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Axiomatic Theory of Probabilistic
Decision Making under Risk
Pavlo R. Blavatskyy
University of Zurich
April 21st, 2007
Outline
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Introduction
Framework
Axioms
Representation Theorem
Implications
Conclusions
Introduction
• Experimental studies of repeated decision
making under risk => individual choices are
often contradictory
– Camerer (1989) reports that 31.6% of subjects
reversed their choices
– Starmer and Sugden (1989) find that 26.5% of all
choices are reversed
– Hey and Orme (1994) report an inconsistency rate of
25%
– Wu (1994) finds that 5% to 45% of choice decisions
are reversed
– Ballinger and Wilcox (1997) report a median switching
rate of 20.8%
Introduction continued
• Majority of decision theories are deterministic
– Exception Machina (1985) and Chew et al. (1991)
• They predict that repeated choice is always
consistent (except for decision problems where
an individual is exactly indifferent)
• Common approach is to embed a deterministic
decision theory into a model of stochastic choice
– tremble model of Harless and Camerer (1994)
– Fechner model of random errors (e.g. Hey and Orme,
1994)
– Random utility model (e.g. Loomes and Sugden, 1995)
This Paper
• Individuals do not have a unique
preference relation on the set of risky
lotteries
• Individuals possess a probability measure
that captures the likelihood of one lottery
being chosen over another lottery
• A related axiomatization of choice
probabilities
– Debreu (1958)
– Fishburn (1978)
Framework
• A finite set of all possible outcomes
(consequences) X  x1 ,..., xn 
– Outcomes are not necessarily monetary
payoffs
• A risky lottery is a probability distribution
L p1 ,..., pn 
on X
• A compound lottery L1  1   L2
• The set of all risky lotteries is denoted by Λ
Framework, continued
• An individual possesses a probability measure
on   

  
– Choice probability Pr L1 , L2  0,1 denotes a
likelihood that an individual chooses L1 over L2 in a
repeated binary choice
• A deterministic preference relation can be easily
converted into a choice probability
– If an individual strictly prefers L1 over L2, then PrL1 , L2   1
– If an individual strictly prefers L2 over L1, then PrL1 , L2   0
– If an individual is exactly indifferent, then PrL1 , L2   1 2
Axioms
• Axiom 1 (Completeness) For any two
lotteries L1 , L2   there exist a choice
probability PrL1 , L2   0,1 and a choice
probability PrL2 , L1   1  PrL1 , L2 
– PrL, L  1 2 for any L  
– Only two events are possible: either choose
L1 or choose L2
Axioms, continued
• Axiom 2 (Strong Stochastic Transitivity)
For any three lotteries
if
L1 , L2 , L3   and
PrL1 , L2   1 2
PrLthen
2 , L3   1 2
PrL1 , L3   max PrL1 , L2 , PrL2 , L3 
• Axiom 3 (Continuity) For any three
lotteries L1 , L2 , L3   the sets
  0,1 PrL1  1   L2 , L3   1 2 and
  0,1 PrL1  1   L2 , L3   1 2 are closed
Axioms, continued
• Axiom 4 (Common Consequence
Independence) For any four lotteries
L1 , L2 , L3 , L4   and any probability   0,1 :
PrL1  1   L3 , L2  1   L3  
 PrL1  1   L4 ,L2  1   L4 
• If two risky lotteries yield identical chances of the
same outcome (or, more generally, if two
compound lotteries yield identical chances of the
same risky lottery) this common consequence
does not affect the choice probability
Axioms, continued
• Axiom 5 (Interchangeability) For any
three lotteries L1 , L2 , L3   if
PrL1 , L2   PrL2 , L1   1 2 then
PrL1 , L3   PrL2 , L3 
• If an individual chooses between two
lotteries at random then he or she does
not mind which of the two lotteries is
involved in another decision problem
Representation Theorem
• Theorem 1 (Stochastic Utility Theorem)
Probability measure on    satisfies
Axioms 1-5 if and only if there exist an
assignment of real numbers u i to every
outcome xi , i  1,..., n , and there exist a
non-decreasing function  : R  0,1 such
that for any two risky lotteries
L1  p1 ,..., pn , L2 q1 ,..., qn    :

PrL1 , L2    i 1 ui pi  i 1 ui qi
n
n

Implications
• Function . has to satisfy a restriction
x  1   x for every x  R , which
immediately implies that 0  1 2
• If a vector U  u1 ,..., u n  and function .
represent a probability measure on   
then a vector U   aU  b and a function
represent the same

.  . a
probability measure for any two real
numbers a and b, a  0
Special cases
• Fechner model of random errors
– function  . is a cumulative distribution
function of the normal distribution with mean
zero and constant standard deviation   0
• Luce choice model
– function  . is a cumulative distribution
function of the logistic distribution
x  1 1  exp  x where   0 is constant
• Tremble model of Harless and Camerer (1994)
x0
 p
– function  . is the step function x    1 2 x  0

1  p x  0

Empirical paradoxes
• Unlike expected utility theory, stochastic
utility theory is consistent with systematic
violations of betweenness and a common
ratio effect
• …but cannot explain a common
consequence effect
Conclusions
• Individuals often make contradictory choices
– Either individuals have multiple preference relations
on Λ (random utility model)
– or individuals have a probability measure on   
• Choice probabilities admit a stochastic utility
representation if and only if they are complete,
strongly transitive, continuous, independent of
common consequences and interchangeable
• Special cases: Fechner model of random errors,
Luce choice model and a tremble model of
Harless and Camerer (1994)