Q&A: Parfit
Parfit
Question
Hello.
I had a story of divine providence (I believe in that…). I have a very brilliant friend; just last week I argued with him about whether there is any point in going to vote in elections, and in any activity whose benefit is conditional on collectivity. He made exactly Parfit’s argument that you brought in column 252, and only now did I see that you discussed it at length, very nicely. I have a few questions, mainly about the argument and your claim.
A. Parfit assumes that the large statistical benefit is equal to the small hassle of voting. My friend proved it like this: if the certainty that a given candidate will win the election is worth the hassle of 100,000 people (for example)—and everyone understands that (that if everyone votes, it’s worth it)—then a probability of 1 in 100,000 is worth the hassle of one person. I argued: just as I don’t buy a lottery ticket with a 1 in a million chance of winning $2,000,000 even if the price is $1, even though I would buy a ticket that would certainly win even for $1,900,000 or more. He argued that the very uncertainty reduces the value of the money—that besides the statistical probability, which is 1 in a million, this statistical money is actually worth less, because for example I can’t buy anything with this money since it is uncertain; I can’t go eat at a restaurant now on the assumption that I’ll pay when I win. So the uncertainty itself reduces the value of the money. (I thought this was an answer to the St. Petersburg paradox, but it isn’t, because if the money I could win is infinite, that explains nothing.) If so, can I make the same claim regarding elections? So he answered that it depends: if, in the case where my candidate doesn’t win, I lose nothing and merely fail to gain, then I am right. But if on the other side I would lose something (say, the amount I failed to gain), then this is not a correct consideration, because the uncertain benefit is indeed worth less than the certain benefit, but there is an uncertain loss against it. Is all this correct? And if not, where is the mistake?
B. Is this whole argument of Parfit’s (and also Pascal’s) correct? When I have two options—either to save one person for sure, or a 50% chance to save two people—are they exactly equal? Or does certainty also have value?
C. Regarding your claim that if the chance is negligible and the benefit is not anticipated, then one should no longer think in terms of the distribution (as I understood it: when the person does not really think he will win)—is there a clear definition of this? And is there also a sharp boundary, only one that is hard to place, or does it work gradually? For example, with a 20% chance, a person presumably thinks, “I probably won’t win,” but of course the benefit is still anticipated. Is it worth investing 20% of what it would be worth to invest in a certainty, or does it decline gradually (for example, for 20% chance one invests 15%, for 10% chance 5%)? I would be very glad if you would address my boring, ignorant arguments.
Answer
First, why aren’t you posting this as a talkback on that column? After all, my responses are everywhere (I fill the world).
A. I don’t understand his argument or your question. In elections we are not talking about a small influence but about a tiny chance of influencing. So St. Petersburg applies here too.
B. I don’t know how to answer that. At first glance it seems like the same thing.
C. Of course there is no sharp boundary.
Discussion on Answer
B. I didn’t understand. I answered you. Beyond that, isn’t that what I showed from St. Petersburg? That expected value is not necessarily a good criterion. Maybe explain the question better.
C. I don’t see any logic in that apart from St. Petersburg. It could be another explanation of why expected value is not a criterion. So what is the novelty?
A. I didn’t post there because I thought that column’s time had already passed, since it was written a few years ago (I really didn’t know that you fill the world and your hands range over everything, and that you trouble yourself to answer everyone who posts a talkback).
B. What you wrote, that at first glance it seems like the same thing—I wasn’t asking whether it is the same thing or not. I want to know whether one can question the basic argument that the investment is equal to the statistical distribution, from which Pascal and Parfit proceed on that assumption. I seem to recall seeing in that column that you hinted at this, but I couldn’t find it now.
C. At the beginning of my remarks, what you wrote—that St. Petersburg applies here too—of course. I am asking: if we do not accept your view and assume, like Pascal and Parfit, that one does not distinguish between an anticipated benefit and one that is not anticipated, still statistical probability is not the whole picture, since in the end uncertain money really is worth less than certain money, and therefore besides the probability one must take another consideration into account.
In any case, thank you very much for taking the time to reply to me.