Consider four gambles: A,B, C, D. If $AB$ and $CD$ and my preferences satisfy the independence axiom, then what must be true for any $p \in[0,1]$ ?
A) $p A+(1-p) Bp A+(1-p) C$
B) $AC$
C) $p C+(1-p) Bp D+(1-p) B$
D) $p C+(1-p) B(1-p) D+p B$

Question

Consider four gambles: A,B, C, D. If $A>B$ and $C>D$ and my preferences satisfy the independence axiom, then what must be true for any $p \in[0,1]$ ?
A) $p A+(1-p) B>p A+(1-p) C$
B) $A>C$
C) $p C+(1-p) B>p D+(1-p) B$
D) $p C+(1-p) B>(1-p) D+p B$

Quizplus · Accepted Answer

The independence axiom of expected utility theory states that if a person prefers option A over B, they will also prefer a lottery that offers A with probability p and some other outcome with probability 1-p over a lottery that offers B with the same probability p and the same other outcome with probability 1-p. Given $A>B$ and $C>D$, the correct application of the independence axiom is shown in option C: $p C+(1-p) B>p D+(1-p) B$. This is because if $C>D$, then a combination of C with any other gamble B should be preferred over the same combination of D with B, regardless of the probability p.

Consider Four Gambles: A,B, C, D A>BA>BA>B And C>DC>DC>D And My Preferences Satisfy the Independence Axiom, Then What Must

Consider Four Gambles: A,B, C, D $A>B$ And $C>D$ And My Preferences Satisfy the Independence Axiom, Then What Must