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Quiz 4 : Minimax

Quiz 4 : Minimax. Minimax is a paranoid algorithm. True Using alpha-beta pruning does not affect the solution. True Every game is also a general search problem. False Every general search problem is also a game. True Typically the best algorithm for game playing is DFS. False

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Quiz 4 : Minimax

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  1. Quiz 4 : Minimax • Minimax is a paranoid algorithm.True • Using alpha-beta pruning does not affect the solution. True • Every game is also a general search problem. False • Every general search problem is also a game. True • Typically the best algorithm for game playing is DFS. False • Evaluation functions still guarantee optimal outcomes. False • It is always best to assume the worst case. False • Checkers is still an open problem. False • Why did Pacman starve?

  2. CSE511a: Artificial IntelligenceSpring 2012 Lecture 7: Expectimax Search 02/05/2012 Kilian Weinberger Many slides over the course adapted from either Dan Klein, Stuart Russell or Andrew Moore

  3. Deterministic Games • Many possible formalizations, one is: • States: S (start at s0) • Players: P={1...N} (usually take turns) • Actions: A (may depend on player / state) • Transition Function: SxA  S • Terminal Test: S  {t,f} • Terminal Utilities: SxP  R • Solution for a player is a policy: S  A

  4. Adversarial Games • Deterministic, zero-sum games: • Tic-tac-toe, chess, checkers • One player maximizes result • The other minimizes result • Minimax search: • A state-space search tree • Players alternate turns • Each node has a minimax value: best achievable utility against a rational adversary max 5 Minimax values:computed recursively min 5 2 8 2 5 6 Terminal values:part of the game

  5. Computing Minimax Values • Two recursive functions: • max-value maxes the values of successors • min-value mins the values of successors def value(state): If the state is a terminal state: return the state’s utility If the next agent is MAX: return max-value(state) If the next agent is MIN: return min-value(state) def max-value(state): Initialize max = -∞ For each successor of state: Compute value(successor) Update max accordingly Return max

  6. Recap: Resource Limits • Cannot search to leaves • Depth-limited search • Instead, search a limited depth of tree • Replace terminal utilities with an eval function for non-terminal positions • Guarantee of optimal play is gone • Replanning agents: • Search to choose next action • Replan each new turn in response to new state max 4 min min -2 4 -1 -2 4 9 ? ? ? ?

  7. Alpha-Beta Pruning Example 3 ≤2 ≤1 3 3 12 2 14 5 1 ≥8 a is MAX’s best alternative here or above b is MIN’s best alternative here or above 8

  8. Alpha-Beta Pruning Properties • This pruning has no effect on final result at the root • Values of intermediate nodes might be wrong! • Good child ordering improves effectiveness of pruning • With “perfect ordering”: • Time complexity drops to O(bm/2) • Doubles solvable depth! • Full search of, e.g. chess, is still hopeless…

  9. Expectimax Search Trees • What if we don’t know what the result of an action will be? E.g., • In solitaire, next card is unknown • In backgammon, die roll unknown. • In minesweeper, mine locations • In pacman, the ghosts act randomly • Can do expectimax search • Chance nodes, like min nodes, except the outcome is uncertain • Calculate expected utilities • Max nodes as in minimax search • Chance nodes take average (expectation) of value of children • Later, we’ll learn how to formalize the underlying problem as a Markov Decision Process max chance 10 4 5 7

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