Buying and Selling Prices under Risk, Ambiguity and Conflict

1. Buying and Selling Prices under Risk, Ambiguity and Conflict Michael Smithson The Australian National University Paul D. Campbell Australian Bureau of Statistics

2. 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.

3. Preference ordering hypothesis For mid-range probabilities, Ellsberg (1961) and many others since have found that people tend to prefer risk to ambiguity. 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.

4. 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 gambles.

5. 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 gambles.

6. Method 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}.

7. Method 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.

8. 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.

9. Relative Valuation Results Hypothesis 1 is contradicted. The conflictive gambles are valued more than the ambiguous gambles, relative to EU-equivalent 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 random-effects parameter estimates in the choice model. These results contradict Hypothesis 2.

10. Conclusions 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 probabilities.

11. 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 functions. • Loss frames have yet to be studied. • The effects of ambiguous versus conflicting utility assessments have yet to be investigated.

12. Thanks!