Question

Why are violations of stochastic dominance considered irrational behaviour rather than taste of risk? How can violations of stochastic dominance be observed in subjective probability weighting curves?

Answer 1

the results of experiments designed to test whether individuals and groups abide by monotonicity with respect to first-order stochastic dominance and Bayesian updating, when making decisions under risk. The results indicate a significant number of violations of both principles. The violation rate when groups make decisions is substantially lower, and decreasing with group size, suggesting that social interaction improves the decision-making process. Greater transparency of the decvision task reduce the violation rate, suggesting that these violations are due to judgment errors rather than the preference structure. In one treatment, however, less complex decisions result in higher error rate.

Violations of stochastic dominance can only be
avoided if the subjective probability weighting depends in a quite specific manner on the entire
vector of probabilities for a prospect. Assume that the probabilities of some prospect are sorted
from most desirable to least desirable according to the values of their corresponding outcomes(x1 ≥ x2 ≥ x3 ≥ · · ·). Next, define the cumulative probability, or rank [28], as
θi =
X
i
j=1
pj , (12)
where θ0 = 0. θi
is the probability of receiving an outcome that is as good or better than i. The
probability weighting on outcome i is given by
ρi = w(θi) − w(θi−1), (13)
where w is the weighting function defined on ranks. w is used rather than W to emphasize that
w is defined on probability ranks, whereas W is defined on probabilities. w maps the interval
[0, 1] onto itself and must be an increasing function of θ. The rank dependent expected utility
is
U =
X
i
ρiu(xi).