1	Adversarial Search Chapter 6 Sections 1 – 4
2	Outline Optimal decisions α-β pruning Imperfect, real-time decisions
3	Games vs. search problems “Unpredictable” opponent à specifying a move for every possible opponent reply Time limits à unlikely to find goal, must approximate Hmm: Is hex a game or a search problem by this definition?
4	Let’s play! Two players: Max Min
5	Game tree (2-player, deterministic, turns)
6	Example : Game of NIM Several piles of sticks are given. We represent the configuration of the piles by a monotone sequence of integers, such as (1,3,5). A player may remove, in one turn, any number of sticks from one pile. Thus, (1,3,5) would become (1,1,3) if the player were to remove 4 sticks from the last pile. The player who takes the last stick loses. Represent the NIM game (1, 2, 2) as a game tree.
7	Minimax Perfect play for deterministic games Idea: choose move to position with highest minimax value = best achievable payoff against best play E.g., 2-ply game:
8	Minimax algorithm
9	Properties of minimax Complete? Yes (if tree is finite) Optimal? Yes (against an optimal opponent) Time complexity? O(b^m) Space complexity? O(bm) (depth-first exploration) For chess, b ≈ 35, m ≈ 100 for “reasonable” games à exact solution completely infeasible What can we do? Pruning!
10	α-β pruning example
11	α-β pruning example
12	α-β pruning example
13	α-β pruning example
14	α-β pruning example
15	Properties of α-β Pruning does not affect final result Good move ordering improves effectiveness of pruning With “perfect ordering”, time complexity = O(b^m/2) à doubles depth of search What’s the worse and average case time complexity? Does it make sense then to have good heuristics for which nodes to expand first? A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)
16	Why is it called α-β? α is the value of the best (i.e., highest-value) choice found so far at any choice point along the path for max If v is worse than α, max will avoid it à prune that branch Define β similarly for min
17	The α-β algorithm
18	The α-β algorithm
19	Resource limits The big problem is that the search space in typical games is very large. Suppose we have 100 secs, explore 10⁴ nodes/sec à 10⁶ nodes per move Standard approach: cutoff test: e.g., depth limit (perhaps add quiescence search) evaluation function = estimated desirability of position
20	Evaluation functions For chess, typically linear weighted sum of features Eval(s) = w₁ f₁(s) + w₂ f₂(s) + … + w_n f_n(s) e.g., w₁ = 9 with f₁(s) = (number of white queens) – (number of black queens), etc. Caveat: assumes independence of the features Bishops in chess better at endgame Unmoved king and rook needed for castling Should model the expected utility value states with the same feature values lead to.
21	Expected utility value A utility value may map to many states, each of which may lead to different terminal states Want utility values to model likelihood of better utility states.
22	Cutting off search MinimaxCutoff is identical to MinimaxValue except Terminal? is replaced by Cutoff? Utility is replaced by Eval Does it work in practice? b^m = 10⁶, b=35 à m=4 4-ply lookahead is a hopeless chess player! 4-ply ≈ human novice 8-ply ≈ typical PC, human master 12-ply ≈ Deep Blue, Kasparov
23	Deterministic games in practice Checkers: Chinook ended 40-year-reign of human world champion Marion Tinsley in 1994. Used a precomputed endgame database defining perfect play for all positions involving 8 or fewer pieces on the board, a total of 444 billion positions. Chess: Deep Blue defeated human world champion Garry 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 some lines of search up to 40 ply. Othello: human champions refuse to compete against computers, who are too good. Go: human champions refuse to compete against computers, who are too bad. In go, b > 300, so most programs use pattern knowledge bases to suggest plausible moves.
24	Summary Games are fun to work on! They illustrate several important points about AI Perfection is unattainable à must approximate Good idea to think about what to think about
25	Min, go over Hex! What does the web page say? http://www.comp.nus.edu.sg/~cs3243/
26	What do you need to do Implement Minimax Implement Pruning (optional) Implement an evaluation function Input: board, selected grid location Output: continuous value (really optional) use state