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# Schedule - Parallel Session 6 - Experiments and Ambiguity 4

Engineering F1.10 (note the change of room from F105-106 to F1.10) - 15:40 - 17:10

#### Ambiguity Attitudes Towards Imprecise Probabilities

Emmanuel Kemel; Mohammed Abdellaoui; Thomas Astebro; Corina Paraschiv

#### Abstract

Although the standard model of rational choice under ambiguity, i.e. subjective expected utility, suggested using subjective probabilities to measure uncertainty, it is nowadays common knowledge that this claim is contradicted by Ellsberg’s paradoxes and subsequent experiments. In his two-color paradox, Ellsberg argued that most decision makers prefer a risky option giving a prize with probability p=0.5 to an ambiguous option giving the same prize with p in [0, 1], i.e. the winning probability is somewhere between 0 and 1. Many subsequent Ellsberg-like experiments refined the initial two-color example by focusing on the general case where the winning probability p belongs to subintervals [p-; p+]. The present paper reports the results of an experimental investigation that aims at understanding how decision makers evaluate probability-interval-based ambiguous bets within an Ellsberg-like setup. Ambiguous bets are not explicitly interpreted in terms of second-order risk as in many multiple-prior-based models of ambiguity. Instead, we primarily consider objects of choice (x,[p-,p+]y) where the decision maker knows that he will get x with a winning probability lying somewhere between p- and p+, and y otherwise, (i.e., the probability of receiving y is not within that interval). Additionally, we postulate that decision makers evaluate ambiguous bets (x,[p-,p+]y) by subjectively combining the values of envelope (extreme) lotteries L+ = (x,p+;y) and L- = (x,p-;y) . The weight assigned to the upper (lower) envelope depends on the decision makers optimism/pessimism. Specifically, we assume that (i) the decision maker evaluates individual lotteries using rank-dependent utility (RDU); and that (ii) the value of an ambiguous bet (x,[p-,p+]y) is given by the convex combination of RDU values of the envelope lotteries, i.e., alphaRDU+(L+) + (1 -alpha)RDU-(L-) where notation RDU+ and RDU- means appealing to a weighting function for upper bound probabilities and a possibly different weighting function for lower bound probabilities respectively. Ambiguity attitude is captured by the coefficient alpha. It reflects decision makers optimism/ pessimism. We elicited this model in a laboratory experiment involving 62 subjects. All components of the models are estimated at the individual level using econometric modelling. Our results are consistent with previous research on ambiguity attitudes: subjects exhibit ambiguity aversion in the standard Ellsberg case, and their ambiguity attitudes vary with the size and location of the interval of probabilities. In terms of our model, we observe that probability weighting of the upper bound is radically different from probability weighting of the lower bound: the former is concave whereas the later is convex. These pattern receives the following psychological interpretation. Attitudes towards the upper bound carries the certainty effect, while attitudes towards the lower bound carries the possibility effect.

Emmanuel Kemel

Researcher, GREGHEC (CNRS, HEC Paris)

#### Ellsberg Meets Allais: An Experimental Study of Ambiguity Attitudes

Bin Miao; Songfa Zhong

#### Abstract

BACKGROUND. Ambiguity attitude along with the Ellsberg paradox have widely studied in both theoretical and experimental literature. Recently, extending Ellsberg paradox, Machina (2011; 2014) provides some thought experiments to consider situations where there are three or more possible outcomes. His examples suggest that people exhibit Allais-type behavior in the domain of ambiguity. This study aims to provide a systematic examination of Allais-type behavior under ambiguity in an experimental setting. EXPERIMENT & RESULTS. Our experiment consists of two treatments. Treatment 1 concerns two urns each containing 90 balls that are red, black or white. Urn 1 contains n red balls, for the rest of 90-n balls, half of them are black and the other half white. Urn 2 contains n red balls, 90-n balls that are either black or white with the exact composition unknown. Subjects are asked to choose which urn to place a bet on. For example, when n=80, subjects are asked to choose between Bet 1 and Bet 2 as follows. BET 1: Draw a ball from Urn 1 containing 80 red balls, 5 black balls and 5 white balls, and get paid \$x if the ball is red, \$50 if it is black and 0 if it is white. BET 2: Draw a ball from Urn 2 containing 80 red balls, 10 black and white balls, and get paid \$x if the ball is red, \$50 if it is black and 0 if it is white. In the experiment, we find significant proportion of subjects exhibiting a switch from Bet 1 to Bet 2 as x decreases from 50 to 0. Treatment 2 concerns one envelop with 100 tickets, each ticket numbered from 1 to 100, while the exact composition of the numbers is unknown. Subjects bet on the number on one ticket drawn from the envelope, and they can choose different sets of numbers to place different stakes on. For example, they can choose n different numbers to bet for \$x and another m different numbers to bet for \$y, while they receive nothing if the drawn number matches the rest of 100-n-m numbers. In a representative choice problem, subjects are asked to choose between the following two bets. BET 3: 20 numbers for \$x, 10 numbers for \$25, and 70 numbers for 0. BET 4: 20 numbers for \$x, 5 numbers for \$50, and 75 numbers for 0. Similarly, we find significant proportion of subjects switching from Bet 4 to Bet 3 when x changes from 55 to 50. IMPLICATION. In treatment 1, choosing Bet 1 over Bet 2 at x=50 and reversing the choice at x=0 violates the independence property. As the state (red) with varying outcomes has known probabilities, such a reversal is incompatible with both subjective expected utility and maxmin expected utility. For treatment 2, choosing Bet 3 over Bet 4 at x=50 and reversing the choice at x=55 violates a weaker version of independence property named commonotonic independence. Thus, such a reversal is incompatible with subjective expected utility, Choquet expected utility and recursive rank-dependent utility.

Bin Miao

Assistant Professor, Shanghai University of Finance and Economics

#### Decisions with Compound Lotteries

Yuyu Fan; David V. Budescu; Enrico Diecidue

#### Abstract

The Reducibility of Compound Lottery axiom (ROCL) states that a multi-stage compound lottery can be reduced to its equivalent simple form. Violations of ROCL were documented, but they are not fully understood, and only few descriptive models of DMs’ decisions in such cases, were offered. To study systematically the factors that lead to violations of ROCL and evaluate their importance and prevalence, we created two types of lotteries, simple one-stage binary lotteries (16 in total) with various winning probabilities and payoffs, and compound lotteries (32 in total) obtained by manipulating (1)the number of stages (2, 3), (2)the global winning probability (.09, .18) paired with the payoff (\$20, \$10), (3)the resolution mechanism (simultaneous, sequential), (4)equality, or inequality, of the probabilities of the various stages, (5)the descending/ascending order of the stage probabilities in sequential lotteries, (6)the magnitude of the differences between the probabilities of adjacent stages. Six groups of DMs (125 students, 39% females, mean age = 22) evaluated different partially overlapping subsets of these lotteries by providing Certainty Equivalents (CEs) via a two-stage choice procedure. All participants evaluated the single stage lotteries and some 2-stage and 3-stage lotteries in various orders. We confirmed the existence of the systematic violations of ROCL and tested the effect of each factor. Half of the DMs preferred the compound lotteries, and only one-fifth preferred the equivalent one-stage lotteries. The effect of each factor was examined via multi-level modeling. We found that the number of stages and the global probability were the major drivers of the CEs. We also inferred pair-wise choices from the CEs and used them to fit the Bradley-Terry (BT) model. For the most part, the scale values from the BT model were consistent with the multi-level models. Prospect Theory (PT) is mute with respect to compound lotteries and does not explain how their probabilities are aggregated. To model the DMs’ CEs of compound lotteries, we developed three classes of models (16 in total) based on PT: (1)DMs aggregate stage probabilities first, and then generate a decision weight; (2)DMs weight stage probabilities first, and then aggregate them; (3)DMs anchor on one of the stage probabilities and then apply the weighting. Model fits were compared in terms of RMSE and pair-wise tournaments. Overall, the best fitting model was the one that assumes that DMs anchor on the minimal stage probability and then apply the weighting function. We assume that even DMs with minimal knowledge of probability theory recognize that the global winning probability cannot exceed any of the stage probabilities, so they use the minimal stage probability as an anchor. Results also showed that, in general, the “aggregate first and weigh second” outperformed the “weight first and aggregate second” models, presumably because it is cognitively easier and more natural.

Yuyu Fan

Student, Fordham University

Sara le Roux

#### Abstract

We report the results from a set of experiments conducted to test the effect of ambiguity on individual behaviour in games of strategic complements and strategic substitutes. We test whether subjects’ perception of ambiguity differs when faced by a local opponent as opposed to a foreign one. Interestingly, though subjects often choose an ambiguity safe strategy (not part of a Nash equilibrium), we do not find much difference in the ambiguity levels when faced by foreign subjects.

Sara le Roux

Senior Lecturer in Ecoomics, Oxford Brookes University