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

1. 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

2. 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.

3. 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).

4. 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 fancyGood, but more usual
ESHHS 2011

5. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Now you know your preference
15% to win
85% to win
15% to win
85% to win
Lottery A Lottery B
ESHHS 2011

6. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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
ESHHS 2011
≻

7. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
More principles…
For any p, q , r:
If p≽q and q≽r, then p≽r.
ESHHS 2011
If you prefer red to blue, blue to yellow, then you prefer red to yellow.

8. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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.
ESHHS 2011

9. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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.
ESHHS 2011

10. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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):
ESHHS 2011

11. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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
. . .
ESHHS 2011
1728
Gabriel Cramer
“… men of good sense
(estimate money) in proportion
to the usage that they
may make of it”

12. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Now you know your preference
Lottery A Lottery B
99% to win
1% to win
99% to win
1% to win
ESHHS 2011

13. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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.
ESHHS 2011

14. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
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%
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%
Experiment 1 Experiment 2
A B A B
Experiment 1. Experiment 2.
ESHHS 2011

15. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
The Common Consequence Paradox describes a failure of the Independence axiom.
ESHHS 2011
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

16. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
The Common Ratio Paradox
ESHHS 2011
Win $ 6000 with p = .45
otherwise nothing
Win $ 3000 with p = .90
otherwise nothing
Experiment 1.
Win $ 6000 with p = .001
otherwise nothing
Win $ 3000 with p = .001
otherwise nothing
Experiment 2.
(Kahneman & Tversky, 1979)
Again, a failure of the Independence axiom.

17. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Loss Aversion
ESHHS 2011
Win 5 EUR
Lose 5 EUR

18. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Instability/inconclusiveness of experimental findings
ESHHS 2011
• 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).

19. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Experimental fragmentation of theoretical concepts
ESHHS 2011
• 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.

20. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Reaction towards the discovery of the violations of the normative theory:
New mathematical formalizations of choice under risk
ESHHS 2011
• 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.

21. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
ESHHS 2011
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

22. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Characteristic of the new formalizations of choice under risk:
as theories get more complex, the axioms become less and less intuitive.
ESHHS 2011
• 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.

23. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Illustration: Tradeoff Consistency (TOC), the central axiom of RDU
and Cumulative Prospect Theory
ESHHS 2011
(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)≽

24. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Illustration: Tradeoff Consistency (TOC), the central axiom of RDU
and Cumulative Prospect Theory
ESHHS 2011
(p1,x1, .., pi,α, .., pn,xn) (p1,y1, .., pi,β, .., pn,yn)≽
(p1,x1, .., pi,γ, .., pn,xn) (p1,y1, .., pi,δ, .., pn,yn)≼

25. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Illustration: Tradeoff Consistency (TOC), the central axiom of RDU
and Cumulative Prospect Theory
ESHHS 2011
(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.

26. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Illustration: Tradeoff Consistency (TOC), the central axiom of RDU
and Cumulative Prospect Theory
ESHHS 2011
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.

27. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Research Lifecycle in The Formalization of Choice Under Risk
ESHHS 2011
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

28. G. S. MILOVANOVIĆ: FORMALIZATIONS OF RATIONAL CHOICE IN THE 20TH CENTURY
Recognizing the value of intuition
ESHHS 2011
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 .