In the St Petersburg game, you keep on tossing a coin until you get heads, and you get a payoff of 2n units (e.g, 2n days of fun) if you tossed n tails. Your expected payoff is:
- (1/2) ⋅ 1 + (1/4) ⋅ 2 + (1/8) ⋅ 4 + ⋯ = ∞.
This infinite payoff leads to a variety of paradoxes (e.g., this).
But note that the infinite payoff is by itself paradoxical. If the expected payoff is infinite, it seems that it’s worth being tortured for a decade by the most effective torturers in the world for the sake of playing the game. And yet this is paradoxical!
However, the paradox here is not actually a paradox of infinity. For there will be some cut-off version of the game—a version where you get to toss the coin some predetermined finite number of times and if you don’t get to heads then you get nothing—where the value of the cut-off game exceeds the disvalue of the decade of torture. And it’s even more paradoxical to think that in the cut-off game it makes sense to pay that much to play, since the cut-off game is dominated by the infinite game.
This line of thought supports Monton’s thesis that we should neglect small probabilities.
3 comments:
Monton’s paper explicitly mentions a truncated St Petersburg. He truncates to a sure payout, not to zero payout, so he doesn’t strictly get the domination that your version does. But otherwise, he makes the same point.
At a minimum, the thesis that we should neglect small probabilities needs some refinement. By subdividing outcomes, you can make the probability of any particular outcome arbitrarily small. But that would not be a reason to ignore all the subdivided outcomes. (For example, you might receive an amount chosen randomly and uniformly in cents from one cent to 10 billion dollars. The probability of any particular outcome would be one in a trillion.)
The problem with the linked paper is that "A is twice as valuable as B" has only one meaning.
It means a 50% chance of A is exactly worth a 100% chance of B.
Without that, "twice as valuable" has no more meaning than "twice as pretty". A figure of speech, but no mathematical meaning.
This means that you cannot discount small probabilities down to zero; if you do, you are violating the meaning of saying they have high utility.
The correct answer is the one he dismissed near the beginning; all values have diminishing marginal utility, and there is an absolute numerical cap to utility (given any particular scale.)
This is the only answer that both preserves the meaning of statements relating utilities with proportions, and admits that you shouldn't do anything stupid.
eu: That is *a* way of defining "twice as valuable". But one can also take "twice as valuable" to be primitive. We intuitively and immediately judge that it's twice as valuable to rescue two drowning people as one, and that two bouts of an hour of torture followed by memory erasure of the torture are twice as bad as one bout. On the other hand, if we ask whether we should be indifferent between a 50% chance of rescuing two vs. a 100% chance of rescuing one, or a 50% chance of the two bouts of torture vs. a 100% chance of one, we do not immediately judge indifference. We are thus confident of the twice-as-bad judgment but not of the 50%-vs.-100% judgment, which is some evidence that these are different judgments.
Ian: Of course "neglect small probabilities" is meant to be shorthand for a more complex procedure, such as: https://alexanderpruss.blogspot.com/2022/11/how-to-discount-small-probabilities.html
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