## Prospect Theory and Expected Value Theory

The subfield of judgment/decision making (JDM) in psychology studies the processes that humans undergo in order to make a decision. A particular area of interest is choosing under uncertainty. The purpose of this essay is to define and compare two of the most important theories that cover decision making with risks. The theories are expected utility theory and prospect theory.

### Expected Value Theory

 Daniel Bernoulli
The first theory that will be covered is expected utility theory. This was developed by Daniel Bernoulli, a Swiss physicist and mathematician ("Daniel Bernoulli," n.d.). Before this theory existed, expected value theory was commonly used. Expected value argues that in order to choose the best outcome on problems that involve money and risk we have to multiply the value of an outcome times their probability (Hardman, 2009). For example, if I bet fifty dollars that if I toss a coin it would land on heads, then, the expected value would be calculated by multiplying 50, which is the probability it would land on heads, times 50, which is the monetary value. The expected value would become two thousand five-hundred dollars.

However, Daniel Bernoulli found a problem with this theory. In order to show the error of expected value theory he developed a game known as St. Petersburg paradox. The game is described as follows: A man is going to toss a coin, if the coin lands on heads, then you win \$5, if it does not, then the individual will toss it again until it falls on heads. If the coin falls on heads on the second try, then you win \$10, if it falls on the third toss, then you win \$20, and the prizes double up until the coin lands on heads (Hardman, 2009). The second part of this game consists of asking a participant what is fair to pay in order to play the game. The error lies in that there is the possibility that the game never ends, therefore, according to expected value theory, the fee to enter the game should also be infinite. However, there is a point in which participants are not willing to pay a certain amount of money, thus, showing that expected value theory cannot be applied to every decision making process.
Another error in the theory is explained by the following example. Imagine that a man is offering you two options. The first one consists of a hundred percent of probability that you will obtain one million dollars. The second option involves fifty percent of probability that you will win three million dollars. Expected value supports the second choice as the better one; however, people usually pick the first option. This does not happen because people avoid gambles. This becomes evident when we see that people prefer a ninety-nine percent of probability of winning one million dollars than just obtaining one dollar without risk. The real reason is because people do not prefer three million dollars three times more than one million ("The problem with expected value theory," n.d.). Thus, Bernoulli argued that choices are not made by their outcomes, but rather by how much an outcome was valued by the person making the decision.

### Expected Utility Theory

The solution for the errors that Bernoulli found in expected value theory is called expected utility theory. For the title of the theory, utility refers to added happiness or worth of an outcome. Utility theory asserts that the value of an outcome should be compared against the probability of that outcome occurring instead of only focusing on the likelihood of that result (Briggs, 2014). Bernoulli argued that people prefer a choice that had a greater psychological value to them instead of an economic one. The following example will show another error in expected value theory that is not found on expected utility theory. Two men with twenty dollars each bet on the toss of a coin. Both of them bet ten bucks. Therefore, the expected outcome would be the same, because the procedure would be thirty dollars plus ten dollars (the thirty dollars is the outcome of winning the bet, and the ten dollars is the outcome of winning it) divided by two (the amount of outcomes) resulting in twenty dollars. Expected value says there is no change in the current monetary position each man has, but by using the concept of loss aversion, it becomes evident that a risk averse person, which is the most common type of individual there is, will not accept the bet.
This risk aversion is explained in terms of utility. For example, let us say that a woman earns one million dollars and adds them to her account each year. The first time she receives her second million and adds it to the bank account that contains her first million, she sees an increase of twenty utility units. Nevertheless, by the time she adds her third million, the increase of utility units is only eighteen. This is important because it demonstrates that people consider the difference between an added small amount of money to a great amount, in terms of subjective values, insignificant. Moreover, it explains why people are risk averse with large sums of money.
If we use the same scale that was used before we would obtain the following results one million dollars equals ten utility units, two million equal thirty, three equals forty-eight utility units, four millions equal sixty, etc. Thus, to explore loss aversion in expected utility theory we could observe the following example provided by psychologist Kahneman (2011). Option A offers an equal chance to win one million or seven million dollars and option “B” ensures us winning four million without risk. If the decision taken is based on the utility scale mentioned above, then, we would see that winning the four million with certainty corresponds with having sixty utility units. However, for option “A” we would obtain ten utility units, which correspond to the one million dollars, and we would add eighty-four units, which belong to the seven million and we would divide them by two. Thus, obtaining a utility of forty-seven units.

### Flaws in Expected Utility Theory

Nevertheless, expected utility theory has also several flaws and that is when prospect theory comes in. Prospect theory will be first described, then, compared to expected utility, and in the end, its flaws will be pointed out. Prospect theory also focuses on choices made under uncertainty. Daniel Kahneman and Amos Tversky developed this theory. Both of them are psychologists. One of the main differences between this theory and the other two mentioned above is that Kahneman and Tversky think of man as an irrational thinker. Expected utility and expected value treat man as rational. However, both psychologists mentioned above have demonstrated that the opposite is true by a series of experiments. Expected utility theory works in two ways as a descriptive model and a prescriptive model. Descriptive indicates how humans make their choices and prescriptive indicates how they should make them. However, prospect theory is only a descriptive model. In other words, this theory by taking into account that humans are irrational beings, it describes the behavior that is probable in situations that involve uncertainty.
 Kahneman
 Tversky
Kahneman created a scenario that showed a fault of utility theory. This consists of giving as an example two people that have the same amount of money. Let us say five million dollars. According to Bernoulli, both of them should have the same utility because they have the same amount of money. However, if one person had less than five million the week before and the other person had more than five million, then, the former is happy and the latter is not. This is one of the concepts that utility theory does not incorporate, which is change in wealth that takes into account a reference point.

Moreover, prospect theory has three main aspects attached to it. They are the adaptive level, sensitivity to dimensions, and loss aversion.
The adaptive level is the change from the reference point. For example, if someone has their left hand in cold water and their right hand in hot water and they put both of them at the same time in a bowl full of lukewarm water, then, the left hand will feel the water in the bowl as hotter, but the hand in right hand will feel it colder. This means that even though the outcome is the same, in this case lukewarm water, the change is different because the reference point is different. In this sense, gaining fifty dollars is good if we had the probability of gaining nothing; however, it is not seen as good if the other probable outcome was gaining two thousand dollars

### Prospect Theory: Sensitivity to Dimensions

Sensitivity to dimensions is the difference between two sets of money. This difference gets lower if the amount of money is large and vice versa. This could also be applied to physical changes such as a small light being turned on in a dark room, compared to in a well-lit one. When this concept is applied to money, the subjective difference between the amounts of money becomes evident. For example, even though the difference between nothing and twenty dollars and the difference between one thousand and one thousand dollars is the same in terms of monetary value, it is not in terms of individual worth.

### Prospect Theory: Loss Aversion

The third cognitive feature of prospect theory is loss aversion. This means that losses have a bigger effect than gains (Saka, 2011). According to Kahneman (2011), loss aversion can be seen as an evolutionary trait because people who avoided risks rather than take a chance at opportunities had a better probability of surviving any danger and, then reproducing, thus passing down along that trait. In order to understand loss aversion we have to balance out the subjective gain against the subjective lose. The loss aversion ratio is commonly set in the range of 1.5 to 2.5 (Kahneman, 2011).
In his book “Thinking Fast and Slow,” Kahneman describes two systems that operate within the mind. System one is automatic and system two is deliberate. System one takes care of unconscious decisions, it is the one that operates with heuristics, but it is not good with complex operations such as logical reasoning. System two takes more effort, it has to be focused on one thing when it is processing information, but it does solve complex operations such as analyzing and compiling data.
He adds to the concept of loss aversion that the rejection of a gamble is deliberate, therefore, people are using their system two; however, the refusal of a gamble is usually motivated by an emotional input that is developed by system one. He also adds that the impact of losses depends heavily on each person. For example, if someone is a professional risk taker, they become less risk averse. An interesting point is that the person does not have to be a risk taker, but rather to think like one. In an experiment, an experimental group was told to think like a trader and the results of the experiment were that their physical responses to losses had been greatly reduced.

### Flaws in Prospect Theory

One of the short comings of prospect theory is that if the reference point is zero and the probability of a positive outcome is large, then, the high probability becomes the reference point (Kahneman, 2011). For example, consider the following gamble: in the first case, there is a one percent chance that an individual wins one million dollars. In the second case, there is a ninety percent chance of probability of winning twenty dollars and a ten percent probability that the person does not win anything, and in the final case, there is a ten percent of winning nothing and a ninety-one of winning a million. In the three cases, prospect theory assigns an equal value to winning nothing, however, the third case has a bigger psychological impact because the high probability of winning makes losing disappointing and seen now as a loss instead of no change.
Thus, prospect theory has a problem with changing the worth of a consequence when the other choice has a lot of value or when the outcome is not probable.

#### References

Briggs, R. (2014, August 8). Normative Theories of Rational Choice: Expected Utility. Retrieved December 9, 2015, from http://plato.stanford.edu/entries/rationality-normative-utility/

Daniel Bernoulli. (n.d.). Retrieved December 8, 2015, from http://www.famousscientists.org/daniel-bernoulli/

Hardman, D. (2009). Judgment and decision making. Malden, MA: Wiley-Blackwell.

Kahneman, D. (2011). Thinking, fast and slow

Saka, G. (2011, August 31). Loss Aversion: Why Do We Hang on to Things for No Reason? Retrieved December 9, 2015, from https://www.psychologytoday.com/blog/the-decision-lab/201108/loss-aversion-why-do-we-hang-things-no-reason

The problem with expected value theory. (n.d.). Retrieved December 9, 2015, from http://www.calculemus.org/logica/log-ekon/exprobab-zad.html