## Tuesday, September 09, 2003

Now what was the last post about ? It is a response to an article by Paul Samuelson remarkable because he wrote “I have made my point and, but for the last, with words of just on syllable.” His point was that it is not rational for an extremely long lived investor to invest each period to maximize the one period geometric mean return on his/her portfolio. Latané had argued that it was rational because, by the law of large numbers, the very long horizon payoff to such a strategy would almost surely be greater than the payoff to any other. Samuelson’s point is that almost surely is not surely and for many utility functions of very long run payoffs, extremely unlikely events can be so painful (or pleasant) to matter. Thinking of this I was reminded of the Saint Petersburg paradox. I didn’t mention it because Petersburg is 3 syllables long. This was noted by Blaise Pascal (whose parents were kind enough to give him a one syllable first name). Blaise was not sure it was a paradox. It is exactly the argument that extremely unlikely events can be extremely important. Most people (including Pascal most of the time) think it is silly. A way of understanding this widespread view is to assume that utility functions are bounded above and below (the below part was explained to me by Peter Mollgaard). That is to assume we are capable of only a finite amount of pleasure or pain. Oddly standard utility functions are typically unbounded either above or below and so Petersburg paradoxical. Samuelson recognised this problem in a related context, but conventiently forgot it when dumping on Latané.

So let’s say that there is a highest possible happiness U and a lowest possible misery u. Events which occur with probability espsilon if we follow Latané’s strategy but not Samuelson can cost us only at most epsilon(U-u). As espsilon goes to 0, this term, the “if you lose you could lose real big” term goes to zero. So why isn’t Latané right for utility functions which are bounded both above and below ? Hmmm. Hard to answer that one using words of just one syllable.