Let A ≥ B mean "A is (weakly) preferred to B," and let A = B indicate indifference between A and B for alternatives A, B, C, etc. in set S. Further, let (ApB) represent a lottery in which A occurs with probability p and B occurs with probability 1 − p. For each of the patterns of choices given below, indicate which of the listed axioms of utility theory is violated. If none is violated (i.e., the decision is rational), select the option for "none".
a. (ApB) > (BpC)
b. B ≥ C Comparability
c. None; this is rational
d. Solvability
e. Independence
f. Transitivity