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Buying and Selling Prices under Risk, Ambiguity
and Conflict
Michael Smithson
The Australian National University
Paul D. Campbell
Australian Bureau of Statistics
We report an empirical study of buying and selling prices for
three kinds of gambles:
• Risky (with known probabilities),
• Ambiguous (with lower and upper probabilities), and
• Conflictive (with disagreeing probability assessments).
We infer preferences among gambles from people’s buying
and selling prices in two ways:
• Valuation: Using the “raw” prices, and
• Relative valuation: Comparison of a price for an ambiguous
or conflictive gamble with the price for a risky gamble
having an equivalent expected utility.
Preference ordering hypothesis
For mid-range probabilities, Ellsberg (1961) and many others
since have found that people tend to prefer risk to
Smithson (1999) and Cabantous (2007) found that people
prefer ambiguity to conflict.
Hypothesis 1: For mid-range probabilities, both valuation
and relative valuation will be lowest for conflictive
gambles, second lowest for ambiguous gambles, and
highest for risky gambles.
Correlated orientations hypothesis
Several researchers have investigated whether attitudes
towards risk and ambiguity are correlated.
An early study by Curley et al. (1986) found no significant
correlation, but later more nuanced investigations by
Lauiola and his colleagues did find a positive correlation
(2001, 2007).
Pushkarskaya et al. (2009) found no correlation between
orientations towards conflictive gambles and orientations
towards the other two kinds.
Hypothesis 2: Valuation and relative valuation of risky and
ambiguous gambles will be positively correlated, but
neither will be correlated with valuation of conflictive
Endowment effect hypothesis
In a well-known violation of subjective expected utility
known as the endowment effect, people tend to offer
higher selling than buying prices for risky gambles.
The standard betting interpretation of lower and upper
probabilities also stipulates a higher selling than buying
price for ambiguous gambles.
However, there appears to be no similar standard
interpretation for conflictive gambles.
Hypothesis 3: For mid-range probabilities, the difference
between buying and selling prices will be higher for
ambiguous and conflictive gambles than for risky
Experimental Design:
88 volunteers randomly assigned to one of two conditions:
• Vendor, asked for a minimum selling price for each
gamble, or
• Purchaser, asked for a maximum buying price.
Card Games (comparable to Ellsberg’s 1961 2-colour task)
Risky gambles. Proportions of winning cards were
.25, .4, .5, .6, and .75.
Ambiguous gambles. Proportions were interval-valued:
[.3, .7] , [.15, .85], and [0, 1].
Conflictive gambles. Proportions were given by two equally
credible sources: {.4, .6} , {.3, .7} , and {.2, .8}.
Expected utilities for all ambiguous and conflictive gambles
were 0.5*$10.
The variance of the probabilities associated with each
conflictive gamble was approximately equal to
the variance in a corresponding ambiguous gamble.
All of the valuations were analyzed with a 2-level choice
model (see poster).
The model was estimated via Bayesian MCMC.
Valuation Results
Hypothesis 1 receives only partial support. The risky
gambles are valued more highly than the ambiguous and
conflictive gambles, but the ambiguous and conflictive
valuation means do not significantly differ.
Hypothesis 3 is well-supported. There are greater differences
between buying and selling prices (i.e., the endowment
effect) for the ambiguous and conflictive gambles than for
risky gambles.
The effect of variance in the probabilities on valuation was
negative for valuation of conflictive gambles. However, this
effect did not emerge for ambiguous gambles.
Relative Valuation Results
Hypothesis 1 is contradicted. The conflictive gambles are
valued more than the ambiguous gambles, relative to EUequivalent risky gambles.
Hypothesis 2 receives partial support. There were no
discernible differences in the strength of correlations
between the different types of gambles. Hypothesis 2 was
further tested by examining correlations between randomeffects parameter estimates in the choice model. These
results contradict Hypothesis 2.
Conflictive and ambiguous gambles were valued less than
expected-utility-equivalent risky gambles, but relative
valuation favoured conflictive over ambiguous gambles.
This latter finding conflicts with Smithson (1999) and
Cabantous (2007) and is difficult to explain.
The endowment effect was decidedly stronger for conflictive
and ambiguous gambles than for risky ones. However,
the standard betting interpretation of lower and upper
probabilities does not seem to explain this effect.
The endowment effect is enhanced equally for ambiguous and
conflictive gambles. Respondents appear to devalue both
types of gamble as if they perceive a feature that makes
both of them inferior to gambles with known
Four Suggestions for Future Research
Include alternative response modes (forced choice versus
direct comparison versus rating or pricing), to look for
preference effects or even reversals.
Systematically varying the monetary amounts and
locations of probability centroids would enable separate
estimation of probability weighting and subjective utility
Loss frames have yet to be studied.
The effects of ambiguous versus conflicting utility
assessments have yet to be investigated.