Every economist surely knows the St. Petersburg Paradox described by Daniel Bernoulli in 1738 in a paper which can fairly claim to be the first piece of theoretical economics. Consider a casino offering a game of sequential coinflips that pays 2^(n-1) as a payoff if the first heads arrives on the nth flip of the coin. That is, if there is a heads on the first flip, you receive 1. If there is a tails on the first flip, and a heads on the second, you receive 2, and 4 if TTH, and 8 if TTTH, and so on. It is quite immediate that this game has expected payoff of infinity. Yet, Bernoulli points out, no one would pay anywhere near infinity for such a game. Why not? Perhaps they have what we would now call logarithmic utility, in which case I value the gamble at .5*ln(1)+.25*ln(2)+.125*ln(4)+…, a finite sum.

Now, here’s the interesting bit. Karl Menger proved in the 1927 that the standard response to the St. Petersburg paradox is insufficient (note that Karl with a K is the mathematically inclined son and mentor to Morganstern, rather than the relatively qualitative father, Carl, who somewhat undeservingly joined Walras and Jevons on the Mt. Rushmore of Marginal Utility). For instance, if the casino pays out e^(2^n-1) rather than 2^(n-1), then even an agent with logarithmic utility have infinite expected utility from such a gamble. This, nearly 200 years after Bernoulli’s original paper! Indeed, such a construction is possible for *any* unbounded utility function; let the casino pay out U^-1(2^(n-1)) when the first heads arrives on the nth flip, where U^-1 is inverse utility.

Things are worse, Menger points out. One can construct a thought experiment where, for any finite amount C and an arbitrarily small probability p, there is a bounded utility function where an agent will prefer the gamble to win some finite amount D with probability p to getting a sure thing of C [Sentence edited as suggested in the comments.] So bounding the utility function does not kill off all paradoxes of this type.

The 1927 lecture and its response are discussed in length in Rob Leonard’s “Von Neumann, Morganstern, and the Creation of Game Theory.” Apparently, Oskar Morganstern was at the Vienna *Kreis* where Menger first presented this result, and was quite taken with it, a fact surely interesting given Morganstern’s later development of expected utility theory. Indeed, one of Machina’s stated aims in his famous paper on EU with the Independence Axiom is providing a way around Menger’s result while salvaging EU analysis. If you are unfamiliar with Machina’s paper, one of the most cited in decision theory in the past 30 years, it may be worthwhile to read the New School HET description of the “fanning out” hypothesis which relates Machina to vN-M expected utility.

http://www.springerlink.com/content/m7q803520757q700/fulltext.pdf (Unfortunately, the paper above is both gated, and in German, as the original publication was in the formerly-famous journal Zeitschrift fur Nationalokonomie. The first English translation is in Shubik’s festschrift for Morganstern published in 1967, but I don’t see any online availability.)