# Schedule - Parallel Session 1 - Experiments and Ambiguity 1

IMC Boardroom 2 - 11:00 - 12:30#### Allais at the Horse Race: Testing Models of Ambiguity Aversion

Florian H. Schneider; Martin Schonger

#### Abstract

Since Ellsberg pioneered the concept of ambiguity aversion, both theorists and experimentalists have taken a keen interest in the concept. Ambiguity aversion is usually studied in the Anscombe-Aumann framework. Anscombe-Aumann proposed a Monotonicity axiom, which is widely used in models of ambiguity aversion. Indeed, apart from basic choice theoretic axioms like Transitivity and Continuity, Monotonicity seems to be the most common axiom models of ambiguity aversion satisfy. Within the Anscombe-Aumann framework, models of ambiguity aversion that satisfy Monotonicity include Multiple priors (Gilboa/Schmeidler, 1989), Choquet Model (Schmeidler, 1989), Smooth ambiguity preferences (Klibanoff et al., 2005), Variational preferences (Maccheroni et al., 2006), Vector Expected Utility (Siniscalchi, 2009), MBA-preferences (Cerreia-Vioglio et al., 2011), and Hedging preferences (Dean/ Ortoleva, forth.). Models that do not satisfy Monotonicity include the recursive models (Segal, 1987; Abdellaoui/Zank, 2015), and mean-dispersion preferences (Grant/Polak, 2013). The descriptive validity of the Monotonicity axiom is the focus of this paper. We know of no prior method to test the Monotonicity axiom, let alone experimental work that actually does so. This paper provides a thought experiment that fills this gap. Our thought experiment adapts the classical Allais paradox to a setting where there are both objective and subjective sources of uncertainty. The so-modified Allais paradox tests Monotonicity, and we call it the Allais Horse Race. The Allais Horse Race serves two roles: first it allows for introspective “testing” of Monotonicity, and second, it paves the way for experimental tests of Monotonicity. In the incentivized experiment we find that about half of all participants violate Monotonicity, and overwhelmingly do so in a specific, non-random way. The hypothesis that violations are due to random error is easily rejected. Therefore models of ambiguity aversion that satisfy Monotonicity cannot describe the behavior of about half of all subjects. In the experiment we also confront participants with the original Allais paradox. It turns out that violating Independence strongly predicts violating Monotonicity. To provide some insight into this empirical correlation, we establish theoretically that for probabilistically sophisticated decision-makers, Monotonicity and Independence are equivalent. This might explain why decision-makers in general tend to either satisfy both Independence and Monotonicity or violate both. For most decision-makers the axioms, while not equivalent, are probably similar. The working paper is available at: www.econ.uzh.ch/static/wp/econwp207.pdf

**Florian H. Schneider**

#### Imprecise Beliefs and Belief Updating Under Ambiguity

Zhihua Li; Andrea Isoni; Graham Loomes

#### Abstract

Beliefs about probabilities of uncertain events are central to decision making under ambiguity. Different methods have been proposed to better and more accurately elicit beliefs in an incentive compatible way. However, elicited beliefs have often been found to be unreliable predictors of choice behaviour. This paper addresses this issue by studying whether people have stable/precise beliefs to be elicited at the first place. If beliefs are very noisy or unstable, incentive compatible methods based on the assumption that beliefs are precisely-defined and stable will not work as intended. We also investigate whether imprecise/unstable beliefs can be improved by experience or additional information. Our experiment implements ambiguity using devices similar to two-colour Ellsberg urns. Participants make decisions not just about ambiguous bets based on the colour of a randomly drawn ball (eliciting their matching probabilities), but also about ambiguous bets on the composition of the urn (eliciting their priors about the composition of the ambiguous urns). These data have the potential to shed light on various theoretical models of decision making under ambiguity (maxmin model, ï¡-maxmin model, variational model, prospect theory and smooth model). Our experimental design also allows us to test probability sophistication, elicit ambiguity attitudes, and derive imprecise intervals for subjective beliefs about uncertain events. We then examine the interaction effect of prior knowledge and new additional information on belief updating, with different prior knowledge of uncertain events given to decision makers in different experimental treatments, to compare how beliefs are updated under conditions of full ambiguity versus partial ambiguity.

**Zhihua Li**

#### Zooming in on Ambiguity Attitude: Rare Events Matter

Aysil Emirmahmutoglu; Aurelien Baillon

#### Abstract

Kahneman & Tversky (1979) pointed out that rare events are either completely neglected or overweighted. For decision making under risk, the common view in the literature is that small probability events are overweighted (e.g., Tversky & Kahneman, 1992; Gonzalez & Wu, 1999). However, the picture is not that clear when we consider decision making under uncertainty. Recent research in psychology has shown that if unlikely events are not described but experienced by agents, they tend to be partly neglected or underweighted (e.g., Barron & Erev, 2003; Hertwig et al, 2004; Hertwig & Erev, 2009). On the other hand, there is also evidence that rare events are overweighted, but less so under experience-based decisions than description-based when the events concern gains, and they are overweighted similarly in both decisions when they concern losses (Abdellaoui et al., 2011). Understanding the attitude towards rare events helps to explain certain economic activities. For example, policies to prevent or cope with environmental catastrophes concern such events with losses. When rare events relate to gains, overweighting can explain betting behavior while underweighting might hinder entrepreneurship. In this paper we measure ambiguity attitude for very unlikely events in an Ellsberg-like experiment. Despite the fact that decision making for ambiguous rare events is crucial, there are no other studies that empirically examine ambiguity attitude with events as unlikely as ours. This paper is the first study of ambiguity attitude for very unlikely events, more than 100 times smaller than in typical ambiguity studies. Additionally, the literature on ambiguity focuses mostly on gains, while there is little known about losses (Trautmann & van de Kuilen, 2014). We measure ambiguity attitude both for gains and losses, and therefore contribute to the existing literature by being one of the few studies that uncover the loss domain, providing opportunities for direct comparison with gains. We deal with two main challenges of measuring ambiguity attitude. The first one relates to identifying ambiguity attitude generated on top of risk attitude when we move from decision making under risk to decision making under uncertainty. We use the solution proposed by Dimmock et al. (2015), namely matching probabilities, to deal with this issue. The second challenge relates to controlling for beliefs, for which we compare matching probabilities with themselves and study their internal consistency, as Baillon et al. (2015) suggested. We do this by the use of additivity measures. We find that very unlikely events are overweighted and loom larger in isolation, more under ambiguity than under risk. This corresponds to ambiguity seeking for unlikely gains and ambiguity aversion for unlikely losses. Additionally, we observe more overweighting of unlikely losses than unlikely gains. Therefore, ambiguity aversion for catastrophes is stronger than ambiguity seeking for betting behavior.

**Aysil Emirmahmutoglu**

#### Precise Versus Imprecise Datasets: Revisiting Ambiguity Attitudes in the Ellsberg Paradox

Roxane Bricet

#### Abstract

The famous Ellsberg paradox (1961) was a first experimental attempt to illustrate the failure of the expected utility hypothesis to predict individual behavior in an uncertain environment. Consider two urns: one urn is filled with 50 red and 50 black balls while the other urn is filled with 100 balls in an unknown composition of black and red balls. Whether the bet is on black or on red, most people prefer betting on the known urn. This behavior provides evidence of ambiguity aversion. While the classic Ellsberg experiment has been replicated several times (Camerer and Weber, 1992), this research deals with the question of decision making in ambiguous situations when the information is provided in the form of datasets. Indeed, decision makers usually observe data generated by the process at hand and have to make a decision based on more or less precise data sets (Gilboa and Schmeidler, 2001). Therefore, we draw on the Ellsberg experiment with two urns and two colors of balls and we describe both urns by sets of data. We investigate the Ellsberg paradox in this context of “partial ambiguity”. For our purpose, we design a short experiment of simple binary questions on preferences over pairs of bags containing 200 balls (blue and red – in unknown proportions). Each bag is described by a dataset which can be either relatively precise (100 draws) or relatively imprecise (10 draws). The participant is asked to chose his preferred bag to bet on blue and to bet on red, when the datasets exhibit different but very similar frequencies. A frequentist (or a Bayesian with a prior 1/2, 1/2) would choose the bag with the highest frequency of desirable outcomes for the respective bet. In contrast, a pessimist (i.e. an ambiguity-averse subject) might prefer to bet on the bag with more draws regardless of the color of the ball, suggesting that the difference in frequencies does not compensate for the precision of the datasets. Similarly, subjects who choose the less precise bag for both bets exhibit an optimistic (i.e. ambiguity-loving) attitude. Such preferences are axiomatized in Eichberger and Guerdjikova, 2013. The slight difference between the frequencies of the two bags serves as a robustness test for these preferences. We ran 6 experimental sessions in the LEEP (Laboratoire d’Economie Experimentale de Paris). We recruited 91 participants and the average payment was above 13 euros for a 30 minutes experiment. Among answers satisfying monotonicity (87%), 2/3 of choices are ambiguity-neutral (frequentists and bayesians). The remaining 1/3 of answers contradict the expected utility hypothesis and can be interpreted as an expression of non-neutral ambiguity attitude. Among them, 2/3 displays ambiguity-aversion. We also calculate an individual score of pessimism. Although our results suggest a significant bias towards ambiguity-aversion, it is weaker than in the relevant experimental literature (Arad, Gayer, 2009 ; Baillon et al., 2013 ; Chew et al., 2013).

**Roxane Bricet**