# Expected Utility with Random Optimism and Pessimism EUROPE

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```					   Expected Utility with Random
Optimism and Pessimism
(EUROPE)

Pavlo Blavatskyy        Ulrich Schmidt
University of Zurich   University of Kiel &
Kiel Institute for the
World Economy
Motivation
• Evidence on errors in risky choice [Harless
and Camerer (1994), Hey and Orme
(1994)]
• Expected utility plus noise
Vc(p) = i u(xi)pi+ 
= i [u(xi) + ]pi
• Noise is added to the utility scale, i.e. the
evaluation of outcomes is noisy
Motivation
• Our model: evaluation of probabilities is noisy:
– People are less familiar with probabilities than with
monetary amounts
– Psychological studies: people in good mood are
optimistic while people in bad mood are pessimistic
[Johnson and Tversky (1983), Wright and Bower
(1992), Mayer and Hanson (1995)]
– Optimism and pessimism influence perception of
probabilities
– If moods are not modelled endogenously they can be
taken as random
Motivation
• Vc(p) = i u(xi)[pi+ i]
• p i = pi + i is perceived probability of xi
*

• Our goal:
– Analyze this model as simple as possible
– Take some basic requirements of rationalitiy
into account:
(i) 0  pi + i  1,
(ii) i (pi + i) = 1,
(iii) i = 0 if pi = 0
The Model
• X = (x1, x2, ..., xn)
• p = (p1, p2, ..., pn)
• True preferences satisfy axioms of EU
• (u1, u2, ..., un) such that V(p) = i uipi
• Choice is subject to noise in the probability
perception
• (1, 2, ..., n)
• V c(p) =  u p *    with p i = [pi+ i]
*
i i i
The Model
• What is the simplest formulation of the
model such that the three rationality
requirements are satisfied?
• Errors i cannot be independently drawn if
they should always sum up to zero
• Independently drawn primary distortions:
(´1, ´2, ..., ´n)
p i  ε
• p 
*              i

1   j1 ε j
i            n
The Model
• Each ´i is drawn from a symmetric
distribution with zero mean and support
[-, ]
• 0 ≤  < 0.5
• Lemma 1:
If perceived probabilities should satisfy
0  p i  1, we must have pi + ´i  0 for all
*

i.
The Model
• What happens if pi < ?
• Realizations of ´i which lead to pi + ´i < 0 have
to be ruled out
• Distribution of primary distortions has to be
truncated
• Lemma 2:
(i) E(´i) = 0 if pi  
(ii) E(´i) > 0 if pi < 
(iii) E(´i) > E(´j) if pi < pj and pi < .
The Model
• Proposition 1:
(i) E(p i ) > pi if pi < pj for all j  i.
*

(ii) E(p i ) < pi if pi >  and there exists an
*

outcome xj with pj < .

 p*  p i
(iii) E *  
i
p  p     for all pi < pj with pi < .
 j     j
Behavioral Implications
• Common consequence and common ratio
effect
• Reflection effect
• Fourfold pattern of risk attitudes
• Event-splitting effects
• Violations of betweenness
Conclusions
• Simple model of probability distortions
leads to overweighting of small and
underweighting of large probabilities
• Psychological foundation for weighting
function in OPT
• Extensions to uncertainty and cumulative
probabilities

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 views: 3 posted: 3/22/2012 language: pages: 11