Wednesday, November 2, 2022

Must we accept free stuff?

Suppose someone offers you, at no cost whatsoever, something of specified positive value. However small that value, it seems irrational to refuse it.

But what if someone offers you a random amount of positive value for free. Strict dominance principles say it’s irrational to refuse it. But I am not completely sure.

Imagine a lottery where some positive integer n is picked at random, with all numbers equally likely, and if n is picked, then you get 1/n units of value. Should you play this lottery for free?

The expected value of the lottery is zero with respect to any finitely-additive real-valued probability measure that fits the description (i.e., assign equal probablity to each number). And for any positive number x, the probability that you will get less than x is one. It’s not clear to me that it’s worth going for this.

If you like infinitesimals, you might say that the expected value of the lottery is infinitesimal and the probability of getting less than some positive number x is 1 − α for an infinitesimal α. That makes it sound like a better deal, but it’s not all that clear.

Of course, infinite fair lotteries are dubious. So I don’t set much store by this example.


James Reilly said...

Professor Pruss,

Here's a theological/philosophical question that I thought you might be interested in. Speaking to the multitudes, John the Baptist says that "God is able from these stones to raise up children to Abraham" (Luke 3:8). Setting aside any questions of historicity (e.g. whether Abraham was a historical figure, and so on), it can be safely assumed that John the Baptist (and those to whom he was speaking) regarded the Judeans as literal, biological descendants of Abraham. So it seems that, if taken at face value, he is saying that God could turn the stones into literal, biological descendants of Abraham. I wonder how this might bear on, for instance, essentiality of origins.

Of course, I think the solution is to avoid this sort of strict literalism, but either way, it's fun to think about.

Alexander R Pruss said...

Converts to Judaism are children of Abraham, too.

James Reilly said...

True. Even still, one wonders what it would mean for God to turn a stone into a child of Abraham; it wouldn't be a literal biological descendent, nor would it have undergone a conversion. I suppose just creating a person and declaring them to be under the Abrahamic covenant could suffice?

Pointless overthinking, of course.