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Making Decisions Under Uncertainty

Making Decisions Under Uncertainty. To judge what one must do to obtain a good or avoid an evil, it is necessary to consider not only the good and the evil in itself, but also the probability that it happens or does not happen; and to view geometrically the proportion that all these

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Making Decisions Under Uncertainty

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  1. Making Decisions Under Uncertainty

  2. To judge what one must do to obtain a good or avoid an evil, it is necessary to consider not only the good and the evil in itself, but also the probability that it happens or does not happen; and to view geometrically the proportion that all these things have in common. -- French philosopher Arnauld, 1662

  3. Ideally, RESULT(s0, a) is the deterministic outcome of taking action a in state s0. But the agent often does not know the current state, so we can consider RESULT(a) to be a random variable.

  4. Ideally, RESULT(s0, a) is the deterministic outcome of taking action a in state s0. But the agent often does not know the current state, so we can consider RESULT(a) to be a random variable. P(RESULT(a) = s’ | a, e) s‘ is the outcome e is evidence/observations

  5. The agent’s preferences are captured by a “utility function” U(s), which maps each state s to a number expressing the state’s desirability.

  6. The agent’s preferences are captured by a “utility function” U(s), which maps each state s to a number expressing the state’s desirability. EU(a | e) = Sum P(RESULT(a) = s’ | a, e) U(s’) s’ EU – expected utility, weighted average of each utility value U, weighted by the probability that the outcome occurs.

  7. EU(a | e) = Sum P(RESULT(a) = s’ | a, e) U(s’) s’ Example: you are playing Irvine rules poker. You hold 4, 5, K, 7, 8. You’ve seen that one of the four 6 cards is in someone else‘s hand. You can fold, and lose $3, or you can pay $1 to draw one more card. If you draw a 6 card, you will win $50, otherwise you will lose $3 + $1 = $4.

  8. EU(a | e) = Sum P(RESULT(a) = s’ | a, e) U(s’) s’ Example: you are playing Irvine rules poker. You hold 4, 5, K, 7, 8. You’ve seen that one of the four 6 cards is in someone else‘s hand. You can fold, and lose $3, or you can pay $1 to draw one more card. If you draw a 6 card, you will win $50, otherwise you will lose $3 + $1 = $4. EU(fold | e) = -$3

  9. EU(a | e) = Sum P(RESULT(a) = s’ | a, e) U(s’) s’ Example: you are playing Irvine rules poker. You hold 4, 5, K, 7, 8. You’ve seen that one of the four 6 cards is in someone else‘s hand. You can fold, and lose $3, or you can pay $1 to draw one more card. If you draw a 6 card, you will win $50, otherwise you will lose $3 + $1 = $4. EU(fold | e) = -$3 EU(draw | e) = $50 * 3 / 46 + -$4 * 43/46

  10. EU(a | e) = Sum P(RESULT(a) = s’ | a, e) U(s’) s’ Example: you are playing Irvine rules poker. You hold 4, 5, K, 7, 8. You’ve seen that one of the four 6 cards is in someone else‘s hand. You can fold, and lose $3, or you can pay $1 to draw one more card. If you draw a 6 card, you will win $50, otherwise you will lose $3 + $1 = $4. EU(fold | e) = -$3 EU(draw | e) = $50 * 3 / 46 = $3.26 + -$4 * 43/46 = -$3.74 -$0.48

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