Game PlayingOutlineState of the artChinookRatings of human and computer chess championsSlide 7Typical caseHow to play a gameEvaluation functionEvaluation function examplesGame treesMinimax procedureMinimax AlgorithmPartial Game Tree for Tic-Tac-ToeMinimax TreeAlpha-beta pruningSlide 18Alpha-beta exampleAlpha-beta algorithmEffectiveness of alpha-betaGames of chanceGame trees with chance nodesMeaning of the evaluation functionGame PlayingGame PlayingChapter 6CS 63CS 63Adapted from materials by Tim Finin, Marie desJardins, and Charles R. DyerOutline•Game playing–State of the art and resources–Framework•Game trees–Minimax–Alpha-beta pruning–Adding randomnessState of the art•How good are computer game players?–Chess: •Deep Blue beat Gary Kasparov in 1997•Garry Kasparav vs. Deep Junior (Feb 2003): tie! •Kasparov vs. X3D Fritz (November 2003): tie! –Checkers: Chinook (an AI program with a very large endgame database) is the world champion. Checkers has been solved exactly – it’s a draw!–Go: Computer players are decent, at best–Bridge: “Expert” computer players exist (but no world champions yet!)•Good place to learn more: http://www.cs.ualberta.ca/~games/Chinook•Chinook is the World Man-Machine Checkers Champion, developed by researchers at the University of Alberta.•It earned this title by competing in human tournaments, winning the right to play for the (human) world championship, and eventually defeating the best players in the world. •Visit http://www.cs.ualberta.ca/~chinook/ to play a version of Chinook over the Internet.•The developers have fully analyzed the game of checkers and have the complete game tree for it.–Perfect play on both sides results in a tie.•“One Jump Ahead: Challenging Human Supremacy in Checkers” Jonathan Schaeffer, University of Alberta (496 pages, Springer. $34.95, 1998).Ratings of human and computer chess championsTypical case•2-person game•Players alternate moves •Zero-sum: one player’s loss is the other’s gain•Perfect information: both players have access to complete information about the state of the game. No information is hidden from either player.•No chance (e.g., using dice) involved •Examples: Tic-Tac-Toe, Checkers, Chess, Go, Nim, Othello•Not: Bridge, Solitaire, Backgammon, ...How to play a game•A way to play such a game is to:–Consider all the legal moves you can make–Compute the new position resulting from each move–Evaluate each resulting position and determine which is best–Make that move–Wait for your opponent to move and repeat•Key problems are:–Representing the “board”–Generating all legal next boards–Evaluating a positionEvaluation function•Evaluation function or static evaluator is used to evaluate the “goodness” of a game position.–Contrast with heuristic search where the evaluation function was a non-negative estimate of the cost from the start node to a goal and passing through the given node•The zero-sum assumption allows us to use a single evaluation function to describe the goodness of a board with respect to both players. –f(n) >> 0: position n good for me and bad for you–f(n) << 0: position n bad for me and good for you–f(n) near 0: position n is a neutral position–f(n) = +infinity: win for me–f(n) = -infinity: win for youEvaluation function examples•Example of an evaluation function for Tic-Tac-Toe: f(n) = [# of 3-lengths open for me] - [# of 3-lengths open for you] where a 3-length is a complete row, column, or diagonal•Alan Turing’s function for chess–f(n) = w(n)/b(n) where w(n) = sum of the point value of white’s pieces and b(n) = sum of black’s•Most evaluation functions are specified as a weighted sum of position features:f(n) = w1*feat1(n) + w2*feat2(n) + ... + wn*featk(n) •Example features for chess are piece count, piece placement, squares controlled, etc. •Deep Blue had over 8000 features in its evaluation functionGame trees•Problem spaces for typical games are represented as trees•Root node represents the current board configuration; player must decide the best single move to make next•Static evaluator function rates a board position. f(board) = real number withf>0 “white” (me), f<0 for black (you)•Arcs represent the possible legal moves for a player •If it is my turn to move, then the root is labeled a "MAX" node; otherwise it is labeled a "MIN" node, indicating my opponent's turn. •Each level of the tree has nodes that are all MAX or all MIN; nodes at level i are of the opposite kind from those at level i+1Minimax procedure•Create start node as a MAX node with current board configuration •Expand nodes down to some depth (a.k.a. ply) of lookahead in the game•Apply the evaluation function at each of the leaf nodes •“Back up” values for each of the non-leaf nodes until a value is computed for the root node–At MIN nodes, the backed-up value is the minimum of the values associated with its children. –At MAX nodes, the backed-up value is the maximum of the values associated with its children. •Pick the operator associated with the child node whose backed-up value determined the value at the rootMinimax Algorithm2 7 18MAXMIN2 7 18212 7 182 122 7 182 12This is the moveselected by minimaxStatic evaluator valuePartial Game Tree for Tic-Tac-Toe•f(n) = +1 if the position is a win for X.•f(n) = -1 if the position is a win for O.•f(n) = 0 if the position is a draw.Minimax TreeMAX nodeMIN nodef valuevalue computed by minimaxAlpha-beta pruning•We can improve on the performance of the minimax algorithm through alpha-beta pruning•Basic idea: “If you have an idea that is surely bad, don't take the time to see how truly awful it is.” -- Pat Winston 2 7 1=2>=2<=1?•We don’t need to compute the value at this node.•No matter what it is, it can’t affect the value of the root node.MAXMAXMINAlpha-beta pruning•Traverse the search tree in depth-first order •At each MAX node n, alpha(n) = maximum value found so far•At each MIN node n, beta(n) = minimum value found so far–Note: The alpha values start at -infinity and only increase, while beta values start at +infinity and only decrease. •Beta cutoff: Given a MAX node n, cut off the search below n (i.e., don’t generate or examine any more of n’s children) if alpha(n) >= beta(i) for some MIN node ancestor i of n. •Alpha cutoff: stop searching below MIN