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1 !! Game Trees and Heuristics Announcements! HW5 Due Monday 11:59 pm : Fundamental Data Structures and Algorithms Margaret Reid-Miller 7 April X O X O X O X O O Games Game Properties!!Two! players! Deterministic! no randomness, e.g., dice! Perfect information! No hidden cards or hidden Chess pieces! Finite! Game must end in finite number of moves! Zero sum! Total winnings of all players 4 Mini Algorithm! Player A (us) maximize goodness.! Player B (opponent) minimizes goodness Player A 0 maximize (draw) Player B -1 0 minimize (lose, draw) Player A maximize (lose, win) -1 1! At a leaf (a terminal position) A wins, loses, or draws.! Assign a score: 1 for win; 0 for draw; -1 for lose.! At max layers, take node score as maximum of the child node scores! At min layers, take nodes score as minimum of the child node scores 5 Heuristics! A heuristic is an approximation that is typically fast and used to aid in optimization problems.! In this context, heuristics are used to rate board positions based on local information.! For example, in Chess I can rate a position by examining who has more pieces. The difference in black s and white s pieces would be the evaluation of the position. 6

2 Heuristics and Minimax Heuristics and Minimax! When dealing with game trees, the heuristic function is generally referred to as the evaluation function, or the static evaluator.! The static evaluation takes in a board position, and gives it a score.! The higher the score, the better it is for you, the lower, the better for the opponent. 7! Each layer of the tree is called a ply.! We cut off the game tree at a certain maximum depth, d. Called a d-ply search.! At the bottom nodes of the tree, we apply the heuristic function to those positions.! Now instead of just Win, Loss, Tie, we have a score. 8 Evaluation Function Minimax in action! Guesses the outcome of an incomplete search (Player A) 10! Eval(Position) =! i w i * f i (Position)! Weights w i may depend on the phase of the game! Features for chess f i :! #of Pawns (material terms)! Centrality! Square control! Mobility Min (Player B) (Player A) Evaluation function applied to the leaves!! Pawn structure 9 10 How fast?! Minimax is pretty slow even for a modest depth.! It is basically a brute force search.! What is the running time?! Each level of the tree has some average b moves per level. We have d levels. So the running time is O(b d ). 11 Alpha Beta Pruning! Idea: Track window of expectations. Two variables: " Best score so far at a max node: increases. # Score can be forced on our opponent. \$ Best score so far at a min node: decreases # Opponent can force a situation no worse than this score. Any move with better score is too good for opponent to allow Either case: If " % \$: # Stop searching further subtrees of that child. Opponent won t let you get that high a score.! Start the process with an infinite window (" = -&, \$ = &). 12

3 alphabeta (", \$) Alpha Beta Example The top level call: Return alphabeta (- &, &)) alphabeta (", \$):! At leaf level (depth limit is reached):! Assigns estimator function value (in the range (- &.. &) to the leaf node. "! \$!! Return this value.! At a min level (opponent moves): Min 10 \$ =10! For each child, until " % \$:! Set \$ = min(\$, alphabeta (", \$))! Return \$ " = 12! At a max level (our move):! For each child, until " % \$:! Set " = max(", alphabeta (", \$)) Min ! Return " Alpha Beta Example Alpha Beta speedup! Alpha Beta is always a win: 10 " = 10 "! \$!! Returns the same result as Minimax,! Is minor modification to Minimax Min 10 \$ = ! Claim: The optimal Alpha Beta search tree is O(b d/2 ) nodes or the square root of the number of nodes in the regular Minimax tree.! Can enable twice the depth Min ! In chess branching is about 38. In practice Alpha Beta reduces it to about 6 and enables 10 ply searches on a PC. 16 Heuristic search techniques Move Ordering Heuristic = aid to problem-solving! Alpha-beta is one way to prune the game tree 17! Explore decisions in order of likely success to get early alpha beta pruning! Guide search with estimator functions that correlate with likely search outcomes or! track which moves tend to cause beta cutoff! Heuristic estimate of the cost (time) of searching one move versus another! Search the easiest move first 18

4 Timed moves! Not uncommon to have a limited time to make a move. May need to produce a move in say 2 minutes.! How do we ensure that we have a good move before the timer goes off? Iterative Deepening! Evaluate moves to successively deeper and deeper depths:! Start with 1-ply search and get best move(s). Fast! Next do 2-ply search using the previous best moves to order the search.! Continue to increased depth of search.! If some depth takes too long, fall back to previous results (timed moves) Iterative Deepening Transposition Tables! Save best moves of shallower searches to control move ordering.! Need to search the same part of the tree multiple times but improved move ordering more than makes up for this redundancy! Difficulty:! Time control: each iteration needs about 6x more time! What to do when time runs out? 21! Minimax and Alpha Beta (implicitly) build a tree. But what is the underlying structure of a game?! Different sequences of moves can lead to the same position.! Several game positions may be functionally equivalent (e.g. symmetric positions). 22 Nim Game Graph Transposition Tables (1,2,3) (2,3) (1,1,3) (1,2,2) (1,3) (1,1,2) (1,2) (2,2) (1,2) (2) (1,3) (3) (1,1,2) (1,1,1) (1,1) (1) (3) (2) (1) win (1,1) (1,2) (1,1,1) (2) (1) loss (1,1) (1) win loss Us Them Us Them Us Them 23Us! Memoize: hash table of board positions to get! Value for the node! Upper Bound, Lower Bound, or Exact Value! Be extremely careful with alpha beta as may only know a bound at that position.! Best move at the position! Useful for move ordering for greater pruning!! Which positions to save?! Sometimes obvious from context! Ones in which more search time has been invested! Collisions: Simply overwrite 24

5 Limited Depth Search Problems! Horizon effect: push bad news over the search depth! Quiescence search: can t simply stop in the middle of an exchange Alpha Beta Pruning Theorem: Let v(p) be the value of a position P. Let X be the value of a call to AB(", \$). Then one of the following holds: v(p) " " and X " " " < v(p) < \$ and X = v(p) \$ " v(p) and X! \$ Suppose we take a chance and reduce the size of the infinite window. What might happen? Aspiration Window: Tricks! Suppose you had information that value of a position was probably close to 2 (say from the result of shallower search)! Instead of starting with an infinite window, start with an aspiration window around 2 (e.g., (1.5, 2.5)) to get more pruning.! If the result is in that range you are done.! If outside the range you don t know the exact value, only a bound. Repeat with a different range.! How might this technique be use for parallel evaluation? 27! Many tricks and heuristics have been added to chess program, including this tiny subset:! Opening Books! Avoiding mistakes from earlier games! Endgame databases (Ken Thompson)! Singular Extensions! Think ahead! Contempt factor! Strategic time control 28 Game of Amazons Game of Amazons! Invented in 1988 by Argentinian Walter Zamkauskas.! Several programming competitions and yearly championships.! Distantly related to Go.! Active area in combinatorial game theory. 30

6 Amazons Board! Chess board 10 x 10! 4 White and 4 Black chess Queens (Amazons) and Arrows! Starting configuration! White moves first Amazons Rules! Each move consists of two steps: 1. Move an amazon of own color. 2. This amazon has to throw an arrow to an empty square where it stays.! Amazons and arrows move as a chess Queen as long as no obstacle blocks the way (amazon or arrow)! Players alternate moves.! Player who makes last move wins Amazons challenges! Absence of opening theory! Branching factor of more than 1000! Often 20 reasonable moves! Need for deep variations! Opening book >30,000 moves AMAZONG! World s best computer player by Jens Lieberum! Distinguishes three phases of the game:! Opening at the beginning! Middle game! Filling phase at the end See Amazons Opening Amazons Filling Phase! Main goals in the opening! Obtain a good distribution of amazons! Build large regions of potential territory! Trap opponent s amazons 35! Filling phase starts when empty squares can be reached by only one player.! Happens usually after 50 moves.! Goal is to have access to more empty squares than the other player! Outcome of game determined by counting number of moves left for each player.! But 36

7 Defective Territories! K-defective territory provides k fewer moves than empty squares Zugzwang! Seems that black has access to 3 empty squares! But if black moves first then can only use two More Complex Zugzwang Chiptest, Deep Thought Timeline Player who moves first must either! take their own region and give region C to the opponent, or! take region C and block off their own region 39 Story IBM! A VLSI design class project evolves into F.H.Hsu move-generator chip! A productive CS-TG results in ChipTest, about 6 weeks before the ACM computer chess championship! Chiptest participates and plays interesting (illegal) moves! Chiptest-M wins ACM CCC! Redesign becomes DT! DT participates in human chess championships (in addition to CCC)! DT wins second Fredkin Prize (\$10K)! DT is wiped out by Kasparov 40 Opinions: Is Computer Chess AI? So how does Kasparov win? From Hans Moravec s Book Robot 41! Even the best Chess grandmasters say they only look 4 or 5 moves ahead each turn. Deep Junior looks up about moves ahead. How does it lose!?! Kasparov has an unbelievable evaluation function. He is able to assess strategic advantages much better than programs can (although this is getting less true).! The moral, the evaluation function plays a large role in how well your program can play. 42

8 State-of-the-art: Backgammon State-of-the-art: Backgammon! Gerald Tesauro (IBM)! Wrote a program which became overnight the best player in the world! Not easy! 43! Learned the evaluation function by playing 1,500,000 games against itself! Temporal credit assignment using reinforcement learning! Used Neural Network to learn the evaluation function c a b 9 7 d Neural Net Learn! Predict! 9 44 State-of-the-art: Go! Average branching factor 360! Regular search methods go bust!! People use higher level strategies! Systems use vast knowledge bases of rules some hope, but still play poorly! \$2,000,000 for first program to defeat a top-level player 45

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