Breathing New Life Into Old Paradoxes

While you may think you're familiar with all the nuances of the unexpected hanging paradox, a recently revised paper by Timothy Chow shows that the issues raised by it continue to be of real interest in the here and now, and not just to puzzle addicts and antinomy-collectors.

In a similar vein, it is often assumed by students of economics that the Saint Petersburg paradox is easily settled by the adoption of the notion of diminishing marginal utility, yet not only do humans clearly and consistently violate the assumption that they operate according to such a theory, but even if they did, it still would not provide any sort of resolution of the problem, as the paradox can easily be restated to cause problems even under such a scenario: in fact, hardly any plausible assumption one could make suffices to save the day, as the preceding link makes clear.

I don't know about the rest of you lot, but personally I obtain a certain degree of satisfaction from the thought that even well-mined issues like these might still be capable of yielding up new and worthwhile insights.

Comments

[bbartlog]

The restatement of the St Petersburg paradox in the link you provide is itself not entirely satisfactory, since (in order to compensate for decreasing marginal utility) it provides for monetary prizes in excess of all the wealth on Earth. The paradox is actually easily explained IMO if we assume that there is only a finite amount of utility a person can gain, even if non-monetary prizes are added to the payout schedule. In that case it becomes trivial to resolve the problem.

[Abiola Lapite]

"The paradox is actually easily explained IMO if we assume that there is only a finite amount of utility a person can gain"

But is this true, though? As the article points out, the veracity of this assumption is far from obvious: no matter how happy I am, I can always imagine myself being just that bit happier, if not through rewards garnered for myself, then through some deed which gives happiness to others.

[Andrew]

Maybe your happiness approaches an asymptote?

[Abiola Lapite]

Here we enter into the realm of analysis: are the terms of the utility function a cauchy sequence? If utility were to increase in increments of 1, 1/2, 1/3, etc. (harmonic sequence), we'd still have a nonfinite sum, even though the terms would clearly head to zero.

Basically, my point is that I don't see why one has to assume that real world utility diminishes rapidly enough for the paradox to vanish: for all one knows, one might hit some extremely low but non-zero limit at which each additional life bettered brings as much pleasure as the ones before, and I don't see any basis for presuming this can't possibly be true for anyone, even if it isn't true for everyone.

[bbartlog]

Fair enough, but even if you are so philanthropically motivated that you derive the same satisfaction from elevating others that you did from gaining wealth yourself, the number of people you can help is still finite (only 6 billion of us after all).
In general I agree with the idea that it is (almost) always possible to be a little bit better off, but that's an assumption born of experiences with normal life which I think might be violated in this case. We can readily increase the rewards offered beyond mere monetary ones by assuming that some all-powerful entity can offer as a payout immortality, oracular knowledge, or whatever flavors of dominion, enlightenment, benevolence or hedonism you like; but at some point you have it all, so to speak, and it becomes difficult to see how the next payout can be twice as good as the previous.
Anyway, I also think that the real issue here is reification: utility as a quantity is a model construct, useful in some cases for predicting human or market behavior, but the paradox relies on this scalar-utility model being very real and applicable in all imaginable circumstances.