G.Milovanović, 2011. FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY: FROM AXIOMS TO PREFERENCE CONDITIONS

FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY:

FROM AXIOMS TO PREFERENCE CONDITIONS

The Story of

Goran S. Milovanovid

Faculty of Philosophy, University of Belgrade

ESHHS 30th Annual Conference Belgrade, Serbia

FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY: FROM AXIOMS TO PREFERENCE CONDITIONS

The Story of

ESHHS 30th Annual Conference Belgrade, Serbia, 2011.

The problem of rational choice under risk is ubiquitous in social and human sciences.

Economics, psychology, political science, management sciences… Mathematicians, economists, psychologists, neurobiologists…

The mathematical formalization of choice under risk is probably the single most important problem of (mathematical) social sciences.

At the deepest fundamental level of theoretical analysis, all systematic experimental study of human behavior can be described as or constrained to a problem of choice.

FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY: FROM AXIOMS TO PREFERENCE CONDITIONS

The Story of

ESHHS 30th Annual Conference Belgrade, Serbia, 2011.

Our goal

To provide a characterization of the advances in formal theories of choice under risk in the second half of the 20th century, in respect to their explanatory power in human sciences (experimental psychology in the first place).

G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY

On Basic Principles

Super fancy

Good, but more usual

Super fancy Good, but more usual

Super fancy Good, but more usual

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Now you know your preference

15% to win

85% to win

15% to win

85% to win

Lottery A Lottery B

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AXIOM OF INDEPENDENCE

IF p q

THEN for any r, λ ∊ (0,1):

λp + (1- λ)r λq + (1- λ)r

any outcome any probability

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≻


More principles…

For any p, q , r:

If p≽q and q≽r, then p≽r.

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If you prefer red to blue, blue to yellow, then you prefer red to yellow.


Why is it important?

BECAUSE EVERYTHING IN THE EXPERIMENTAL STUDY OF HUMAN BEHAVIOR IS ABOUT CHOICE.

Likert-type scales

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

1 2 3 4 5

O1

O2

O3

Rank 1

Rank 2

Rank 3

Ranking Choice

A B

? C D

ANY SYSTEMATIC METHOD TO STUDY HUMAN BEHAVIOR CAN BE FORMULATED AS AND COSTRAINED TO A PROBLEM OF CHOICE.

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Why mathematics? Enter history…

BECAUSE THE AXIOMATIZATION OF CHOICE PROVIDES A THEORY OF MEASUREMENT FOR HUMAN AND SOCIAL SCIENCES.

John von Neumann Oskar Morgenstern

1700 1900 2000

. . .

1948

IF observable preferences satisfy the axioms of rational choice THEN these preferences can be represented by a real-valued function.

= We can assign numbers to objects of choice

= We can measure human preferences.

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Why mathematics?

IF observable preferences satisfy the axioms of rational choice THEN these preferences can be represented by a real-valued function.

One’s preferences satisfy the Axioms of

Rational Choice

One behaves

as if…

O1 O2 O4 O6

U1

U2

U6

U4

Utility Function: X R

Utility functions measure preferences by assigning a real

number to each object of choice.

Utility functions represent the

preferences: THE REPRESENTATIONAL

THEORY OF MEASUREMENT.

THE FUNDAMENTAL RESULT of von Neumann & Morgenstern (1948):

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Historical Review: Formalization of Choice Under Risk

1700 1900 2000

. . .

1948

Leonard J. Savage „Foundations of Statistics“ Subjective Expected Utility Theory

John von Neumann Oskar Morgenstern „Theory of Games and Economic Behavior“ (2nd edition) Expected Utility Theory

1738

1954

Daniel Bernoulli „Specimen theoriae novae

de mensura sortis “ Proposed the existence of

utility functions

Frank P. Ramsey „Truth and Probability“

Eliciting subjective beliefs about probabilities

1926

1931

Bruno de Finetti „Dutch book arguments“,

Coherence, de Finetti’s game: Eliciting subjective beliefs

about probabilities

. . .

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1728

Gabriel Cramer “… men of good sense

(estimate money) in proportion to the usage that they

may make of it”


Now you know your preference

Lottery A Lottery B

99% to win

1% to win

99% to win

1% to win

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Maurice Allais (1953): a famous experiment

Lottery A 11% you win 1 milion $, 89% you win nothing.

Lottery B

10% you win 5 milion $, 90% you win nothing.

Lottery A 100% you win 1 milion $

Lottery B

89% you win 1 milion $, 1% you win nothing,

10% you win 5 milion $.

Experiment 1. Experiment 2.

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Maurice Allais (1953): a famous paradox

Value p Value p Value p Value p

$ 1M 100% $ 1M 89% $0 89% $0 90%

$0 1% $ 1M 11% $ 5M 10%

$ 5M 10%

Experiment 1 Experiment 2

A B A B

Value p Value p Value p Value p

$ 1M 89% $ 1M 89% $0 89% $0 89%

$ 1M 11% $0 1% $ 1M 11% $0 1%

$ 5M 10% $ 5M 10%


A B A B

Value P Value P Value P Value P

$ 1M 89% $ 1M 89% $0 89% $0 89%

$ 1M 11% $0 1% $ 1M 11% $0 1%

$ 5M 10% $ 5M 10%


A B A B

Experiment 1. Experiment 2.

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The Common Consequence Paradox describes a failure of the Independence axiom.

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The discovery of this paradox was only the beginning of an immense production of experimental findings that falsify the normative theory of rational choice.

Maurice Allais


The Common Ratio Paradox

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Win $ 6000 with p = .45 otherwise nothing


Experiment 1.



Experiment 2.

(Kahneman & Tversky, 1979)

Again, a failure of the Independence axiom.


Loss Aversion

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Win 5 EUR

Lose 5 EUR


Instability/inconclusiveness of experimental findings

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• Varying the format of the presentation of stimuli (reformulating the lotteries used to elicit the Allais’ common consequence paradox) can be used to show that people do not always exhibit failures of the independence axiom (Conlisk, 1989, Reed, 2009).

• Other important findings, like loss aversion (which was incorporated into Kahneman &

Tversky’s Prospect Theory, 1979, 1992), are not robust in the sense of being observed in any experiment with every participant. On the contrary, there are systematic experimental studies (Schmidt & Traub, 2001) that find loss aversion in only 1/3 of participants.

• Even in studies confirming the phenomena it is admitted that the degree of loss aversion

varies with the theoretical definition used (Abdellaoui, Bleichrodt & Paraschiv, 2007).


Experimental fragmentation of theoretical concepts

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• Experimental findings are inconclusive

• There is something like loss aversion out there, but no one is certain about its true nature

• People sometimes violate independence, but it is not clear under which conditions and how exactly

• Some empirical phenomena tend to disappear and reappear in an unsystematic fashion in the experimental practice

Each different method used to elicit a particular empirical

phenomenon…

… brings about a new, different instance of that phenomenon…

…while we still keep the same name, and, presumably, the same concept, fixed for that

fragmented empirical phenomenon.

Our methodological operations do not converge onto single theoretical concepts.


Reaction towards the discovery of the violations of the normative theory: New mathematical formalizations of choice under risk

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• A plentitude of new formal theories of choice: attempts to provide a plausible formalization of choice under risk able to incorporate the observed violations of the normative theory.


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Historical Review: Formalization of Choice Under Risk 1953 – 2000. note: this is a selected review only (based on Schmidt, 2004)

1953 2000

. . .

1992

1991

1979 1982

. . .

Allais Paradox

. . .

Chew & MacCrimmon Weighted Utility Theory

1979

Kahneman & Tversky Prospect Theory

Gull Disappointment Aversion Theory

Tversky & Kahneman Cumulative Prospect Theory (CPT)

Quiggin First axiomatization of a Rank-Dependent Utility Model

1987 Yaari Dual Expected Utility Another RDU axiomatization

1984 - 1989 Segal, Green, Jullien General Rank-Dependent Model

1993

Chateauneuf & Wakker CPT Axioms for risk

1999

Wakker & Tversky CPT Axioms for uncertainty


Characteristic of the new formalizations of choice under risk: as theories get more complex, the axioms become less and less intuitive.

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• In order to axiomatically characterize Rank-Dependent Utility theories, like the most popular Cumulative Prospect Theory, one needs to start with quite complicated formal constructions.


Illustration: Tradeoff Consistency (TOC), the central axiom of RDU and Cumulative Prospect Theory

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(p1,x1, .., pi,xi, .., pn,xn)

α

(p1,x1, .., pi,α, .., pn,xn)

(p1,y1, .., pi,yi, .., pn,yn)

β

(p1,y1, .., pi,β, .., pn,yn) ≽



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(p1,x1, .., pi,α, .., pn,xn) (p1,y1, .., pi,β, .., pn,yn) ≽

(p1,x1, .., pi,γ, .., pn,xn) (p1,y1, .., pi,δ, .., pn,yn) ≼



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(p1,x1, .., pi,α, .., pn,xn) (p1,y1, .., pi,β, .., pn,yn) ≽

(p1,x1, .., pi,γ, .., pn,xn) (p1,y1, .., pi,δ, .., pn,yn) ≼

(q1,v1, .., qi,α, .., qn,vn) (q1,w1, .., qi,β, .., qn,wn)

(q1,v1, .., qi,γ, .., qn,vn) (q1,w1, .., qi,δ, .., qn,wn) ≽

≺

If one does not encounter this situation in his choices then his behavior satisfies the TOC axiom.



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Only after further mathematical inferences it becomes obvious that TOC guarantees that the following will never happen:

U(α) – U(β) ≥ U(γ) – U(δ)

and

U(α) – U(β) < U(γ) – U(δ)

The ordering of differences in utilities is invariant under the transformations described in the

formulation of the TOC axiom.


Research Lifecycle in The Formalization of Choice Under Risk

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FORMAL THEORY OF CHOICE

EMPIRICAL VIOLATIONS

EXPERIMENTAL FRAGMENTATION OF

THEORETICAL CONCEPTS

DEVELOPMENT OF MORE COMPLEX THEORIES

MORE COMPLICATED FORMAL INFERENCES/EMPIRICAL

PREDICTIONS

INCREASED PROBABILITY OF

EMPIRICAL VIOLATIONS

Enter here


Recognizing the value of intuition

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Daniel Bernoulli „Specimen theoriae novae

de mensura sortis“ 1738.

Bernoulli obviously got it wrong back in 1738.

Still he was able to explain the most robust empirical phenomena relying on a very small number of empirical intuitions only .

Education

G.Milovanović, 2011. FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY: FROM AXIOMS TO PREFERENCE CONDITIONS