14 January 2013

Prospect Theory a “Descriptive Theory of Decision Making”


Prospect theory is one of the foundation stones of behavioral economics and behavior sciences overall. It was developed in the 1970s by Amos Tversky and Daniel Kahneman. For this work Daniel Kahneman received the Nobel Prize for Economics. Unfortunately Amos Tversky passed away a few years before the prize was awarded. 

The revolutionary nature of Prospect Theory came from the fact that it challenged a centuries old assumption, namely perfect rationality. It criticized the traditional (normative) model of decision making and replaced it with a positive one (how people actually make decisions).

In order to better understand what are the true innovations of Prospect Theory we should first take a look at the normative model, namely at the Expected Utility Model developed by Daniel Bernoulli in the XVIII-th century.

Before describing the Expected Utility Model, let’s make clear what Utility means. Utility can be perceived as “the amount of pleasure” or “the benefit” given by a certain amount of money. The main characteristic of Utility is that it is not linear. For small and moderate amounts of money it is in fact linear, but after a certain sum the utility per monetary unit starts to decrease.

For example, the utility of 20 Euros is twice the utility of 10 Euros. However, the utility of 2 million Euros is not twice the utility of 1 million Euros. The utility of 2 million Euros is smaller than twice the utility of 1 million Euros.

In mathematical expressions it would go like this:

U(20)=U(10) + U(10)

U(2.000.000) < U(1.000.000) + U(1.000.000)

The Expected Utility Model is straight-forward: The value of an uncertain outcome is the Utility multiplied by the probability of occurring.

To better understand, let’s assume that an alien comes from the sky and sits on your shoulder. The alien says it’s you lucky day and you get to play a series of games with him.  The alien takes out an intergalactic coin that is very similar to the earthly ones, namely it has two sides. If the side with the space ship comes up, you win 100 Euros. If the side with the “2 intergalactic credits” comes up, you win nothing. The coin is not tricked, namely there is a 50-50 chance of either side to come up.

The value of the alien’s proposition can be translated like this:

Expected Value = 0.5*U(100) + 0.5*U(0).

Another example would be if the alien would offer to throw a dice and if the side with “1” comes up you get 20 Euros, if the side with “2” comes up you get 2 Euros, if the side with “3” comes up you lose 10 Euros and if any other side comes up, nothing happens. The value of this proposition is:

E.V. = 1/6*U(20) + 1/6*U(2) + 1/6*U(-10) + ½*U(0).

I gave examples with “gambles” because they are the easiest, but the reality is that we make decisions about uncertain events all the time. For example, what are the chances of getting that nice job and what would the benefit (utility) be? Should I take an umbrella with me today or not? Will it be safer to go on the trip by car or by train?

I assume that you got the main idea of the Expected Value model. Before prospect theory (and even nowadays in some areas of science) this was assumed to be the way people make judgments about uncertain events.

Moreover, this is the RATIONAL way of making decisions. This is HOW WE SHOULD MAKE DECISIONS.

Since in the above paragraph you read “should” it implies that it is not the way we make decisions. Tversky and Kahneman developed prospect theory which describes how people actually make decisions under uncertainty.
  
Prospect theory keeps the “Utility function” or in other words it also uses the concept of utility. The benefit that money brings is not linear in both theory and practice. At the same time prospect theory brings two major new insights into the Expected value model.

First, prospect theory makes a very important distinction between gains and losses. The reality is that we humans do not think (perceive) the same way about what we gain and what we lose. Moreover, gains or losses are relative concepts. What is a gain or what is a loss depends on the reference against which we judge it.

To better understand consider that you are in the following situation. Your salary is 25.000 Euros per year and you have put in a lot of effort in the last year. Now you are up for a raise and in a couple of days the HR department will tell you what your salary will be next year. You can’t help and think about how much you would deserve and you come up with the number 32.000 Euros per year. When you have the meeting with the HR person, she tells you that you will get a raise of 5000 Euros, meaning that your salary next year will be 30.000 Euros.

Will you be happy about the raise? Most likely you will not be unhappy, but neither you will be happy. The explanation is that your reference point was the salary you believed that you deserve – 32.000 euros. In comparison with this amount, the 30.000 you were offered represents a loss of 2000 Euros.

At the same time, you actually get more money than last year and if you change the reference to your current salary, you will in fact experience a gain of 5000 euros.

The main implication of distinguishing between gains and losses is that people “suffer” from loss aversion. In more simple words, we hate to lose and we would put in more effort to avoid a loss than to achieve an equivalent gain.

Let’s illustrate this with the following example. The alien suggests the following bet:

50% chance of gaining 100 Euros and a 50% chance of losing 100 Euros.

What is your attitude about this gamble? Would you be willing to take it? It’s a 50-50 chance of either winning or losing 100 Euros. Will you take it?

Most likely your answer is a definite “NO” and most people in the world would be giving the same answer. From a strictly rational (normative economics) perspective, this answer is at least weird. The reason is that the objective value of this gamble is 100*0.5 + (-100)*0.5=0. If the objective expected value is zero, then you should be indifferent to the gamble and take it (or at least more people would take it). But you and most (normal) people in the world would not even consider taking this gamble.

The idea of being indifferent to a gamble basically means that you are equally inclined to take and to not taking it.

Now the alien can understand to a certain extent humans and he suggests the following bet:

50% chance of gaining 110 Euros and a 50% chance of losing 100 Euros.

What is your attitude about this gamble? Would you be willing to take it? It’s a 50-50 chance of either winning 110 Euros or losing 100 Euros. Will you take it?

Most likely your answer is still “No”. At the same time you have already learned how to compute the objective value of the gamble and by applying the very simple formula of 110*0.5 + (-100)*0.5=5. As you can see this gamble has a positive objective expected value. From a normative economics (rational) perspective everyone should be willing to take this gamble, but again most people are not willing to take it.

For more on loss aversion please check this post: Loss Aversion and its Implications


Roughly we perceive losses as twice as bad as an equivalent gain. In other words, the pain (dis-utility) of losing 100 Euros is (roughly) twice as large as the pleasure (utility) of gaining 100 Euros.

This would translate in a mathematical expression like this:

|U(-100)| = 2*U(100)

The loss aversion coefficient which in theory is called (lambda) λ is estimated to be approximately 2. This is a general estimate established after averaging values from large sample of people. What this means is that some people might have a loss aversion coefficient smaller or larger than “2”. To what extent this is relevant for your work remains to be established, the key idea is that there can be individual differences in the value of (lambda) λ.

The utility function has similar characteristics in both the gains and losses areas when it comes to linearity. In other words, as in the case of gains, when it comes to losses the (dis)utility of small and moderate amounts is linear. When it comes to large amounts, however, the (dis)utility becomes nonlinear.

To illustrate this, a loss of 20 Euros is equal to twice the loss of 10 Euros. However, a loss of 100.000 euros is smaller than twice the loss of 50.000 Euros.

In Mathematical expressions it goes like this:

U(-20)=U(-10) + U(-10)

U(-100.000) < U(-50.000) + U(-50.000)

The graphic illustration of the utility function is:


The image is a “screen shot” from the original paper introducing Prospect Theory by Tversky and Kahneman.  


To sum up the first part, Prospect theory keeps the concept of utility and its characteristic (non-linearity for large amounts). It makes a difference between gains and losses; in the area of losses the utility function has a steeper slope illustrating loss aversion (losses loom larger than equivalent gains). What is perceived as a gain or a loss is dependent on a reference point.

A major implication of distinguishing between gains and losses is the existence of a differentiated attitude towards risk. There can be two attitudes towards risk, namely Risk Seeking and Risk Averse (avoiding).

Prospect theory has concluded that in the area of gains, people are “Risk averse”, namely that we would reject a risky option if a certain one would be available.

Let’s illustrate this with an example. Imagine that you have to choose from the following options:

Option A. 50% chance of winning 100 Euros and 50% of wining nothing.
Option B. 50 Euros for sure.

Which would you prefer? Most likely you prefer option B and it is normal to do so. The expected value of option A is 50 Euros but it involves a risk, whereas option B has the same expected value, but it has no risk.

Now consider the following options:

Option C. 50% chance of winning 100 Euros and 50% of wining nothing.
Option D. 45 Euros for sure.

Which would you prefer? Most likely you prefer option D. In this case, however, the expected value of the preferred option is smaller than the expected value of the rejected option (C). In other words you would trade 5 euros for having certainty (a sure gain).  

In the area of losses things are different. When it comes to losses people are risk seeking, namely they would take a risk of losing a larger amount than accepting a sure loss.

Let’s illustrate this with an example. Imagine that you have to choose from the following options:

Option E. 50% chance of losing 100 Euros and 50% of losing nothing.
Option F. A sure loss of 50 Euros.

Which would you prefer? Most likely you prefer option E. In this case you would take a 50-50 risk of losing 100 Euros instead of accepting a sure loss of 50 Euros. From the point of view of expected value you should be indifferent between the two options.

Now consider the following options:

Option G. 50% chance of losing 100 Euros and 50% of losing nothing.
Option H. A sure loss of 45 Euros.

Which would you prefer? Most likely you prefer option G. In this case it is rational (from the point of view of expected value) to choose option H since its expected value is smaller than the expected value of option G. Still, the popular choice is option G, illustrating risk seeking in the area of losses.

To sum up attitudes towards risk, People are risk averse (avoiding) when it comes to gains and risk seeking when it comes to losses.



Second, prospect theory questions the way people judge probabilities. From a mathematical perspective probabilities are straight-forward. Probabilities are always between (including) 0 and 1 and they always add up to 1. People perceive probabilities in a different way than the mathematical view. Prospect theory introduces something called a weighting function for probabilities.

To make this a bit more clear, when judging on an uncertain event, even if we know the (real) probabilities we don’t actually take them into account to their “full value”. Instead we use the “probability weights” in making decisions.

Rephrasing this, there are objective probabilities (the real ones) and there are subjective probabilities (the weights we give to probabilities). The subjective probabilities are what we perceive the probability to be.

Having established that people do not perceive (use in judgment) probabilities as they are, the question is how do we perceive probabilities? The answer goes like this.

First, the “0” and “1” probabilities are perceived as they are. In other words certainty and impossibility are perceived as actual certainty and impossibility.

Second, small probabilities are either over-estimated, or ignored. The probability of a rare event that is salient in our mind will be overestimated. For example if you go for the first time by airplane, you will overestimate the probability of something going wrong. At the same time, probabilities of rare events that are not salient in our minds are simply ignored. For example if you take the subway every day to work, most likely you will not even consider the risk of dying in a subway accident.  

If you would like to learn more on this please check out “Certainty and Possibility Effects” 

Third, medium and large probabilities are underestimated. This means that we tend to perceive medium and large probabilities are smaller than they actually are. For example an objective (real) probability of 90% (or 0.9) is perceived (weighted) as a subjective probability of approximately 70% (or 0.7).

A graphic representation of the probability weighting function is presented in the next picture. This is in fact a screen-shot from the original paper introducing Prospect Theory byTversky and Kahneman. 


One issue is rather unclear in the graph above, namely up to what value are probabilities small and subsequently overestimated and from which value are probabilities medium and subsequently underestimated? Looking at the graph, one could say that the “critical value” is 10% (or 0.1).

In his book “Thinking Fast and Slow” Daniel Kahneman leaves this question open. However, in chapter 29 he presents a set of data that suggests that probabilities of 20% (or 0.2) are also overestimated.

In a paper published about 13 years later than the one introducing prospect theory, Tversky and Kahneman (1992)click here for the paper  the authors refined their description of the probability weighting function and presented the following graph. 



From this graph we can deduce that the “critical value” is around 30% or 0.3.

The exact value of probability that marks the shift from over to under estimating probabilities can be important in some areas of activity such as insurance, while in many other areas is less relevant.

For your general knowledge, it is important to remember that small probabilities are overestimated while medium and large probabilities are underestimated. If, however, you are interested in more mathematical details on the probability weighting function, check out this article.  

To sum up on probabilities, probabilities of 0 and 1 are perceived as such; medium and large probabilities are underestimated; small probabilities are either overestimated or ignored. 


Before ending, I would like to give an advice... if you want to understand and work in behavioral economics, the following picture should be forever imprinted in your mind:



2 comments:

Tudor Jitianu said...

Dear Nicolae,

I think the article it is very interesting and want to add something to it.

Our subjective perceived utility not only depends on the amount but on time too. The interesting study about the overall happiness of those loosing a limb or those winning 1M USD at the lottery. It seems that after 6 months their happiness restores to the initial level.

I would certainly mention Dan Ariely which documents some of the anomalies especially about the free things. On top of that decisions are highly influenced (according to the same Dan Ariely) by the context in which they are made especially if the respective action has some moral charge.

I would also like to see something pointing to the critique of this theory that is SO mathematical.

Warm regards,

Tudor

Nicolae Naumof said...

Nice to hear (read) from you Tudor!

In this post I had described Prospect Theory which does not take context into account, at least not directly. What Dan Ariely says is true and it is captured in the 4 Dimensions Model of Behavior.

With regard to adaptation after a certain amount of time, it is again very true. Prospect theory does not take time into account since it focuses on decisions made now. However, one major element of prospect theory is that reference points are not fixed. So wining 1 million at the lottery is today a very large gain, but we tend to adapt and not before long the 1 million will be the status quo, namely the reference point.

Indeed you are right that the theory is not perfect. Moreover, I agree it is a bit uncomfortable due to the amount of math integrated in it. At the same time, Prospect theory was one the foundation stones for less mathematical discoveries...

For me Prospect theory is one of the "sacred" pieces of information one must have in order to understand decision making. It is not pleasant, it is not "fluffy" and it has no "fireworks". Without it, however, our understanding of human nature would be significantly impoverished.

You can check out other posts on this blog to see less mathematical insights.

Warm regards

Nicolae