Prospect Theory: A Descriptive Theory of Human Behaviors As we have seen in the previous post, people’s actual behaviors do not always conform to normative decision making rules. In fact, there is a wide range of decisions in which decision makers violate the assumptions of the Expected Utility Theory, the foundation of normative decision making. In the present essay, we will discuss Prospect Theory, a cousin of Expected Utility Theory. Prospect Theory (PT) is developed by Kahneman and Tversky, and it plays an important role in Kahneman’s Nobel Prize (Tversky passed away a few years before; otherwise he would have been awarded as well). As we will see, PT is better than EU in explaining people’s actual judgments and decision making processes. In this essay, we will briefly go over a few key components of the theory as outlined below (expanded concepts of PT will be discussed in subsequent posts): •
Altered concept of risk attitude (risk aversion and risk seeking)
•
Nonlinear probability weighting
•
Relativity of value depending on the perception of gains or losses [Readers interested in the details of PT should review the classic article: D. Kahneman, A. Tversky, Prospect theory: an analysis of decision under risk, Econometrica 4 (1979) 263–291]
1. Risk perception Logically and realistically, when someone says you have a 1% chance of winning a jackpot, you should code the probability of becoming a millionaire as 0.01. However, in reality, people do not perceive the 1% chance of winning the jackpot as 0.01 but they typically
exaggerate this probability. This is just one example of how the EU fails to explain actual behaviors because EU assumes that people perceive probability linearly. PT depicts a more complex picture. The theory proposes that people are risk seeking and risk averse in certain situations. First let’s define what risk seeking and risk averse are. Consider a gamble where you have to choose between 2 options: gain $50 for sure, ($50, 1) or playing a game where you have 50% of wining $100, (100, 0.5; 0, 0.5). You are said to be risk averse when you choose the sure gain, and risk seeking when you choose to play the game. On the other hand, according to the EU theory, it does not matter which options you choose because the expected value of both options is the same U = $50. Another example is your decisions when you choose to buy car insurance and gambles at Vegas. You are risk averse when you buy car insurance although you know that the added up cost of paying for the insurance premiums outweigh the cost of damage you have to pay in an accident when you do not purchase such insurance. You are risk averse because you inflate the likelihood of getting involved in a car accident, which makes you concern that you may not have enough money to cover for such accident (this is not to say that you should not buy car insurance. If you are a poor student like me, you definitely want to buy car insurance because you and I have low tolerance for financial loss, which means we will be severely affected if such unfortunate event happens). On the other hand, you are risk seeking when you “burn” your money in Las Vegas casinos when you know (or do not know) that the odd of winning almost always against you. You are risk seeking because you inflate the probability of winning than it actually is (there is a host of psychological factors that influence your perception of winning probability such as your sense of gamble mastery and self-control, but we will discuss this issue in a different post).
When will people are risk averse or risk seeking? PT states that in general people are risk averse in the domain of gains and risk seeking in the domain of. In addition, subsequent research by behavioral scientists depicts a more complex picture. Within the domain of gain, people switch from risk seeking to risk averse when the probability of gain increases from small to medium and large [usually, as the probability of gaining increases, the amount of utility you receive generally decreases. For example, it is easy to win $5 at Vegas (because the probability is high) than to win $5000 (because the probability is low)]. Within the domain of loss, there is a reverse pattern, people switch from risk averse to risk seeing as the probability of loss increases. Table 1 summarizes these patterns
Small Probabilities
Medium to Large Probabilities
Risk-Seeking
Risk Averse
(buy lottery tickets)
(playing gambles for gains)
Risk-Averse
Risk-Seeking
(buy insurance)
(playing gambles for losses)
Gains
Losses
The following examples will illustrate the patterns above. (w.r.t = with respect to) o
People are risk seeking w.r.t. buying lottery tickets (small probability of large gain).
o
People are risk averse w.r.t. gambles for large gains. (Prefer a gamble with 100% chance to win $5,000 over a gamble in which one has 50-50 chance of $10,010 or $0)
o
People are risk averse w.r.t. buying home insurance (small probability of large loss).
o
People are risk seeking w.r.t. gambles for large losses. (Prefer a gamble in which one has 75% chance of losing (-$1,000) over $0 over a gamble in which on has 100% to lose -$700)
The fourfold pattern of risk perceptions can also be illustrated by the graph below:
Value
Value Function in Prospect Theory
Probability axis as well
Losses
Gains
Without showing mathematical proof, we can derive the fourfold pattern with our intuitions. First, since the proportion of gain (or loss) is inversed with its associated probability, we can conceive the x-axis as if it shows the probability of an outcome. In the domain of “Gains”, when the amount of gain is small (hence, high probability), people perceive this amount of gain larger than it is, evidence in the concave shape of the function. Since they perceive the
(small) amount of gain to be larger than it should be, people are more likely to choose an option that leads to this outcome, winning a small amount of good with a large probability. On the other hand, when the amount of gain is large (hence probability is low), people perceive this amount of gain as smaller than it is, evidence in the flattened shape of the curve. This makes people less likely to choose an option that leads to this outcome, winning a large amount of good with a small probability. On the other hand, in the domain of “Losses”, when there is a small loss, people exaggerate their perception of this small loss, which makes them less likely to choose the option that leads to this unfavorable outcome. The convex shape of the curve near the center in the loss domain indicates this distorted perception. However, when the amount of loss is large, people perceive this large amount of loss to be smaller than it is, and they are more likely to choose the option that leads to this option. Of course, this intuitive interpretation helps us understand the patterns conceptually but it will not suffice to explain for some specific observations. 2. Non linear perception of probability This concept has been discussed earlier. Essentially, people do not perceive objective probabilities as they are. The important point we should know is that people consistently overweight small probability and underweight large probability (see figure below). However, there is a unique difference in very small and very big probability. People tend to overweigh very small probability and under weigh big probability. For example, it is a big difference between a probability of 0 and 0.01 to lose $100,000. You want to get every single (minimal) chance to avoid such a large loss. The different here is no longer quantitative but qualitative. Nevertheless, if for some reason, you have to choose between two options in which you have 89% and 89.01% chance to win $100,000, respectively (this is contradictory with the usual example of large gain
associated with small probability). There is not much difference between these two options and it should be easy for you to make a choice. The graph below depicts people’s perception of nonlinear probability.
0.6 0.4 0.0
0.2
Probability Weight
0.8
1.0
Probability Weighting Function in 1979 Prospect Theory
0.0
0.2
0.4
0.6
0.8
1.0
Stated Probability
3. Reference Point and Relative Value Another important contribution of PT is its nomination of a new concept called “reference point”. In EU, it is assumed that 0 is the reference point for all decision. For example, whether you gain $200 or $300 does not have a huge difference because you will be happy in either case, regardless to contextual situations. However, PT suggests that if you expect to win $250, there will be a huge difference in your emotion when you win $200 compared to $300. The idea here is that your expectation serves as a reference point for your comparison. In the current example, since you expect to win $250, if you win only $200, you will code this as a loss because the money you win is less than the money you expect. On the other hand, since $300 is larger than $250, you will code this as again, and you will be much more enjoyable compared to when you win $200.
In social psychology research, the concept of reference point has been referred to by many researchers as social comparison. For example, in an interesting study, researchers found that when comparing their awful situations with those who had even worse circumstances, people generally felt better. This suggests that the concept of “reference point” does not refer uniquely to monetary value but it can be conceptualized as a psychological concept. The concept of reference point is important because it implies that the same outcome could be coded as either gain or loss depending on how the decision maker perceives the outcome. As a result, generally speaking, for the same outcome, when the decision maker codes it as a loss, he tends to be more risk seeking, and when he codes the outcome as a gain, he tends to be more risk averse (this is true when ignoring the cut-off points between low and high probability in the fourfold pattern). In next posts, we will see how the fourfold pattern operates in the real world.