World Othello Champion 1997: The Logistello software crushed Murakami by 6 games to 0
Go: Goemate vs. ?? Name: Chen Zhixing Profession: Retired Computer skills: self-taught programmer Author of Goemate (arguably the best Go program available today) Gave Goemate a 9 stone handicap and still easily beat the program, thereby winning $15,000
Go: Goemate vs. ?? Name: Chen Zhixing Profession: Retired Computer skills: self-taught programmer Author of Goemate (arguably the strongest Go programs) Gave Goemate a 9 stone handicap and still easily beat the program, thereby winning $15,000 Jonathan Schaeffer Go has too high a branching factor for existing search techniques Current and future software must rely on huge databases and pattern- recognition techniques Go has too high a branching factor for existing search techniques Current and future software must rely on huge databases and pattern- recognition techniques
Backgammon 1995 TD-Gammon by Gerald Thesauro won world championship on 1995 BGBlitz won 2008 computer backgammon olympiad
Secrets Many 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 configurations The 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
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 McCarthy Saying 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
Other Types of Games Multi-player games, with alliances or not Games with randomness in successor function (e.g., rolling a dice) Expectminimax algorithm Games with partially observable states (e.g., card games) Search of belief state spaces See R&N p. 175-180