I have a tiny technical nitpick -- you make a subtle error when you say "by the Law of Large Numbers, since the X_n are independent, identically distributed random variables, this sum converges almost surely to n times the expected value". This is actually not true! The proper statement of the strong LLN is "this sum divided by n converges almost surely to the expected value". These two statements seem like they should be equivalent, but because of some subtleties around limits and convergence, they actually are not. It's fun to think about why!
One thing that might help show why your original statement must be false, is to note that by the Central Limit Theorem, the distribution of the random variable corresponding to the sum of the X_n will approach a normal distribution centered at n * E[X], which clearly means it doesn't converge in probability (or almost surely) to any one value.
> But these zero-probability scenarios are still factored into expected value calculations, which allows expected value to be maximized by the all-in betting strategy.
If the set of these as a whole has probability zero, then it won't contribute to expected value. So that can't be what's going on; it has to be something else.
In the scenario with N double-or-nothing flips of a slightly tilted (51% win) coin, the only scenario where "all-in" betting doesn't give you $0 is the one where you get N consecutive coin flips. This is the only event that contributes a positive term to the expected value of the betting strategy.
For finite N, the probability of this "win scenario" is nonzero; it's 2^(-N).
In the limit as N goes to infinity, though, the probability that you are in the "win condition" goes to zero.
OK, but that's different from what you wrote above. Are you considering an infinite series of flips, or are you considering a finite sequence of flips and taking limits of associated statistics?
Like, if you are looking at an actual infinite series of flips, that is where the law of large numbers applies. And in that setting -- or any fixed setting! -- a set of measure zero does not affect the expected value.
Now in your reply above you describe a sequence of probabilities that approach zero, and expected values that approach infinity, but that's not the same thing. Not everything is continuous. I mean with the probabilities sure, but evidently not with the utilities. Perhaps that's what's actually going wrong here and causing the discrepancy you discuss? That because the convergence isn't uniform (or dominated or monotone or whatever), expected value doesn't act continuously? Because what you wrote doesn't seem to make any sense...
Yeah, I think the point is that we don't have a single random variable on the space of infinite sequences of coin flips with expected value infinity and measure zero success event, here; we have a sequence of random variables with expected value converging to infinity and the measure of success converging to zero. (These random variables converge in distribution to the always-zero random variable.)These aren't quite the same thing but I don't think it affects the argument.
Note: Kelly criterion is only optimal in the sense you mean if you expect to be offered an infinite sequence of positive expected value bets on which you can bet any percentage (up to 100%) of your current bankroll. If you expect to have only a fixed, finite number of such opportunities, optimizing EV may well be the way to go.
The most of your utility is concentrated on some event with vanishingly small probability objection is also somewhat mitigated if you are part of a large EA movement and are making bets that are independent of those made by other people in the movement.
I think section two is playing fast and loose with words, and I'm not convinced that the derivation is correct.
> You have repeated opportunities to make a bet which is a binary random variable; with probability p you get a return of b times the amount you bet, and with probability (1-p) you lose and get zero.
> Each bet is an independent random event.
> You have to stop betting when your bankroll runs out
> You care about your long-run total money, after “many” repeated bets.
What is "many" here? I tend to take "many" to mean some large $n$. The post jumps between this being some large value and it being infinity.
In particular, it derives that the value is
> W_0 (e^(E[log(X)])(1 + o(1)))^n
and says
> As n becomes large, the small-o terms disappear, and you just get W_n ~ W_0 (e^(E[log(X)])^n)
No you don't. I'm not sure that turning the o(n) term into an o(1) term that doesn't depend on $n$ was valid to begin with (I'm not sure that using big-o itself was a good idea), but it's in the exponent, and you can't just drop it. If you want to see what this converges to as n approaches infinity, you have to take a limit.
The reality is that this process, like everything else in the real world, does not go to infinity. There is no event of measure zero to discuss. The expected value of betting everything is p^n * b^n for however many betting opportunities show up. And it's easy to show that any deviation from that lowers your expect value. It's also easy to see that p^n is going to be very small if p < 0.9 and n > 100, so you will probably (not certainly!) lose all your money.
Really, this could be rolled into the St. Petersburg paradox discussion – I don't think this is adding anything except for an unnecessary discussion of infinities (and thereby almost sure convergence and measure zero).
The resolution to the St. Petersburg Paradox is, in banking parlance, risk management, or more specifically, credit risk management or counterparty credit risk. Can your counterparty afford to pay out $1 million? If so, you might be willing to pay up to a maximum of around $20 to play. How about $1 billion? Now the value of the game might be as high as $30 if you're confident the wager is legally enforceable. 1 Trillion? 40 bucks. But saying the EV is infinite is just wrong, because it ignores the reality that no entity is good for the amounts that would be required to keep that series infinite.
Log utility can also be applied, but is generally unnecessary unless a large bank or casino is offering the wager. If your friend is selling you a game and you don't trust them to pay out huge sums or that your wager is legally enforceable, you might cap the value of the wager at something closer to $5 or $10 without log utility coming into play at all.
In context, the argument about starting FTX with a low probability of success was that opportunity costs of not focusing on Alameda were high. Launching a startup with low odds of success while your existing startup is doing well is a tricky bet and does depend on risk tolerance.
And the argument about taking a 10,000x bet that pays off 10% of the time is explicitly framed as a one-shot, so the Kelly derivation doesn't actually apply. And even there, he's saying he'd bet half his bankroll, which is over Kelly but is far from risk neutral (where you'd bet 100%).
I hope the reintroduction of Kelly is obvious: pad out that one-shot opportunity with an arbitrarily large number of "do I put on pants today?" bets, and maximize over the long-run results of this inhomogeneous thing. Arguably, this matches the observations of ordinary risky behaviors of the "drunk teens go driving" kind.
Donating to charity doesn't make the money/economic activity disappear though. If you somehow managed to donate $2 trillion through GiveDirectly, you'd just be shifting resources to the recipients. There would probably be shocks as suddenly whatever-poor-people-want jobs increasing hiring and investment and whatever-the-alternative-was jobs lose investment, but the net effect should still be similar amounts of economic activity.
Nice post!
I have a tiny technical nitpick -- you make a subtle error when you say "by the Law of Large Numbers, since the X_n are independent, identically distributed random variables, this sum converges almost surely to n times the expected value". This is actually not true! The proper statement of the strong LLN is "this sum divided by n converges almost surely to the expected value". These two statements seem like they should be equivalent, but because of some subtleties around limits and convergence, they actually are not. It's fun to think about why!
One thing that might help show why your original statement must be false, is to note that by the Central Limit Theorem, the distribution of the random variable corresponding to the sum of the X_n will approach a normal distribution centered at n * E[X], which clearly means it doesn't converge in probability (or almost surely) to any one value.
> But these zero-probability scenarios are still factored into expected value calculations, which allows expected value to be maximized by the all-in betting strategy.
If the set of these as a whole has probability zero, then it won't contribute to expected value. So that can't be what's going on; it has to be something else.
In the scenario with N double-or-nothing flips of a slightly tilted (51% win) coin, the only scenario where "all-in" betting doesn't give you $0 is the one where you get N consecutive coin flips. This is the only event that contributes a positive term to the expected value of the betting strategy.
For finite N, the probability of this "win scenario" is nonzero; it's 2^(-N).
In the limit as N goes to infinity, though, the probability that you are in the "win condition" goes to zero.
OK, but that's different from what you wrote above. Are you considering an infinite series of flips, or are you considering a finite sequence of flips and taking limits of associated statistics?
Like, if you are looking at an actual infinite series of flips, that is where the law of large numbers applies. And in that setting -- or any fixed setting! -- a set of measure zero does not affect the expected value.
Now in your reply above you describe a sequence of probabilities that approach zero, and expected values that approach infinity, but that's not the same thing. Not everything is continuous. I mean with the probabilities sure, but evidently not with the utilities. Perhaps that's what's actually going wrong here and causing the discrepancy you discuss? That because the convergence isn't uniform (or dominated or monotone or whatever), expected value doesn't act continuously? Because what you wrote doesn't seem to make any sense...
Yeah, I think the point is that we don't have a single random variable on the space of infinite sequences of coin flips with expected value infinity and measure zero success event, here; we have a sequence of random variables with expected value converging to infinity and the measure of success converging to zero. (These random variables converge in distribution to the always-zero random variable.)These aren't quite the same thing but I don't think it affects the argument.
control-f "Kelly Criterion" :)
Note: Kelly criterion is only optimal in the sense you mean if you expect to be offered an infinite sequence of positive expected value bets on which you can bet any percentage (up to 100%) of your current bankroll. If you expect to have only a fixed, finite number of such opportunities, optimizing EV may well be the way to go.
The most of your utility is concentrated on some event with vanishingly small probability objection is also somewhat mitigated if you are part of a large EA movement and are making bets that are independent of those made by other people in the movement.
I think section two is playing fast and loose with words, and I'm not convinced that the derivation is correct.
> You have repeated opportunities to make a bet which is a binary random variable; with probability p you get a return of b times the amount you bet, and with probability (1-p) you lose and get zero.
> Each bet is an independent random event.
> You have to stop betting when your bankroll runs out
> You care about your long-run total money, after “many” repeated bets.
What is "many" here? I tend to take "many" to mean some large $n$. The post jumps between this being some large value and it being infinity.
In particular, it derives that the value is
> W_0 (e^(E[log(X)])(1 + o(1)))^n
and says
> As n becomes large, the small-o terms disappear, and you just get W_n ~ W_0 (e^(E[log(X)])^n)
No you don't. I'm not sure that turning the o(n) term into an o(1) term that doesn't depend on $n$ was valid to begin with (I'm not sure that using big-o itself was a good idea), but it's in the exponent, and you can't just drop it. If you want to see what this converges to as n approaches infinity, you have to take a limit.
The reality is that this process, like everything else in the real world, does not go to infinity. There is no event of measure zero to discuss. The expected value of betting everything is p^n * b^n for however many betting opportunities show up. And it's easy to show that any deviation from that lowers your expect value. It's also easy to see that p^n is going to be very small if p < 0.9 and n > 100, so you will probably (not certainly!) lose all your money.
Really, this could be rolled into the St. Petersburg paradox discussion – I don't think this is adding anything except for an unnecessary discussion of infinities (and thereby almost sure convergence and measure zero).
The resolution to the St. Petersburg Paradox is, in banking parlance, risk management, or more specifically, credit risk management or counterparty credit risk. Can your counterparty afford to pay out $1 million? If so, you might be willing to pay up to a maximum of around $20 to play. How about $1 billion? Now the value of the game might be as high as $30 if you're confident the wager is legally enforceable. 1 Trillion? 40 bucks. But saying the EV is infinite is just wrong, because it ignores the reality that no entity is good for the amounts that would be required to keep that series infinite.
Log utility can also be applied, but is generally unnecessary unless a large bank or casino is offering the wager. If your friend is selling you a game and you don't trust them to pay out huge sums or that your wager is legally enforceable, you might cap the value of the wager at something closer to $5 or $10 without log utility coming into play at all.
In context, the argument about starting FTX with a low probability of success was that opportunity costs of not focusing on Alameda were high. Launching a startup with low odds of success while your existing startup is doing well is a tricky bet and does depend on risk tolerance.
And the argument about taking a 10,000x bet that pays off 10% of the time is explicitly framed as a one-shot, so the Kelly derivation doesn't actually apply. And even there, he's saying he'd bet half his bankroll, which is over Kelly but is far from risk neutral (where you'd bet 100%).
I hope the reintroduction of Kelly is obvious: pad out that one-shot opportunity with an arbitrarily large number of "do I put on pants today?" bets, and maximize over the long-run results of this inhomogeneous thing. Arguably, this matches the observations of ordinary risky behaviors of the "drunk teens go driving" kind.
Great explanation! Thanks!
Donating to charity doesn't make the money/economic activity disappear though. If you somehow managed to donate $2 trillion through GiveDirectly, you'd just be shifting resources to the recipients. There would probably be shocks as suddenly whatever-poor-people-want jobs increasing hiring and investment and whatever-the-alternative-was jobs lose investment, but the net effect should still be similar amounts of economic activity.