death as a cost in decision theory

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?

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I might argue the risk of death is an individual paramater, that each person might decide the value of on their own. If I already have £10,000,000 I might not take the 1 in a million chance of death. If you're currently a soldier fighting in a war, you're daily life might be 1 in 1000, meaning you may take a 1 in 100 chance at 10,000,000. Each person would decide their own value of C. – mwjohnson Oct 1 '13 at 1:09
Let us suppose then that you are shipwrecked. Now the currency of life may be calories. With the limited resources you have, you might consider you could live two weeks and hope a ship or plane passes by which you could signal for rescue. Alternatively, you could risk swimming to the wreck which brings with it a probability of early death, but the potential benefit of, let's say, two months of supplies, giving a greater chance of rescue. Now we have removed the financial aspect, or at least made it a known quantity. – stevemarvell Oct 1 '13 at 1:31
Your equation and my suggestion still hold true. This only (slightly further) obfuscates the (potential true) value of the parameters. One would weigh the perceived chance at rescue - the reward versus the risk multiplied by the perceived cost. You perception of C might be based on your perceived ability to swim, whether or not you've seen sharks, there is a hurricane, etc. Your equation is a good one, I'll remember it if I'm ever stranded on an island. =) (1-P)C will always be based on the perception the individual has of their situation, thus being case by case. – mwjohnson Oct 1 '13 at 1:39
Thanks for the participation. Let us suppose that the probability of rescue is linear with time, the amount of supplies is 10 days and the amount of supplies on the ship is 100 days. Assuming that our ability to swim does not change and the weather is no affecting our decision, we might consider swimming on the last day of our current supplies. Let us further simplify by saying that we definitely will get the supplies if we don't die (let's say with a $\frac{1}{100}$ probability). Clearly, we definitely swim, since we're going to die anyway. This is the simple case of course. – stevemarvell Oct 1 '13 at 1:47

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).

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I rather suspect that the unit of life can be measured in at least days and that it may be better to convert all the other units to that rather than try to convert the days to the units of the task. – stevemarvell Oct 1 '13 at 1:59
I have read the linked (fascinating) article and am further driven to write a system to determine what should be done rather than model what is done. – stevemarvell Oct 1 '13 at 2:05
Well, the issue is that you have it in dollars, so now you have a conversion from dollars to time, which is easily just as artificial. There's really no skirting around the issue that the pieces are incommensurate. I'd be interested to see what you find though. – Vernepator Cur Oct 1 '13 at 13:27
I'm planning to take financial gain out of the equation to begin with since it's causing a units problem as mentioned. Instead, I'm going to concentrate on the shipwreck context. – stevemarvell Oct 1 '13 at 13:49

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".

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Your approach to this answer related to one of my thoughts over the survival oriented version of the question. Life is bounded anyway, so one might take into consideration the length and possibly quality of life remaining after potential rescue. – stevemarvell Oct 1 '13 at 1:52
And this is exactly what is widely used in Health Care in order to assess the effectiveness of different medical treatments - the QALY's ("quality adjusted life years" - that the patient will have under the one or the other treatment). – Alecos Papadopoulos Oct 1 '13 at 2:02
All of a sudden I can see the 80 year old homeless guy with TB saying "you know what, this island isn't so bad after all, even if it might only be for a couple of weeks". Thanks for the term, I'll be using that. – stevemarvell Oct 1 '13 at 2:08
In addition, I note that the survival case is likely to be related to the "Standard Gamble" aspect of decision theory with respect to medical intervention. – stevemarvell Oct 1 '13 at 2:15

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.

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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.

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