Defining Ambiguity and Ambiguity Attitude

According to the well-known distinction attributed to Knight (1921), there are two kinds of uncertainty. The first, called “risk,” corresponds to situations in which all events relevant to decision making are associated with obvious probability assignments (which every decision maker agrees to). The second, called “(Knightian) uncertainty” or (following Ellsberg (1961)) “ambiguity,” corresponds to situations in which some events do not have an obvious, unanimously agreeable, probability assignment. As Chapter 1 makes clear, this collection focuses on the issues related to decision making under ambiguity. In this Chapter, I briefly discuss the issue of the formal definition of ambiguity and ambiguity attitude. In his seminal paper on the CEU model (1989), Schmeidler proposed a behavioral definition of ambiguity aversion, showing that it is represented mathematically by the convexity of the decision maker’s capacity v. The property he proposed can be understood by means of the example of the two coins used in Chapter 1. Assume that the decision maker places bets that depend on the result of two coin flips, the first of a coin that she is very familiar with, the second of a coin provided by somebody else. Given that she is not familiar with the second coin, it is possible that she would consider“ambiguous” all the bets whose payoff depends on the result of the second flip. (For instance, a bet that pays $1 if the second coin lands with heads up, or equivalently if the event {HH, TH} obtains.) If she is averse to ambiguity, she may therefore see such bets as somewhat less desirable that bets that are “unambiguous,” i.e., only depend on the result of the first flip. (For instance, a bet that pays $1 if the first coin lands with heads up, or equivalently if the event {HH, HT} obtains.) However, suppose that we give the decision maker the possibility of buying shares of each bet. Then, if she is offered a bet that pays $0.50 on {HH} and $0.50 on {HT}, she may prefer it to either of the two ambiguous bets. In fact, such a bet has the same contingent payoffs as a bet which pays $0.50 if the first coin lands with heads up, which is unambiguous. That is, a decision maker who is averse to ambiguity may prefer the equal-probability “mixture” of two ambiguous acts to either of the acts. In contrast, a decision maker who is attracted to ambiguity may prefer to choose one of the ambiguous acts. Formally, Schmeidler called ambiguity averse a decision maker who prefers the even mixture1 (1/2)f+(1/2)g of two acts that she finds indifferent to either of the two acts. That is, (1/2)f+ ∗Dipartimento di Matematica Applicata and ICER, Università di Torino. Recall that Schmeidler used the Anscombe-Aumann setting, in which mixtures of acts can be defined stateby-state. Also, he used the term “uncertainty” averse rather than ambiguity averse.