Learn, Think & Do

Introduction

Brief Discussion on the Expected Utility Theory (EUT)

The Prospect Theory

Limitations of the Prospect Theory

Introduction

Let’s start with the following question:

Which do you prefer, based on a coin toss (assumed to be fair unless mentioned otherwise)?

1. If it comes up heads you win $100, and if it comes up tails you win nothing.
2. Get $46 for sure.

If you are like the most people, you chose the second option, the sure option of getting $46 even though the expected value (for the expected value look at this blog) of the first option is higher ($50). According to the Expected Utility Theory, the rational choice is to choose the first option. The behavioral economist Richard Thaler calls a rational person (as described in economics) as Econs while the real persons who may not always make rational choice as Humans. That is the difference between Humans and Econs. In my opinion, Humans choose second option because they think probability as one off event where sure thing makes much more sense than something which is only probable. You can perform your own thought experiment where you keep on reducing the value in second option to see at what value you stop making second choice. 40? 35? 30? 25? Or you can do thought experiment for the first option where you increase your wining amount so that gamble becomes attractive to the second option. What would be your choice? 110? 120? 140? 180? 200? Such problems and experiments are essential tools in the behavioral economics, a domain further developed by Daniel Kahneman and Amos Tversky through their work on the prospect theory.

Brief Discussion on the Expected Utility Theory (EUT)

Before going into the discussion about the prospect theory, let’s talk briefly about the expected utility theory (EUT) which is widely used as the rational-agent model in economics and social studies (why social studies are called “sciences” and people studying them as “scientists” I have no clue). In EUT, we calculate the expected utility by adding utility function of each outcome with its probability, that is, let be the outcomes with corresponding probabilities such that and corresponding utility functions then the expected utility is given by where, . There are certain axioms this theory has to satisfy, we won’t get into much technical details about that. Nevertheless I would show you descriptive examples of them from Thinking, Fast and Slow by Daniel Kahneman.

First person to think of money in terms of psychological value or desirability of money (now called utility) was Daniel Bernoulli (there were many mathematicians/scientists in Bernoulli family, a good kind of dynasty). Keep in mind this utility is not limited only to money, for example for a person who is earning well, the utility of $500 might be less than the utility of the few days of vacation with the family. Bernoulli proposed that a gift of 10 bucks has the same utility to a person with 100 bucks as a gift of 20 bucks to a person with 200 bucks, that is percentage changes were more apt for comparison than absolute changes. This can be thought in the modern world as raise in salary is calculated in terms of percentage raise rather than absolute values. This shows that utility of $1 million to a person who has a wealth of $1 million is far more than the person whose current wealth is $10 million. This in today’s language is called as the law of diminishing marginal utility. Bernoulli’s argument was that this is what explains risk aversion as we saw in the first example in the start, people prefer sure thing over a favorable gamble of equal or slightly higher expected value. In EUT one of the axioms can be explained as follows, if you prefer an apple to a banana, then you also prefer a 10% chance to win an apple to a 10% chance to win a banana. There are many drawbacks in this model which was studied by Daniel Kahneman and Amos Tversky, so let’s talk about the Prospect Theory.

The Prospect Theory

Consider the following problems:

3. Which do you choose? Get $900 for sure OR 90% chance to get $1000.
4. Which do you choose? Lose $900 for sure OR 90% chance to lose $1000.

If you are like most people, you chose sure thing in 3 this is because in the EUT being risk averse is a rational choice. So your choice matches with the EUT. But in the 4th problem you probably chose the gamble over losing the $900 for sure. The risk seeking choice is the mirror image of the risk averse choice people make but this phenomenon could not be explained using the EUT. The duo Kahneman and Tversky employed gains and losses as their thinking apparatus for the Prospect Theory. In the EUT only the final states or outcomes mattered not the initial starting point, called as the reference point by the duo. Consider the following problem:

5. Today Jack and Jill each have a wealth of $5 million. Yesterday, Jack had $1million and Jill had $9 million. Are they equally happy? (Do they have the same utility?)

Bernoulli’s theory assumes that the utility of their wealth is what makes people more or less happy. Jack and Jill have the same wealth, so they should both be equally happy but you know that Jack is far more happier at this point than Jill. Because in this case for Jack the reference point is $1 million while for Jill the reference point is $9 million, hence the reason for their delight and sadness respectively. The duo explains their idea in the following manner, suppose you have three bowls in front of you, on the left you have bowl with warm water, on the right with cold water and in the middle bowl with water at the room temperature. Put your left hand in the left bowl and right hand in the right bowl, then after a while put your both hands in the middle bowl, how do you think would be your experience? Here you can think of middle bowl as the reference point while feelings of hot and cold as gains and losses respectively. This is what makes people happy or sad in life, for example if your reference point was happiness of the past and currently you are not happy as much as you were you feel sad whereas someone who was sad earlier but now is in the same position as you, might be feeling happy. Consider the next two problems:

6. In addition to whatever you own, you have been given $1000. You are now asked to choose one of these options: 50% chance to win $1000 or get $500 for sure.
7. In addition to whatever you own, you have been given $2000. You are now asked to choose one of these options: 50% chance to lose $1000 or lose $500 for sure.

From the perspective of the utility theory both the problems 6 and 7 are identical. In each case, you have the choice to become richer by $1500 with certainty, or accept a gamble in which you have equal chances to be richer by $1000 or by $2000. So by EUT, one should make similar choices in both the cases. But in the 6th problem majority of respondents preferred the sure thing while for the 7th problem a large majority preferred the gamble. Because in the 6th problem the reference point was moved higher by $1000 while for the 7th it was moved higher by $2000, therefore being richer by $1500 in the 6th problem is a gain while it is a loss of $500 in 7th case. This gives way to the concept of loss aversion, that is, the response to losses is stronger than the response to corresponding gains. In other words, when compared losses loom larger than gains. This also has an evolutionary history, in the sense that organisms who treat threats as more urgent than opportunities have a better chance to survive and reproduce.

The following figure is what the duo called as the image representing the flag of the prospect theory if it had one. The graph shows the psychological value of gains and losses, which are the “carriers” of value in prospect theory. The graph is S-shaped and it shows reduced sensitivity for both the gains and losses but the slope changes abruptly at the reference point, in this graph represented by the origin, for losses it is slightly sharper than for the gains which should be clear from our earlier discussions.

Limitations of the Prospect Theory

Just like EUT has some flaws this prospect theory suffers from some flaws too. Consider the assumption of the prospect theory in which the reference point is generally assumed to have a value of zero. The assumption seems reasonable but it can lead to some weird outcomes. Consider the following three scenarios.

8. One chance in a million to win $1 million.
9. 10% chance to win $15 and 90% chance to win nothing.
10. 90% chance to win $1 million and 10% chance to win nothing.

In the 8th and 9th cases winning nothing is not a significant event but in the 10th case winning nothing is a huge disappointment. So assigning the value zero for winning nothing in 10th case is problematic, as is done in the prospect theory. So the prospect theory is unable to incorporate these situations. In real life you can consider example of a child prodigy who everyone thought was going to be hugely successful, let’s say he has 90% chance of being hugely successful and 10% chance of being mildly successful like average person. And his life turn out to be mildly successful, so of course he feels disappointed in life. Or you can think of this phenomenon as a salary raise that was promised but ultimately was not given. You naturally feel disappointed. Now consider the following scenarios.

11. Choose between 90% chance to win $1 million or $50 with certainty.
12. Choose between 90% chance to win $1 million or $150,000 with certainty.

Now if you chose the gamble in the 11 (that is what most would do) and you lost , do you feel any regret? Most probably not. But the same is not true in the 12, if you chose gamble and lost, you are going to regret your decision because you had a chance to get $150,000 surely. These kinds of regret scenarios are also not captured by both, the prospect theory and utility theory. But these are the kind of choices one has to make in real life especially in finance and business. For example, two persons choose to go for a start-up, one person has no stable income or job but another one is earning well and has a stable job, where chances of success is huge but ultimately start-up fails, then the obvious feeling for the second person is regret whereas the first person most probably wouldn’t mind the decision.

All the examples and pics for this post have been taken from Thinking, Fast and Slow and pic for the blog is the cover of the book. For more technical details one can refer Prospect Theory: An Analysis of Decision under Risk, the pdf for which is freely available.

References:

Daniel Kahneman, Thinking, Fast and Slow