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 Ellsbergs 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 that gives the same prize with a winning probability lying 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 belongs to subintervals [p, q].

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. We consider objects of choice x[p,q]y where the decision maker knows that she will get x with a winning probability lying somewhere between p and q, and y otherwise. Additionally, we propose a model that postulates that decision makers evaluate ambiguous bets x[p,q]y by subjectively combining the values of envelope (extreme) lotteries xpy and xqy . 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,q]y is given by the convex combination of RDU values of the envelope lotteries.

We elicited this model in a laboratory experiment involving 62 subjects. All the components of the model are estimated at the individual level using discrete choice modeling. 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 intervals 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. This pattern receives the following psychological interpretation. Upper-probability weighting carries the possibility effect, whereas lower-probability weighting carries the certainty effect. Eventually, the convex combination of these two functions allows to recover the inverse-S shape probability weighting generally observed for risk. Therefore, our model not only explains ambiguity attitudes but also offers a new insight to understand the shape of probability weighting under risk.