death as a cost in decision theory

Question

If I presented with an optional task for which I have an outcome independent investment $I$, a probability of success $P$ and a reward for success $R$, then I chose to undertake this task iff $PR > I$.

If I presented with an optional task for which I have an outcome independent investment $I$, a probability of success $P$, a reward for success $R$ and a cost of failure $C$, then I chose to undertake this task iff $PR - (1-P)C > I$.

What if the undertaking of this task has a risk of death? How does one begin to determine the cost? Is the cost of death infinite?

In a situation where there is a one in a million chance of death and a fifty fifty chance of getting £10,000,000, I don't know many people who wouldn't give that a go, but that would imply that the cost of death is finite or no mathematician would ever undertake the task.

I suspect the answer is that the problem required context. Let us suppose you leave the handbrake off the car and it rolls down a steep cliff. In this car is a big sack of lovely cash which you were taking the bank. The tide below is rising and you are certain that you have no time to get help so you are faced with the decision as to whether or not you should attempt to climb down to retrieve the cash. There is a chance you could fail (and lose the money) and there is a chance you could die.

How would you quantify death in context?

Answer 1

Something that the comments to your post have hinted at and that you do have to grapple with here, is that there is first and foremost an issue of units.

Your problem as it starts has an inequality in units of money. You want to consider death, which is not obviously in units of money. So, you have to cook up a conversion factor for how many dollars a life is worth. There is certainly no standard one. From some perspectives, it's easier. From an insurance perspective, for example, it's easier to define as there are losses only paid on death.

If you considered it more narrowly than that but more widely than yourself, you could consider the costs to your family and friends of your demise, which could give you a number, but it's not obviously reliable.

If you want to consider just yourself, however... That's a hard problem, and one that is not obviously solved by observing human behavior. There is one possible solution, though. If you consider the amount of money you could potentially make (over your lifetime, say) by staying alive for sure and not risking your life, you could use this as a cost. Some research indicates that this should not be discounted using the typical time value of money, as we seem instinctively to discount hyperbolically rather than exponentially (http://en.wikipedia.org/wiki/Hyperbolic_discounting).

Answer 2

Although @mwjohnson comment about the subjectivity of such an evaluation is totally valid, I take it that you are trying to examine the case in abstract.

So think of the following: if the "cost of death" is infinity (understood as negative), then the "reward of life" must be (plus) infinity. Is it? All experience up to our time says that human beings are in all aspects (including psychological and the like) bounded - so nothing related to human beings can be infinity. So the cost of death must be quantifiable. How?

Although we don't know how long are we going to live, an average expected life length does exist, and for various subcategories of a population. So the philosophical concept of "life" for our purposes reduces to "remaining years of being alive". Now one has some views about one's future. Imperfect, biased perhaps, inaccurate, uncertain, but existent. These views include a life style that to a large extent can be quantified in monetary terms: for example income generated during those "remaining years of being alive". Then this is the monetary value of the life one expects to live, call it $L$.

Is this what you are going to lose if you die before reaching your expected life length? No -because $L$ includes also the necessary resource absorption for mere biological survival which is a prerequisite to enjoy all the rest. I mean the absolute minimum of food, water, clothing, shelter, some expected costs for health care, for the human body to continue to function for the duration. Call the monetary value of these necessities $S$. Then $L-S$ is the monetary net value of your remaining life, and so it is the opportunity cost of death: the value you lose if you "choose" death. So $C= L-S$.

Obviously calculating $L$ and $S$ is not that simple (for example matters of discounting enter the picture as another answer mentions), but in principle, this is how one can approach the quantification of the "cost of death".

Answer 3

Consider that £10,000,000 can surely be used to prolong my life. Depending on your age, there's a one in a million chance to lose let's say 50 years, against the 99.9999% chance of gaining say one year. Now the choice is easy.

Answer 4

The generally accepted model for the cost of death or the value of saving a life (which is fundamentally equivalent) is QALY - i.e., instead of putting a fixed value on death, you measure the amount of life that is lost/gained.

This is generally used to compare different lives/deaths; but if you'd need to compare a [risk of] death against a monetary value, then you can get rather solid evaluations on how much QALY can be 'purchased' by investing in medicine and prevention in some country, or you can estimate a similar estimate for a specific individual.