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Introduction to Adversarial Search in AI: Understanding Game Theory and Minimax Algorithm

This lecture explores adversarial search in artificial intelligence, focusing on multi-agent, non-cooperative environments. We analyze classic games like Chess and Go, discussing the Minimax algorithm and its properties, including completeness and optimality against optimal opponents. The lecture highlights the challenges of planning ahead in unpredictable scenarios and introduces α-β pruning to improve efficiency. Real-life applications in games demonstrate the importance of evaluation functions and approximation methods when perfect solutions are unattainable. Join us in learning how AI navigates competitive settings!

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Introduction to Adversarial Search in AI: Understanding Game Theory and Minimax Algorithm

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  1. An Introduction to Artificial Intelligence Lecture VI: Adversarial Search (Games) Ramin Halavati (halavati@ce.sharif.edu) In which we examine problems that arise when we try to plan ahead in a world were other agents are playing against us.

  2. Overview

  3. Primary Assumptions • “Game” in AI: • A multi-agent, non-cooperative environment • Zero Sum Result. • Turn Taking. • Deterministic. • Two Player • Real Problems vs. Toy Problems: • Chess: b=35 , d = 100  Tree Size: ~10154 • Go: b=1000 (!) • Time Limit / Unpredictable Opponent

  4. Game tree (2-player, deterministic, turns)

  5. Minimax Algorithm

  6. Minimax algorithm

  7. Properties of minimax • Complete? Yes (if tree is finite) • Optimal? Yes (against an optimal opponent) • Time complexity? O(bm) • Space complexity? O(bm) (depth-first exploration) • For chess, b ≈ 35, m ≈100 for "reasonable" games exact solution completely infeasible

  8. α-β pruning example

  9. α-β pruning example

  10. α-β pruning example

  11. α-β pruning example

  12. α-β pruning example

  13. Properties of α-β • Pruning does not affect final result • Good move ordering improves effectiveness of pruning • With "perfect ordering," time complexity = O(bm/2) doubles depth of search • A simple example of the value of reasoning about which computations are relevant (a form of metareasoning)

  14. 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  prune that branch Define β similarly for min Why is it called α-β?

  15. The α-β algorithm

  16. The α-β algorithm

  17. Resource limits Suppose we have 100 secs, explore 104 nodes/sec106nodes per move Standard approach: • cutoff test: e.g., depth limit (perhaps add quiescence search) • evaluation function = estimated desirability of position

  18. Evaluation functions • For chess, typically linear weighted sum of features Eval(s) = w1 f1(s) + w2 f2(s) + … + wn fn(s) • e.g., w1 = 9 with f1(s) = (number of white queens) – (number of black queens), etc.

  19. Cutting off search MinimaxCutoff is identical to MinimaxValue except • Terminal? is replaced by Cutoff? • Utility is replaced by Eval Does it work in practice? bm = 106, b=35  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

  20. 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.

  21. Summary • Games are fun to work on! • They illustrate several important points about AI • perfection is unattainable  must approximate • good idea to think about what to think about

  22. Exercise Excercise 6.16Send to n_ghanbari@ce.sharif.eduSubject: AIEX-C616

  23. Project Proposals: Choose a gamin, compose a group of rival agents, implement agents to compete. 1st Choice: Backgammon (Takhteh Nard) - refer to Mr.Esfandiar's call for participants. 2nd Choice: Choose a board game such as DOOZ, AVALANGE, etc. 3rd Choice: A card game, such as HOKM or BiDel.

  24. Essay Proposals 1-What was the "King and Rock vs. King" story, stated in page 186 of book. 2-What are other general puropose heuristics such as null-move? 3-What is B* algorithm? (See Page 188, for clue) 4-What is MGSS* algorithm? (See Page 188, for clue) 5-What is SSS* algorithm? (See Page 188, for clue) 6-What is Alpha-Beta pruning with probability? (See Page 189, for clue)

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