Lecture 2 - Prospect Theory, Framing and Mental Accounting

Prospect theory was developed by two psychologists, Kahneman and Tversky (1979) based on observing actual behaviour (aka. Positive model).
Experimental evidence suggests that individuals frequently deviate from the behavioural predictions of expected utility theory (Normative model).
Prospect theory has a solid mathematical basis.
Unlike expected utility theory which concerns itself with how decisions under uncertainty should be made (a prescriptive approach), prospect theory concerns itself with how decisions are actually made (a descriptive approach)

Make decisions to maximise their expected utility

behave against expected utility theory

not behaving the way we should, observing actual behaviour, can still quantify and model

Still based on mathematical theory

expected utility is more a normative model

Recap:

Expected Utility can be defined as:

$E(U) = \sum^{N}_{i=1}P_{i}U(w_{i})$

Where:

$P_{i}$ - probability of each outcome
$U(w_{i})$ is the utility derived from each outcome ( $w_{i}$ = wealth)

Example:

If you face this gambling opportunity: 50% win $105, 50% lose $100. Take it?
If you are an expected utility maximizer with wealth W, you should take the gamble if current utility < expected utility
- expected outcome - $2.5
Von Neumann and Morgenstern
- if you follow certain axioms, you must maximize “expected utility.”

$U(W) < 0.5 \times U(+\$105) + 0.5 \times U(-\$100)$

Loss Aversion vs. Risk Aversion

Loss Aversion

Strong preference to avoid losses rather than acquiring equivalent gains.
Feelings of loss are psychologically about twice as powerful as feelings of gain.
May lead to taking bigger risks to avoid losses.

Strong aversion to avoid losses

may make the same decision, but for a different reason

Preference for lower levels of risk and uncertainty.
Choice of guaranteed outcomes, even if less profitable.
Drives decisions towards certainty over uncertainty.

Based on the uncertainty you are facing

will sacrifice higher rewards for something more certain

Development of prospect theory

The prospect theory was proposed in 1979 to address certain empirical observations that Expected Utility Theory failed to explain.
- Value vs. Utility & Difference in Wealth vs. Total Wealth

Three Key characteristics of the value function include:

Reference Dependence: Value perception is relative to a specific reference point, typically the status quo.
Loss Aversion: Losses impact utility more than equivalent gains.

$v(x) < -v(-x)$

Diminishing Sensitivity: Sensitivity to wealth changes decreases as the magnitude of gain or loss increases.

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use value function, not utility

look at the difference in wealth

losses, more risk seeking, gains, more risk averse

These 3 aspects make Prospect Theory more fitting than Expected Utility Theory in high-risk and uncertain situations.

Common value functional form of Prospect Theory

Positive Domain

$v(z) = z^{\alpha} \; \text{for} \; z≥0, \; \; 0< \alpha <1$

$z$ is change in wealth (horizontal line)
Value function (not utility) so $v$ is used.

Value functions

alpha, determines the curvature of your graph

smaller, the more curved it will be

Negative Domain

$v(z) = -\lambda(-z)^{\beta} \; \text{for} \; z<0, \lambda >1, \;\; 0<\beta<1$

The factor $\lambda$ , which is greater than 1, introduces loss aversion
The parameter β (0<β<1) controls the curvature of the function on the loss side, similar to α on the gain side (see Appendix).
Kink at origin.

Beta determines curvature of graph

lambda; people are loss averse

Common ratio effect

Prospect pair 4 – choose between:

A: (.9, $2000)
B: (.45, $4000) Most people opt for? - A

positive domain, tend to be more risk averse

Prospect pair 5 – choose between:

A: (.002, $2000)
B: (.001, $4000) Most people opt for? - B

risk-seeking attitude

Invoke linear transformation rule:

Prospect pair 4:

$0.9u(2000) > 0.45u(4000)$

$2u(2000) > u(4000) 0.45$

This reflects the fact that individuals prioritize the high probability of winning a smaller amount ($2000) over the lower probability of winning a larger amount ($4000).

Prospect pair 5

$0.001u(4000) > 0.002u(2000)$

$u(4000) > 2u(2000)$

This shows the contradiction where people are choosing a smaller expected utility when it comes to small probabilities, opting for the chance to win a larger amount ($4000) despite the lower likelihood.

more people chose spin the wheel

overall utility for A should be more than B

Reconciliation Through Prospect Theory:

The answer lies in its nonlinear weighting function. In Prospect Theory, probabilities are not treated linearly. Small probabilities are overweighted, which means they seem larger than they are, and moderate to high probabilities are underweighted, appearing smaller than they are.

This distortion of probabilities can make option B in Prospect Pair 5 seem more attractive than it would be under expected utility theory, thus reconciling the apparent contradiction.

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non-linear weighting function

not how you believe the probability is

the weight you believe 'in your heart'

Overweighting for small - scratch ticket

Lottery effect

Prospect pair 6 -- choose between:

A: (0.001, $5,000)
B: ($5)

Most prefer A which is inconsistent with risk aversion.

People seem to overweight low-probability events (which is why people buy lottery tickets)

Insurance

Prospect pair 7 -- choose between:

A: (0.001, -$5,000)
B: (-$5)

Most prefer B which is inconsistent with risk seeking.

Insurance need
Once again, people seem to overweight low-probability events (which is why people buy insurance)

Overweight low probability

Certainty effect

Prospect pair 8 -- choose between:

A: (0.2, $4000) → E(w) = 800
B: (0.25, $3000) → E(w) = 750

Prospect pair 9 – choose between:

A: (0.8, $4,000) → E(w) = 3200
B: ($3000) → E(w) = 3000

Most choose 8A and 9B, but they shouldn’t.

Certainty is accorded high weight relative to near-certainty

prospect 9 - higher expected outcome

something with higher probability vs guaranteed for sure - we prefer certainty

Prospect Theory: More than two outcomes

Prospect theory assumes people maximize a “weighted sum of utilities,” although the weights are not the same as the true probabilities, and the “utilities” are determined by a value function rather than a utility function:

Max: E(v) = σ 𝝅(𝒑𝒊) × (𝒙𝒊 − 𝒓)

Where 𝝅 is a non-linear weighting function,
v( xi - r) is the value function evaluated with respect to the reference point, r [Change in Wealth]

$r$ is your reference point

Weighting function

The decision weighting function is non-linear with the probability p.
- That is, decision weights are not probability.
- They do not obey the probability axioms.
- They should not be interpreted as measures of belief.
Properties of the weighting function:
- 𝝅′ 𝐩 > 𝟎 is an increasing function of p - 𝝅 (0) = 0: outcomes contingent on an impossible event are ignored - 𝝅 (1) = 1: scale is normalised so the certainty event has a π value of 1. For small p, 𝝅 (r*p) > r 𝝅 (p) where $0<r<1$ :𝝅 is subadditive
- Ex: 𝝅 (0.001)/ 𝝅 (0.002) > 1/2
- Meaning: when winning is possible but not probable (i.e., small p), most people choose the prospect that offers larger gain : V(6,000, 0.001) > V(3,000, 0.002)

weight is not probability

probability of a sample space is equal to 1

not the case for weighting - objective feeling

Increasing function

weighting function is monotonically increasing

Weighting function notes

This (displayed) mathematical function is

$\pi (pr) = pr^{\gamma} / [pr^{\gamma} + (1- pr)^{\gamma} ] (1/^{\gamma}) , \text{where} \;\; \gamma = .65$

Weighting function for losses can vary from weighting function for gains.
Low probabilities are given relatively higher weights than more probable events.
And certainty is weighted highly vs. near-certainty.
Using functions like this solves some earlier puzzles.

expected utility function is a special case where they treat prospect theory like expected utility function

Valuing prospects under prospect theory

$V(P) = \pi(pr_{A}) * v(z_{A}) + \pi(1 - pr_{A}) * v(z_{B})$

Steps:

Convert probabilities to decision weights
Calculate values of wealth differences
Use the above formula

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Frames

Essential condition for a theory of choice is principle of invariance: Different representations of same problem should yield same preference.

Mental accounting

Related to prospect theory and frames.

Accounting is process of categorizing money, spending and financial events.
Mental accounting is a description of way people intuitively do these things, and how it impacts financial decision-making.
Often tendency to use mental accounting leads to odd and suboptimal decisions.

Mental Accounting: This is a concept where people treat money differently depending on where it comes from, where it is kept, or how it is spent. For example, people may view a tax refund as a "bonus" or "windfall" and may be more inclined to spend it on a luxury or save it, rather than using it to pay off debts.

In your mind, create different accounts for money

Prospect theory, mental accounting and prior outcomes

Problem with prospect theory is that it was set up to deal with one-shot gamble – but what if there have been prior gains or losses?

Do we go back to zero (segregation), or move along curve (integration)?

Integration or segregation

Segregation: This refers to the idea that each new gamble is evaluated in isolation, starting from the reference point of zero (as if we "reset" after each gamble). The idea is that people have short memories, or they "start fresh" with each new gamble.
Integration: This suggests that people keep track of their gains and losses over time and use this cumulative total as their reference point. In this case, prior outcomes influence the valuation of current prospects.

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prior gains/losses effects One game, start at 0

previous game or loss - fresh start or affected?

segregated or integrated

Segregated

value each game in isolation

Silver lining effect:

They tend to mentally combine wins but separate losses to make them feel less impactful.

House money effect:

People tend to be more willing to gamble with money that they have won (house money) than with their initial stake or with money won in earlier rounds of betting.

Which of these approach individuals take can depend on the specific context and individual psychological factors

House money effect - you win, more likely to gamble more

also more risk-seeking

Theater ticket problems

Imagine you have decided to see a play where admission is $10. As you enter theatre you discover that you have lost a $10 bill. Would you still pay $10 for a ticket to the play?
Imagine that you have decided to see a play and paid the admission price of $10 per ticket. As you enter the theatre you discover that you have lost the ticket. The seat was not marked, and the ticket cannot be recovered. Would you pay $10 for another ticket?

Nothing is really different about the problems, however:

Of respondents given first question, 88% said they would buy a ticket.
Of respondents given second question, 54% said they would not buy a ticket.

In 2nd question, integration is more likely because both lost ticket and new ticket would be from same “account.”

Integration might suggest that $20 is too much for the ticket.

Example of how integration + mental accounting affect decision-making process

Opening and Closing accounts

Once an “account” is closed, you go back to zero. Evidence that people avoid closing accounts at a loss:

Selling a stock at a loss is painful: disposition effect (to be discussed).

Managers do the same thing as well.

Companies rarely have low negative earnings but often have low positive earnings:
- They manage earnings by either pushing things to a low positive,
- Or they “take a bath” and move to a high negative.

When firms went IPO, price of stock should increase

price set by agency

normally agents undervalue the company

Managers try and move the books a little and try to have a positive earnings

if they can't push the books, have a really bad year for bigger positives in the future