Precommitment is an interesting aspect of game theory that ties in well with lukeprog's how to beat procrastination.

Here is the course description:

"An examination of how rational people cooperate and compete. Game theory explores situations in which everyone's actions affect everyone else, and everyone knows this and takes it into account when determining their own actions. Business, military and dating strategies will be examined."

When teaching economics I strive to relate all the material to my students' lives and concerns, rather than the type of abstract mathematical concepts that often capture economists' interests.

Particularly when it comes to public policy.

That would require being able to predict the results of public policy decisions with a reasonable degree of accuracy.

Back to Newcomb's problem: Say that brown-haired people almost always one-box, and people with other hair colors almost always two-box. Omega predicts on the basis of hair color: both boxes are filled iff you have brown hair. I'd two-box, even though I have brown hair. It would be logically inconsistent for me to find that one of the boxes is empty, since everyone with brown hair has both boxes filled. But this could be true of any attribute Omega uses to predict.

If the agent filling the boxes follows a consistent, predictable pattern you're outside of, you can certainly use that information to do this. In Newcomb's Problem though, Omega follows a consistent, predictable pattern you're inside of. It's logically inconsistent for you to two box and find they both contain money, or pick one box and find it's empty.

I agree that changing my decision conveys information about what is in the boxes and changes my guess of what is in the boxes... but doesn't change the boxes.

Why is whether your decision actually changes the boxes important to you? If you know that picking one box will result in your receiving a million dollars, and picking two boxes will result in getting a thousand dollars, do you have any concern that overrides making the choice that you expect to make you more money?

A decision process of "at all times, do whatever I expect to have the best results" will, at worst, reduce to exactly the same behavior as "at all times, do whatever I think will have a causal relationship with the best results." In some cases, such as Newcomb's problem, it has better results. What do you think the concern with causality actually does for you?

We don't always agree here on what decision theories get the best results (as you can see by observing the offshoot of this conversation between Wedrifid and myself,) but what we do generally agree on here is that the quality of decision theories is determined by their results. If you argue yourself into a decision theory that doesn't serve you well, you've only managed to shoot yourself in the foot.

Ok, but what if Ann's mom is right 99% of the time about how you would choose when playing her?

I would one-box. I gave the relevant numbers on this in my previous comment; one-boxing has an expected value of $990,000,000 to the expected $10,001,000 if you two-box.

I agree that one-boxers make more money, with the numbers you used, but I don't think that those are the appropriate expected values to consider. Conditional on the fact that the boxes have already been filled, two-boxing has a $1000 higher expected value. If I know only one box is filled, I should take both. If I know both boxes are filled, I should take both. If I know I'm in one of those situations but not sure of which it is, I should still take both.

When you're dealing with a problem involving an effective predictor of your own mental processes (it's not necessary for such a predictor to be perfect for this reasoning to become salient, it just makes the problems simpler,) your expectation of what the predictor will do or already have done will be at least partly dependent on what you intend to do yourself. You know that either the opaque box is filled, or it is not, but the probability you assign to the box being filled depends on whether you intend to open it or not.

Let's try a somewhat different scenario. Suppose I have a time machine that allows me to travel back a day in the past. Doing so creates a stable time loop, like the time turners in Harry Potter or HPMoR (on a side note, our current models of relativity suggest that such loops are possible, if very difficult to contrive.) You're angry at me because I've insulted your hypothetical scenario, and are considering hitting me in retaliation. But you happen to know that I retaliate against people who hit me by going back in time and stealing from them, which I always get away with due to having perfect alibis (the police don't believe in my time machine.) You do not know whether I've stolen from you or not, but if I have, it's already happened. You would feel satisfied by hitting me, but it's not worth being stolen from. Do you choose to hit me or not?

Another analogous situation would be that you walk into an exam, and the professor (who is a perfect or near-perfect predictor) announces that he has written down a list of people whom he has predicted will get fewer than half the questions right. If you are on that list, he will add 100 points to your score at the end. The people who get fewer than half of the questions right get higher scores, but you should still try to get questions right on the test... right? If not, does the answer change if the professor posts the list on the board?

If the professor is a perfect predictor, then I would deliberately get most of the problems wrong, thereby all but guaranteeing a score of over 100 points. I would have to be very confident that I would get a score below fifty even if I weren't trying to on purpose before trying to get all the questions right would give me a higher expected score than trying to get most of the questions wrong.

If the professor posts the list on the board, then of course it should affect the answer. If my name isn't on the list, then he's not going to add the 100 points to my test in any case, so my only recourse to maximizing my grade is to try my best on the test. If my name is on the list, then he's already predicted that I'm going to score below 50, so whether he's a perfect predictor or not, I should try to do well so that he's adding 100 points to as high a score as I can manage.

The difference between the scenario where he writes the names on the board and the scenario where he doesn't is that in the former, my expectations of his actions don't vary according to my own, whereas in the latter, they do.

Yes, then, following the utility function you specified, I would gladly risk $100 for an even chance at $10000. Since Omega's omniscient, I'd be honest about it, too, and cough up the money if I lost.

If it's rational to do this when Omega asks you in advance, isn't it also rational to make such a commitment right now? Whether you make the commitment in response to Omega's notification, or on a whim when considering the thought experiment in response to a blog post makes no difference to the payoff. If you now commit to a "if this exact situation comes up, I will commit to paying the $100 if I lose the coinflip", and p(x) is the probability of this situation occurring, you will achieve a net gain of $4950*p(x) over a non-committer (a very small number admittedly given that p(x) is tiny, but for the sake of the thought experiment all that matters is that it's positive.)

Given that someone who makes such a precommitment comes out ahead of someone who doesn't - shouldn't you make such a commitment right now? Extend this and make a precommitment to always make the decision to perform the action that would maximise your average returns in all such newcombelike situations and you're going to come off even better on average.

You choose the boxes according to the expected value of each box choice. For a 99% accurate predictor, the expected value of one-boxing is $990,000,000 (you get a billion 99% of the time, and nothing 1% of the time,) while the expected value of two-boxing is $10,001,000 (you get a thousand 99% of the time, and one billion and one thousand 1% of the time.)

The difference between this scenario and the one you posited before, where Ann's mom makes her prediction by reading your philosophy essays, is that she's presumably predicting on the basis of how she would expect you to choose if you were playing Omega. If you're playing against an agent who you know will fill the boxes according to how you would choose if you were playing Omega (we'll call it Omega-1,) then you should always two-box (if you would one-box against Omega, both boxes will contain money, so you get the contents of both. If you would two-box against Omega, only one box would contain money, and if you one-box you'll get the empty one.)

An imperfect predictor with random error is a different proposition from an imperfect predictor with nonrandom error.

Of course, if I were dealing with this dilemma in real life, my choice would be heavily influenced by considerations such as how likely it is that Ann's mom really has billions of dollars to give away.

You've basically come up with four criteria that describe the use of the word "signal" in a highly specific context - traits that exist for pure signalling purposes in evolution or game theory - and then decided, arbitrarily, that this is the one true meaning of "signal." I do not think you have provided adequate evidence or argument to back this claim up.

If everyone around me is a Republican and I am not, it might make sense that I would do things that would signal that I am a Republican, even if these are very cheap and have obvious positive returns. Your definition would not allow this - if it is cheap and has obvious positive returns, it is not "signaling" to you. What you're saying is that if I send a birthday card to a coworker I hate, then I am not "signaling" that I like that person because it's too cheap to send the card.

It may make sense to speak of weak or strong signals, or reliable or unreliable or misleading signals. But you've arbitrarily said that the word applies only when a certain arbitrary threshold is crossed (your 2 and 4).

Incidentally, your theory might actually work if 4 were eliminated and 2 read "the behaviour is more likely to occur if you possess a certain characteristic than if you do not." This would cover my birthday card example - it's cheap, but I'm more likely to do it if I like the person, so it does signal liking the person. But this change would also fix the counter-productive manager. She's doing things that she is more likely to do if she is decisive and in charge. Since she's being evaluated on those criteria, and not "good manager-ness" - which is not generally observable - it would make sense that she would choose to give those signals rather than not. But revising the theory appropriately seems to nullify most or all of your objections.

Other factors that could explain:

The difference is not as large, either because of the file drawer effect, or because someone selected / massaged the data to make the difference look bigger (the researcher or the journalist).

Selection effects: men and women may go into economics for different reasons; for example (as a bit of a caricature); men who want to get obscenely rich study economics to get into business, and women who want to get obscenely rich try to marry into money, and money-grabiness is correlated with pro-free market views.

Differences in peer groups: there seem to be more men than women majoring in economics, so assuming one's views are influenced by peers of the same sex, it seems likely female students will have more non-economist peers.

Differences in conformity: women may conform a bit more to widespread social views (at least, to views of "their social class") and/or compartimentalize more between what they learn about a specific topic and their general views. This would mean female scientists would be slightly less likely to be atheists in religious countries, female theology students would be slightly less likely to be fanatics in not-that-fanatical societies, etc.

Changes in major: I don't know how frequent changes of major are, but if they are frequent it seems likely you'd see more women than men coming from social sciences in economics (and more men coming from mathematics).

Different subfields in economics: Maybe "economics" shouldn't be considered one big blob - there may be some subfields that have more in common with other social sciences (and thus have a more female student body, and a more "liberal" outlook), and some more in common with maths and business.

"You go to visit your friend Ann, and her mom pulls you into the kitchen, where two boxes are sitting on a table. She tells you that box A has either $1 billion or $0, and box B has $1,000. She says you can take both boxes or just A, and that if she predicted you take box B she didn't put anything in A. She has done this to 100 of Anne's friends and has only been wrong for one of them. She is a great predictor because she has been spying on your philosophy class and reading your essays."

To be properly isomorphic to the Newcomb's problem, the chance of the predictor being wrong should approximate to zero.

If I thought that the chance of my friend's mother being wrong approximated to zero, I would of course choose to one-box. If I expected her to be an imperfect predictor who assumed I would behave as if I were in the real Newcomb's problem with a perfect predictor, then I would choose to two-box.

In Newcomb's Problem, if you choose on the basis of which choice is consistent with a higher expected return, then you would choose to one-box. You know that your choice doesn't cause the box to be filled, but given the knowledge that whether the money is in the box or not is contingent on a perfect predictor's assessment of whether or not you were likely to one-box, you should assign different probabilities to the box containing the money depending on whether you one-box or two-box. Since your own mental disposition is evidence of whether the money is in the box or not, you can behave as if the contents were determined by your choice.