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Buying and Selling Prices under Risk, Ambiguity and Conflict
Michael Smithson, The Australian National University and Paul D. Campbell, Australian Bureau of Statistics
Introduction
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.
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.
Hypothesis 2: Valuation and relative valuation of risky and
ambiguous gambles will be positively correlated, but neither
will be correlated with valuation of 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 gambles.
Method
Experimental Design:
88 volunteers were randomly assigned
to one of two conditions:
Vendor, where they were asked for a minimum selling price for
each gamble, or
Purchaser, where they were asked for a maximum buying price
for each gamble.
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.
Results
A minority of participants’ valuations were equivalent
to the expected utilities (EU’s) of the gambles.
In the Purchaser condition there were 13 EU responses for risky
gambles, 13 for ambiguous gambles and 14 for conflictive
gambles.
In the Vendor condition, there were 5, 3, and 9 EU responses
respectively.
A two-level logistic regression found that the difference
between the Vendor and Purchaser conditions was significant (p
= .031), but found no difference among the three types of
gambles.
Choice Model
All of the valuations were analyzed with a 2-level
choice model without a weighting parameter for probabilities,
to ensure model identifiability:
yij ~ N (mij, s2) .
The μij are defined as subjective expected utilities:
mij = Uijpi,
where pi is the probability for the ith gamble and jth subject, and
Uij is the subjective utility estimated by a 2-level choice model:
Uij = b0j + b1j x1i + (b2j + b22jx1i)z1i
+ (b3j + b33j)x1i z2i + (b4j + b44jz1i)x2i,
with predictors
x1 = 0 for the purchaser condition and 1 for the vendor
condition,
x2 is the variance of the probability in the gamble,
z1 = 0 for a precise or conflictive probability and 1 for an
ambiguous probability, and
z2 = 0 for a precise or ambiguous probability and 1 for a
conflictive probability.
The random-effects coefficients are defined as follows:
bkj = nk + ukj, with ukj ~ N (0, skj2) .
The model was estimated via Bayesian MCMC, in a 2-chain
model with a burn-in of 5,000 iterations and estimations based
on a subsequent 10,000 iterations.
Results
Table 1: Fixed-Effect Parameter Estimates
lower
param. estimate
se
credib.
n0
9.298
0.177
8.954
n1
-0.772
0.290
-1.341
n2
-1.462
0.201
-1.856
n22
-0.782
0.290
-1.347
n3
-1.317
0.200
-1.709
n33
-0.520
0.296
-1.100
n4
0.092
0.024
0.044
n44
-0.088
0.033
-0.153
Results
upper
credib.
9.651
-0.205
-1.071
-0.208
-0.924
0.063
0.139
-0.022
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. Both n22 and n33 are negative
and not significantly different from each other, reflecting
greater differences between buying and selling prices (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 3 is not testable for relative valuation. However,
again the endowment effect is present but does not differ
between ambiguous and conflictive gambles.
This time the effect of variance in the probabilities on valuation
was negative for both conflictive and ambiguous gambles.
Hypothesis 2 receives partial support. There were no
discernible differences in the strength of correlations
between the different types of gambles. The correlations of
valuations among gambles were relatively high, ranging
from .625 to .950, with RMS r = .786.
Relative Valuation Results
Hypothesis 2 was further tested by examining correlations
between random-effects parameter estimates in the choice
model. These results contradict Hypothesis 2.
Risky
Ambiguous
Conflictive
Contra Hypoth. 2
Conclusion
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.
Response mode (forced choice versus direct comparison versus
rating or pricing) has been shown to affect preferences, so
this should be the next step.
The endowment effect was decidedly stronger for conflictive
and ambiguous gambles than for risky ones. However, in
our study 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 probabilities.
These findings are compatible with studies showing that people
simply regard options with missing information as inferior
to those with complete information.
Four Suggestions for Future Research
1. Include alternative response modes (forced choice versus
direct comparison versus rating or pricing), to look for
preference effects or even reversals.
2. Systematically varying the monetary amounts and expected
values of the imprecise probabilities would enable separate
estimation of probability weighting and subjective utility
functions.
3. Loss frames need to be studied as well as gain frames.
4. The effects of ambiguous versus conflicting utility
assessments have yet to be investigated, perhaps along lines
suggested by Cooman and Walley’s work.