Dutch Book Theorem problem with probability and Von Neumann Morgenstern axiom

- Suppose Ann assigns the following probabilities:

P(A) = 0.25P(B) = 0.25P(A ∪ B) = 0.5P(A ∩ B) = 0.1

Construct a Dutch book against Ann. Show that no matter what happens Ann willlose money.

- Consider Julie who is very afraid of taking gambles. Julie uses minimax to comparetwo different gambles. She starts with a preference relation ≻ over the prizes in Z, andthen compares two different lotteries using this preference relation. She looks at theworst prize (according to ≻) that has non-zero probability in each lottery and comparesthem. If one lottery, p, gives a better worst prize than another, q, she chooses p. If theyare tied she looks at the second worst prize (according to ≻)—she uses lexicographicminimax. Does Julie violate any of the axioms of von Neumann-Morgenstern? If so,which ones? Illustrate any violations that you might find.