<p><strong>(1) Satisfying completeness, independence, and continuity but not transitivity:</strong></p>
<p>Take two outcomes, <span class="math-container">$\{0,1\}$</span>, and the associated lottery space <span class="math-container">$[0,1]$</span>. Consider the preference relation <span class="math-container">$\succsim$</span> where</p>
<ul>
<li><span class="math-container">$x\sim y$</span> if and only if <span class="math-container">$x=y$</span>, <span class="math-container">$x=0$</span> and <span class="math-container">$y=1$</span> or vice versa,</li>
<li><span class="math-container">$x\succ y$</span> if and only if <span class="math-container">$x&gt;y$</span>, except for <span class="math-container">$x=0$</span> and <span class="math-container">$y=1$</span> or vice versa.</li>
</ul>
<p>Transitivity is violated because <span class="math-container">$1\succ 0.5 \succ 0$</span> but <span class="math-container">$0\sim 1$</span>.</p>
<p><strong>(2) Satisfying transitivity, independence, and continuity but not completeness:</strong></p>
<p>Take the empty relation on any lottery space. This relation is clearly incomplete but the other axioms are vacuously satisfied.</p>
<p><strong>(3) Satisfying completeness, transitivity, and independence but not continuity:</strong></p>
<p>Take three outcomes, <span class="math-container">$\{A,B,C\}$</span>, and lexicographic preferences over the associated lottery space. That is, the decision-maker strictly prefers lottery <span class="math-container">$M$</span> over lottery <span class="math-container">$N$</span> iff (1) <span class="math-container">$P_M(A)&gt;P_N(A)$</span> or (2) <span class="math-container">$P_M(A)=P_N(A)$</span> and <span class="math-container">$P_M(B)&gt;P_N(B)$</span>.</p>
<p>To see why continuity is violated, consider lotteries <span class="math-container">$(Pr(A),Pr(B),Pr(C))=(1,0,0)$</span>, <span class="math-container">$(0,1,0)$</span>, and <span class="math-container">$(0,0,1)$</span>.</p>
<p><strong>(4) Satisfying completeness, transitivity, continuity but not independence:</strong>
Take three outcomes, <span class="math-container">$\{A,B,C\}$</span>. Consider the preference relation induced by the utility function
<span class="math-container">$$u(Pr(A),Pr(B),Pr(C))=Pr(A)+2Pr(B)(1-2Pr(C)).$$</span></p>
<p>To see why Independence is violated, consider lotteries <span class="math-container">$(1,0,0)$</span> and <span class="math-container">$(0,1,0)$</span> versus <span class="math-container">$(0.5,0,0.5)$</span> and <span class="math-container">$(0,0.5,0.5)$</span>.</p>

CC BY-SA 4.0
2021-10-12T08:05:39.510Z
Condition (3) in point (3) is redundant, however.
VARulle
CC BY-SA 4.0
2021-10-11T21:30:06.797Z
2021-10-12T09:17:00.753Z
2021-10-12T09:17:00.753Z
axelniemeyer
<p>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?</p>
<p>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.</p>

CC BY-SA 4.0
2021-10-11T12:51:29.510Z
Can be satisfied by what?
Herr K.
CC BY-SA 4.0
2021-10-11T13:19:38.623Z
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?
anonymous
<microeconomics><game-theory><utility><preferences><decision-theory>
Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?
anonymous
# Can any three of the four vNM axioms (of expected utility theory) be satisfied without satisfying the fourth?

(Asked by: username_0 on 2021-10-11)

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.

**Tags:** microeconomics, game-theory, utility, preferences, decision-theory

> **Comments:**
> **username_1**: Can be satisfied by what?  
> **username_0**: 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?  

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## 1 answers

### Answer by username_2 on 2021-10-11. Score: 5

**(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 $(0,0.5,0.5)$.

> **Comments:**
> **username_3**: Condition (3) in point (3) is redundant, however.  

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