Components of decision making Criteria for decision making Utility theory Decision trees Posterior probabilities using Bayes’ rule The Monty Hall problem
Outline 2
A decision problem may be represented by a model in terms of the following elements:
1. Components of decision making 3 The decision maker who is responsible for making the decision. Alternative courses of action: given that the alternatives are specified, the decision involves a choice among the alternative course action. Events: the scenarios or states of the environment not under control of the decision maker that may occur. Consequences: measures of the net benefit, or payoff received by the decision maker. They can be conveniently summarised in a payoff or decision matrix.
Alternative course of action
Components of decision making 4 Given alternatives are specified, the decision involves a choice among the alternatives course of action. When the opportunity to acquire information is available, the decision maker’s problem is to choose a best information source and a best overall strategy. A strategy is a set of decision rules indicating which action should be taken contingent on a specific observation received from the chosen information source.
Events
Components of decision making 5 The events are defined to be mutually exclusive and collectively exhaustive. Uncertainty of an event is measured in terms of the probability assigned to this event. A characteristic of decision analysis is that the probabilities of events can be: subjective (reflecting the decision maker’s state of knowledge or beliefs) or objective (theoretically or empirically determined) or both.
2. Criteria for decision making 6 Optimistic (Maximax) : maximize the maximum possible profit. Pessimistic (Maximin): maximize the minimum possible profit. Minimax-regret criterion: minimize the regret for not having chosen the best alternative. Regret = (profit from the best decision) - (profit from the nonoptimal decision) The expected value criterion: choose the action that yields the largest expected rewards.
Criteria for decision making 7 Example. Ah Beng sells newspapers at a bus interchange. He pays the company 20 cents and sells the paper for 25 cents. Newspapers that are unsold at the end of the day are worthless. He knows that he can sell between 6 and 10 papers a day. Payoff matrix: Papers Ordered Papers demanded 6 7 8 9 10 6 30 30 30 30 30 7 10 35 35 35 35 8 -10 15 40 40 40 9 -30 -5 20 45 45 10 -50 -25 0 25 50 decision
Criteria for decision making 8 Maximax criterion. Papers Ordered Papers demanded 6 7 8 9 10 6 30 30 30 30 30 7 10 35 35 35 35 8 -10 15 40 40 40 9 -30 -5 20 45 45 10 -50 -25 0 25 50 Papers ordered States with best outcome Best outcome 6 6,7,8,9,10 30 cents 7 7,8,9,10 35 cents 8 8,9,10 40 cents 9 9,10 45 cents 10 10 50 cents Decision: Order 10 papers for a potential profit of 50 cents.
Criteria for decision making 9 Maximin criterion. Papers Ordered Papers demanded 6 7 8 9 10 6 30 30 30 30 30 7 10 35 35 35 35 8 -10 15 40 40 40 9 -30 -5 20 45 45 10 -50 -25 0 25 50 Papers ordered Worst state Reward in worst state 6 6,7,8,9,10 30 cents 7 6 10 cents 8 6 -10 cents 9 6 -30 cents 10 6 -50 cents Decision: Order 6 papers and earn a profit of at least 30 cents.
Criteria for decision making 11 The expected value criterion. Papers Ordered Papers demanded 6 7 8 9 10 6 30 30 30 30 30 7 10 35 35 35 35 8 -10 15 40 40 40 9 -30 -5 20 45 45 10 -50 -25 0 25 50 Decision: Order 6 or 7 papers to maximize the expected reward Note: Equal probability of demands is assumed Expected value: Papers ordered Expected reward 6 0.2(30+30+30+30+30) = 30 7 0.2(10+35+35+35+35) = 30 8 0.2(-10+15+40+40+40) = 25 9 0.2(-30-5+20+45+45) = 15 10 0.2(-50-25+0+25+50) = 0
3. Utility theory 12 Utility function is a formula or a method for converting any profit of a decision maker to an associated utility. Suppose you are asked to choose between two lotteries L1 and L2 . With certainty, Lottery L1 yields $10,000. Lottery L2 consists of tossing a coin. Head, $30,000 and tail $0. L1 yields an expected reward of $10,000 and L2 yields an expected reward of 0.5(30000 + 0) = $15,000. Although L2 has larger expected value than L1, most people prefer L1 to L2.
Utility theory 13 Converting profit into utilities Suppose that there are 9 possible profit levels. In decreasing order: 15, 7, 5, 4, 3, 2, -1, -1, and -5 millions. We assign a utility of 100 to the highest profit and 0 to the lowest profit. Consider the profit of 7 millions. Would you prefer a lottery that offers a potential winning of 15 with probability of 0.05 and winning of -5 with probability of 0.95 or a guaranteed profit of 7. Would you prefer a lottery that offers a potential winning of 15 with probability of 0.95 and winning of -5 with probability of 0.05 or a guaranteed profit? The goal is to determine a probability p in the lottery for winning 15 and (1-p) probability for winning -5 so that you will be indifferent to playing the lottery or taking the guaranteed profit of 7.
Utility theory 14 Converting profit into utilities Suppose p = 0.40, that is, you are indifferent between playing a lottery that offers 15 with p = 0.40, and -5 with probability 1 - p = 0.60 or taking a guaranteed profit of 7. With p = 0.40, the expected profit is 0.4015 + 0.6(-5) = 3. -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 p x: profit 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (15,1.0) (-5,0.0) Risk neutral: u = 0.05 x + 0.25 (7,0.6) (7,0.4) Risk seeking: u = 0.4 Exp. profit = 3 < 7
Utility theory 15 Converting profit into utilities Suppose p = 0.40, that is, you are indifferent between playing a lottery that offers 15 with p = 0.40, or -5 with probability 1 - p = 0.60 or taking a guaranteed profit of 7. With p = 0.40, the expected profit is 0.4015 + 0.6(-5) = 3. You are willing to pay 7 to play a lottery whose expected profit is only 3. This is because you value the opportunity of making 15 relatively more. Hence for profit = 7, p = 0.40 and utility = p100 = 40. We must now repeat the interview process to find the utilities for each of the remaining profit. A risk-seeking decision maker has a utility function that indicates preference for taking risks (Convex utility function). Risk-averse decision maker (Concave utility function). Risk-neutral decision maker (Linear utility function).
Utility theory 17 Computing utility function: an example. If you are risk-averse and have assigned the following two endpoints on your utility function: U(-30) = 0 U( 70) = 1 what is a lower bound on U(30)? Answer: p(70) + (1-p)(-30) 30 100 p 60 or p 0.6
Utility theory 18 Computing utility function: a second example. If Joe is risk-averse , which of the following lotteries he prefers? L1: with probability .10 Joe loses $100 with probability .90 Joe wins $0 L2: with probability .10 Joe loses $190 with probability .90 Joe wins $10 Answer: Straight line equation: y = 0.5x + 95 L1: 0.10 U(-100) + 0.90 U(0) > 0.10 (45) + 0.90 (95) = 90 L2: 0.10 U(-190) + 0.90 U(10) = 0.10(0) + 0.90(100) = 90 L1 is preferred, its utility is strictly greater than 90. (10,100) (-190,0) 45 When payoff is -100, the utility is more than 45 for Joe When payoff is 0, the utility is more than 95 for Joe
Utility theory 19 Utility function: a paradox. Suppose we are offered a choice between 2 lotteries: L1: with probability 1, we receive $1 million. L2: with probability .10, we receive $5 million. with probability .89, we receive $1 million. with probability .01, we receive $0. Which lottery do we prefer? Answer: L1 U(1M) > 0.10 U(5M) + 0.89 U(1M) + 0.01 U(0M) or 0.11 U(1M) > 0.10 U(5M) + 0.01 U(0M)
Utility theory 20 Utility function: a paradox - continued. Suppose we are offered a choice between 2 lotteries: L3: with probability .11, we receive $1 million. with probability .89, we receive $0. L4: with probability .10, we receive $5 million. with probability .90, we receive $0. Which lottery do we prefer? Answer: L4 0.11 U(1M) + 0.89 U(0M) < 0.10 U(5M) + 0.90 U(0M) or 0.11 U(1M) < 0.10 U(5M) + 0.01 U(0M) compare with previous slide!
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