## 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.

### St. Petersburg Paradox

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.

### Prospect Theory: Adaptive Level

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