The first step in using utility theory is to assign utility values to each monetary value in a given situation. It is convenient to begin utility assessment by assigning the worst outcome a utility of 0 and the best outcome a utility of 1. Although any values may be used as long as the utility for the best outcome is greater than the utility for the worst outcome, using 0 and 1 has some benefits. Because we have chosen to use 0 and 1, all other outcomes will have a utility value between 0 and 1. In determining the utilities of all outcomes, other than the best or worst outcome, a standard gamble is considered. This gamble is shown in Figure 3.7.

Figure 3.6 Your Decision Tree for the Lottery Ticket

In Figure 3.7, p is the probability of obtaining the best outcome, and $\left(1-p\right)$ is the probability of obtaining the worst outcome. Assessing the utility of any other outcome involves determining the probability (p) that makes you indifferent between alternative 1, which is the gamble between the best and worst outcomes, and alternative 2, which is obtaining the other outcome for sure. When you are indifferent between alternatives 1 and 2, the expected utilities for these two alternatives must be equal. This relationship is shown as

Now all you have to do is to determine the value of the probability (p) that makes you indifferent between alternatives 1 and 2. In setting the probability, you should be aware that utility assessment is completely subjective. It’s a value set by the decision maker that can’t be measured on an objective scale. Let’s take a look at an example.

Jane Dickson would like to construct a utility curve revealing her preference for money between $0 and $10,000. A utility curve is a graph that plots utility value versus monetary value. She can invest her money either in a bank savings account or in a real estate deal.

If the money is invested in the bank, in 3 years Jane would have $5,000. If she invested in the real estate, after 3 years she could either have nothing or have $10,000. Jane, however, is very conservative. Unless there is an 80% chance of getting $10,000 from the real estate deal, Jane would prefer to have her money in the bank, where it is safe. What Jane has done here is to assess her utility for $5,000. When there is an 80% chance (this means that p is 0.8) of getting $10,000, Jane is indifferent between putting her money in real estate and putting it in the bank. Jane’s utility for $5,000 is thus equal to 0.8, which is the same as the value for p. This utility assessment is shown in Figure 3.8.

Other utility values can be assessed in the same way. For example, what is Jane’s utility for $7,000? What value of p would make Jane indifferent between $7,000 and the gamble that would result in either $10,000 or $0? For Jane, there must be a 90% chance of getting the $10,000. Otherwise, she would prefer the $7,000 for sure. Thus, her utility for $7,000 is 0.90. Jane’s utility for $3,000 can be determined in the same way. If there were a 50% chance of obtaining the $10,000, Jane would be indifferent between having $3,000 for sure and taking the gamble of either winning the $10,000 or getting nothing. Thus, the utility of $3,000 for Jane is 0.5. Of course, this process can be continued until Jane has assessed her utility for as many monetary values as she wants. These assessments, however, are enough to get an idea of Jane’s feelings toward risk. In fact, we can plot these points in a utility curve, as is done in Figure 3.9. In the figure, the assessed utility points of $3,000, $5,000, and $7,000 are shown by dots, and the rest of the curve is inferred from these.

Figure 3.9 Utility Curve for Jane Dickson

Jane’s utility curve is typical of a risk avoider. A risk avoider is a decision maker who gets less utility or pleasure from a greater risk and tends to avoid situations in which high losses might occur. As monetary value increases on her utility curve, the utility increases at a slower rate.

Figure 3.10 illustrates that a person who is a risk seeker has an opposite-shaped utility curve. This decision maker gets more utility from a greater risk and higher potential payoff. As monetary value increases on his or her utility curve, the utility increases at an increasing rate. A person who is indifferent to risk has a utility curve that is a straight line. The shape of a person’s utility curve depends on the specific decision being considered, the monetary values involved in the situation, the person’s psychological frame of mind, and how the person feels about the future. It may well be that you have one utility curve for some situations you face and completely different curves for others.

After a utility curve has been determined, the utility values from the curve are used in making decisions. Monetary outcomes or values are replaced with the appropriate utility values, and then decision analysis is performed as usual. The expected utility for each alternative is computed instead of the EMV. Let’s take a look at an example in which a decision tree is used and expected utility values are computed in selecting the best alternative.

Figure 3.10 Preferences for Risk

Mark Simkin loves to gamble. He decides to play a game that involves tossing thumbtacks in the air. If the point on the thumbtack is facing up after it lands, Mark wins $10,000. If the point on the thumbtack is down, Mark loses $10,000. Should Mark play the game (alternative 1), or should he not play the game (alternative 2)?

Alternatives 1 and 2 are displayed in the tree shown in Figure 3.11. As can be seen, alternative 1 is to play the game. Mark believes that there is a 45% chance of winning $10,000 and a 55% chance of suffering the $10,000 loss. Alternative 2 is not to gamble. What should Mark do? Of course, this depends on Mark’s utility for money. As stated previously, he likes to gamble. Using the procedure just outlined, Mark was able to construct a utility curve showing his preference for money. Mark has a total of $20,000 to gamble, so he has constructed the utility curve based on a best payoff of $20,000 and a worst payoff of a $20,000 loss. This curve appears in Figure 3.12.

We see that Mark’s utility for –$10,000 is 0.05, his utility for not playing ($0) is 0.15, and his utility for $10,000 is 0.30. These values can now be used in the decision tree. Mark’s objective is to maximize his expected utility, which can be done as follows:

1. Step 1.

   $$\begin{array}{r}\hfill U\text{(}-\text{\$10,000) = 0}\text{.05}\\ \hfill U\text{(\$0) = 0}\text{.15}\\ \hfill U\text{(\$10,000) = 0}\text{.30}\end{array}$$

2. Step 2. Replace monetary values with utility values. Refer to Figure 3.13 . Here are the expected utilities for alternatives 1 and 2:

   

   Figure 3.11 Decision Facing Mark Simkin

   Figure 3.11 Full Alternative Text

   

   Figure 3.12 Utility Curve for Mark Simkin

   Figure 3.12 Full Alternative Text

   

   Figure 3.13 Using Expected Utilities in Decision Making

   Figure 3.13 Full Alternative Text
   $$\begin{array}{lll}\hfill E\text{(alternative 1: play the game)}& =\hfill & \text{(0}\text{.45)(0}\text{.30) + (0}\text{.55)(0}\text{.05)}\hfill \\ \hfill & =\hfill & \text{0}\text{.135 + 0}\text{.027 = 0}\text{.162}\hfill \\ \hfill E\text{(alternative 2: don't play the game)}& =\hfill & \text{0}\text{.15}\hfill \end{array}$$

   Therefore, alternative 1 is the best strategy using utility as the decision criterion. If EMV had been used, alternative 2 would have been the best strategy. The utility curve is a risk-seeker utility curve, and the choice of playing the game certainly reflects this preference for risk.