I was just thinking “ugh, I’m having that specific frustration where I’m getting bored with a book, yet feel a neurotic compulsion not to start any other book until finishing it”
Then I realized that what I’m thinking of a “book” in this case is really just 1/6 of a single 6-volume book (Knausgaard’s My Struggle)
…well, I guess that settles it, there’s no way I’m making it through the whole thing then. Good to have clarity, I guess
It turns out that “a 3000-page, nonlinear autobiography that feels like being inside the author’s head as he remembers things free-associatively, and – like one’s own memories – includes a huge amount of quotidian material that anyone would leave out if they were trying to tell a story” is, while kind of cool in just the way it sounds, also kind of boring in just the way it sounds
lol, and here i am now reading volume 3… i love this book. really grows on you
frank’s been getting a huge number of generic goncharov asks
she’s already been asked some variant of “what did you think of goncharov (1973)” like 10+ times. it keeps happening. if you sent one of these and didn’t know, it’s fine, but now you know
i’ve put “goncharov” on a word list that makes this stuff get flagged for my review before frank can post it, and i’ve started deleting most of it.
if you can think of a new, creative, and/or funny goncharov thing to send frank, go ahead. i’m only deleting the repetitive, generic stuff
Another way to look at the Kelly criterion is to think about betting on a variable number of independent things at once.
If you make a single bet repeatedly, and you use the Kelly criterion, then over time, your log(wealth) is a sum of IID random variables.
So the Law of Large Numbers and Central Limit Theorem hold…
asymptotically, as time passes
for log(wealth)
Now imagine that instead, you diversify your wealth across many identical and independent bets. (And you use Kelly to decide how to bet on each one, given the fraction of wealth assigned to it.)
Here, the limit theorems hold…
asymptotically, as the number of simultaneous bets grows
for wealth
which is better in both respects. You control the number of bets, so you can just set “n” to a large number immediately rather than having to wait. And the convergence is faster and tighter in term of real money, because the thing that converges doesn’t get magnified by an exp() at the end.
This is regular diversification, which is very familiar. And then, making sequential independent bets turns out to be kind of like “diversifying across time,” because they’re independent. But it’s not as nice as what happens in regular diversification.
In fact, the familiar knee-jerk intuition “never go all-in, bet less than everything!” comes from this distinction, rather than from any result about how to bet on a single random variable if forced to do so.
In the real world, you’re not stuck in an eternal casino with exactly one machine. If you keep some money held back from your bet, it doesn’t just sit there unused. Money you hold back from a bet can be used for things, including other independent bets.
(The Kelly criterion holds money back so it can be used on future rounds of the same bet, which are a special case of “other independent bets.”)
Of course, if you have linear utility (i.e. no risk aversion), you should still go all-in on whichever bet has highest expected return individually. If this were really true, your life would be so simple that most of finance would be irrelevant to it (and vice versa). You’d just put 100% in whichever asset you thought was best at any given time.
I’ve been off Tumblr for a little while, but there’s apparently been some discussion about the Kelly criterion, a concept in probability, in relation to some things Sam Bankman-Fried said about it and how that relates to risk aversion. I’m going to do what I can to explain some aspects of the math as I understand them.
The Kelly criterion is a way of choosing how much to invest in a favorable bet, i.e. one where the expected value is positive. The Kelly criterion gives the “best” amount for a bunch of different senses of “best” in a bunch of different scenarios, but I’m going to restrict to one of the simplest ones.
Suppose you have some bet where you can bet whatever amount of money you want, you have probability p of winning, and you gain b times the amount you bet if you win. (Of course, if you lose, you lose the amount you bet.) Also suppose you get the opportunity to make this bet some large number n of times in a row, you have the same probabilities and payoff rules for each of them, and they’re independent events from each other. The assumption that all of the bets in the sequence have the same probabilities and payoff rules is made here to simplify the discussion; the basic concepts can still hold when there are a mix of different bets, but it’s a lot messier to state things and reason about them.
Also suppose that your strategies are limited to choosing a single quantity f between 0 and 1 and always betting f times your total wealth at every step. This is a pretty big restriction, and it too can be relaxed at the cost of making things much messier. But even with this restriction we’ll be able to compare the strategy prescribed by the Kelly criterion to the “all-in” strategy of always betting all of your money.
So what is the best choice of f? The Kelly criterion gives an answer, but the sense in which it’s the “best” is one that it’s not obvious should apply to any choice of f. I’ll state it here but keep in mind that until we’ve done some more calculation, we shouldn’t assume that that there is any choice of f which is the best in this sense.
The Kelly criterion gives a choice of f such that, for any other choice of f, the Kelly criterion produces a better result than the other choice with high probability. Here “high probability” means that the probability that the Kelly choice outperforms the other one goes to 1 as n goes to infinity.
So why is this possible?
Let Xi be the random variable representing the ratio of the money you have after the ith bet to the amount you had before it. So your final wealth is equal to your starting wealth times the product of the Xi for i from 1 to n. Also these
Xi
are independent identically distributed variables. (We can describe their distribution in terms of p, b, and f but the exact details aren’t too important to the concepts I want to communicate.) Sums of random variables have some nicer things that can be said about them than products, so we take the logarithm. The logarithm of your final wealth is the log of your starting wealth plus a sum of n independent variables log(Xi).
Now, the expected value of that sum is n times the expected value of one of the individual summands, and the (weak) law of large numbers tells us that with high probability the actual value of the sum will be close to that. (To be rigorous about this: For any constant C, the probability that the sum will be further than Cn away from its expected value goes to 0 as n goes to infinity.) So for any betting strategy f, define r(f) to be the expected value of log(Xi). So if we have any two strategies f and f’, the log of your final wealth following strategy f minus the log of your final wealth following strategy f’ will be about r(f)n-r(f’)n, and so will be positive with high probability if r(f)>r(f’). (If you understood the rigorous definition in the previous parenthetical, you should be able to make this argument rigorous as well.) Thus with high probability the log of your final wealth will be greater using strategy f than strategy f’. Since log is an increasing function, this is equivalent to saying that with high probability, f will result in a greater final wealth than f’.
Then if you pick f such that r(f) is maximized, then for each other choice of f, you’ll outperform that choice with high probability. This is what the Kelly criterion says to do. Maximizing r(f) can be equivalently described by saying that at each bet, you bet the amount that maximizes the expectation of the logarithm of the amount you’ll have after the bet.
A pitfall to avoid here: Although the log of the final wealth can be said to be “about” a certain value with high probability, we can’t really say that the final wealth is guaranteed to be “about” anything in particular. Differences that we can consider to be negligibly small when we’re looking at the logarithm can balloon to very large differences when we’re looking at the actual value, and it is very possible for one experimental trial using a given strategy to yield something many times larger than another trial using the same strategy where you’re a little less lucky.
The Kelly criterion is not the strategy that maximizes the expected amount of money you have at the end. The best strategy for that goal is that is the one where you put all of your money in on every bet. This isn’t inconsistent with the previously stated results; in almost all cases the Kelly criterion outperforms the all-in strategy (because the all-in strategy loses at some point and ends up with no money). But in the very unlikely event that you win every single one of your bets, you end up with an extremely large amount of money, so large that even when you multiply it by that very small probability you get something that’s larger than the expected value of any other strategy.
What if, instead of trying to maximize the expected dollar payoff, you have some utility function of wealth, and you’re trying to maximize the expected value of that? Well, it depends what your utility function is. If your utility function is the logarithm of your wealth, the Kelly criterion maximizes your expected utility; in fact, in this case we don’t even need to assume n is large or invoke the law of large numbers. But going back to the case of large n, there are a lot of other utility functions where the Kelly criterion is also optimal. Think about it like this: the Kelly strategy outperforms any other strategy in almost all cases; the only situation where you might still prefer the other strategy is if in the tiny chance that you get a better outcome, your outcome is so much better than it makes up for losing out the vast majority of the time. So if your utility function grows slower than the logarithm, you care even less about that tiny chance of vast riches than you would if you had a logarithmic utility function, so the Kelly criterion continues to be optimal. More generally, I think it can be shown that when comparing the Kelly criterion to some other strategy, the probability of that other strategy doing better than it decays exponentially in n. Since the amount the other strategy can obtain in that tail situation grows at most exponentially in n, this implies that as long as your utility function grows slower than nε for all ε>0, you won’t care about the tail so the Kelly criterion is still optimal. If your utility function grows faster than that, i.e. if there is some
ε>0 such that your utility function grows faster than
nε, then I think for sufficiently favorable bets, all-in comes out ahead again.
Okay but how does this all of this apply in the real world? Honestly I’m not sure. If your utility function is your individual well-being, it seems very likely to me that that grows logarithmically or slower; if what you care about is maximizing the amount of good you do for the world by charitable donations, I think there is some merit to SBF’s argument that you should treat that utility as a linear function of money, at least up to a certain point. But even he acknowledged that it drops off significantly once you get into the trillions, and since the reasons for potentially preferring riskier strategies over the Kelly criterion hinged on exponentially small probabilities of exponentially large payoffs, I think that that trillion-dollar regime might actually be pretty relevant to the computation.
Really any utility function should be eventually constant, but in that case the Kelly criterion ceases to be optimal in the way discussed before. For large enough n, it will get you all the money you could want, but so will any other strategy other than all-in and “never bet anything”. Obviously this is not a good model of how the world works. To repair this we probably want to introduce time-discounting, but to make sense of that we need to have some money getting spent before the end of the experiment rather than all of it available for reinvesting, and by this point things have gotten far enough away from the original scenario that it’s hard to tell how relevant the conclusions from it even are. It seems like it’s a useful heuristic in a pretty wide range of scenarios? But I have no idea whether SBF was right that he was not in one of them.
To be clear, none of this is to excuse his actions; whether or not he should have been applying the Kelly criterion, I think “committed billions of dollars of fraud” does a better job of capturing what he did wrong than “was insufficiently risk-averse”.
OK yeah, that thing I was talking to @raginrayguns about is way simpler than I thought
The Kelly criterion maximizes the rate of exponential growth, which is just
log(final / initial)
up to a constant.
Like if you have w(t) = exp(rate * t) , and you end at t=T, then
rate = 1/T log(w(T) / w(0))
and T is a constant.
So the Kelly criterion really is nothing but maximizing log wealth, only phrased equivalently as “maximizing exponential growth rate.”
And this phrasing is confusing, because “maximizing exponential growth rate” sounds sort of generically good. Like why wouldn’t you want that?
But the equivalence goes both ways: it’s the same thing as maximizing log wealth, and it’s easy to see you may not want that.
—-
I made a mistake in my original post about geometric averages – I linked to a twitter thread about the Kelly criterion, and a blog post by the same person, as if they were making the same point.
The thread was how I found the post. But in fact, the thread is both wrong and not really about geometric averages being confusing. The post, however, is mostly good and doesn’t mention Kelly at all.
Why did the thread link back to the post, then? The author is conflating several things.
Here are some things you can compute:
The expected growth in wealth from n sequential bets, E[ w_n / w_0 ]. This is what you want to maximize if you have linear utility.
The expected arithmetic average over the growth in wealth from the individual bets.
This is E[ (w_1 / w_0) + (w_2 / w_1) + … + (w_n / w_{n-1}) ] / n.
This is meaningless, there’s no reason to do this. However, this gets reported in financial news all the time, I’ve seen in the WSJ for example.
The expected geometric average over the growth in wealth from the individual bets.
This is E[ ((w_1 / w_0) * (w_2 / w_1) * … )^1/n ], or after cancelling, E[ (w_n / w_0)^1/n ]. So this is (1.), but with a power of 1/n inside the E[].
Like (3.), but with a logarithm inside the E[]: E[ log((w_n / w_0)^1/n) ]. This is the exponential growth rate.
Everything except (1.) has dubious importance at best, IMO.
(1.) is for linear utility, but you have nonlinear utility U, you would just maximize a variant of #1, E[ U(w_n / w_0) ] instead.
In the blog post, Hollerbach is essentially talking about the confusing relationship between (1.) and terms like (w_1 / w_0). You have to multiply these terms to get (1.), and multiplication is confusing.
However, in the post he conflates this product (1.) with the geometric average (3.). They’re not equivalent because the power doesn’t commute with expectation. But I guess they both involve multiplication, and multiplication is confusing.
In the twitter thread, he sort of conflates the geometric average (3.) with the exponential growth rate (4.). Then he pits these against the arithmetic average (2.), which is bad, but is not what SBF was advocating.
Then, since the blog post has already conflated the geometric average with the expected wealth growth, he ends up conflating together everything except the bad one, (2.). In fact, all four are different. And only (1.), or a nonlinear-utility variant of it, is what matters.
