Let L be the difference in utility between living-and-not-paying and dying. Fix one of the scenarios — say, the first one*, where you can pay to have no bullets. For each positive number X, consider the following decision problem:
Let the difference in utility between living-and-paying and living-and-not-paying be X. (Dying is assumed to have the same utility regardless of whether you paid.) Should you pay to change the probability of dying as described? For each X, answering this is just a matter of computing the expected utilities of paying and not-paying, respectively.
Now determine the maximum value of X (in terms of L) such that you decide to pay.
Now repeat the above for the other scenario.
It turns out that, in both scenarios, the maximum value of X such that you decide to pay is the same: X = 1⁄3 L. That is the meaning of the claim that “you should pay the same amount in both cases”.
* … as enumerated at the Landsburg link, not in the OP …
I see. So the problem should be not “How much you’d pay to remove bullets?”, but “How much you’d precommit to paying if you survive, to remove bullets?”
Here is how I understood the problem:
Let L be the difference in utility between living-and-not-paying and dying. Fix one of the scenarios — say, the first one*, where you can pay to have no bullets. For each positive number X, consider the following decision problem:
Let the difference in utility between living-and-paying and living-and-not-paying be X. (Dying is assumed to have the same utility regardless of whether you paid.) Should you pay to change the probability of dying as described? For each X, answering this is just a matter of computing the expected utilities of paying and not-paying, respectively.
Now determine the maximum value of X (in terms of L) such that you decide to pay.
Now repeat the above for the other scenario.
It turns out that, in both scenarios, the maximum value of X such that you decide to pay is the same: X = 1⁄3 L. That is the meaning of the claim that “you should pay the same amount in both cases”.
* … as enumerated at the Landsburg link, not in the OP …
I see. So the problem should be not “How much you’d pay to remove bullets?”, but “How much you’d precommit to paying if you survive, to remove bullets?”
Yes. It’s assumed that you have no control over the value of what happens if you die.