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Lecture 9 of 14

Lecture 9 of 14. Game Tree Search: Minimax and Alpha-Beta. Friday, 10 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Chapter 6, Russell and Norvig 2e. Lecture Outline. Today’s Reading

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Lecture 9 of 14

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  1. Lecture 9 of 14 Game Tree Search: Minimax and Alpha-Beta Friday, 10 September 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Chapter 6, Russell and Norvig 2e

  2. Lecture Outline • Today’s Reading • Sections 6.1-6.4, Russell and Norvig 2e • Recommended references: Rich and Knight, Winston • Reading for Next Class: Sections 6.5-6.8, Russell and Norvig • Games as Search Problems • Frameworks: two-player, multi-player; zero-sum; perfect information • Minimax algorithm • Perfect decisions • Imperfect decisions (based upon static evaluation function) • Issues • Quiescence • Horizon effect • Need for pruning • Next Lecture: Alpha-Beta Pruning, Expectiminimax, Current “Hot” Problems • Next Week: Knowledge Representation – Logics and Production Systems

  3. Overview • Perfect Play • General framework(s) • What could agent do with perfect info? • Resource Limits • Search ply • Static evaluation: from heuristic search to heuristic game tree search • Examples • Tic-tac-toe, connect four, checkers, connect-five / Go-Moku / wu3 zi3 qi2 • Chess, go • Games with Uncertainty • Explicit: games of chance (e.g., backgammon, Monopoly) • Implicit: see project suggestions! Adapted from slides by S. Russell, UC Berkeley

  4. Games versus Search Problems • Unpredictable Opponent • Solution is contingency plan • Time limits • Unlikely to find goal • Must approximate • Plan of Attack • Algorithm for perfect play (J. von Neumann, 1944) • Finite horizon, approximate evaluation (C. Zuse, 1945; C. Shannon, 1950, A. Samuel, 1952-1957) • Pruning to reduce costs (J. McCarthy, 1956) Adapted from slides by S. Russell, UC Berkeley

  5. Types of Games • Information: Can Know (Observe) • … outcomes of actions / moves? • … moves committed by opponent? • Uncertainty • Deterministic vs. nondeterministic outcomes • Thought exercise: sources of nondeterminism? Adapted from slides by S. Russell, UC Berkeley

  6. Minimax Figure 5.2 p. 125 R&N Adapted from slides by S. Russell, UC Berkeley

  7. Minimax Algorithm:Decision and Evaluation  what’s this?  what’s this? Figure 5.3 p. 126 R&N Adapted from slides by S. Russell, UC Berkeley

  8. Properties of Minimax • Complete? • … yes, provided following are finite: • Number of possible legal moves (generative breadth of tree) • “Length of game” (depth of tree) – more specifically? • Perfect vs. imperfect information? • Q: What search is perfect minimax analogous to? • A: Bottom-up breadth-first • Optimal? • … yes, provided perfect info (evaluation function) and opponent is optimal! • … otherwise, guaranteed if evaluation function is correct • Time Complexity? • Depth of tree: m • Legal moves at each point: b • O(bm) – NB, m 100, b 35 for chess! • Space Complexity? O(bm) – why? Adapted from slides by S. Russell, UC Berkeley

  9. Resource Limits Adapted from slides by S. Russell, UC Berkeley

  10. Static Evaluation Function Example:Chess Figure 5.4(c), (d) p. 128 R&N Adapted from slides by S. Russell, UC Berkeley

  11. Do Exact Values Matter? Adapted from slides by S. Russell, UC Berkeley

  12. Cutting Off Search [1] Adapted from slides by S. Russell, UC Berkeley

  13. Cutting Off Search [2] • Issues • Quiescence • Play has “settled down” • Evaluation function unlikely to exhibit wild swings in value in near future • Horizon effect • “Stalling for time” • Postpones inevitable win or damaging move by opponent • See: Figure 5.5 R&N • Solutions? • Quiescence search: expand non-quiescent positions further • “No general solution to horizon problem at present Adapted from slides by S. Russell, UC Berkeley

  14. Why Prune? Figure 5.6 p. 131 R&N Adapted from slides by S. Russell, UC Berkeley

  15. Summary Points • Introduction to Games as Search Problems • Frameworks • Two-player versus multi-player • Zero-sum versus cooperative • Perfect information versus partially-observable (hidden state) • Concepts • Utility and representations (e.g., static evaluation function) • Reinforcements: possible role for machine learning • Game tree • Family of Algorithms for Game Trees: Minimax • Propagation of credit • Imperfect decisions • Issues • Quiescence • Horizon effect • Need for pruning

  16. Terminology • Game Graph Search • Frameworks • Two-player versus multi-player • Zero-sum versus cooperative • Perfect information versus partially-observable (hidden state) • Concepts • Utility and representations (e.g., static evaluation function) • Reinforcements: possible role for machine learning • Game tree: node/move correspondence, search ply • Family of Algorithms for Game Trees: Minimax • Propagation of credit • Imperfect decisions • Issues • Quiescence • Horizon effect • Need for (alpha-beta) pruning

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