Prospect Theory
Kahneman, D., Tversky, A. (1979)
Econometrica 47(2), 263-292
Introduction
• This section will present the results of a research paper by
Kahneman & Tversky (1979)
• These results show that people do not take decisions in a way
that is compatible to the axioms of traditional Expected Utility
Theory (EUT)
• Kahneman & Tversky, based on these results, developed and
presented an alternative theory, Prospect Theory (PT)
• This theory describes the way people take decisions under
uncertainty
Introduction
• In the following experimental results the monetary unit is
Israeli pound (the first experiment took place in Israel)
• In order to realize the magnitude of monetary outcomes note
that, at the time of the experiment, the average net family
income in Israel was approximately 3000 pounds.
• Note also that this initial experiment has been repeated in other
countries and Universities (e.g. University of Michigan,
Stockholm University, among others) and the results have
been qualitatively the same.
Certainty, probability, and possibility
• In EUT the utilities of outcomes are weighted by their
probabilities
• In this section Kahneman & Tversky show how people
systematically violate this rule
• They show that people overweigh outcomes that are
considered certain, relative to outcomes that are merely
probable
• They call this phenomenon the certainty effect
Certainty, probability, and possibility
N=the number of subjects that answered this question
[] = in brackets the percentage that choose this prospect
* = denotes that the percentage that choose prospect is statistically significant
Certainty, probability, and possibility
• In Problem 1 the choice is between two prospects
• In Prospect A the subjects have the chance to win 2500 with probability
33%, 2400 with probability 66%, and nothing with probability 1%
• In Prospect B they have the chance to win 2400 with probability 100%
• Of the 72 people that participated in the experiment,
→ 18% chose Prospect A and
→ 82% (statistically significant) chose Prospect B
• Then, the researchers ask the subjects the following Problem 2:
Certainty, probability, and possibility
Certainty, probability, and possibility
• In Problem 2 the choice is between two prospects
• In Prospect C the subjects have the chance to win 2500 with probability
33%, and nothing with probability 67%
• In Prospect D they have the chance to win 2400 with probability 34%, and
nothing with probability 66%
• Of the 72 people that participated in the experiment,
→ 83% chose Prospect C (statistically significant) and
→ 17% chose Prospect D
Certainty, probability, and possibility
Have another look at the Problems
How do they differ?
Certainty, probability, and possibility
• According to EUT the choice in Problem 1 suggests that the utility of
winning 2400 with probability 100%,
→ u(2400)
• Is higher than the utility of winning 2500 with probability 33%, 2400 with
probability 66%, and nothing with probability 1%,
→ 0.33u(2500) + 0.66u(2400) + 0.01u(0)
• In other words:
u(2400) > 0.33u(2500) + 0.66u(2400) + 0.01u(0)
Certainty, probability, and possibility
• Since u(0) = 0 the choice in Problem 1 implies:
→
u(2400) > 0.33u(2500) + 0.66u(2400)
• Now, deduct 0.66u(2400) from BOTH sides of the inequality:
→
0.34u(2400) > 0.33u(2500)
• Note that the choice in Problem 2 implies the REVERSE inequality:
→
0.34u(2400) < 0.33u(2500)
Certainty, probability, and possibility
• In other words, Problem 1 is the same to Problem 2, if we deduct
0.66u(2400) from BOTH prospects A and B.
• That is, Problem 2 is obtained from Problem 1 if we eliminate 66% chance
of winning 2400 from both prospects under consideration
• This change resulted to a CHANGE IN PREFERENCES, something that
is not compatible with EUT
• When the nature of a prospect is changed from a certain gain to a possible
gain it results to a greater reduction in preferences, compared to the case
where both prospects are uncertain
Certainty, probability, and possibility
Certainty, probability, and possibility
• In Problem 3 the choice is between two prospects
• In Prospect A the subjects have the chance to win 4000 with probability
80%, and nothing with probability 20%
• In Prospect B they have the chance to win 3000 with probability 100%
• Of the 72 people that participated in the experiment,
→ 20% chose Prospect A and
→ 80% (statistically significant) chose Prospect B
Certainty, probability, and possibility
• In Problem 4 the choice is between two prospects
• In Prospect C the subjects have the chance to win 4000 with probability
20%, and nothing with probability 80%
• In Prospect D they have the chance to win 3000 with probability 25%, and
nothing with probability 75%
• Of the 72 people that participated in the experiment,
→ 65% (statistically significant) chose Prospect C and
→ 35% chose Prospect D
Certainty, probability, and possibility
• Note that the choice in Problem 3 (Prospect B) implies
→
→
1u(3000) > 0.80u(4000) + 0.20u(0)
i.e. u(3000) > 0.80u(4000)
• If we divide BOTH sides of the inequality with u(4000) we obtain
u(3000) / u(4000) > 0.80 > 4/5
• Note that the choice in Problem 4 (Prospect C) implies
→
→
0.20u(4000) + 0.80u(0) > 0.25u(3000) + 0.75u(0)
i.e. 0.20u(4000) > 0.25u(3000)
If we re-arrange:
u(3000) / u(4000) < (0.20/0.25) < 4/5
Certainty, probability, and possibility
•
In other words, the choice in Problem 3 suggests the REVERSE inequality from the
choice in Problem 4.
•
Also note that:
•
•
•
•
Prospect A can be written as (4000, 0.80)
Prospect B can be written as (3000)
Prospect C can be written as (4000, 0.20) = (A, 0.25)
Prospect D can be written as (3000, 0.25) = (B, 0.25)
•
However, the substitution axiom of EUT implies that if agents
→ prefer Prospect B to Prospect A, then
→ they will prefer any probability mixture (B,p) to (A,p)
•
The subjects did not obey to this axiom.
Certainty, probability, and possibility
What about non-monetary outcomes?
Certainty, probability, and possibility
• Problems 5 & 6 illustrate that agents behave in a similar manner when the
prospects are expressed in non-monetary outcomes
• Note that Problem 6 is equal to Problem 5 with the probabilities being
reduced by an equal amount in all relevant Prospects:
→
→
from 50% to 5% in A and C
from 100% to 10% in B and D
• Prospect C can be written as (A, 0.10)
• Prospect D can be written as (B, 0.10)
• The certainty effect violates the substitution axiom even when nonmonetary outcomes are involved
Certainty, probability, and possibility
What about very small probabilities?
Certainty, probability, and possibility
• In Problem 7 the probabilities of winning are substantial (45% and 90%)
• In Problem 8 the probabilities of winning the same amount are extremely
small (1% and 2%), i.e. winning is possible but not probable
• When probabilities are high people select the most probable prospect
• When probabilities are low people select the prospect with the higher gain
Certainty, probability, and possibility
• The results, so far, suggest the following empirical generalization, as
regards to the violation of the substitution axiom:
• If (y,pq) is equal to (x,p) then (y,pqr) is preferred to (x,pr)
• For 0 < p,q,r < 1
• Kahneman & Tversky incorporate this finding to their theory
The Reflection Effect
• What about losses?
• Kahneman & Tversky, in what follows, pose a series of Problems to
subjects only this time they change the sign
• That is, the Problems are the same as the Problems presented above but
instead of gains we have losses
• The results are presented below; the left Panel of the following Table
presents the results so far (gains), while the right Panel presents the results
(i.e. preferences) when gains are replaced by losses
The Reflection Effect
The Reflection Effect
• Note that the change in sign REVERSES all inequalities
• In Problem 3 the choice is between Prospect A where agents have the
chance to win 4000 with probability 80%, and nothing with probability
20%, and Prospect B where they have the chance to win 3000 with
probability 100%.
→ 80% (statistically significant) chose Prospect B
• In Problem 3’ the choice is between Prospect A where agents have the
chance to loose 4000 with probability 80%, and nothing with probability
20%, and Prospect B where they have the chance to loose 3000 with
probability 100%.
→ 92% (statistically significant) chose Prospect A
• The same happens with all Problems
The Reflection Effect
• The reflection of Prospects around zero REVERSES preferences
• Three significant implications:
• Firstly, in the positive domain (gains) people are risk averse while in the
negative domain (losses) people are risk seeking
• For instance, in Problem 3’ the majority (92%) prefer to undertake the risk
of loosing 4000 with probability 80% than the certainty of a smaller loss of
3000
• Note that the expected value of the preferred choice is also smaller
→ (-4000, 0.80) = (-3200)
The Reflection Effect
• Secondly, preferences in the negative domain violate EUT in exactly the
same way as in the positive domain
• For instance, in Problems 3’ and 4’ the certain prospects have a larger
weight (are over-weighted) in preferences
• In the positive domain, the Certainty Effect leads to risk aversion (where
the certain profit is preferred to a larger gain that is merely possible)
• In the negative domain, the Certainty Effect leads to risk seeking (where
the larger loss that is merely possible is preferred to a smaller but certain
loss)
• Thus, the same psychological principle (larger weight to certainty) leads to
risk aversion in the positive domain and risk seeking in the negative
domain
The Reflection Effect
• Thirdly, the Reflection Effect shows that the aversion for uncertainty (i.e.
volatility) cannot explain the Certainty Effect
• For example, the Prospect of winning 3000 (3000) has zero volatility
compared to the Prospect of winning 4000 with probability 80%
(4000,0.80) and could be preferred
(and it WAS)
• However, the Prospect of losing 3000 (-3000) has BOTH zero volatility
and a higher expected value compared to the Prospect of losing 4000 with
probability 80% (-4000,0.80) [this has an expected value of (-3200)] and
should be preferred
(but it WAS NOT )
The Isolation Effect
The Isolation Effect
→
→
78% chose the certain payoff in the second stage
22% chose the (higher) uncertain payoff in the second stage
•
Problem 10: there is a 20% probability (25% times 80%) to win 4000 and a 25%
probability (25% times 100%) to win 3000, i.e. (4000, 0.20) or (3000, 0.25)
•
In terms of outcomes and probabilities it is equal to Problem 4
•
Note that the SECOND STAGE is equal to Problem 3 i.e. (4000, 0.80) or (3000)
•
Recall that in Problem 3 the majority chose (3000)
•
Recall that in Problem 4 the majority chose (4000, 0.20)
•
Although Problem 10 is EQUAL to Problem 4, people answered as in Problem 3
•
That is, they IGNORED the first stage
The Isolation Effect
The Isolation Effect
• Note that in Problem 11 the majority prefers B while in Problem 12 the
majority prefers C; consistent with the Reflection Effect.
• Also note that Problem 12 is equal to Problem 11 if we add 1000 to the
initial sum and subtract 1000 from all outcomes.
• Also, if we view all prospects as final states:
•
•
•
•
Α = (2000, 0.50 or 1000, 0.50)
Β = (1500)
C = (2000, 0.50 or 1000, 0.50)
D = (1500)
• That is, A = C and B = D
The Isolation Effect
•
This pattern is inconsistent with EUT, where the same utility is assigned to a
wealth of, say, 100000 irrespective of whether it was reachedd from a prior wealth
of 95000 or a prior wealth of 105000
•
As a result, the choice between:
→ 100000 with certainty and
→ 50% chance of 95000 and 50% chance of 100000
•
Should be independent of whether someone currently owns the smaller or the larger
amount
•
With the added assumption of risk aversion in EUT the certainty of 100000 should
always be preferred to the gamble.
•
The responses to Problem 12 indicate that this will happen only if someone owns
the smaller amount.
The Isolation Effect
• A final, but extremely important, observation is that the initial
bonus (common to both prospects) was neglected
• That is, agents did not consider the final asset position that
includes the current wealth
• Instead, they assigned value (utility) to:
→ CHANGES IN WEALTH
• This is the cornerstone of Prospect Theory
PROSPECT THEORY
• Based on the above experiments (and many similar)
Kahneman & Tversky developed their theory
• They argued that there are two phases in the decision making
process
• During the first phase there is an initial editing, simplification,
and analysis of prospects; this phase has four stages: coding,
combination, segregation, cancellation
• During the second phase there is an evaluation of prospects;
in this phase the prospect with the highest value is chosen)
PROSPECT THEORY
• 1st phase: Coding
• As discussed above, people perceive prospects as changes in wealth rather
than final wealth
• More specifically, they examine gains and losses
• Gains and losses, of course, are defined relative to a REFERECE POINT,
which is usually the current wealth
• The exact position of the reference point and the subsequent coding of
gains and losses may be affected by the framing of the offered prospects
and the expectations of the decision maker
PROSPECT THEORY
• 1st phase: Combination
• At this stage prospects may be simplified with the combination of
probabilities of similar prospects
• For example, the prospect
(200, 0.25 and 200, 0.25)
• may be simplified as
(200, 0.50)
• and be evaluated in this form
PROSPECT THEORY
• 1st phase: Segregation
• Many prospects often contain a riskless element that may be segregated in
the editing phase
• For example, the prospect
(300, 0.80 and 200, 0.20)
• May be segregated
→ to a sure gain of (200) and
→ an uncertain prospect (100, 0.80)
PROSPECT THEORY
• 1st phase: Segregation
• Often people will isolate prospects and/or ignore elements; recall the
reaction to Problem 10 where the first stage was ignored.
• For instance, the choice between:
→ (200, 0.20; 100, 0.50; -50, 0.30) and
→ (200, 0.20; 150, 0.50; -100, 0.30)
• May be reduced by cancellation to
→ (100, 0.50; -50, 0.30) and
→ (150, 0.50; -100, 0.30)
PROSPECT THEORY
• 1st phase: Segregation
• Two additional operations that take place at this stage is the simplification
of prospects and the detection of dominance
• For example, rounding of probabilities and outcomes may take place:
• The prospect (101, 0.49) is possible to be rounded as (100, 0.50) and be
evaluated in this form
• Or a prospect that is considered extremely unlikely may be ignored
• The prospect (500, 0.20; 101, 0.49) will dominate to (500, 0.15; 99, 0.51)
→ if the second constituents of prospects are simplified to (100, 0.50).
PROSPECT THEORY
•
In the 2nd phase the evaluation takes place
•
Let us denote the overall value of an edited prospect as V
•
V is expressed in terms of two scales π and u
•
The first scale, π, associates with each probability, p, a decision weight, π(p), which
reflects the impact of p on V
•
Note that π(p) is NOT a probability
•
The second scale, u, assigns to each outcome x a number u(x) which reflects the
subjective value of the outcome.
•
Recall that outcomes are defined relative to a reference point; thus, u(x) captures
the value of deviations from the reference point, i.e. gains and losses.
PROSPECT THEORY
• The following formulation is concerned with simple prospects of the form:
(x, p; y, q), i.e. a gain of x with probability p, a gain of y with probability q,
or nothing with probability (1-p-q), (p+q)≤1
• A prospect is strictly positive if x, y > 0, p+q=1
• A prospect is strictly negative if x, y < 0, p+q=1
• A prospect is regular if neither is strictly positive nor strictly negative
• For regular prospects, Kahneman and Tversky suggest that (3.1):
V(x, p; y, q) = π(p)u(x) + π(q)u(y)
Where u(0)=0, π(0)=0, π(1)=1
PROSPECT THEORY
• As in EUT,
→
V is defined on prospects
→
u is defined on outcomes
→
V(x,1)=V(x)=u(x)
It generalizes EUT by relaxing the expectation principle
PROSPECT THEORY
• If (x, p; y, q) is strictly positive or strictly negative then in the editing phase
prospects are segregated in the riskless component (minimum gain or loss)
and the risky component i.e. the additional gain or loss.
• If (p+q)=1 and either x>y>0 or x<y<0, then:
V(x, p; y, q) = u(y) + π(p) [u(x) - u(y)]
• For example,
V(400, 0.25; 100, 0.75) = u(100) + π(0.25) [u(400) - u(100)]
The Value Function
The Value Function
• Of course we must take under consideration the fact that special
circumstances may have an impact on individual Value Functions.
• For example, the Value function of a person that needs 60000 to buy a
house may have a steeper slope around the critical value
• Similarly, the aversion for losses could increase suddenly around a loss
level tat would lead a person to sell his/her house and move to a less
desirable neighborhood.
• Thus, the Value Function for each person does not always reflect clear
positions on monetary outcomes since it is affected by additional
consequences
The Value Function
• A significant issue in Value changes is that people consider losses more
important than gains (Loss Aversion)
• This refers to the observation that the regret from losing a certain sum of
money is greater than the happiness from earning the same sum of money
• Most people find symmetric prospects of the type (x, 0.50; –x, 0.50) not
attractive with the loss aversion increasing with the magnitude of the
relative sum of money
• For instance, if x>y≥0 then people will prefer
→ (y, 0.50; –y, 0.50)
than
→ (x, 0.50; –x, 0.50).
The Value Function
• Loss aversion refers to people's tendency to strongly prefer avoiding losses
to acquiring gains.
• Most studies suggest that losses are twice (2.25) as powerful,
psychologically, as gains
• Loss aversion implies that one who loses $100 will lose more satisfaction
than another person will gain satisfaction from a $100 gain.
• Note that whether a transaction is framed as a loss or as a gain is very
important to this calculation: would you rather get a $5 discount, or avoid a
$5 surcharge?
• The same change in price framed differently has a significant effect on
behavior
The Value Function
• Based on the above, Kahneman & Tversky suggest that the Value Function
will be:
→
Defined based on a reference point
→
Concave for gains
→
Convex for loses
→
Steeper in the negative domain
• That is, it sill be S-shaped
The Value Function
The Value Function
• In Kahneman and Tversky (1979) the functional form that satisfies these
properties is (λ = the loss aversion coefficient):
• In the special case where V is linear (α⁺= α⁻=1):
The Weighting Function
• Probability Distortion
• As discussed above (see also below) it has been observed that PT investors
do not weight outcomes linearly
• They tend to underestimate large and moderate probabilities and
overestimate small probabilities instead
• Based on this observation, Kanheman and Tversky (1979, 1992) proposed
to replace the objective probabilities by decision weights using a nonlinear, continuous and strictly increasing probability weighting function
The Weighting Function
• In PT the value of each outcome is multiplied with a decision weight
• The decision weight is obtained from choices between prospects and
subjective probabilities BUT IS NOT a probability and does not follow the
principles of Probability Theory
• Think of it like this: a gamble where you can gain 1000 or nothing
depending on the toss of a coin
• For every reasonable person the probability of a gain is 50% but as we will
show below the decision weight π(p), may be less than 0.50.
The Weighting Function
• The decision weight captures the impact of events on the preference for the
prospects
• Not just the likelihood of events
• π is a function of probability
The Weighting Function
•
Observe that the preferences in Problems 8 and 8΄ imply that for small values of p,
π is a sub-additive function of p, i.e. π(rp) > rπ(p) για 0 < r < 1.
•
In Problem 8 the prospect (6000, 0.01) is preferred to (3000, 0.02)
•
In PT terms:
π(0.01) u(6000) > π(0.02) u(3000)
•
Thus,
•
The results from the Reflection Effect lead to the same conclusion
•
However, the results in Problems 7 and 7΄ show that the sub-additivity does not
necessarily hold for large values of p
[π(0.01)/π(0.02)] > [u(3000)/u(6000)] > [1/2]
The Weighting Function
• Kahneman Tversky suggest that very low probabilities are, on average,
overweighed, i.e. π(p)>p, for a small p.
• Look at the answers to the following Problems:
• Problem 14. Choose between prospects A & B
A: (5000, 0.001)
(72%)*
B: (5)
(28%)
• Problem 14’. Choose between prospects A & B
A: (-5000, 0.001) (17%)
B” (-5)
(83%)*
The Weighting Function
• In Problem 14 people prefer a lottery that may yield a profit with very low
probability than the expected value of this gain.
• In Problem 14’ people prefer a small loss (e.g. an insurance premium) than
a very small probability of a larger loss.
• Problem 14 implies, in PT terms, that:
→
→
π(0.001) u(5000) > u(5)
π(0.001) > u(5) / u(5000) > 0.001
• Assuming the Value Function for gains is concave.
• The readiness to pay insurance in Problem 14’ implies the same, assuming
the Value Function for losses is convex.
The Weighting Function
• It is important to make a distinction between
→
over-weighting
(refers to a property of decision weights)
→
over-estimation
(refers to the evaluation of the probability of rare events)
• In many real situations both may operate to increase the impact of rare
events.
The Weighting Function
• Although π(p) > p, for low probabilities, the data show that:
π(p) + π(1-p) < 1, for all 0 < p <1.
• This property is called sub-certainty (see the answer to Problems 1 and 2)
• In PT terms, for Problem 1:
u(2400) > π(0.66) u(2400) + π(0.33) u(2500)
[1-π(0.66)] u(2400) > π(0.33) u(2500)
• In PT terms, for Problem 2:
π(0.33) u(2500) > π(0.34) u(2400)
1-π(0.66) > π(0.34)
π(0.66) + π(0.34) < 1
The Weighting Function
• The slope of π in the interval (0,1) can be viewed as a measure of the
sensitivity of preferences to changes in probability.
• Sub-certainty implies that π is regressive with respect to p, i.e. that
preferences are, on average, less sensitive to variations in probability than
the expectation principle (EUT) suggests.
• People are limited in their ability to evaluate extreme probabilities and thus
highly unlikely events are either ignored or overweighed and the difference
between high probability and certainty is either neglected or exaggerated.
• As a result π is not well behaved near the end points.
The Weighting Function
(Kahneman & Tversky, 1979, page 283)
The Weighting Function
• Tversky and Kahneman (1992) propose the following specification for the
inverse S-shaped function probability distortion.
• The parameter γ ≈ 0.65 determines the overweighting of small probabilities
and the underweighting of large probabilities.
• This functional form gives the subjective probabilities that agents give to
each event and it is in accordance with the observation that investors tend
to overweight small objective probabilities and underweight moderate and
large objective probabilities
The Weighting Function
• Prelec (1998) proposes a different probability weighting function.
• As the risk aversion parameter α decreases the function becomes more
regressive, more sub-proportional and more S-shaped while for α = 1 the
weighting function is the identity one.
The Weighting Function
•
In Prelec (1998) the parameters determine the overweighting of small probabilities
and the underweighting of large probabilities.
•
Such a transformation inflates the large probabilities and deflates the small
probabilities.
•
This functional form satisfies all the properties of a well-behaved probability
weighting function.
•
It is about a one-parameter sub-proportional functional form with a fixed and an
inflection point at.
•
It is also a regressive function that drives to the well-known "four-fold pattern of
risk attitudes" (risk-seeking for large-probability losses and small-probability gains
and risk-aversion for large-probability gains and small-probability losses,
introduced by Kahneman and Tversky, 1992).
•
One can also set β = 1 and γ = 0.74 (as in Wu and Gonzalez, 1996).
The Weighting Function
• Prelec’s probability weighting function is a well-behaved probability
distortion function which is regressive, asymmetric, reflective, and Sshaped; it has a fixed point at 0.33 which means that the function is
asymmetric.
• The fact that the fixed point is below the probability 0.5 drives to a
decrease of an uncertain outcome's weight relatively to the certain
outcomes' weights and thus makes investors more risk-averse for gains and
more risk-seeking for losses, because of the reflection.
• A reflective function gives the same weight to a given loss-probability and
to a given gain-probability.
• This S-shaped property is strictly related with the psychological principle of
diminishing sensitivity according to which as one moves away from the
bounds of the probability intervals the impact of probability changes is
reduced.
CONCLUSION
•
If we
change the way outcomes and probabilities are presented people may
change their preferences (Framing Effects)
• Utility is not measured by people in final states of wealth but with changes
in wealth
• The value function is S-shaped; defined based on a reference point;
Concave for gains and convex for loses; Steeper in the negative domain
• The value of each outcome is weighted by a decision weight
• Sub-certainty implies that preferences are less sensitive to changes in
probability than what the principles of EUT assume.
References
•
Kahneman, D., Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk.
Econometrica, 47(2), 263-292.
•
Markowitz, H., (1952). The Utility of Wealth. Journal of Political Economy, 60(2), 151-158.
•
Tversky, A., Kahneman, D. (1992). Advances in Prospect Theory: Cumulative Representation of
Uncertainty. Journal of Risk and Uncertainty, 5(4), 297–323.
•
Prelec, D., (1998). The Probability Weighting Function. Econometrica, 66(3), 497-527.
•
He, X.D., Zhou X.Y. (2011) Portfolio Choice under Cumulative Prospect Theory: An Analytical
Treatment. Management Science, 57(2), 315 – 331.
•
Wu, G., Gonzalez, R. (1996). Curvature of the Probability Weighting Function. Management
Science, 42(12), 1676-1690.
•
De Giorgi, E., Hens, T., Mayer, J. (2007). Computational Aspects of Prospect Theory and Asset
Pricing Applications. Computational Economics, 29(3-4), 267-281.
•
Rubinstein, R.Y. (1982). Generating random vectors uniformly distributed inside and on the
surface of different regions. European Journal of Operational Research, 10(2), 205–209.