Four studies with 3440 participants investigated four new paradoxes that refute rank-dependent utility and cumulative prospect theories of risky decision making. All four paradoxes can be interpreted as violations of coalescing, the assumption that branches leading to the same consequence can be combined by adding their probabilities. These studies explored if there is some format in which coalescing and cumulative prospect theory would be satisfied. Three variables were manipulated: probability format, branch splitting, and event-framing. Probability was displayed via text, pie charts, natural frequencies, and lists of equally likely consequences. Probability format and event framing had minimal effects, but branch splitting had large effects. In all 12 conditions of format and framing, splitting created majority violations of stochastic dominance and a second round of splitting reversed preferences, restoring majority satisfaction of stochastic dominance. Two cumulative independence properties were violated in 47 of 48 new tests. Birnbaum's TAX model, in which the relative weight of each probability-consequence branch depends on probability and rank of its consequence, correctly predicted the main trends. In this model, splitting the branch with the lowest consequence can make a gamble worse, and splitting the branch with the highest consequence can make a gamble better. (C) 2004 Elsevier Inc. All rights reserved.
