Download A Survey on Prospect Theory

A Survey on Prospect Theory
Suhansanu Kumar
Department of Computer Science,
University of Illinois at Urbana Champaign,
{skumar56}@illinois.edu
Abstract. In this survey project, we focus on the biasness people have
with regard to payoffs and the probabilities and their cumulative effect on
our decision making. Previous works on probability theory were focused
on the expected payoffs and considered the cost perceived by everyone
same, also called as the utility theory. However, over the course of experimentation and surveys it was found with resounding repetition that
human’s decision making is not completely logic due to our cognitive
biases and other behavioral attributes. Prospect theory was the first attempt to quantize this decision making considering all the aspects into
a single succinct model : prospect theory.
Keywords: prospect theory, utility theory
1
Introduction
Almost all analytically study across various disciplines have been made using
expected utility theory. This theory has been accepted as a normative model of
ration choice and has been used to describe the economic behavior [5] of people in
behavioral economics. This is the reasoning used in most common business and
personal decisions. It is even more paramount to understand when the decisions
are inter-dependent on one another and therefore understanding the decisions
of opponents is need to make the optimal decision. Under such circumstances,
we note that the utility theory is unable to explain several decision phenomenon
in financial and insurance markets. The Israeli authors, Kahneman and Tversky
in their most celebrated work on prospect theory explained for the first time
the different biases in the decision through the utility function and the decision
weight function - also called as the prospect theory [7]. In the light of these
observations we argue that utility theory, as it is commonly interpreted and
applied, is not an adequate descriptive model and the authors propose alternative
account of choice under risk also popularly called as the prospect theory.
In the following sections, we introduce the utility theory, the shortcomings of
the utility theory, the prospect theory, followed by sections on application and
limitation of application of utility theory.
2
2
Suhansanu Kumar
Utility theory
According to Wikipedia, utility in economics is a measure of preferences over
some set of goods and services. The concept is an important underpinning of
rational choice theory in economics and game theory, because it represents satisfaction experienced by the consumer of a good. A good is something that satisfies
human wants. As a natural extension, expected utility hypothesis is a hypothesis concerning choices people make in regard to gamble or games with uncertain
outcomes. Expected utility theory states that the utility would be the sum of the
product probabilities and the payoff of the different outcomes. In the presence
of risky outcomes, a decision maker could use the expected value criterion as a
rule of choice: higher expected value investments are simply the preferred ones.
Consider a gamble,
(xm , pm ; xm+1 , pm+1 ; ...; x0 , p0 ; ...; xn1 , pn1 ; xn , pn ),
where the notation should be read as gain xm with probability pm , xm+1 with
probability pm+1 , and so on, where the outcomes are arranged in increasing
order, so that xi < xj for i ¡ j, and where x0 = 0.
P Under expected utility theory, an individual evaluates the above gamble as,
pi × U (W + xi ),
Therefore, in this model it is assumed that the people are able to perceive
the probability exactly and the payoff for each person is the same irrespective
of his/her wealth. The von NeumannMorgenstern axioms that define a rational
decision maker namely completeness, transitivity, independence and continuity.
Completeness assumes that an individual has well defined preferences and can
always decide between any two alternatives. Transitivity assumes that, as an individual decides according to the completeness axiom, the individual also decides
consistently. Independence also pertains to well-defined preferences and assumes
that two gambles mixed with a third one maintain the same preference order as
when the two are presented independently of the third one. The independence
axiom is the most controversial one. Continuity assumes that when there are
three lotteries (A, B and C) and the individual prefers A to B and B to C, then
there should be a possible combination of A and C in which the individual is
then indifferent between this mix and the lottery B. Therefore theses properties
as were given by Neumann in his seminal paper [15]. However as noted by the
famous works by Allias [1] and Kahneman and Twersky [7] noted that there are
several discrepancies in the present day expected utility theory.
3
Shortcomings of the utility theory
A person is risk averse if he prefers the certain prospect (x) to any risky prospect
with expected value x. In expected utility theory, risk aversion is equivalent to
the concavity of the utility function. This phenomenon can be shown by the
following experiment as conducted by Kahneman and Tversky on 2700 participants. One shortcoming may be noted is that the authors are given hypothetical
A Survey on Prospect Theory
3
prospects to choose from but their results are very applicable and seen in other
fields as well.
3.1
Effect of probability weighting
Experiment 1 : 50% chance to win 1,000, and 50% chance to win nothing; as
the first prospect A
The second prospect B being getting 450 for sure.
As expected, people did not want to choose the first prospect even though the
expected utility suggests to take the first option. People are risk averse and want
to rely on the certainty of the event. The weight function of the probability actually captures this phenomenon, that is the weight of 0.5 is less than 0.5 and
causes the expected utility to therefore not work. Another glaring experiment
as done by French economist Maurice Allais in 1953 [1] also proofs the lack of
information in the expected utility theory by the following experiment.
Experiment 2 Given a choice between two prospects, A (2500 with probability
0.33; 0 with probability 0.67) and B (2400 with probability 0.34, 0 with probability 0.66), the people choose prospect A with 82% thereby showing the mid
range probabilities are not that significantly different. That is the perceiving of
the probability between 0.99 and 1.0 and between 0.33 and 0.33 is not the same
in people’s mind. This gives us the idea of the curve of the weight function.
Consider the above two experiments, and if we just use the expected utility
theory, it causes contradiction in formula. From the experiment 1, u(2400) >
0.33u(2500) + 0.66u(2400) and 0.34u(2400) > 0.33u(2500) therefore by utility
theory people should have chosen the prospect B. People however when the probabilities are mid range(comparable) they treat the probabilities equally and the
second experiment suggests 0.34u(2400) < 0.33u(2500).
Experiment 3 However, people have different perspective at low probabilities
as well different from the larger probabilities and the mid-probabilities.
Problem 7: Consider two prospects given to you, A (6000, 0.45) B (3000, 0.9),
as expected people choose the B with 86% majority. However, if you reduce
the probabilities of A and B to 0.001 and 0.002, people behavior choose altogether. They don’t consider the difference anymore and treat both the probability weights as negligent. This also shows a deviant from the expected utility theory. The above three experiments have focused on just the probability
weights.
3.2
Effect of the payoff
People’s initial wealth also matter a lot while making the decisions. Consider
playing a game of gaining a amount of 5 dollars on winning at 0.5 probability
and losing 5 dollars with half probability. We are more uncomfortable for playing
this game however, when the playoffs are just one dollars, people are less hesitant
to play. This is because the payoff function on losing and gaining is different and
4
Suhansanu Kumar
the change at the wealth value is very dramatic as well will see in the later
sections.
3.3
Isolation effect
People disregard the components that prospects share and only consider the
components that distinguish them. The isolation effect refers to the phenomenon
whereby people value a thing differently depending on whether it is placed in
isolation and whether it is placed next to an alternative. In particular, a certain
choice can be made to look more attractive if it is placed next to an alternative
relative to which it is distinctively better in some respect.
Experiment 4: Example, consider a two stage game 1st stage of 0.75 of ending
game without anything, 2nd stage there is a choice A (4000, 0.8) or B (3000 with
certainty). People went for the prospect B completely ignoring the fact of the
presence of the 1st stage. However, when people were presented with the same
problem, however the probability values of both the stages combined into one,
prospect A(4000, 0.8 * 0.25) = A (4000, 0.2) and prospect B (3000, 1 * 0.25) =
B (3000, 0.25), people’s choice was now B with probability of 65%. As shown in
the figure ??, the decision making is done at the square position in the figure.
This is probably because of the cognitive bias in humans – people try to simplify
things by removing the same part but this doesn’t work when the things are
multiplicative and not additive.
Table 1. Isolation effect
3.4
Reflection effect
The authors conducted the same experiments but reversed the value of gaining
to losing and observed the behavior of people just reversed. The same people
who were risk averse to the gain were now risk taking under the loss. This effect
coined by them was the reflection effect which is also contradicting the expected
theory in similar way as the previous way. However the symmetry about the origin in the payoff function does not exist because people don’t take 5 dollars loss
and 5 dollars gain in the same way. The absolute difference created in the payoffs
perceived is different in the two cases. People take the loss much more severely
A Survey on Prospect Theory
5
and are more unhappy than the happiness they get when gaining the same
amount of 5 dollars. The reflection property can be summarized in the figure
.
The occurrence of this reflection effect was previously observed by by Markowitz
et al [12] and Williams et al [16] where a translation of outcomes produces a
dramatic shift from risk aversion to risk seeking.
4
Prospect theory
Daniel Kahneman and Amos Tversky in their work [7] [14] which got recognition
from the financial world and had profound impact on the economics of the world
received the noble prize and more than forty thousand citations for their work.
They combined all the previous works and discrepancies observed in the previous
sections, by a single measure by replacing the payoff functions with the value
function and the probability weighing function. Before evaluating the payoffs
from the weighting and the value function, the editing phase removes all the
human biases and comparisons to simplify the decision making.
–
–
–
–
–
–
4.1
Coding Gains and loses are set relative to a reference point
Combination (200, 0.25; 200, 0.25) = (200, 0.5)
Segregation Riskless components are removed. (300,0.8;200,0.2) = (100,0.8)
Cancellation Removing the common components(Isolation effect)
Simplification Rounding of probabilities or outcomes
Detection of dominance removal of dominated prospects
Probability weighing function
The probability weighing function as depicted in the figure captures the exact way how people perceive on the y-axis as against the different probability
values. At the lower values, we have over-estimated the probabilities and underestimated at the higher probabilities. In the mid range, the probability function
is not that dramatic and people don’t observe the difference between 0.25 and
0.35 as much as the low values (0 and 0.05) and high values (0.95 and 1.0). This
is exactly captured in the weighing function.
6
Suhansanu Kumar
Although there is a direct mapping between the probability values and the
weights, the decision weights are not probabilities: they do not obey the probability axioms and they should not be interpreted as measures of degree or
belief. As shown from the experiments 2 and 3, v(2, 400) > π(.66)v(2, 400) +
π(.33)v(2, 500), i.e.,
[1 − π(.66)]v(2, 400) > π(.33)v(2, 500) and π(.33)v(2, 500) > π(.34)v(2, 400);
hence,
1 − π(.66) > π(.34), orπ(.66) + π(.34) < 1.
Therefore the weighing function, is sub-additive in nature. Further, the slope
of $pi in the interval (0, 1) can be viewed as a measure of the sensitivity of
preferences to changes in probability. Sub-certainty entails that π is regressive
with respect to p, i.e., that preferences are generally less sensitive to variations
of probability than the expectation principle would dictate
4.2
Utility value function
An essential feature of the present theory is that the carriers of value are changes
in wealth or welfare, rather than final states. However, the value function has
to be calculated in terms of the gain or loss from the wealth of a person. The
same situation can be seen in different ways by different persons. For example, a loss to the company can effect the decision making on the later decision
making of the CEO if he still incorporates the previous loss as less loss and
he will be risk seeking, however if he doesn’t take into account the last loss,
A Survey on Prospect Theory
7
he will be risk averse. Also, we saw that loss and gain are not reflective in
their magnitude. This is exactly captured in this model. People see the gain as
concave. A change from 5 to 10 dollar is much more significant than a change
from 100 to 105 dollars. Therefore the overall value function can be calculated
keeping in mind - the asset position that serves as reference point, and the magnitude of the change (positive or negative) from that reference point. Shown
in the figure is the deviation of the value function from the expected value.
.
5
Application of Prospect Theory
There are several phenomenons in insurance and financial markets that have
been very succinctly captured by the Prospect Theory which was previously not
easy to explain from the expected utility function. In the paper [2], the authors
explain the thirty years history of the prospect theory and places where it has
found application and has failed at.
5.1
Pseudo-certainty effect
In prospect theory, the pseudo-certainty effect is people’s tendency to perceive
an outcome as certain while in fact it is uncertain. It is observed in multi-stage
decisions, in which evaluation of outcomes in previous decision stage is discarded
when making an option in subsequent stages. This is called from the isolation
8
Suhansanu Kumar
effect which we studied earlier and is captured in the editing phase of the prospect
theory.
5.2
Disposition effect
This is behavioral anomaly often observed in financial markets all around the
world that people sell the shares whose price rise while keep the ones that fall.
This is due to the reflection theory, when the stock prices rise, people become risk
averse and see the stocks. However when it falls, people become risk seeking since
loss is happening in this situation. The paper [4] shows that irrational behavior
causes the the future performance of equity to be unrelated to its purchase price.
5.3
Finance (Stock)
The financial world phenomenons like cross section of average returns why
some financial assets have higher average returns than others is captured by
prospect theory driven models like Capital Asset Pricing Model (CAPM) [3].
The despostion effect from prospect theory is able to show quantitative and
qualitative reasons for the average return of US stock market for being greater
than the average return of Treasury bills.
5.4
Insurance (Over-insuring modest risks)
Insurance companies use the fact the people’s value function is very dramatic
near the low values and they take the loss very seriously than the gains. Therefore
they design some of their insurances such that people are encouraged to take
insurances, although by expected utility theory they shouldn’t take it. Syndor
et al []sydnor2010over studied 50 thousand customer of casualty insurance. They
studied two types of insurances of deductible $500 and $1000 with premium $715
and $615 respectively. It was found that households chose higher premium and
lower deductible. The risk is only 5% chance of paying $500 in the event of
claim but they chose to pay extra $100 every year for this. This is due to the
over-weighing of the causality (tail event) or low probability events.
5.5
Endowment effect
Knetsch [8] studied the endowment effect by distributed mug and cup worth $5
which was uniformly accepted by the subjects. However when given a chance to
exchange the prize, 89% didnt want exchange. Cost of acceptance for exchange
was way higher to $5.75 than cost of willingness to pay for exchange which was
just $2.25. Loss aversion after a change of reference point (wealth) explains why
the exchange becomes unattractive. There was no loss in the earlier part of the
experiment and the people were not differentiating between cup and mug since
the payoff function subsidies at large payoffs. However exchange means, they are
getting loss or gain by exchange, and not the gain of 5 dollars, therefore they
are hesitant to exchange.
A Survey on Prospect Theory
5.6
9
Others
There are several others application of the prospect theory such as marketing
of the products, consumption- savings decision, Industrial Organization (pricing
strategy and pricing strategy of monopolists), annuitization puzzle, etc. where
the applicability of prospect theory is very profound.
6
Limitation and Challenges of applying Prospect theory
Even though the prospect theory has been extensively used in the financial and
market analysis and for insurance, it has still not gained much application in
other fields although the utility theory has been under application at lots of
places. This is because some of the limitations of the theory.
1. One of the most subjective and ambiguous questions is the reference point of
the customers to use this theory in a favorable way. Are they gains and losses
in overall wealth, in the value of total stock market holdings, or in the value
of specific stocks? is one such reference point questions. Some researchers
have tried in the recent past to address this problem [9–11].
2. Another question is the applicability outside the lab. Most of the work and
the functions are based on the surveys conducted in labs where only hypothetical cases can be captured. People have tried doing the experiments
in real world in poor countries where a US researcher’s budget represents a
large amount of money have found that prospect theory continues to provide
a good description [6][13].
3. It is critiqued from the field of psychology argued that even if Prospect
Theory arose as a descriptive model, it offers no psychological explanations
for the processes stated in it. Furthermore, factors that are equally important
to decision making processes, have not been included in the model, such as
emotion like anger, hate, love, etc.
4. In the last part, we tried to capture the social effects and the previous experiences with the gamble. A person who has won a gamble (of 0.5 probability)
would be more interested in gambling again than one that has lost in the
previous games.
We can try to capture the effect of social network or social influence on the
decision making using the influence propagation theory. I hypothesised
that
P
(w(ni ))
it would be the product of the social influence (S) Sv = sum(w(n
,
that
is
v ))
the affect of uncertainty reported and seen in their friend circle would bias
the probabilities. There is higher chance of me buying an insurance is most
of my friends have bought it. My best friend/family (high w) or someone
whom I follow will have greater impact on their decision :
Wnew = (1 − a) ∗ Wold ∗ (1 − Sv ) + a ∗ Wold ∗ (Sv ) where a > 0.5(0.75),
Using the recommendation model of products and reviews to gauge the expectation and the value of the products from customer buying pattern.
10
7
Suhansanu Kumar
Conclusion
In this part, we see that K&T were very brilliant in explaining various components of anomalies into a single model – Prospect Theory. This is very compact
model and follows intuition and is therefore able to explain the different phenomenons in a very easy way. But this is still a descriptive model and is different
for different persons, doesn’t take into account social norms, rules, physiological
conditions, etc. into mind for the decision making. Therefore it makes this theory
very difficult to implement in the real life.
References
1. M. Allais. Le comportement de l’homme rationnel devant le risque: critique des
postulats et axiomes de l’école américaine. Econometrica: Journal of the Econometric Society, pages 503–546, 1953.
2. N. C. Barberis. Thirty years of prospect theory in economics: A review and assessment. Technical report, National Bureau of Economic Research, 2012.
3. M. J. Brennan. Capital asset pricing model. In Finance, pages 91–102. Springer,
1989.
4. C. F. Camerer. Prospect theory in the wild: Evidence from the field. Advances in
behavioral economics, pages 148–161, 2004.
5. M. Friedman and L. J. Savage. The utility analysis of choices involving risk. The
journal of political economy, pages 279–304, 1948.
6. S. J. Kachelmeier and M. Shehata. Examining risk preferences under high monetary incentives: Experimental evidence from the people’s republic of china. The
American Economic Review, pages 1120–1141, 1992.
7. D. Kahneman and A. Tversky. Prospect theory: An analysis of decision under risk.
Econometrica: Journal of the Econometric Society, pages 263–291, 1979.
8. J. L. Knetsch. The endowment effect and evidence of nonreversible indifference
curves. The american Economic review, 79(5):1277–1284, 1989.
9. B. Kőszegi and M. Rabin. A model of reference-dependent preferences. The Quarterly Journal of Economics, pages 1133–1165, 2006.
10. B. Kőszegi and M. Rabin. Reference-dependent risk attitudes. The American
Economic Review, pages 1047–1073, 2007.
11. B. Kőszegi and M. Rabin. Reference-dependent consumption plans. The American
Economic Review, 99(3):909–936, 2009.
12. H. Markowitz. The utility of wealth. The Journal of Political Economy, pages
151–158, 1952.
13. T. Post, M. J. Van den Assem, G. Baltussen, and R. H. Thaler. Deal or no deal?
decision making under risk in a large-payoff game show. The American economic
review, pages 38–71, 2008.
14. A. Tversky and D. Kahneman. Advances in prospect theory: Cumulative representation of uncertainty. Journal of Risk and uncertainty, 5(4):297–323, 1992.
15. J. Von Neumann and O. Morgenstern. Theory of games and economic behavior.
Princeton university press, 2007.
16. C. A. Williams. Attitudes toward speculative risks as an indicator of attitudes
toward pure risks. The Journal of Risk and Insurance, 33(4):577–586, 1966.