Making Decisions under Uncertainty

How people come to a decision in the presence of uncer- tainty. Much of this research has been cast in terms of how people choose between gambles. Sometimes, the choices that we have to make are easy. If we are offered the choice of a gamble where we have a 25% chance of winning $100 and another gamble where we have a 50% chance of winning $1,000, most of us would not have much difficulty in figuring out which to accept. However, if we were faced with the choice of a certainty of $400 but only a 50% chance of $1,000, which would we select then? Something like this situation might arise if we inherited a risky stock that we could cash in for $400 or that we could hold on to and see whether the company takes offs or folds. A great deal of research on decision making under uncertainty requires participants to make choices among gambles. For instance, a participant might be asked to choose between the following two gambles:
A. $8 with a probability of 1∕3
B. $3 with a probability of 5∕6
In some cases, participants are just asked for their opinions; in other cases, they actually play the gamble that they choose. As an example of the latter possibility, a participant might roll a die and win in case A if he gets a 5 or 6 and win in case B if he gets a number other than 1. Which gamble would you choose?
As in the other domains of reasoning, such decision making has its own standard prescriptive theory for the way that people should behave in such situ- ations (von Neumann & Morgenstern, 1944). This theory says that they should choose the alternative with highest expected value. The expected value of an al- ternative is to be calculated by multiplying the probability by the value. Thus, the expected value of alternative A is $8 3 1∕3 5 $2.67, whereas the expected value of alternative B is $3 3 5∕6 5 $2.50. Thus, the normative theory says that partic- ipants should select gamble A. However, most participants will select gamble B.
As a perhaps more extreme example of the same result, suppose you are given a choice between
A. $1 million with a probability of 1
B. $2.5 million with a probability of 1∕2
Maybe, in this case, you are on a game show and are offered a choice between this great wealth with certainty or the opportunity to toss a coin and get even more. I (and I assume you) would take the money ($1 million) and run, but in fact, if we do the expected value calculations, we should prefer the second choice because its expected value is .5 3 $2.5 million 5 $1.25 million. Are we really behaving irrationally?
Most people, when asked to justify their behavior in such situations, will argue that there comes a point when one has enough money (if we could only convince CEOs of this notion!) and that there really isn’t much difference for them between $1 million and $2.5 million. This idea has been formal- ized in the terms of what is referred to as subjective utility—the value that we place on money is not linear with the face value of the money. Figure 11.7, which shows a typical function proposed for the relation of subjective utility to money (Kahneman & Tversky, 1984), has two interesting properties. The first is that it curves in such a way that the amount of money must more than double in order to double its utility. Thus, in the preceding example, we may value $2.5 million only 20% more than $1 million. Let us say that the subjec- tive utility of $1 million is U. The subjective utility of $2.5 million can then be expressed as 1.2U. In this case, then, the expected value of gamble A is 1 3 U 5 U, and the expected value of gamble B is 1∕2 3 1.2U 5 .6U. Thus, in terms of subjective utility, gamble A is more valuable and is to be preferred.
The second property of this utility function is that it is steeper in the loss region than in the gain region. For example, participants might be given the following choice of gambles
A. Gain $10 with 1∕2 probability and lose $10 with 1∕2 probability
B. Nothing with certainty
and most would prefer B because they weigh the loss of $10 more strongly than the gain of $10.
Kahneman and Tversky (1984) also argued that, as with subjective utility, people associate a subjective probability with an event that is not identical with the objective probability. They proposed the function in Figure 11.8 to relate subjective probability to objective probability. According to this function, very low probabilities are overweighted relative to high probabilities, producing a bowing in the function. Thus, a participant might prefer a 1% chance of $400 to a 2% chance of $200 because 1% is not represented as half of 2%. Kahneman and Tversky (1979) showed that a great deal of human decision making can be explained by assuming that participants are responding in terms of these subjective utilities and subjective probabilities.
1.0
Subjective probability .50
As we get more money, getting even more seems less and less important. Certainly, the amount of happiness that a billion dollars can buy is not 1,000 times the amount of happiness that a million dollars can buy. One can imagine someone needing $10,000 for an important medical procedure. Then, all sums less than $10,000 would be rather useless, and all sums greater than $10,000 would be about equally good. Thus, such a person would have a very large step in the utility function at $10,000.
I (J. R. Anderson, 1990) have argued that it might actually make sense to treat very low probabilities as if they were a bit higher, like that function does. The argument is that, sometimes when we are told that probabilities are extreme, we are being misinformed. However, there is little consensus in the field about how to evaluate the subjective probability function.
■ People make decisions under uncertainty in terms of subjective utilities and subjective probabilities.