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In areas such as insurance pricing and government policy analysis, it is often necessary to assign human life a monetary amount in order to compare it with other monetary amounts. So economists have a measure called the statistical value of life, which in some sense quantifies how much a person values his own life. It's usually calculated to be about 10 million dollars for most people. Now this isn't literally the dollar amount a person puts on his life, because that amount is usually infinity; it's possible that no amount of money would convince the average person to give up his own life, and the average person would be willing to spend any amount of money to save his own life. So the technical definition is trickier: the statistical value of a person's life is the dollar amount $X$ such that for all probabilities $p$, or at least all values of $p$ relatively close to 0 the person would be indifferent between a situation where their chance of dying is $p$, and a situation where their chance of losing $X$ dollars is $p$. (An equivalent definition can be given in terms of reducing your chance of death and gaining money.)

My question is not about why this concept is useful; I understand its utility. (No pun intended.) My question is, why should the statistical value of life exist at all? That is to say, why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$?

Let's discuss this more formally. Let $A$ is the set of possible preferences, and let $G(A)$ be the set of "gambles" or "lotteries" over $A$. Then the von Neumann-Morgenstern theorem states that if a person's preference ordering over $G(A)$ satisfies certain rationality axioms, then the person's preferences can be represented by a utility function $u: A → ℝ$. That means that the value that a person puts on any lottery $L$ is the expected value of$u$ under the probability distribution of $L$.

So I would not be surprised at all if a person was indifferent between a 1 percent chance of getting 10 dollars and a 1 percent chance of getting a chocolate sundae, and was also indifferent between a 2 percent chance of getting 10 dollars and a 2 percent chance of getting a chocolate sundae; that would just indicate to me that the person's preferences satisfy the von Neumann-Morgenstern rationality axioms. But I don't understand why, if a person was indifferent between a 1 percent chance of losing 10 million dollars and a 1 percent chance of dying, they would necessarily also be indifferent between a 2% chance of losing 10 million dollars and a 2% chance of dying. That's because living and dying do not comport with von Neumann Morgenstern axioms; the average places the utility of survival at infinity, and yet they assign finite values to small risks of dying. So I see no reason why lotteries involving risks of living and dying should obey the von Neumann-Morgenstern axioms.

And yet empirically, it seems that studies have found that the statistical value of life is a well-defined and measurable quantity, at least for sufficiently small values of $p$. So what is the reason for this? What is the reason that lotteries involving small risks of dying obey the von Neumann-Morgenstern axioms, when living and dying themselves do not?

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    $\begingroup$ Do you have data or literature to back up the claim that human beings assign infinite utility to survival? $\endgroup$ – Alecos Papadopoulos Jun 24 '16 at 0:50
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    $\begingroup$ The difference between the 1% chance and 2% chance scenarios that you describe would be different for me because of risk aversion, not because I hold infinite value to my life. If I could sacrifice myself to save a certain number of people, I would definitely consider it. $\endgroup$ – Kitsune Cavalry Jun 24 '16 at 1:51
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    $\begingroup$ @KitsuneCavalry Concerning the 1% and 2% chance scenarios, risk aversion is complete irrelevant here; it is perfectly possible for someone to the risk-averse and still obey the von Neumann-Morgenstern rationality axioms; it just means that the shape of their utility function is concave. Risk aversion is about not valuing a bet at the expected dollar value of the bet, risk aversion is not about not valuing a bet at the expected utility of the bet. $\endgroup$ – Keshav Srinivasan Jun 24 '16 at 2:14
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    $\begingroup$ @KitsuneCavalry In any case, tell me this: Suppose you value a chocolate sundae at ten dollars. Then one of the vNM axioms states that for any x, you would be indifferent between an x% chance of getting a chocolate sundae and an x% chance of getting 10 dollars. Why is that? Because when comparing those two scenarios, there is a (100-x)% chance that nothing happens, and then there is an x% chance that you'll be given a choice between the chocolate sundae and ten dollars, which you'll be indifferent about. Do you agree with that reasoning? $\endgroup$ – Keshav Srinivasan Jun 24 '16 at 2:17
  • $\begingroup$ Maybe I'm being imprecise. People's ideas of risk influences them to violate VNM assumptions. See the Zeckhauser paradox. mindyourdecisions.com/blog/2014/07/14/… $\endgroup$ – Kitsune Cavalry Jun 24 '16 at 2:44
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You asked:

why should there exist a single value of $X$ that satisfies this definition for all values of $p$, or even all values of $p$ that are sufficiently close to $0$

There isn't such a value. I would hope that no one claims that there is.

The statistical value of life is a (somewhat lazy) calculation of convenience. Lots of business case protocols need a value for anything that's going into the business case. Changing the probabilities of survival is an outcome of many interventions for which the decision-makers have insisted on business cases, so some method is needed to value these probabilities.

One of the earliest ways to do this, back when relevant research was scarcer than it is today, and computational power was much more limited, was to assign a single value of life, which was calculated using methods that assumed a priori that there existed a single value of $X$ that was an adequate approximation for all values of $p$ that are sufficiently close to $0$.

That method is still used today largely due to institutional inertia.

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"What is the reason that lotteries involving small risks of dying obey the von Neumann-Morgenstern axioms, when living and dying themselves do not?"

I believe that living and dying do obey these axioms. The apparent discrepancy you've seen is because you are applying the biggest assumption of the statistical value of life inconsistently. (Kitsune Cavalry touched on this in a comment already.) That assumption is that human lives and money are interchangeable in terms of utility. Now let's look at your key objection:

It's possible that no amount of money would convince the average person to give up his own life, and the average person would be willing to spend any amount of money to save his own life.

Let's apply the money-lives conversion assumption completely:

It's possible that no amount of lives saved would convince the average person to give up their own life, and the average person would be willing to kill any number of people to save their own life.

Now we can see that this objection no longer holds (at least, I hope so). Therefore, living and dying do seem to obey von Neumann-Morgenstern axioms. They just don't if you try to restrict them to monetary terms on one side of the equation.

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