# Ambiguity aversion

In decision theory and economics, ambiguity aversion (also known as uncertainty aversion) describes a preference for known risks over unknown risks. An ambiguity averse individual would rather choose an alternative where the probability distribution of the outcomes is known over one where the probabilities are unknown. This behavior was first introduced through the Ellsberg paradox (people prefer to bet on the outcome of an urn with 50 red and 50 blue balls to on one with 100 total balls but for which the number of blue or red balls is unknown).

There are a number of choices involving uncertainty and normally they can be classified in two categories: risky and ambiguous events. Risky events have a certain probability distribution over outcome while ambiguous events have some uncertainty over said probability distribution. The reaction is behavioral and still being formalized. Ambiguity aversion can be used to explain incomplete contracts, volatility in stock markets, and selective abstention in elections (Ghirardato & Marinacci, 2001).

The concept is expressed in the English proverb: "Better the devil you know than the devil you don't".

## Ambiguity aversion vs. risk aversion

The distinction between ambiguity aversion and risk aversion is important but subtle. Risk aversion comes from a situation where a probability can be assigned to each possible outcome of a situation and it is defined by the preference between a risky alternative and its expected value. Ambiguity aversion applies to a situation when the probabilities of outcomes are unknown (Epstein 1999) and it is defined through the preference between risky and ambiguous alternatives, after controlling for preferences over risk.

Using the traditional two-urn Ellsberg choice, urn A contains 50 red balls and 50 blue balls while urn B contains 100 total balls (either red or blue) but the number of each is unknown. An individual that prefers a certain payoff strictly smaller than \$10 over a bet that pays \$20 if the color of a ball drawn from urn a is guessed correctly and \$0 otherwise is said to be risk averse but nothing can be said about her preferences over ambiguity. On the other hand, an individual that strictly prefers that same bet if the ball is drawn from urn A over the case where the ball is drawn from urn B is said to be ambiguity averse but not necessarily risk averse.

A real world consequence of increased ambiguity aversion is the increased demand for insurance because the general public are averse to the unknown events that will affect their lives and property (Alary, Treich, and Gollier 2010).

## Causes of ambiguity aversion

Unlike risk aversion, which is primarily attributed to decreasing marginal utility, there is no widely accepted main cause for ambiguity aversion. The many possible explanations include different choice mechanisms, behavioral biases and differential treatment of compound lotteries; this in turn explains the lack of a widespread measure of ambiguity aversion.

### Maxmin Expected Utility

In their 1989 paper, Gilboa and Schmeidler[1] propose an axiomatic representation of preferences that rationalizes ambiguity aversion. An individual that behaves according to these axioms would act as if having multiple prior subjective probability distributions over the set of outcomes and chooses the alternative that maximizes the minimum expected utility over these distributions. In the Ellsberg example, if an individual has a set of subjective prior probabilities of a ball drawn from urn B being red ranging from, for example, 0.4 and 0.6 and applies a maxmin choice rule, she will strictly prefer a bet on urn A over a bet on urn B since the expected utility she assigns to urn A is greater than the one she assigns to urn B.

### Choquet expected utility

David Schmeidler[2] also developed the Choquet expected utility model. It's axiomatization allows for non-additive probabilities and the expected utility of an act is defined using a Choquet integral. This representation also rationalizes ambiguity aversion and has the Maxmin expected utility as a particular case

### Compound lotteries

In Halevy (2007)[3] the experimental results show that ambiguity aversion is correlated with violations of the Reduction of Compound Lotteries axiom (ROCL). This suggests that the effects attributed to ambiguity aversion may be partially explained by an incapacity of reducing compound lotteries to their corresponding simple lottery or some behavioral violation of this axiom.

## Gender difference in ambiguity aversion

Women are more risk averse than men. One potential explanation for gender differences is that risk and ambiguity are related to cognitive and noncognitive traits on which men and women differ. Women initially respond to ambiguity much more favorably than men, but as ambiguity increases, men and women show similar marginal valuations of ambiguity. Psychological traits are strongly associated with risk but not to ambiguity. Adjusting for psychological traits explains why a gender difference exists within risk aversion and why these differences are not a part of ambiguity aversion. Since psychological measures are related to risk but not to ambiguity, risk aversion and ambiguity aversion are distinct traits because they depend on different variables (Borghans, Golsteyn, Heckman, Meijers, 2009.)

## A framework that allows for ambiguity preferences

Smooth ambiguity preferences are represented as:

• s ∈ S set of contingencies or states
• πθ is a probability distribution over S
• f is an “act” yielding state contingent payoffs f (s)
• u is a von Neumann-Morgenstern utility function and represents risk attitude
• φ maps expected utilities and represents ambiguity attitude
• Ambiguity attitude is summarized using measure similar to absolute risk aversion, only absolute ambiguity aversion:
• μ is a subjective probability over θ ∈ Θ; Represents the ambiguous belief – it summarizes the decision-maker’s subjective uncertainty about the “true” πθ, probability distribution over contingencies. (Collar, 2008)

## An examination of ambiguity aversion: Are two heads better than one?

• Ambiguity aversion has been widely observed in individuals' judgments. The experiment examines risky and cautious shifts from individuals' original judgments to their judgments when they are paired up in dyads.
• In the experiment the participants were first asked to specify individually their willingness-to-pay for six monetary gambles. They were then paired at random into dyads, and were asked to specify their willingness-to-pay amount for the same gambles. The dyad's willingness-to-pay amount was to be shared equally by the two individuals. Of the six gambles in our experiment, one involved no ambiguity and the remaining five involved different degrees of ambiguity. It is found that dyads exhibited risk aversion as well as ambiguity aversion. The majority of the dyads exhibited a cautious shift in the face of ambiguity, stating a smaller willingness-to-pay than the two individuals' average.
• People pay less under ambiguous situations when compared to a corresponding unambiguous situation (Scenario 2 vs. Scenario 1). Similarly, they pay less for more ambiguous situations as compared to less ambiguous situations (Scenarios 4 vs. 3 and 6 vs. 5).

## Ambiguity aversion in real options

Real option valuation has traditionally been concerned with investment under project value uncertainty while assuming the agent has perfect confidence in a specific model.[4] The classical model of McDonald and Siegel developed quantitative methods used to analyze the options. They investigate the problem from the approach of derivative pricing and assign the value of the option to invest as The expected value is taken under an appropriate risk-adjusted measure, I is the cost of investing in the project, Pt is the value of the project at time t and T denotes the family of allowed stopping times in [0; T ]. In the European case, the agent may invest in the project only at maturity, in the Bermudan case, the agent may invest at a set of specific times (e.g. monthly), and in the American case, the agent may invest at any time. As such, the problem is in general a free boundary problem in which the optimal strategy is computed simultaneously with the option's value. (Jaimungal)

Note that it is not the same as risk aversion since it is a rejection of types of risk based in part on measures of their certainty, not solely on their magnitude.