# Contradicting the von Neumann–Morgenstern axioms

This is a story taken from Shmuel Zamir - Game theory

"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?

Never tell me the odds

Han Solo in Star Wars: The Empire Strikes Back

In most domains, people appear to be ambiguity averse (Binmore, Stewart, and Voorhoeve (2012), Ellsberg (1961)), 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) 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*.

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.

• 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? Commented Jul 27, 2022 at 21:44

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.

Let $$D_1, D_2$$ be lotteries. Assume $$D_1 \succ D_2$$, then the monotonicity axiom states

$$pD_1 + (1-p)D_2 \succ qD_1 + (1-q)D_2 \iff p > q$$