Tuesday, November 8, 2022

A principle about infinite sequences of decisions

There are many paradoxes of infinite sequences of decisions where the sequence of individual decisions that maximize expected utility is unfortunate. Perhaps the most vivid is Satan’s Apple, where a delicious apple is sliced into infinitely many pieces, and Eve chooses which pieces to eat. But if she greedily takes infinitely many, she is kicked out of paradise, an outcome so bad that the whole apple does not outweigh it. For any set of pieces Eve eats, another piece is only a plus. So she eats them all, and is damned.

Here is a plausible principle:

  1. If at each time you are choosing between a finite number of betting portfolios fixed in advance, with the betting portfolio in each decision being tied to a set of events wholly independent of all the later or earlier events or decisions, with the overall outcome being just the sum or aggregation of the outcomes of the betting portfolios, and with the utility of each portfolio well-defined given your information, then you should at each time maximize utility.

In Satan’s Apple, for instance, the overall outcome is not just the sum of the outcomes of the individual decisions to eat or not to eat, and so Satan’s Apple is not a counterexample to (1). In fact, few of the paradoxes of infinite sequences of decisions are counterexamples to (1).

However, my unbounded expected utility maximization paradox is.

I don’t know if there is something particularly significant about a paradox violating (1). I think there is, but I can’t quite put my finger on it. On the other hand, (1) is such a complex principle that it may just seem ad hoc.


Kritsch said...

"But if she greedily takes infinitely many, she is kicked out of paradise, an outcome so bad that the whole apple does not outweigh it."

How about taking only the slices with even numbers?
Is she then also doomed to leaving paradise, just because she took an infinite amount of slices from Satan's apple, but not exactly the whole (100%) apple?!?
∑n∈ℕ(1/2^n)/2=1/(1-1/2)·1/2=1 (=100% of Satan's apple)
≠ ∑m∈ℕ(1/2^(2m))/2=∑m∈ℕ(1/4^m)/2=1/(1-1/4)·1/2=2/3 (≈66.67% of Satan's apple)

Is Hilbert's Hotel not capable of accommodating new guests, just because all and every room is currently occupied?
From the wiki article for "Hilbert's paradox of the Grand Hotel":
Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms....

And these are the problems/propositions, which philosophers are hung up on these days.

Alexander R Pruss said...

Yup, in the story, taking all the even-numbered slices, or all the prime-numbered slices, or all the power-of-two-numbered slices will get you kicked out of paradise.

Kritsch said...

If so, then I guess, that any and every finite amount of slices will do for Eve, as long as she doesn't go for any amount of slices with cardinality equal to the cardinality of the set of all natural numbers. If she is ought to maximise her utility AND there is no certain bound or limit to that maximisation of a finite amount of slices of Satan's apple, then go figure, what such a "maximum" in this case might be.
As for me I will take an arbitrary amount of percentage below 100% of that pie, I mean, of that "Satan's apple" with an arbitrary finite amount of slices.
Thank you very much.