<p>Define outcomes <span class="math-container">$\Omega = \{ live, die \}$</span>, and the original distribution <span class="math-container">$D_1 = \frac{1}{2}live + \frac{1}{2}die$</span>. Under the new rule we now have <span class="math-container">$D_2 = \frac{3}{4}live + \frac{1}{4}die$</span>. If we assume <span class="math-container">$live \succ die$</span>, then by monotonicity property of the Axioms we should have <span class="math-container">$D_2 \succ D_1$</span> as <span class="math-container">$\frac{3}{4} &gt; \frac{1}{2}$</span>. But this is not the case so, the preferences do not satisfy the Von Neumann and Morgenstern’s axioms.</p>
<p><strong>Monotonicity</strong> <a href="https://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html#:%7E:text=Monotonicity%3A%20This%20big%20ugly%20word,in%20the%20gambles%20remain%20unchanged" rel="nofollow noreferrer">https://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html#:~:text=Monotonicity%3A%20This%20big%20ugly%20word,in%20the%20gambles%20remain%20unchanged</a>.</p>
<p>Let <span class="math-container">$D_1, D_2$</span> be lotteries. Assume <span class="math-container">$D_1 \succ D_2$</span>, then the monotonicity axiom states</p>
<p><span class="math-container">$$ pD_1 + (1-p)D_2 \succ qD_1 + (1-q)D_2 \iff p &gt; q$$</span></p>

CC BY-SA 4.0
2023-10-16T23:21:18.923Z
2023-10-16T23:28:37.323Z
2023-10-16T23:28:37.323Z
Ari
<blockquote>
<p>Never tell me the odds</p>
</blockquote>
<p>Han Solo in Star Wars: The Empire Strikes Back</p>
<p>In most domains, people appear to be <a href="https://www.behavioraleconomics.com/resources/mini-encyclopedia-of-be/ambiguity-uncertainty-aversion/#:%7E:text=Ambiguity%20aversion%2C%20or%20uncertainty%20aversion,known%20risks%20over%20unknown%20risks." rel="nofollow noreferrer">ambiguity averse</a> (<a href="https://link.springer.com/article/10.1007/s11166-012-9155-3" rel="nofollow noreferrer">Binmore, Stewart, and Voorhoeve (2012)</a>, <a href="https://www.jstor.org/stable/1884324#metadata_info_tab_contents" rel="nofollow noreferrer">Ellsberg (1961)</a>), meaning roughly that people would prefer a lottery where the probabilities are known to one where they are unknown, holding fixed the risk.</p>
<p>This story, like that of Han Solo, is a tale of ambiguity loving behavior (ambiguity loving is to ambiguity aversion as risk loving is to risk aversion). The bomber pilots preferring the uncertain (but higher) probabilities of death over multiple runs to the certain (but much lower) probability of death with the one-way bomb runs seems like a classic example of ambiguity loving behavior.</p>
<p>Anything other than ambiguity neutrality is a violation of the assumptions of expected utility theory. In particular, Ellsberg's thought experiments showing ambiguity aversion are violations of Savage's P2 and p4* axioms of as <a href="https://faculty.wcas.northwestern.edu/msi661/Ambiguity-06-29-2013.pdf" rel="nofollow noreferrer">Machina and Siniscalchi (2013)</a> note:</p>
<blockquote>
<p>While most Ellsberg urn examples illustrate violations of both P2 and
P4*, not all do, and it is departures from probabilistic
sophistication (i.e., violations of P4*) which constitute the
phenomena of ambiguity aversion or ambiguity preference.</p>
</blockquote>
<p>So too, I expect that most ambiguity loving behavior is a violation of P2 and p4*.
<a href="https://i.sstatic.net/pP4mA.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/pP4mA.png" alt="P2 axiom" /></a>
<a href="https://i.sstatic.net/6OPBz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/6OPBz.png" alt="p4 star" /></a></p>
<p>Also, for what it is worth, it isn't clear to me that the utility of being dead is a well specified object. You don't exist, so what value should your utility function take?</p>
<p>Update:
I don't have a Von Neumann–Morgenstern related citation on violated axioms, but here is my best guess. Both the Continuity and Independence axioms turn on the existence of a probability existing to establish a preference ordering over lotteries. But Ellsberg showed in his paradox that in cases of ambiguity aversion, no probability beliefs can exist that would rationalize the observed behavior. If that probability doesn't exist, then one or more of the continuity and independence axioms are likely violated.</p>
<p><a href="https://i.sstatic.net/yGGi4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yGGi4.png" alt="VNM Axioms" /></a>
Source: <a href="https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem" rel="nofollow noreferrer">Wikipedia Von Neumann–Morgenstern utility theorem</a></p>

CC BY-SA 4.0
2022-07-27T21:44:22.307Z
Thank you for your answer. In the book he mentions that the Von Neumann–Morgenstern axioms might be violated, continuity and independence axioms.
Do you understand how this might relate to breaking one of them?
Jorge González
CC BY-SA 4.0
2022-07-27T21:24:46.467Z
2022-07-28T00:26:19.417Z
2022-07-28T00:26:19.417Z
BKay
<p>This is a story taken from
<a href="https://www.fpbooks.space/books/35/35-game-theory-(www.tawcer.com).pdf" rel="noreferrer">Shmuel Zamir - Game theory</a></p>
<p>&quot;This is a true event that occurred during the Second World War on
the Pacific front,5 seems to contradict utility theory.</p>
<p>A United States bomber squadron,
charged with bombing Tokyo, was based on the island of Saipan, 3000 kilometers from
the bombers’ targets. Given the vast distance the bombers had to cover, they flew without
fighter-plane accompaniment and carried few bombs, in order to cut down on fuel consumption. Each pilot was scheduled to rotate back to the United States after 30 successful
bombing runs, but Japanese air defenses were so efficient that only half the pilots sent on
the missions managed to survive 30 bombing runs.</p>
<p>Experts in operations research calculated a way to raise the odds of overall pilot
survival by increasing the bomb load carried by each plane – at the cost of placing
only enough fuel in each plane to travel in one direction. The calculations indicated
that increasing the number of bombs per plane would significantly reduce the number
of required bombing runs, enabling three-quarters of the pilots to be rotated back to the
United States immediately, without requiring them to undertake any more missions. The
remaining pilots, however, would face certain death, since they would have no way of
returning to base after dropping their bombs over Tokyo.</p>
<p>Every single pilot rejected the suggested lottery outright. They all preferred their existing
situation. &quot;</p>
<p>Were they contradicting the von Neumann–Morgenstern axioms? Specifically the continuous and independence axioms?</p>

<game-theory><utility>
Contradicting the von Neumann–Morgenstern axioms
Jorge González
# Contradicting the von Neumann–Morgenstern axioms

(Asked by: username_0 on 2022-07-27)

This is a story taken from
[Shmuel Zamir - Game theory](https://www.fpbooks.space/books/35/35-game-theory-(www.tawcer.com).pdf)

"This is a true event that occurred during the Second World War on
the Pacific front,5 seems to contradict utility theory.

A United States bomber squadron,
charged with bombing Tokyo, was based on the island of Saipan, 3000 kilometers from
the bombers’ targets. Given the vast distance the bombers had to cover, they flew without
fighter-plane accompaniment and carried few bombs, in order to cut down on fuel consumption. Each pilot was scheduled to rotate back to the United States after 30 successful
bombing runs, but Japanese air defenses were so efficient that only half the pilots sent on
the missions managed to survive 30 bombing runs.

Experts in operations research calculated a way to raise the odds of overall pilot
survival by increasing the bomb load carried by each plane – at the cost of placing
only enough fuel in each plane to travel in one direction. The calculations indicated
that increasing the number of bombs per plane would significantly reduce the number
of required bombing runs, enabling three-quarters of the pilots to be rotated back to the
United States immediately, without requiring them to undertake any more missions. The
remaining pilots, however, would face certain death, since they would have no way of
returning to base after dropping their bombs over Tokyo.

Every single pilot rejected the suggested lottery outright. They all preferred their existing
situation. "

Were they contradicting the von Neumann–Morgenstern axioms? Specifically the continuous and independence axioms?

**Tags:** game-theory, utility

---

## 2 answers

### Answer by username_2 on 2022-07-27. Score: 3 | Accepted answer

> Never tell me the odds

Han Solo in Star Wars: The Empire Strikes Back

In most domains, people appear to be [ambiguity averse](https://www.behavioraleconomics.com/resources/mini-encyclopedia-of-be/ambiguity-uncertainty-aversion/#:%7E:text=Ambiguity%20aversion%2C%20or%20uncertainty%20aversion,known%20risks%20over%20unknown%20risks.) ([Binmore, Stewart, and Voorhoeve (2012)](https://link.springer.com/article/10.1007/s11166-012-9155-3), [Ellsberg (1961)](https://www.jstor.org/stable/1884324#metadata_info_tab_contents)), meaning roughly that people would prefer a lottery where the probabilities are known to one where they are unknown, holding fixed the risk.

This story, like that of Han Solo, is a tale of ambiguity loving behavior (ambiguity loving is to ambiguity aversion as risk loving is to risk aversion). The bomber pilots preferring the uncertain (but higher) probabilities of death over multiple runs to the certain (but much lower) probability of death with the one-way bomb runs seems like a classic example of ambiguity loving behavior.

Anything other than ambiguity neutrality is a violation of the assumptions of expected utility theory. In particular, Ellsberg's thought experiments showing ambiguity aversion are violations of Savage's P2 and p4\* axioms of as [Machina and Siniscalchi (2013)](https://faculty.wcas.northwestern.edu/msi661/Ambiguity-06-29-2013.pdf) note:

> While most Ellsberg urn examples illustrate violations of both P2 and
> P4\*, not all do, and it is departures from probabilistic
> sophistication (i.e., violations of P4\*) which constitute the
> phenomena of ambiguity aversion or ambiguity preference.

So too, I expect that most ambiguity loving behavior is a violation of P2 and p4\*.
[![P2 axiom](https://i.sstatic.net/pP4mA.png)](https://i.sstatic.net/pP4mA.png)
[![p4 star](https://i.sstatic.net/6OPBz.png)](https://i.sstatic.net/6OPBz.png)

Also, for what it is worth, it isn't clear to me that the utility of being dead is a well specified object. You don't exist, so what value should your utility function take?

Update:
I don't have a Von Neumann–Morgenstern related citation on violated axioms, but here is my best guess. Both the Continuity and Independence axioms turn on the existence of a probability existing to establish a preference ordering over lotteries. But Ellsberg showed in his paradox that in cases of ambiguity aversion, no probability beliefs can exist that would rationalize the observed behavior. If that probability doesn't exist, then one or more of the continuity and independence axioms are likely violated.

[![VNM Axioms](https://i.sstatic.net/yGGi4.png)](https://i.sstatic.net/yGGi4.png)
Source: [Wikipedia Von Neumann–Morgenstern utility theorem](https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem)

> **Comments:**
> **username_0**: Thank you for your answer. In the book he mentions that the Von Neumann–Morgenstern axioms might be violated, continuity and independence axioms. Do you understand how this might relate to breaking one of them?  

---

### Answer by username_1 on 2023-10-16. Score: 1

Define outcomes $\Omega = \{ live, die \}$, and the original distribution $D\_1 = \frac{1}{2}live + \frac{1}{2}die$. Under the new rule we now have $D\_2 = \frac{3}{4}live + \frac{1}{4}die$. If we assume $live \succ die$, then by monotonicity property of the Axioms we should have $D\_2 \succ D\_1$ as $\frac{3}{4} > \frac{1}{2}$. But this is not the case so, the preferences do not satisfy the Von Neumann and Morgenstern’s axioms.

**Monotonicity** [https://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html#:~:text=Monotonicity%3A%20This%20big%20ugly%20word,in%20the%20gambles%20remain%20unchanged](https://www.econport.org/content/handbook/decisions-uncertainty/basic/von.html#:%7E:text=Monotonicity%3A%20This%20big%20ugly%20word,in%20the%20gambles%20remain%20unchanged).

Let $D\_1, D\_2$ be lotteries. Assume $D\_1 \succ D\_2$, then the monotonicity axiom states

$$ pD\_1 + (1-p)D\_2 \succ qD\_1 + (1-q)D\_2 \iff p > q$$


---

