I’ve been wanting to do something like this for a while, so it’s good to see it properly worked out here.
If you wanted to expand this you could look at games which weren’t symmetrical in the players. So you’d have eight variables, W, X, Y and Z, and w, x, y and z. But you’d only have to look at the possible orderings within each set of four, since it’s not necessarily valid to compare utilities between people. You’d also be able to reduce the number of games by using the swap-the-players symmetry.
Right. But also we would want to use a prior that favoured biases which were near fair, since we know that Wolf at least thought they were a normal pair of dice.
Suppose I’m trying to infer probabilities about some set of events by looking at betting markets. My idea was to visualise the possible probability assignments as a high-dimensional space, and then for each bet being offered remove the part of that space for which the bet has positive expected value. The region remaining after doing this for all bets on offer should contain the probability assignment representing the “market’s beliefs”.
My question is about the situation where there is no remaining region. In this situation for every probability assignment there’s some bet with a positive expectation. Is it a theorem that there is always an arbitrage in this case? In other words, can one switch the quantifiers from “for all probability assignments there exists a positive expectation bet” to “there exists a bet such that for all probability assignments the bet has positive expectation”?
I believe you missed one of the rules of Gurkenglas’ game, which was that there are at most 100 rounds. (Although it’s possible I misunderstood what they were trying to say.)
If you assume that play continues until one of the players is bankrupt then in fact there are lots of winning strategies. In particular betting any constant proportion less than 38.9%. The Kelly criterion isn’t unique among them.
My program doesn’t assume anything about the strategy. It just works backwards from the last round and calculates the optimal bet and expected value for each possible amount of money you could have, on the basis of the expected values in the next round which it has already calculated. (Assuming each bet is a whole number of cents.)
If you wager one buck at a time, you win almost certainly.
But that isn’t the Kelly criterion! Kelly would say I should open by betting two bucks.
In games of that form, it seems like you should be more-and-more careful as the amount of bets gets larger. The optimal strategy doesn’t tend to Kelly in the limit.
EDIT: In fact my best opening bet is $0.64, leading to expected winnings of $19.561.
EDIT2: I reran my program with higher precision, and got the answer $0.58 instead. This concerned me so I reran again with infinite precision (rational numbers) and got that the best bet is $0.21. The expected utilities were very similar in each case, which explains the precision problems.
EDIT3: If you always use Kelly, the expected utility is only $18.866.
Can you give a concrete example of such a game?
even if your utility outside of the game is linear, inside of the game it is not.
Are there any games where it’s a wise idea to use the Kelly criterion even though your utility outside the game is linear?
Marginal utility is decreasing, but in practice falls off far less than geometrically.
I think this is only true if you’re planning to give the money to charity or something. If you’re just spending the money on yourself then I think marginal utility is literally zero after a certain point.
Yeah, I think that’s probably right.
I thought of that before but I was a bit worried about it because Löb’s Theorem says that a theory can never prove this axiom schema about itself. But I think we’re safe here because we’re assuming “If T proves φ, then φ” while not actually working in T.
I’m arguing that, for a theory T and Turing machine P, “T is consistent” and “T proves that P halts” aren’t together enough to deduce that P halts. And as I counter example I suggested T = PA + “PA is inconsistent” and P = “search for an inconsistency in PA”. This P doesn’t halt even though T is consistent and proves it halts.
So if it doesn’t work for that T and P, I don’t see why it would work for the original T and P.
Consistency of T isn’t enough, is it? For example the theory (PA + “The program that searches for a contradiction in PA halts”) is consistent, even though that program doesn’t halt.
This is a good point. The Wikipedia pages for other sites, like Reddit, also focus unduly on controversy.
And the fact that situations like that occurred in humanity’s evolution explains why humans have the preference for certainty that they do.
As well as ordinals and cardinals, Eliezer’s construction also needs concepts from the areas of computability and formal logic. A good book to get introduced to these areas is Boolos’ “Computability and Logic”.
being unable to imagine a scenario where something is possible
This isn’t an accurate description of the mind projection fallacy. The mind projection fallacy happens when someone thinks that some phenomenon occurs in the real world but in fact the phenomenon is a part of the way their mind works.
But yes, it’s common to almost all fallacies that they are in fact weak Bayesian evidence for whatever they were supposed to support.
Eliezer made this attempt at naming a large number computable by a small Turing machine. What I’m wondering is exactly what axioms we need to use in order to prove that this Turning machine does indeed halt. The description of the Turing machine uses a large cardinal axiom (“there exists an I0 rank-into-rank cardinal”), but I don’t think that assuming this cardinal is enough to prove that the machine halts. Is it enough to assume that this axiom is consistent? Or is something stronger needed?
games are a specific case where the utility (winning) is well-defined
Lots of board games have badly specified utility functions. The one that springs to mind is Diplomacy; if a stalemate is negotiated then the remaining players “share equally in a draw”. I’d take this to mean that each player gets utility 1/n (where there are n players, and 0 is a loss and 1 is a win). But it could also be argued that they each get 1/(2n), sharing a draw (1/2) between them (to get 1/n each wouldn’t they have to be “sharing equally in a win”?).
Another example is Castle Panic. It’s allegedly a cooperative game. The players all “win” or “lose” together. But in the case of a win one of the players is declared a “Master Slayer”. It’s never stated how much the players should value being the Master Slayer over a mere win.
Interesting situations occur in these games when the players have different opinions about the value of different outcomes. One player cares more about being the Master Slayer than everyone else, so everyone else lets them be the Master Slayer. They think that they’re doing much better that everyone else, but everyone else is happy so long as they all keep winning.
I actually learnt quantum physics from that sequence, and I’m now a mathematician working in Quantum Computing. So it can’t be too bad!
The explanation of quantum physics is the best I’ve seen anywhere. But this might be because it explained it in a style that was particularly suited to me. I really like the way it explains the underlying reality first and only afterwards explains how this corresponds with what we perceive. A lot of other introductions follow the historical discovery of the subject, looking at each of the famous experiments in turn, and only building up the theory in a piecemeal way. Personally I hate that approach, but I’ve seen other people say that those kind of introductions were the only ones that made sense to them.
The sequence is especially good if you don’t want a math-heavy explantation, since it manages to explain exactly what’s going on in a technically correct way, while still not using any equations more complicated than addition and multiplication (as far as I can remember).
The second half of the sequence talks about interpretations of quantum mechanics, and advocates for the “many-worlds” interpretation over “collapse” interpretations. Personally I found it sufficient to convince me that collapse interpretations were bullshit, but it didn’t quite convince me that the many-worlds interpretation is obviously true. I find it plausible that the true interpretation is some third alternative. Either way, the discussion is very interesting and worth reading.
As far as “holding up” goes, I once read through the sequence looking for technical errors and only found one. Eliezer says that the wavefunction can’t become more concentrated because of Liouville’s theorem. This is completely wrong (QM is time-reversible, so if the wavefunction can become more spread out it must also be able to become more concentrated). But I’m inclined to be forgiving to Eliezer on this point because he’s making exactly the mistake that he repeatedly warns us about! He’s confusing the distribution described by the wavefunction (the uncertainty that we would have if we performed a measurment) with the uncertainty we do have about the wavefunction (which is what Liouville’s theorem actually applies to).
Really, the fact that different sizes of moral circle can incentivize coercion is just a trivial corollary of the fact that value differences in general can incentivize coercion.