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A Brief History of Descriptive Theories of Decision Making

Kiel, June 9, 2005Michael H. Birnbaum

California State University, Fullerton

Overview

• This class will review a series of experiments testing critical properties of decision-making

• A “critical property” is a clearly stated implication (theorem) of a descriptive model

• These are sometimes called “paradoxes” when violations contradict a theory that has gained acceptance.

Prescriptive versus Descriptive Theory

• A normative or “prescriptive” theory is a theory that is regarded as “rational” or optimal with respect to some agreed upon rules of rationality. Such theories attempt to define what people should do.

• A descriptive theory is a behavioral theory of what people actually do when facing a decision.

Models to be Reviewed

• Expected Value (EV)• Expected Utility (EU) and Subjectively

Expected Utility (SEU)• Subjectively Weighted Utility (SWU), including

Prospect Theory (OPT)• Rank Dependent Utility (RDU), including Rank-

and Sign-Dependent Utility (RSDU) and Cumulative Prospect Theory (CPT)

• Configural Weighted Utility, including Rank-Affected Multiplicative Weights (RAM) and Transfer of Attention Exchange (TAX) models.

Expected Value Theory

• Let G = (x, p; y, q; z, r)• Where p + q + r = 1• EV = px + qy + rz• Judged Value = f(EV)• The function, f, is strictly monotonic.• Hence, if EV(F) > EV(G), then F is

preferred to G.

Reminder: Functions

• A function is a rule that assigns to each element in the domain one and only one element in the range.

• If y = f(x), then to each x in the domain, there is one and only one value of y.

• A strictly increasing monotonic function has the property that x > z if and only if f(x) > f(z).

Experiments

• Choose between: A: .50 to win $10 B: .50 to win $100 .50 to win $0 .50 to win $0 C: .50 to win $100 D: .70 to win $100 .50 to win $0 .30 to win $0Judge buying prices of A, B, C, D.EV theory handles these data.

Critical Properties # 1 (Classic Paradoxes)

• Risk Aversion (RA) and Risk-Seeking

• St. Petersburg Paradox• Sales and Purchase of gambles

and insurance• These are inconsistent with EV

theory.

Risk Aversion

• Which would you prefer?A: $50 for sure or B: .5 to win

$100 .5 to win $0

Most people prefer A. In fact, most people prefer $45 for sure rather than B. Preference for sure cash rather than gamble with the same or higher EV is called “risk aversion.”

From Bernoulli (1738)Exposition of a new theory on

the measurement of risk

Bernoulli (1738) quotes from a 1728 letter from Gabriel Cramer to Nicolas Bernoulli, addressing a problem (St. Petersburg paradox) Nicolas had posed in 1713 to Montmort:

In Exposition of a new theory on the measurement of risk, Daniel Bernoulli (1738) Quotes Cramer (1728):

"You ask for an explanation of the discrepancy between the mathematical calculation and the vulgar evaluation... in their theory, mathematicians evaluate money in proportion to its quantity while, in practice, people with common sense evaluate money in proportion to the utility they can obtain from it”

Bernoulli (1738)

If a poor man had a lottery ticket that would pay 20,000 ducats or nothing with equal probability, he would NOT be ill-advised to sell it for 9,000 ducats. A rich man would be ill-advised to refuse to buy it for that price.

Expected Utility (EU) Theory

Von Neumann & Morgenstern axioms (1944).Savage (1954) SEU: “sure thing axiom.”

EU(G) pii1

n

u(xi )

The Utility Function

• The idea of distinguishing utility of money from objective measure had effects in both economics and psychology.

• Bernoulli: u(x) = log(x) (later Fechner)

• Cramer: u(x) = axb (later Stevens)

Expected Utility Theory

• Why people would buy and sell gambles

• Sales and purchase of insurance• St. Petersburg Paradox• Risk-Aversion or Risk-Seeking:

u (x) 0 u (x) 0

Classic Paradoxes #2

• Allais Common Consequence Paradox• Allais Common Ratio Paradox• Risk-Seeking and Risk-Aversion in the

same person• Consequence Framing, Reflection

Hypothesis• Preference Reversals: Choice versus

Valuation, • Preference reversals between Buying

versus Selling Prices

Allais (1953) “Constant Consequence” Paradox

Called “paradox” because preferences contradict Expected Utility.

A: $1M for sure B: .10 to win $2M.89 to win $1M

.01 to win $0

C: .11 to win $1M D: .10 to win $2M.89 to win $0 .90 to win $0

Allais contradicts EU

A B u($1M) > .10u($2M) + .89u($1M) +.01u($0) Subtr. .89u($1M): .11u($1M) > .10u($2M)+.01u($0)

Add .89u($0): .11u($1M)+.89u($0) > .10u($2M)+.90u($0)

C D. So, Allais Paradox refutes EU.

EU(G) pii1

n

u(xi )

Allais Common Ratio Paradox

• Which do you choose?A: sure to win $3000 B: .8 to win $4000 .2 to win $0

C: .25 to win $3000 D: .2 to win $4000 .75 to win $0 .8 to win $0People prefer A to B and D over C.

Common Ratio Paradox Violates EU

• According to EU, A preferred to B if and only if C is preferred to D.

A over B: u(3) > .8u(4) + .2u(0)D over C: .25u(3) + .75u(0) < .2u(4) + .8u(0)Subtract .75u(0) from both sides and multiply

both sides by 4: u(3) < .8u(4) + .2u(0).Contradiction means EU is violated.

Risk Seeking and Risk Aversion in the Same Person

• Risk Aversion for moderate to large p:• Which do you prefer?A: $50 for sure B: .5 to win $100 .5 to win $0C: $1 for sure D: .01 to win $100 .99 to win $0Many people choose A over B and D

over C; risk aversion & risk-seeking for small p

Gambles with Losses

• Which do you prefer?A: .5 to get $0 B: lose $50 for

sure .5 to lose $100C: .99 to get $0 D: lose $1 for sure .01 to lose $100Many choose A over B, and D over C,

showing risk-seeking for moderate p and risk-aversion for small p.

Reflection Hypothesis

• Consider gambles of the form, A = (x, p; y), B = (x’, p; y’), where x, y > $0. The reflection hypothesis is the

conjecture that if A is preferred to B, then –B is preferred to –A, where

–A = (-x, p; -y) and –B = (-x’, p; -y’).

Consequence Framing

• Suppose I give you $100, but require you to choose one of these gambles:

A: .5 to get $0 B: lose $50 for sure .5 to lose $100Many people choose A over B, even

though they chose $50 for sure over a fifty-fifty bet to win $100, otherwise $0. But these are objectively the same.

Preference Reversals

Which do you prefer?A: .9 to win $100 B: .1 to win $1000 .1 to win $0 .9 to win $0If you owned these gambles, what is the

least you would charge to sell them?Many people choose A over B, but set a

higher selling price for B than for A.

Buying and Selling Prices

• Buying and Selling Prices are not monotonically related to each other.

• Which would you prefer?A: .5 to win $37 B: .5 to win $100 .5 to win $35 .5 to win $0• Buying and Selling Prices. Buy (A) = $36 > Buy (B) = $30Sell (A) = $36 < Sell (B) = $45

Paradoxes Refute EU

• Allais: Common Consequence & Common Ratio Paradoxes do for EU what the St. Petersburg paradox did for EV.

• Risk-Seeking and Risk-Aversion in the same person (small and large p)

• Consequence Framing, Reflection Hypothesis

• Preference Reversals: Choice versus Valuation

• Preference reversals between Buying versus Selling Prices

Next Lecture: SWU and PT

• In the next lecture, we see how Edwards (1954) subjectively weighted model can handle the Allais paradoxes. Key ideas are preserved in Kahneman and Tversky’s (1979) prospect theory. But PT will not account for all of the phenomena and leads to wrong predictions.

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