Q&A: The St. Petersburg Paradox
The St. Petersburg Paradox
Question
Regarding the St. Petersburg paradox. Your solution, that an extremely tiny probability is not worth its full expected-value distribution, is the reason behind the St. Petersburg paradox, and likewise the reason for rejecting Pascal’s Wager, of blessed memory, and Parfit’s argument for the value of voting. (As I recall, this solution appears on Wikipedia on that paradox under point 4, though in an unclear formulation.)
I thought of a different formulation of the paradox that in effect rules out some of the solutions presented there on Wikipedia. Let us imagine that we can perform some action that carries with it an extremely tiny risk, say one in a trillion, of destroying the entire world. Would we say that if the probability of this happening is less than one in 8 billion, then we are allowed to take that risk, since according to the distribution it would not cost the life of even one person? Or perhaps, since the potential of all the people in the universe is to give rise to an unbounded number of people, we must not do anything that carries even the slightest chance of destroying the universe? On that side of the argument, it would also follow that one may not do any action that involves any risk whatsoever to human life, even the life of one person, since even one person has the potential to produce infinitely many people. And that is clearly not so. So we are forced to give up the assumption that a tiny probability is worth its expected-value distribution.
Unless we say that people who have not yet been born are not taken into account. (And the approach of the Sages is explained in Mishnah Sanhedrin 37a: “his blood and the blood of his descendants.”)
(I was prompted to think of this בעקבות Yuval Steinitz’s science-fiction book There Will Be Nowhere for Us to Return, where he wrote something along these lines—that we cannot take any risk with respect to all of humanity—but he said this only regarding the destruction of all humanity, not what would follow from it regarding the life of one individual.)
Answer
I didn’t understand a thing. In your eagerness to phrase it Talmudically, you lost clarity.
Pharaoh decreed against the males, but Laban sought to destroy everything.
Discussion on Answer
I don’t understand the connection. I wasn’t talking here about murder, so future descendants are irrelevant. Besides, one does not murder future descendants, only prevent their birth. In short, I understand neither the argument nor its connection to our discussion.
Right. We weren’t talking about murder; I brought it as an analogy.
In practice, if the person dies, the consequences will be infinite, won’t they?
So how do we take that risk?
Sorry for the lack of clarity. Hebrew is not my native language, and I write with difficulty in clumsy wording. I’ll try to be as clear as I can:
A. The number of descendants of one person is infinite.
B. An action that carries even the slightest risk even to the life of one person is, by expected value, equivalent to killing infinitely many people (= because even one in a trillion of the infinite descendants that would have come from him is still infinite).
C. The conclusion is either that indeed it is forbidden to do any action that involves even the remotest risk to human life—which is completely absurd—or that a tiny probability is not worth its expected-value distribution, as you wrote.
(Unless we give up the hidden assumption that preventing the birth of descendants is taken into account.) And I cited the words of the Sages, who in effect expressed this idea: when we kill one person, it is as though we killed the infinitely many descendants that would have come from him.
Thank you.