Santtu may be an exception but people are generally only risk neutral until the risk of death becomes high enough (say, 1/10,000). Then they become risk averse, and all bets are off, so to speak. (Well, not literally, but it stops being linear.)
I didn't say that I didn't say it. I said that I didn't say that I said it. I want to make that very clear.
I got surprised by how little the money would matter when I studied it.
If $4M of additional money would make my life twice as good, I could indeed gamble at a rate of $6M/100% of death for a gain of $2M (and 50% increase in total life quality) and a chance of death of 1/3. This would balance out to the same expected total life quality as without gambling.
The maximum expected life quality would be had with $1M and 1/6 chance of death. But this would only give
expected life quality
= chance to live * life quality IF alive
= (5/6)*(5/4)
= 25/24
= 1.0416..
or only 4.16% more than what I'd had to begin with.
What an insignificant improvement - a disappointment, really! This outcome is the weighted average of the value of life when dead (0) and that when winning the gamble (1.25), with the probabilities 1/6 and 5/6 to each outcome.
I would substitute the term "incentive" and see what people choose. Here I've rearranged your opening sentence a bit, but you could work on it some more if it's not clear:
"What is the least incentive that you would require to take one in one billion chance of dying?"
I just get frustrated when I feel I can't participate in an interesting question. It's not easy being a recluse!
I didn't say that I didn't say it. I said that I didn't say that I said it. I want to make that very clear.
You can calculate your own optimal stake, and expected profit.
Lets assume that you can gamble your life at a rate of X$/ 1% chance of death, and you need Y$ to get an improvement of 1% in life quality.
Then you must have Y>X for any gamble to be profitable. Lets denote the y/x ratio by r. We can then calculate the expected payoff as a function of the ratio of life gambled, g, as
expected payoff = (1-g)*(1+r*g)
which is maximal at g=(r-1)/2r for values of r=>1.
This maximum approaches the value g= .5 when r approaches infinity. So the most you'd ever want to gamble is half your life, if you are given near infinite rewards per unit of life gambled.
Here's few plotted values for easy reference!
Here's the "gambling ratio" is the ratio how much you appreciate % increase of life quality versus % chance to live, when using some mean of exchange. For example, if you are willing to accept 1% chance of death for $100,000 and $100,000 will improve the total life quality experienced by 2%, you have a gambling ratio of 2.
You may notice that the unit of exchange cancels out in this model, meaning that something else than money could be used instead.
This model does not take into account the marginally diminishing returns from any bets won.
Disclaimer: the publisher does not encourage people to gamble their lives and takes no responsibility for lives thus lost.