Game Playing ECE457 Applied Artificial Intelligence Spring 2008 Lecture #5

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 2 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 3 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 nodes!

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 4 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 5 Types of Games DeterministicChance Perfect information Chess Checkers Go Backgammon Monopoly Imperfect information Stratego Battleship Bridge Poker Scrabble 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 6 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 7 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 8 Minimax Algorithm XX X X O XO XO XOXXO X XO X … … …

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 9 Minimax Algorithm MAX MIN MAX

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 10 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 (max wins) Minimax Algorithm MAX MIN MAX MIN MAX MIN -1 (max loses) +1 (max wins) The value of each node is the value of the best leaf the current player (MAX or MIN) can reach.

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 12 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 13 Minimax Algorithm Complete, if tree is finite Optimal against a perfect opponent Time complexity = O(b m ) 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 14 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 15 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 16 Alpha-Beta Pruning MAX MIN MAX [- ,  ] 5. [3,  ] 1. [- ,  ] 3. [- , 3] 7. [3,  ] 4. [3,  ] 6. [3, 1] 9. [3, -12] 8. [3, 56] [ ,  ]

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 17 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 18 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(b m/2 ) Random ordering gives roughly O(b 3m/4 ) Minimax is O(b m )

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 19 Minimax Exercise A BC D EFGH JM N O LK I

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page [8, 0] 2.[- ,  ] 5.[5,  ] Pruning Exercise A BC D EFGH JM N O [- ,  ] 3.[- , 6] 4.[- , 5] 6.[5,  ] 7.[- ,  ] 8.[8,  ] 9.[8,  ] LK 11.[5, 8] 12.[- , 8] 13.[9, 8] I 14.[5, 4]

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 21 Imperfect Play Real-time or time constraints Chance Hidden information

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 22 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 23 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, … X X O X OX <

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 24 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 25 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 26 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 27 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 28 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 29 Recall

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 30 Expectiminimax MAX has already rolled the dice and has three possible moves Then, MIN rolls the dice And MIN picks an action based on the roll result There are three possible outcomes to the roll

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 31 Expectiminimax

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 32 Problems with Expectiminimax

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 33 Problems with Expectiminimax Time complexity: O(b m n m ) n is the number of possible outcomes of a chance node Recall: minimax is O(b m ) 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 34 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 35 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) 12 ab

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 36 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 37 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 38 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 39 IBM Deep Blue Searched a minimax tree M 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 40 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 41 Assumptions Utility-based agent Environment Fully observable Deterministic Sequential Static Discrete / Continuous Single agent

ECE457 Applied Artificial Intelligence R. Khoury (2008)Page 42 Assumptions Updated Utility-based agent Environment Fully observable / Partially observable (approximation) Deterministic / Strategic / Stochastic Sequential Static / Semi-dynamic Discrete / Continuous Single agent / Multi-agent