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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.