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Lecture 9 of 14

Game Tree Search: Minimax and Alpha-Beta

Lecture Outline

• Today’s Reading
• 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

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)

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?

Minimax Algorithm:

Decision and Evaluation

← what’s this?

← what’s this?

Figure 5.3 p. 126 R&N

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?

Static Evaluation Function Example:

Chess

Figure 5.4(c), (d) p. 128 R&N

Do Exact Values Matter?

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

Why Prune?

Figure 5.6 p. 131 R&N

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