That isn’t the issue. At the point in time I am talking about, the adversary has already made his non-revealed choice (and he is not telekinetic). There is no other state.
Tails versus Heads is objectively 1:1 resulting from the toss of a fair coin, whereas A versus B has an uncertainty that results from my adversary’s choice. I may not have reason to think that he will choose A over B, so I can still call it 1:1, but there is still a qualitative distinction between uncertainty and randomness, or ambiguity and risk, or objective and subjective probability, or whatever you want to call it, and it is not irrational to take it into account.
I have to admit, this ordering seem reasonable… for the reasons nshepperd suggests. Just saying that he’s not telepathic isn’t enough to say he’s not any sort of predictor—after all, I’m a human, I’m bad at randomizing, maybe he’s played this game before and compiled statistics. Or he just has a good idea how peope tend to think about this sort of thing. So I’m not sure you’re correct in your conclusion that this isn’t the issue.
Then I claim that a non-psychic predictor, no matter how good, is very different from a psychic.
The powers of a non-psychic predictor are entirely natural and causal. Once he has written down his hidden choice, then he becomes irrelevant. If this isn’t clear, then we can make an analogy with the urn example. After the ball is drawn but before its colour is revealed, the contents of the urn are irrelevant. As I pointed out, the urn could even be destroyed before the colour of the ball is revealed, so that the ball’s colour truly is the only state. Similarly, after the predictor writes his choice but before it is revealed, he might accidentally behead himself while shaving.
Now of course your beliefs about the talents of the late predictor might inform your beliefs about his hidden choice. But that’s the only way they can possibly be releveant. The coin and the predictor’s hidden choice on the paper really are the only states of the world now, and your own choice is free and has no effect on the state. So, if you display a strict preference for the coin, then your uncertainty is still not captured by subjective probability. You still violate P2.
To get around this, it seems you would have to posit some residual entanglement between your choice and the external state. To me this sounds like a strange thing to argue. But I suppose you could say your cognition is flawed in a way that is invisible to you, yet was visible to the clever but departed predictor. So, you might argue that, even though there is no actual psychic effect, your choice is not really free, and you have to take into account your internalities in addition to the external states.
My question then would be, does this entanglement prevent you from having a total ordering over all maps from states (internal and external) to outcomes? If yes, then P1 is violated. If no, then can I not just ask you about the ordering of the maps which only depend on the external states, and don’t we just wind up where we were?
Because there might be more to uncertainty than subjective probability.
Let’s take a step back.
Yes, if you assume that uncertainty is entirely captured by subjective probability, then you’re completely right. But if you assume that, then you wouldn’t need the Savage axioms in the first place. The Savage axioms are one way of justifying this assumption (as well as expected utility). So, what justifies the Savage axioms?
One suggestion the original poster made was to use Dutch book arguments, or the like. But now here’s a situation where there does seem to be a qualitative difference between a random event and an uncertain event, where there is a “reasonable” thing to do that violates P2, and where nothing like a Dutch book argument seems to be available to show that it is suboptimal.
I hope that clarifies the context.
EDIT: I put “reasonable” in scare-quotes. It is reasonable, and I am prepared to defend that. But it isn’t necessary to believe it is reasonable to see why this example matters in this context.
That isn’t the issue. At the point in time I am talking about, the adversary has already made his non-revealed choice (and he is not telekinetic). There is no other state.
Tails versus Heads is objectively 1:1 resulting from the toss of a fair coin, whereas A versus B has an uncertainty that results from my adversary’s choice. I may not have reason to think that he will choose A over B, so I can still call it 1:1, but there is still a qualitative distinction between uncertainty and randomness, or ambiguity and risk, or objective and subjective probability, or whatever you want to call it, and it is not irrational to take it into account.
I have to admit, this ordering seem reasonable… for the reasons nshepperd suggests. Just saying that he’s not telepathic isn’t enough to say he’s not any sort of predictor—after all, I’m a human, I’m bad at randomizing, maybe he’s played this game before and compiled statistics. Or he just has a good idea how peope tend to think about this sort of thing. So I’m not sure you’re correct in your conclusion that this isn’t the issue.
Then I claim that a non-psychic predictor, no matter how good, is very different from a psychic.
The powers of a non-psychic predictor are entirely natural and causal. Once he has written down his hidden choice, then he becomes irrelevant. If this isn’t clear, then we can make an analogy with the urn example. After the ball is drawn but before its colour is revealed, the contents of the urn are irrelevant. As I pointed out, the urn could even be destroyed before the colour of the ball is revealed, so that the ball’s colour truly is the only state. Similarly, after the predictor writes his choice but before it is revealed, he might accidentally behead himself while shaving.
Now of course your beliefs about the talents of the late predictor might inform your beliefs about his hidden choice. But that’s the only way they can possibly be releveant. The coin and the predictor’s hidden choice on the paper really are the only states of the world now, and your own choice is free and has no effect on the state. So, if you display a strict preference for the coin, then your uncertainty is still not captured by subjective probability. You still violate P2.
To get around this, it seems you would have to posit some residual entanglement between your choice and the external state. To me this sounds like a strange thing to argue. But I suppose you could say your cognition is flawed in a way that is invisible to you, yet was visible to the clever but departed predictor. So, you might argue that, even though there is no actual psychic effect, your choice is not really free, and you have to take into account your internalities in addition to the external states.
My question then would be, does this entanglement prevent you from having a total ordering over all maps from states (internal and external) to outcomes? If yes, then P1 is violated. If no, then can I not just ask you about the ordering of the maps which only depend on the external states, and don’t we just wind up where we were?
Well, that sounds irrational. Why would you pay to switch from X to U, a change that makes no difference to the probability of you winning?
Because there might be more to uncertainty than subjective probability.
Let’s take a step back.
Yes, if you assume that uncertainty is entirely captured by subjective probability, then you’re completely right. But if you assume that, then you wouldn’t need the Savage axioms in the first place. The Savage axioms are one way of justifying this assumption (as well as expected utility). So, what justifies the Savage axioms?
One suggestion the original poster made was to use Dutch book arguments, or the like. But now here’s a situation where there does seem to be a qualitative difference between a random event and an uncertain event, where there is a “reasonable” thing to do that violates P2, and where nothing like a Dutch book argument seems to be available to show that it is suboptimal.
I hope that clarifies the context.
EDIT: I put “reasonable” in scare-quotes. It is reasonable, and I am prepared to defend that. But it isn’t necessary to believe it is reasonable to see why this example matters in this context.