1. 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.




  2. 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.