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Game Playing. ECE457 Applied Artificial Intelligence Spring 2008 Lecture #5. Outline. Types of games Playing a perfect game Minimax search Alpha-beta pruning Playing an imperfect game Real-time Imperfect information Chance Russell & Norvig, chapter 6 Project #2. Game Problems.

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## Game Playing

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**Game Playing**ECE457 Applied Artificial Intelligence Spring 2008 Lecture #5**Outline**• Types of games • Playing a perfect game • Minimax search • Alpha-beta pruning • Playing an imperfect game • Real-time • Imperfect information • Chance • Russell & Norvig, chapter 6 • Project #2 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 2**Game Problems**• Games are well-defined search problems… • Well-defined board configurations (states) • Limited set of well-defined moves (actions) • Well-defined victory conditions (goal) • Values assigned to pieces, moves, outcomes (cost) • …that are hard to solve by searching • A search tree for chess has an average branching factor of 35 • An average chess game lasts for 50 moves per player (ply) • The average search tree has 35100 nodes! ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 3**Game Problems**• The opponent • He wants to win and make our agent lose • We have no control over his actions • He prevents us from reaching the optimal solution • Introduces uncertainty in the search • We don’t know what moves the opponent will do • We will assume “perfect play” behaviour ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 4**Types of Games**• Zero-sum games: a player’s gains are exactly substracted from another player’s score (chess) • Non-zero-sum games: players can gain or lose without an exact change on others (prisoners’ dilemma) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 5**Game-Playing Strategy**• Our agent and the opponent play sequentially • We assume the opponent plays perfectly • Our agent cannot get to the optimal goal • The opponent won’t allow it • Our agent must find the best achievable goal ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 6**Minimax Algorithm**• Payoff (utility) function assigns a value to each leaf node in the tree • Value then propagates up to non-leaf nodes • Two players • MAX wants to maximise payoff • MIN wants to minimise payoff • MAX is the player currently looking for a move (i.e. at root of tree) • Payoff function • Simple 1 = win / 0 = draw / -1 = lose • Complex for different victory conditions • Win/lose for MAX ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 7**Minimax Algorithm**… … … ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 8**MAX**3 1 -12 MIN MAX 3 18 5 1 15 42 56 -12 -5 Minimax Algorithm 3 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 9**Minimax Algorithm**• Game of Nim • Initial state: 7 matches in a pile • Each player must divide a pile into two non-empty unequal piles • Player who can’t do that, loses • Payoff • +1 win, -1 loss ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 10**3-1-1-1-1**2-1-1-1-1-1 2-2-1-1-1 3-2-1-1 2-2-2-1 5-1-1 4-2-1 3-2-2 4-1-1-1 6-1 5-2 4-3 7 3-3-1 Minimax Algorithm MAX MIN MAX MIN MAX MIN -1 -1 -1 -1 -1 +1 +1 -1 +1 -1 +1 (max wins) The value of each node is the value of the best leaf the current player (MAX or MIN) can reach. +1 -1 (max loses) +1 (max wins) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 11**Minimax Algorithm**• Generate entire game tree • Compute payoff of leaf nodes • For each non-leaf node, from the lowest in the tree to the root • If MAX level, then assign value of the child with maximum payoff • If MIN level, then assign value of the child with minimum payoff • At the root, select action with maximum payoff ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 12**Minimax Algorithm**• Complete, if tree is finite • Optimal against a perfect opponent • Time complexity = O(bm) • Space complexity = O(bm) • But remember, b and m can be huge • For chess, b ≈ 35 and m ≈ 100 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 13**Alpha-Beta Pruning**• MAX take the max of its children • MIN gives each child the min of its children max(min(3,18,5),min(1,15,42),min(56,-12,-5)) • We don’t need to compute the values of all the grandchildren! • Only until we find a value lower than the highest child’s value max(min(3,18,5),min(1,?,?),min(56,-12,?)) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 14**Alpha-Beta Pruning**• Maintain values and • is the maximum value that MAX is assured of at any point in the search • is the minimum value that MIN is assured of at any point in the search • Both computed using payoff propagated through the tree • Start with = - and = • As the search goes on, the number of possible values of and decreases • When • Current path is not the result of best play by both players, so no need to explore further ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 15**MAX**MIN MAX Alpha-Beta Pruning 1. [-, ] [, ] 4. [3, ] 3 7. [3, ] 8. [3, 56] 9. [3, -12] 5. [3, ] 2. [-, ] 1 -12 3 6. [3, 1] 3. [-, 3] 3 18 5 1 56 -12 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 16**Alpha-Beta Pruning**• Called as “rootvalue = Evaluate(root, -, )” Evaluate(node, , ) • If node is leaf • Return payoff • If node is MAX • v = - • For each child of node • v = max( v, Evaluate(child, , ) • Break if v • = max(, v) • Return v • If node is MIN • v = • For each child of node • v = min( v, Evaluate(child, , ) ) • Break if v • = min(, v) • Return v ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 17**Alpha-Beta Pruning**• Efficiency dependant on ordering of children • Will check each of MAX’s children until finding one with a value higher than beta • Will check each of MIN’s children until finding one with a value lower than alpha • Use heuristics to order the nodes to check • Check the highest-value children first for MAX • Check the lowest-value children first for MIN • Good ordering can reduce time complexity to O(bm/2) • Random ordering gives roughly O(b3m/4) • Minimax is O(bm) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 18**Minimax Exercise**5 A B C 2 5 8 9 E F G H I D 6 5 4 2 J K L M 0 8 9 1 0 17 N O ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 19**Pruning Exercise**1.[-, ] A 5.[5, ] 6.[5, ] 11.[5, 8] 2.[-, ] B C 14.[5, 4] 3.[-, 6] 4.[-, 5] 7.[-, ] 12.[-, 8] E F G H I D 8.[8, ] 13.[9, 8] 6 5 4 2 J K L M 9.[8, ] 10.[8, 0] 8 9 14 0 -4 N O ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 20**Imperfect Play**• Real-time or time constraints • Chance • Hidden information ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 21**Real-Time Games**• Sometimes we can’t search the entire tree • Real-time games • Time constraints (playing against a clock) • Tree too big (e.g. chess) ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 22**Real-Time Games**• Evaluation function • Estimate value of a non-leaf node in the tree • Cut off search at a given level • Chess: count value of pieces, available moves, board configurations, … < ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 23**Real-Time Minimax Algorithm**• Generate entire game tree down to maximum number of ply • Evaluate lowest nodes • For each non-leaf node, from the lowest in the tree to the root • If MAX level, then assign value of the child with maximum payoff • If MIN level, then assign value of the child with minimum payoff • At the root, select action with maximum payoff ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 24**Real-Time Alpha-Beta Pruning**• Called as “rootvalue = Evaluate(root, -, )” Evaluate(node, , ) • If node is at lowest level • Return evaluation • If node is MAX • v = - • For each child of node • v = max( v, Evaluate(child, , ) • Break if v • = max(, v) • Return v • If node is MIN • v = • For each child of node • v = min( v, Evaluate(child, , ) ) • Break if v • = min(, v) • Return v ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 25**Real-Time Games: Problems**• Non-quiescent positions • Some state configurations cause value to change wildly • Solved with quiescence search • Expand non-quiescent boards deeper, until you reach stable “quiescent” boards ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 26**Real-Time Games: Problems**• Horizon effect • A “singular” move is considerably better than all others • But a damaging unavoidable move is (or can be pushed) just beyond the search depth limit (the “horizon”) • Solved with singular extension • Expand singular state deeper ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 27**Games of Chance**• Minimax requires planning for upcoming moves • If moves depend on dice rolls, random draws, etc., planning is impossible • We need to add all possible outcomes in the tree! ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 28**3**3 1 -12 3 18 5 1 15 42 56 -12 -5 Recall ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 29**0.05**0.05 0.05 0.8 0.8 0.8 0.15 0.15 0.15 Expectiminimax Then, MIN rolls the dice MAX has already rolled the dice and has three possible moves 4.45 There are three possible outcomes to the roll 4.15 -10.45 4.45 3 16 -7 1 25 -8 -12 -25 58 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 30 And MIN picks an action based on the roll result**Expectiminimax**0.8 0.05 0.15 4.45 4.15 -10.45 4.45 0.15 0.05 0.8 0.15 0.05 0.05 0.8 0.8 0.15 1 25 -8 -12 -25 58 3 16 -7 3 7 12 16 22 -7 -3 4 17 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 31**0.05**0.05 0.05 0.8 0.8 0.8 0.15 0.15 0.15 Problems with Expectiminimax 26.65 4.15 26.65 4.45 3 16 -7 1 25 -8 -12 -25 800 ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 32**Problems with Expectiminimax**• Time complexity: O(bmnm) • n is the number of possible outcomes of a chance node • Recall: minimax is O(bm) • Trees can grow very large very quickly • Minimax & pruning limits search to likely sequences of actions given perfect play • With randomness, there is no likely sequence of actions ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 33**Imperfect Information**• Algorithms so far require knowing everything about the game • In some games, information about the opponent is hidden • Cards in poker, pieces in Stratego, etc. • We could approximate hidden information to random events • The probability that the opponent has a flush, the probability that a piece is a bomb, etc. • Then use expectiminimax to get best action ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 34**Imperfect Information**• List all possible outcomes, then average best action overall • Can lead to irrational behaviour! • Possible cases: • Road 1 leads to money, road 2-a leads to gold, road 2-b leads to death (rational action is road 2, then a) • Road 1 leads to money, road 2-a leads to death, road 2-b leads to gold (rational action is road 2, then b) • But the real situation is: • Road 1 leads to money, road 2 leads to gold or death (rational action is road 1) 1 2 a b ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 35**Imperfect Information**• It’s a useful approximation, but it’s not exact! • Advantages: • Works in many cases • Doesn’t require new techniques to handle information discovery • Disadvantages: • In reality, hidden information is not the same as random events • Can lead to irrational behaviour ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 36**Imperfect Information**• Need to handle information • Gather information • Plan based on what information we will have at a given point in the future • Leads to more rational behaviour • Acting to gain information • Acting to give information to partners • Acting to conceal information from the opponents • We will learn to do that later in the course ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 37**IBM Deep Blue**• First chess computer to defeat a reigning world champion (Garry Kasparov) under normal chess tournament constraints in 1997 • Relied on brute hardware search power • 30 processors for the search • 480 custom VLSI chess processors for move generation and ordering, and leaf node evaluation ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 38**IBM Deep Blue**• Searched a minimax tree • 100-200M states per second, maximum 330M • Average 6 to 16 ply, maximum 40 ply • Decide which moves are worth expanding, giving priority to singular expansion and chess threats • Null-window alpha-beta pruning • Alpha-beta pruning but limited to a “window” of moves rather than the entire tree • Faster and easier to implement on hardware • Approximate, can only returns bounds on the minimax value • Allows for a highly non-uniform, more selective and human-like search of the tree ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 39**IBM Deep Blue**• Two board evaluation heuristics • Fast evaluation to get a quick approximate value • Considers piece position value • Slow evaluation to get an exact value • Considers 8,000 features • Includes common chess concepts and specific Kasparov strategies • Features have programmable weights learned automatically from 700,000 grandmaster games and fine-tuned manually by a chess grandmaster ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 40**Assumptions**• Utility-based agent • Environment • Fully observable • Deterministic • Sequential • Static • Discrete / Continuous • Single agent ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 41**Assumptions Updated**• Utility-based agent • Environment • Fully observable / Partially observable (approximation) • Deterministic / Strategic / Stochastic • Sequential • Static / Semi-dynamic • Discrete / Continuous • Single agent / Multi-agent ECE457 Applied Artificial Intelligence R. Khoury (2008) Page 42

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