Act as if you have log utility and with probability 1 your wealth will converge to infinity.

Sergiu Hart presented this paper at Northwestern last week. Suppose you are going to be presented an infinite sequence of gambles. Each has positive expected return but also a positive probability of a loss. You have to decide which gambles to accept and which gambles to reject. You can also invest purchase fractions of gambles: exposing yourself to some share of its returns. Your wealth accumulates (or depreciates) along the way as you accept gambles and absorb their realized returns.

Here is a simple investment strategy that guarantees infinite wealth. First, for every gamble that appears you calculate the wealth level such that an investor with that as his current wealth and who has logarithmic utility for final wealth would be just indifferent between accepting and rejecting the gamble. Let’s call that critical wealth level . In particular, such an investor strictly prefers to accept if his wealth is higher than and strictly prefers to reject it if his wealth is below that level.

Next, when your wealth level is actually and you are presented gamble , you find the maximum share of the gamble that an investor with logarithmic utility would be willing to take. In particular, you determine the share of such that the critical wealth level of the resulting gamble is exactly . Now the sure-thing strategy for your hedge fund is the following: purchase the share of the gamble , realize its returns, wait for next gamble, repeat.

If you follow this rule then no matter what sequence of gambles appears you will never go bankrupt and your wealth will converge to infinity. What’s more, this is in some sense the most aggressive investment strategy you can take without running the risk of going bankrupt. Foster and Hart show that any investor that is willing to accept some gambles at wealth levels below the critical wealth level there is a sequence of gambles that will drive that investor to bankruptcy. (This last result assumes that the investor is using a “scale free” investment strategy, one whose acceptance decisions scale proportionally with wealth. That’s an unappealing assumption but there is a convincing version of the result without this assumption.)

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May 1, 2012 at 12:54 am

Lones SmithSergiu is one of my favorite theorists! This is known as the Kelly criterion,

http://en.wikipedia.org/wiki/Kelly_criterion

and was used by Thorpe, the author of “Beat the Dealer” and “Beat the Market”, who also inspired the MIT Blackjack team, and ultimately that fun movie “21”. Paul Samuelson in 1969 tried to shoot it down as a smart investment strategy, seemed to think Thorp missed the bigger picture. It is clear that if one is living at all for the here and now, this is far too aggressive. Loosely, one might view this criterion as the optimal solution for delta=1. I have been pondering for some years the insights this has for optimal experimentation. I teach it every year in advanced theory hoping a crystalizing insight will strike me.

Meanwhile, those who want good fireside reading on this might peruse “Fortune’s Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street.”

May 3, 2012 at 2:40 pm

party pooperLones, you are right. If you know more than what you have written, you probably realized that most of this paper points out the well-known fact that if you bet more than “double-Kelly”, you eventually go bankrupt. This has been known for over half a century. (Anon’s post below is just one of many relevant citations.) In fact Theorems 1 and 2 are exactly variations on that fact. Half of Proposition 1 is also well known (e.g. that Kelly is myopic and unit-free). Finally: you can be bolder than log utility, have CRRA, and still get infinite wealth with probability 1. Exercise left to the reader.

May 2, 2012 at 4:35 am

AnonymousBREIMAN, L. (1961): “Optimal gambling systems for favorable games,” in Proceedings

of the Fourth Berkeley Symposium on Mathematical Statistics and

Probability, ed. by J. Neyman, vol. 1, pp. 65–78, Berkeley. University of California

Press.

May 9, 2012 at 11:12 pm

Lones SmithDear party pooper: Since you could forecast a comment yet to arrive, I infer that you have it in your capacity to do much better than Kelly, simply investing the gambles that will win.

May 17, 2012 at 9:26 pm

party pooper???

party pooper: May 3 2:40

Anonymous: May 2 4:35

Just to clarify my obtuse post, I was merely surprised that you simultaneously praise the authors, know about the Kelly criterion (to the extent you are teaching it), but not point out the redundancy of the paper that I pointed out. This isn’t an attack on you, especially if you were unaware of what I pointed out (hence my qualifier “if you know…”). In fact, if you didn’t know these details, that puts you in good company… of at least a few referees!