Games vs. search problems “Unpredictable” opponent ⇒ solution is a strategy specifying a move for every possible opponent reply Time limits ⇒ unlikely to find goal, must approximate Adversarial Search Plan of attack: • Computer considers possible lines of play (Babbage, 1846) Chapter 5 • Algorithm for perfect play (Zermelo, 1912; Von Neumann, 1944) • Finite horizon, approximate evaluation (Zuse, 1945; Wiener, 1948; Shannon, 1950) • First chess program (Turing, 1951) • Machine learning to improve evaluation accuracy (Samuel, 1952–57) • Pruning to allow deeper search (McCarthy, 1956) Chapter 5 1 Chapter 5 3 Outline Types of games ♦ Games deterministic chance ♦ Perfect play perfect information chess, checkers, backgammon – minimax decisions go, othello monopoly – α – β pruning imperfect information battleships, bridge, poker, scrabble ♦ Resource limits and approximate evaluation blind tictactoe nuclear war ♦ Games of chance ♦ Games of imperfect information Chapter 5 2 Chapter 5 4
Minimax algorithm Game tree (2-player, deterministic, turns) MAX (X) function Minimax-Decision ( state ) returns an action inputs : state , current state in game return the a in Actions ( state ) maximizing Min-Value ( Result ( a , state )) X X X MIN (O) X X X X X X function Max-Value ( state ) returns a utility value if Terminal-Test ( state ) then return Utility ( state ) X O X O X . . . MAX (X) O v ← −∞ for a, s in Successors ( state ) do v ← Max ( v , Min-Value ( s )) return v X O X X O X O . . . MIN (O) X X function Min-Value ( state ) returns a utility value if Terminal-Test ( state ) then return Utility ( state ) . . . . . . . . . . . . v ← ∞ for a, s in Successors ( state ) do v ← Min ( v , Max-Value ( s )) . . . X O X X O X X O X return v TERMINAL O X O O X X O X X O X O O Utility −1 0 +1 Chapter 5 5 Chapter 5 7 Minimax Properties of minimax Perfect play for deterministic, perfect-information games Complete?? Idea: choose move to position with highest minimax value = best achievable payoff against best play E.g., 2-ply game: 3 MAX A 1 A 2 A 3 3 2 2 MIN A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 3 12 8 2 4 6 14 5 2 Chapter 5 6 Chapter 5 8
Properties of minimax Properties of minimax Complete?? Only if tree is finite (chess has specific rules for this). Complete?? Yes, if tree is finite (chess has specific rules for this) ps. a finite strategy can exist even in an infinite tree! Optimal?? Yes, against an optimal opponent. Otherwise?? Optimal?? Time complexity?? O ( b m ) Space complexity?? Chapter 5 9 Chapter 5 11 Properties of minimax Properties of minimax Complete?? Yes, if tree is finite (chess has specific rules for this) Complete?? Yes, if tree is finite (chess has specific rules for this) Optimal?? Yes, against an optimal opponent. Otherwise?? Optimal?? Yes, against an optimal opponent. Otherwise?? Time complexity?? O ( b m ) Time complexity?? Space complexity?? O ( bm ) (depth-first exploration) For chess, b ≈ 35 , m ≈ 100 for “reasonable” games ⇒ exact solution completely infeasible But do we need to explore every path? Chapter 5 10 Chapter 5 12
α – β pruning example α – β pruning example 3 3 MAX MAX 3 3 2 14 MIN MIN X X 3 12 8 3 12 8 2 14 Chapter 5 13 Chapter 5 15 α – β pruning example α – β pruning example 3 3 MAX MAX 3 2 3 2 14 5 MIN MIN X X X X 3 12 8 2 14 5 3 12 8 2 Chapter 5 14 Chapter 5 16
α – β pruning example Why is it called α – β ? 3 3 MAX MAX MIN 3 2 14 5 2 MIN .. .. .. MAX X X 3 12 8 2 14 5 2 MIN V α is the best value (to max ) found so far off the current path If V is worse than α , max will avoid it ⇒ prune that branch Define β similarly for min Figure 5.5, p. 168. Chapter 5 17 Chapter 5 19 Why is it called α – β ? The α – β algorithm function Alpha-Beta-Decision ( state ) returns an action [ α , β ] – range: [lowerbound, upperbound] return the a in Actions ( state ) maximizing Min-Value ( Result ( a , state )) [− ∞ , + ∞ ] [− ∞ , + ∞ ] (a) A (b) A [− ∞ , 3 ] [− ∞ , 3 ] function Max-Value ( state , α , β ) returns a utility value B B inputs : state , current state in game α , the value of the best alternative for max along the path to state 3 3 12 β , the value of the best alternative for min along the path to state [ 3 , + ∞ ] [ 3 , + ∞ ] if Terminal-Test ( state ) then return Utility ( state ) (c) (d) A A v ← −∞ [3, 3] B [3, 3] B [− ∞ , 2] C foreach a in Actions ( state ) do v ← Max ( v , Min-Value ( Result ( s,a ), α , β )) if v ≥ β then return v 3 12 8 3 12 8 2 α ← Max ( α , v ) return v [3, 14] [3, 3] (e) (f) A A [3, 3] B [− ∞ , 2 ] C [− ∞ , 14 ] D [3, 3] B [− ∞ , 2] C [2, 2] D function Min-Value ( state , α , β ) returns a utility value same as Max-Value but with roles of α , β reversed 3 12 8 2 14 3 12 8 2 14 5 2 Chapter 5 18 Chapter 5 20
Properties of α – β Evaluation functions Pruning does not affect final result Good move ordering improves effectiveness of pruning With “perfect ordering,” time complexity = O ( b m/ 2 ) ⇒ doubles solvable depth with constant time constraint A simple example of the value of reasoning about which computations are relevant (a form of metareasoning) Unfortunately, 35 50 is still impossible! Black to move White to move White slightly better Black winning For chess, typically linear weighted sum of features Eval ( s ) = w 1 f 1 ( s ) + w 2 f 2 ( s ) + . . . + w n f n ( s ) e.g., w 1 = 9 with f 1 ( s ) = (number of white queens) – (number of black queens), etc. Chapter 5 21 Chapter 5 23 Resource limits Digression: Exact values don’t matter Standard approach: MAX • Use Cutoff-Test instead of Terminal-Test e.g., depth limit (perhaps add quiescence search) MIN 1 2 1 20 • Use Eval instead of Utility i.e., evaluation function that estimates desirability of position 1 2 2 4 1 20 20 400 Suppose we have 100 seconds, explore 10 4 nodes/second ⇒ 10 6 nodes per move ≈ 35 8 / 2 Behaviour is preserved under any monotonic transformation of Eval ⇒ α – β reaches depth 8 ⇒ pretty good chess program Only the order matters : payoff in deterministic games acts as an ordinal utility function Chapter 5 22 Chapter 5 24
Deterministic games in practice Nondeterministic games in general Checkers: Chinook ended 40-year-reign of human world champion Marion In nondeterministic games, chance introduced by dice, card-shuffling Tinsley in 1994. Used an endgame database defining perfect play for all Simplified example with coin-flipping: positions involving 8 or fewer pieces on the board, a total of 443,748,401,247 positions. MAX Chess: Deep Blue defeated human world champion Gary Kasparov in a six- game match in 1997. Deep Blue searches 200 million positions per second, uses very sophisticated evaluation, and undisclosed methods for extending 3 −1 CHANCE some lines of search up to 40 ply. 0.5 0.5 0.5 0.5 Othello: human champions refuse to compete against computers, which are too good. MIN 2 4 0 −2 Go: human champions refuse to compete against computers, which are too bad. In go, b > 300 , so most programs use pattern knowledge bases to suggest plausible moves. 2 4 7 4 6 0 5 −2 Chapter 5 25 Chapter 5 27 Nondeterministic games: backgammon Algorithm for nondeterministic games Expectiminimax gives perfect play 0 1 2 3 4 5 6 7 8 9 10 11 12 Just like Minimax , except we must also handle chance nodes: . . . if state is a Max node then return the highest ExpectiMinimax-Value of Successors ( state ) if state is a Min node then return the lowest ExpectiMinimax-Value of Successors ( state ) if state is a chance node then return average of ExpectiMinimax-Value of Successors ( state ) . . . 25 24 23 22 21 20 19 18 17 16 15 14 13 Chapter 5 26 Chapter 5 28
Nondeterministic games in practice Summary Dice rolls increase b : 21 possible rolls with 2 dice Games are fun to work on! (and dangerous) Backgammon ≈ 20 legal moves (can be 6,000 with 1-1 roll) They illustrate several important points about AI depth 4 = 20 × (21 × 20) 3 ≈ 1 . 2 × 10 9 ♦ perfection is unattainable ⇒ must approximate As depth increases, probability of reaching a given node shrinks ♦ good idea to think about what to think about ⇒ value of lookahead is diminished ♦ uncertainty constrains the assignment of values to states α – β pruning is much less effective ♦ optimal decisions depend on information state, not real state TDGammon uses depth-2 search + very good Eval ≈ world-champion level Games are to AI as grand prix racing is to automobile design Chapter 5 29 Chapter 5 31 Digression: Exact values DO matter MAX 2.1 1.3 21 40.9 DICE .9 .1 .9 .1 .9 .1 .9 .1 MIN 2 3 1 4 20 30 1 400 2 2 3 3 1 1 4 4 20 20 30 30 1 1 400 400 Behaviour is preserved only by positive linear transformation of Eval Hence Eval should be proportional to the expected payoff Chapter 5 30
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