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SMU EMIS 5300/7300 NTU SY-521 -N Systems Analysis Methods Dr. Jerrell T. Stracener, SAE Fellow Utility Theory Applications updated 11. 27. 01 1

Rationality • Never try to explain with a rational model that which is perfectly well explained by stupidity • All judgmental statements require as their basis some assumption regarding the objective of that which is being judged 2

Example - Prom Paul sensed that he was caught in a dilemma. All month he had planned to take his girl to the school’s annual prom. He had even saved up the usual \$20 per couple admission fee. But at the last minute the student council raised the fee to \$30 to cover the unexpected costs of hiring a well-known dance band. That seemed to doom Paul’s plans. Then one of his friends, knowing his dilemma, joking offered him a coin -toss gamble. If the toss showed heads, Paul must pay him the \$20 he had saved, but if it showed tails, he would pay Paul \$12. At the time, Paul had laughed off the gamble since it was so obviously unfair. But now, Paul wondered - perhaps it wasn’t such a foolish gamble after all. 3

Example - Prom States of nature Alternatives 0. 5 Heads Tails Expected payoff (EMV) Accept offer 0 32 16 Reject offer 20 20 20 Maximum Solution: Using the expect payoff approach, Paul should reject his friend’s offer, since the expected value of rejection is higher. Analysis: Before Paul rejects the offer, he should do some thinking. He should consider the enjoyment he will have at the prom, and about his date’s possible reaction. He should probably accept the offer. The reason for this is that the \$20 is of very little use to him, since it cannot get him to the prom; but if he wins the gamble, he will have enough money to pay the admission fee and even buy a soft drink or two. What should actually be compared in this case are the not the monetary values but the benefits or utilities that it has for Paul. 4

Expected Monetary Values Discussion: In the analysis of the decision making under risk, the best alternative is usually selected by calculating and comparing the expected monetary values (EMV) of the various alternatives. However, there at least three situations in which EMV is not likely to be a valid criterion. First, if the decision maker finds difficulty in expressing the values of some of the outcomes of his or her decisions in terms of monetary payoffs, then EMV cannot be used. Second, EMV assumes that the decision maker is willing to risk losing money in the short run as long as he or she is better off in the long run. In reality, however, decision makers frequently act to avoid risk in the short run, particularly if there is any possibility whatsoever of incurring a large initial loss. Finally, EMV assumes a linear relationship between the amount of money and its value (or utility). For example, it is assumed that the value of \$20, 000 is twice that of \$10, 000. In reality, however, it has been observed that with an increase in the amount of wealth accumulated, the value of additional money decreases. (For example, the value of a dollar added to \$10 is larger than that of one added to \$1000) 5

Utility Function • For all these situations there is a need for a measure other than money which better describes how decision makers value possible outcomes. • Each individual has a measurable preference among various choices available in risk situations. This preference is called utility and is measured in arbitrary units called utiles. By suitable questioning, it is possible to determine a person’s utility for various amounts of money. This is called a person’s utility (or preference) function. The graph of this function offers a picture of the individual's attitude towards risk-taking. Usually, a person will choose that alternative which maximizes his or her expected value. 6

Utility 1. Utility can be measured on a cardinal scale. That is, cardinal numbers (1, 2, and so on) can describe how many utiles constitute a payoff. 2. Utilities of different objects can be added together (this is called the additivity assumption) 7

Certainty Equivalent Let us return to Paul's dilemma. Paul has two alternatives: accept the offer with an EMV of \$16 or reject it and save \$20. Let us assume that Paul is indifferent between the two alternatives. That is, he is willing to accept the offer at \$20 but he will not be paying more than \$20. It is clear that the decision of how much to pay for this gamble is subjective. While Paul is willing to pay \$20, someone else may be willing to pay even \$25 while a third person might not be willing to pay more than \$18. We call such a subjective evaluation of a risky situation (a gamble) the certainty equivalent (CE) of the decision maker for that risky situation. 8

Certainty Equivalent The concept of certainty equivalent enables us to classify decision makers into three types. • EMV takers whose certainty equivalent = EMV • Risk takers whose certainty equivalent is larger than the EMV of the gamble (such as Paul) • Risk averters whose certainty equivalent is smaller than the EMV. 9

Risk premium We see that people’s attitudes toward risk are related to the certainty equivalent (CE) and the EMV of the risk. The difference between these two is called the risk premium (RP). RP = EMV - CE For a risk taker the RP is negative since the CE > EMV. For a risk averter the RP is positive. 10

School Prom Using Utilities Let us examine the problem in utility terms: Since the \$20 will not get Paul into the prom, it is of little value to him. We may arbitrarily assume that \$20 is worth 100 utiles. Losing the \$20 will leave him with no money, a situation which is worth 0 utiles. However, the additional \$12 is crucial since he will have \$32 and will be able to go to the prom. Therefore, the \$32 is extremely valuable, worth say, 500 utiles. Now the situation can be reevaluated in terms of utilities. 11

School Prom Using Utilities States of nature Alternatives 0. 5 Heads Tails Expected Utility (EU) Accept offer 0 500 250 Reject offer 100 Maximum 100 Solution: Using the expected utility, Paul should accept the offer, since the expected utility of acceptance is higher. 12

Example: To insure or not to insure? Suppose that management is about to make a decision concerning fire insurance on a plant valued at \$2 million. There is a chance of 1 in 2, 000 (0. 0005) that the fire will destroy the plant during a one-year period. The annual premium for insurance is \$1, 500. Should the company insure or not? States of nature 0. 0005 0. 9995 Fire No Fire Insure \$1, 500 Do not insure \$2, 000 \$0 \$1, 000 Alternatives Expected payoff (EMV) Minimum cost According to the EMV, management should not insure. In the long run it will cost more to insure than not to. 13

Example: To insure or not to insure? However, since the situation is in the realm of uncertainty, there is a chance, although very small, that fire may even occur during the first year. The loss of \$2 million would probably bankrupt the company - a situation that manage could not afford. Therefore, they will buy the insurance even though it has a larger expected monetary cost. 14

Example: To insure or not to insure? If the same situation is analyzed in utility terms, it might look like this: The premium is worth \$1, 500 is worth -1 utile to the company; zero dollars is worth zero utiles. A loss of \$2 million is worth -10, 000 utiles. States of nature 0. 0005 0. 9995 Fire No Fire Insure -1 -1 -1 utile Do not insure -10, 000 0 -5 utiles Alternatives Expected Utility (EU) Maximum utility The results show that the expected utility of insuring is higher than that of not insuring; thus, the company will elect to insure. 15

Utility (preference) Curve Risk Averse Risk Taker when Poor; Risk Averse when Rich Utility Risk Neutral Risk Taker Money (\$) 16