Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?

Is it true that any three of the four vNM axioms (of expected utility theory) can be satisfied without satisfying the fourth? Any examples which support such claim?

Basically I'd like to prove that the 3 axioms form an independent set of axioms, so I am seeking 4 examples of binary relations: (1) satisfying completeness, independence and continuity but not transitivity. (2) satisfying transitivity, independence and continuity but not completeness. (3) satisfying completeness, transitivity and independence but not continuity. (4) satisfying completeness, transitivity, continuity but not independence.

• Can be satisfied by what? Oct 11 at 12:51
• Basically I'd like to prove that the 3 axioms form an independent set of axioms, so I am seeking 4 examples of binary relations: (1) satisfying completeness, independence and continuity but not transitivity. (2) satisfying transitivity, independence and continuity but not completeness. (3) satisfying completeness, transitivity and independence but not continuity. (4) satisfying completeness, transitivity, continuity but not independence. Does it make more sense? Oct 11 at 13:19

1 Answer

(1) Satisfying completeness, independence, and continuity but not transitivity:

Take two outcomes, $$\{0,1\}$$, and the associated lottery space $$[0,1]$$. Consider the preference relation $$\succsim$$ where

• $$x\sim y$$ if and only if $$x=y$$, $$x=0$$ and $$y=1$$ or vice versa,
• $$x\succ y$$ if and only if $$x>y$$, except for $$x=0$$ and $$y=1$$ or vice versa.

Transitivity is violated because $$1\succ 0.5 \succ 0$$ but $$0\sim 1$$.

(2) Satisfying transitivity, independence, and continuity but not completeness:

Take the empty relation on any lottery space. This relation is clearly incomplete but the other axioms are vacuously satisfied.

(3) Satisfying completeness, transitivity, and independence but not continuity:

Take three outcomes, $$\{A,B,C\}$$, and lexicographic preferences over the associated lottery space. That is, the decision-maker strictly prefers lottery $$M$$ over lottery $$N$$ iff (1) $$P_M(A)>P_N(A)$$ or (2) $$P_M(A)=P_N(A)$$ and $$P_M(B)>P_N(B)$$.

To see why continuity is violated, consider lotteries $$(Pr(A),Pr(B),Pr(C))=(1,0,0)$$, $$(0,1,0)$$, and $$(0,0,1)$$.

(4) Satisfying completeness, transitivity, continuity but not independence: Take three outcomes, $$\{A,B,C\}$$. Consider the preference relation induced by the utility function $$u(Pr(A),Pr(B),Pr(C))=Pr(A)+2Pr(B)(1-2Pr(C)).$$

To see why Independence is violated, consider lotteries $$(1,0,0)$$ and $$(0,1,0)$$ versus $$(0.5,0,0.5)$$ and $$(0,0.5,0.5)$$.

• Condition (3) in point (3) is redundant, however. Oct 12 at 8:05
• Sure, thanks. Edited. Oct 12 at 9:16