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Martingale progressions and Kelly

My criticism of the gambling literature on betting is it almost ignores the academic math and finance literature. It often uses obscure references instead of the standard authoritative results. For example, Kelly’s paper merely “reinterpreted” the logarithmic criterion in terms of information transmission. But the log-criterion is not due to Kelly, and is pretty obvious anyway. I wonder who coined the term “Kelly criterion” anyway (probably Thorp).

I would like to help straighten out two fallacies involving limits and expectations. In both fallacies the expectation of the limit is not equal to the limit of the expectation. The first fallacy is that you can/cannot make money with doubling strategies. The second is that everyone “should” follow the Kelly criterion.

Contrary to your article, [Mathematical Proof That Progressions Cannot Overcome Expectation] there are positive expectation strategies that make money betting on a random walk. The classic example is the martingale doubling strategy: bet $1, $2, $4, … until you win. Any proof you can’t make money this way must rule out such strategies. Bounded wealth or bounded borrowing capacity is one way to do this (you can’t double up and lose more than 100 bets in a row). Finite time is another way (you won’t live long enough to lose more than 1,000,000 bets in a row). Doob’s Optional Sampling Theorem says (roughly) if Gambler #1 has zero expectation and Gambler #2 quits before Gambler #1 then Gambler #2 also has zero expectation. So even if you have no predetermined betting limit or time limit, the doubling strategy is prevented by stopping you when your spouse wants to leave (perhaps after going broke playing a different game).

Some people claim limited doubling strategies work in practice because a 1/1,000,000 chance of losing $1,000,000 is worth an almost certain chance of winning $1. They probably don’t buy insurance or wear seat belts either. The limiting case of a doubling strategy is winning $1 with certainty. But the expected value of finite doubling strategies is still zero: the limit of the expectation (0) is not equal to the expectation of the limit ($1).

A similar fallacy is involves logarithmic growth. With proportional betting strategies, the multiplicative effect of repeated wagers translates to an additive effect on log-wealth. With reasonable independence a central limit theorem applies to the growth of log-wealth: your results will eventually approach your expected hourly growth rate. Consequently a person (“Kelly”) who maximizes this growth rate will eventually be richer than a person who doesn’t. In the limit Kelly is richest with certainty. But “Kelly” betting doesn’t maximize everyone’s expected utility because the limit of the expected utility is not the expected utility of the limit.

One more thing: I really like Stanford Wong‘s definition of bankroll: the amount of money you are willing to lose at gambling. Below are two Usenet posts responding to Abdul Jalib. The Samuelson article cites previous results about optimal long-run betting.

. . . see Nobel Laureate Paul A. Samuelson, “Why We Should Not Make Mean Log Of Wealth Big Though Years To Act Are Long” Journal of Banking and Finance, 1979, v3(4), 305-308.

My blackjack references incorrectly apply “Kelly” betting to an artificially defined bankroll, instead of applying it to total wealth. If you are a real card counter with log utility who owns a house, you should often bet thousands of dollars per hand.

With independent sequential gambles, it is optimal to bet a fixed fraction of wealth for power utility functions of the form U(W) = (W^a-1)/a, for a > 1. This approaches ln(W) as the the coefficient a gets small. Log-utility is the only “myopic” utility function that bets a fixed fraction of W with dependent sequential gambles. For example, sequential hands are slightly negatively correlated in blackjack. If you lose the current hand, you will probably have a bigger advantage on the next hand. This effect is too slight to be practical, but so is the advantage of card counting with $5 chips.

[response by Abdul]
That’s not the issue for me at all. The issue
for me is the purely mathematical question of whether Kelly betting
is the best for an idealized gambling game (say, biased coin tossing)
and whether log utility is the One True Utility Function. The question
has nothing to do with blackjack … .
I’m not all that well versed in utility theory, so I am not sure
my position is correct, and I’m looking for you and others
to point out the flaws. Thus far, I don’t see a big problem
with my assertions, at least if one lives forever, but
I’m still not completely convinced myself. The point others
brought up about being able to beat Kelly for a particular
utility function if you break the fixed fraction assumption are
familiar to me and I don’t see it as a strong counterargument.
The point about bet-it-all with linear utility being superior
is flawed; after an infinite number of bets, Kelly will prevail.

It would probably be worth digging up Professor Cover’s proof.

Dear Abdul:
Despite your many correct contributions on blackjack, you are wrong on this issue. I am a finance professor, and am familiar with Cover’s article in _Mathematical Finance_ volume 1, issue 1. Paul Samuelson is also pretty familiar with utility theory. :-).

You believe a common fallacy that that everyone should maximize the Kelly criterion because a Kelly bettor will eventually be richer than a non-Kelly bettor. This fallacy often involves incorrect application of limits. You can’t just say that an immortal Kelly bettor will become infinitely rich, because there are many investment strategies that will also become infinitely rich. The only way to compare strategies is to use a finite horizon and then take limits.

If I maximize the expected square-root of wealth and you maximize expected log of wealth, then after 10 years you will be richer 90% of the time. But so what, because I will be _much_ richer the remaining 10% of the time. After 20 years, you will be richer 99% of the time, but I will be fantastically richer the remaining 1% of the time. It’s not enough for your conservative log utility function, but it make my square-root utility higher in expectation.

Conversely, an investor who maximizes utility U(W) = -1/W will be slightly poorer than you 99% of the time, but will be insured against your biggest losses 1% of the time. We all get infinitely rich, but we get infinite (expected) utility at different rates.

The goal is to maximize the mean utility of wealth, not the median. If we maximized the median, then all (monotonic) utility functions would be equivalent. If that were the case, then Von Neumann and Morganstern would not have invented them!

There are two essential points.

The first is blackjack guys should cite mathematicians and economists more. The martingale results are in textbooks on stochastic processes (including Doob’s Optional Sampling Theorem). Economists Von Neumann and Morgenstern invented expected utility and Paul Samuelson cites authoritative work from the first issue of the Journal of Financial Economics on long-run investment strategy. Cover and Kelly are not appropriate references except as illustrations of fallacies.

The second point is the limit of the expectation is not the expectation of the limit. Martingale doubling or Kelly strategies are good if you can gamble literally forever. But they are not necessarily good if you can gamble for a “long” time.

Kim Lee

P.S.: Von Neuman and Morganstern’s expected utility theory doesn’t assume a utility function. It merely assumes axioms about choice under uncertainty. The axioms are sufficient to derive a utility function. For example, transitivity says if you prefer gamble A to gamble B, and gamble B to gamble C, then you will prefer gamble A to gamble C. The crucial axiom is that preferences are linear in probability. If one lottery ticket is worth a dollar then two lottery tickets are worth two dollars. But a lottery ticket with a double prize would be worth less.

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