Question 1: Prove/show that the Allais paradox violates EU theory regardless of whether preferences are risk-averse, risk-neutral, or risk-preferring.
In Section IIV from the class notes, the following two gambles were presented where people take the second choice in the first gamble and the first choice in the second gamble:
Gamble 1: $27,500 w/p=.33, $24,000 w/p=.66, $0 w/p=.01 or $24,000 w/p=1
Gamble 2: $27,500 w/p=.33, $0 w/p=.67 or $24,000 w/p=.34, $0 w/p=.66
Converting to expected utility terms:
Gamble 1: \(.33(EU($27500))+.66(EU($24000))+.01(EU($0)) <1(EU($24000))\)
Gamble 2: \(.33(EU($27500))+.67(EU($0))>.34(EU($24000))+.66(EU(0))\)
Reducing Gamble 1:
\(.33(EU($27500))+.01(EU($0)) <1(EU($24000))-.66(EU($24000))\)
\(.33(EU($27500))+.01(EU($0)) <(EU($24000))(1-.66)\)
\(.33(EU($27500))+.01(EU($0)) <.34(EU($24000))\)
Reducing Gamble 2:
\(.33(EU($27500))+.67(EU($0))-.66(EU($0))>.34(EU($24000))\)
\(.33(EU($27500))+(.67-.66)(EU($0))>.34(EU($24000))\)
\(.33(EU($27500))+.01(EU($0))>.34(EU($24000))\)
The result is that Gamble 1 and Gamble 2 contradict each other showing the Allais paradox violates Expected Utility theory.