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572 CHAPTER 13 DECISION ANALYSIS

lottery and the E30000 guaranteed payoff as the risk premium that the president would
be willing to pay to avoid the 5 per cent chance of losing E50000.
To calculate the utility associated with a payoff of E20 000, we must ask
Swofford’s president to state a preference between a guaranteed E20 000 payoff
and the opportunity to engage in the following lottery.
Lottery: Swofford obtains a payoff of E50 000 with probability p
and a payoff of E50 000 with probabilityð1 pÞ:
Note that it is exactly the same lottery we used to establish the utility of a payoff of
E30 000. In fact, this lottery will be used to establish the utility for any monetary
value in the Swofford payoff table. Using this lottery, then, we must ask the
president to state the value of p that provides an indifference between a guaranteed
payoff of E20 000 and the lottery. For example, we might begin by asking the
president to choose between a certain loss of E20 000 and the lottery with a payoff
of E50 000 with probability p ¼ 0.90 and a payoff of E50 000 with probability
(1 p) ¼ 0.10. What answer do you think we would get? Surely, with this high
probability of obtaining a payoff of E50 000, the president would select the lottery.
Next, we might ask if p ¼ 0.85 would result in indifference between the loss of
E20 000 for certain and the lottery. Again, the president might tell us that the lottery
would be preferred. Suppose that we continue in this fashion until we get to
p ¼ 0.55, where we find that with this value of p, the president is indifferent between
the payoff of E20 000 and the lottery. That is, for any value of p less than 0.55, the
president would rather take a loss of E20 000 for certain than risk the potential loss
of E50 000 with the lottery; for any value of p above 0.55, the president would select
the lottery. Thus, the utility assigned to a payoff of E20 000 is:

Uð E20 000Þ¼ pUðE50 000Þþð1 pÞUð E50 000Þ
¼ 0:55ð10Þþ 0:45ð0Þ
¼ 5:5
Again, let us examine the significance of this assignment as compared with the
expected value approach. When p ¼ 0.55, the expected value of the lottery is:
EVðLotteryÞ¼ 0:55ðE50 000Þþ 0:45ð E50 000Þ
¼ E27 500 E22 500
¼ E5 000
Thus, the president would just as soon absorb a loss of E20 000 for certain as take
the lottery, even though the expected value of the lottery is E5000. Once again we
see the conservative, or risk-avoiding, point of view of Swofford’s president.
In the two preceding examples where we calculated the utility for a specific monetary
payoff, M, we first found the probability p where the decision maker was indifferent
between a guaranteed payoff of M and a lottery with a payoff of E50000 with proba-
bility p and E50000 with probability (1 p). The utility of M was then calculated as:
UðMÞ¼ pUðE50 000Þþð1 pÞUð E50 000Þ
¼ pð10Þþð1 pÞ0
¼ 10p
Using this procedure, utility values for the rest of the payoffs in Swofford’s problem
were developed. The results are presented in Table 13.10.
After we determine the utility value of each of the possible monetary values, we
can write the original payoff table in terms of utility values. Table 13.11 shows the
utility for the various outcomes in the Swofford problem. The notation we use for

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