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Decision-Making Under Uncertainty: Understanding Expected Utility in Poker

This article explores the philosophy and mathematics behind making decisions under uncertainty, particularly in the context of gambling, like poker. It discusses how to weigh the potential outcomes of various actions, considering both their desirability—expressed through a utility function—and the probabilities of those outcomes. Using a practical example from poker, the article illustrates how to calculate expected utility to inform decision-making when faced with risks and unknowns. This analytical framework helps agents navigate uncertainty effectively.

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Decision-Making Under Uncertainty: Understanding Expected Utility in Poker

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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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