Consider the following pairs of gambles:

A : 100% chance of $3,000 versus B :

80% chance of $4,000 20% chance of $0 C :

25% chance of $3,000 75% chance of $0 versus D :

20% chance of $4,000 80% chance of $0 .

(a) Show that an expected utility maximizer who prefers A to B must also prefer C to D.

(b) Show that the preferences A B and D C violate the independence axiom by showing that C = αA +(1− α)Q and D = αB+ (1−α)Q for some 0 <α< 1 and some gamble Q.

(c) Plot the gambles A, B, C, and D in the probability simplex of Figure 20.1, taking p1 to be the probability of $0 and p3 to be the probability of

$4,000. Show that the line connecting A with B and the line connecting C with D are parallel.

Note: The preferences A B and D C are common. This example is due to Allais (1953) and is a special case of the common ratio effect. See, for example, Starmer (2000).