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It would kind of use assumption 3 inside step 1, but inside the syntax, rather than in the metalanguage. That is, step 1 involves checking that the number encoding "this proof" does in fact encode a proof of C. This can't be done if you never end up proving C.

One thing that might help make clear what's going on is that you can follow the same proof strategy, but replace "this proof" with "the usual proof of Lob's theorem", and get another valid proof of Lob's theorem, that goes like this: Suppose you can prove that []C->C, and let n be the number encoding a proof of C via the usual proof of Lob's theorem. Now we can prove C a different way like so:

  1. n encodes a proof of C.
  2. Therefore []C.
  3. By assumption, []C->C.
  4. Therefore C.

Step 1 can't be correctly made precise if it isn't true that n encodes a proof of C.

The revelation that he spent maybe 10x as much on villas for his girlfriends as EA cause areas


The idea that he was trying to distance himself from EA to protect EA doesn't hold together because he didn't actually distance himself from EA at all in that interview. He said ethics is fake, but it was clear from context that he meant ordinary ethics, not utilitarianism.

"Having been handed this enormous prize, how do I maximize the probability that I max out on utility?" Hm, but that actually doesn't give back any specific criterion, since basically any strategy that never bets your whole stack will win.

That's not quite true. If you bet more than double Kelly, your wealth decreases. But yes, Kelly betting isn't unique in growing your wealth to infinity in the limit as number of bets increases.

If the number of bets is very large, but due to some combination of low starting wealth relative to the utility bound and slow growth rate, it is not possible to get close to maximum utility, then Kelly betting should be optimal.

I basically endorse what kh said. I do think it's wrong to think you can fit enormous amounts of expected value or disvalue into arbitrarily tiny probabilities.

It is true that in practice, there's a finite amount of credit you can get, and credit has a cost, limiting the practical applicability of a model with unlimited access to free credit, if the optimal strategy according to the model would end up likely making use of credit which you couldn't realistically get cheaply. None of this seems important to me. The easiest way to understand the optimal strategy when maximum bet sizes are much smaller than your wealth is that it maximizes expected wealth on each step, rather than that it maximizes expected log wealth on each step. This is especially true if you don't already understand why following the Kelly criterion is instrumentally useful, and I hadn't yet gotten to the section where I explained that, and in fact used the linear model in order to show that Kelly betting is optimal by showing that it's just the linear model on a log scale.

One could similarly object that since currency is discrete, you can't go below 1 unit of currency and continue to make bets, so you need to maintain a log-scale bankroll where you prevent your log wealth from going negative, and you should really be maximizing your expected log log wealth, which happens to give you the same results when your wealth is a large enough number of currency units that the discretization doesn't make a difference. Like, sure, I guess, but it's still useful to model currency as continuous, so I see no need to account for its discreteness in a model. Similarly, in situations where the limitations on funds available to place bets with don't end up affecting you, I don't think it needs to be explicitly included in the model.

Access to credit. In the logarithmic model, you never make bets that could make your net worth zero or negative.

Again, the max being a small portion of your net worth isn't the assumption behind the model; the assumption is just that you don't get constrained by lack of funds, so it is a different model. It's true that if the reason you don't get constrained by lack of funds is that the maximum bets are small relative to your net worth, then this is also consistent with maximizing log wealth on each step. But this isn't relevant to what I brought it up for, which was to use it as a step in explaining the reason for the Kelly criterion in the section after it.

No. The point of the model where acting like your utility is linear is optimal wasn't that this is a more realistic model than the assumptions behind the Kelly criterion; it's just another simplified model, which is slightly easier to analyze, so I was using it as a step in showing why you should follow the Kelly criterion when it is your wealth that constrains the bet sizes you can make. It's also not true that the linear-utility model I described is still just maximizing log wealth; for instance, if the reason that you're never constrained by available funds is that you have access to credit, then your wealth could go negative, and then its log wouldn't even be defined.

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