I think the interesting thing is what does AB+CD actually means? If we treat the fraction of decisions at X as the probability here is X, same for Y, then AB+CD should be the expected payoff of this decision. Typically the best decision should be derived by maximizing it. But clearly, that leads to wrong results such as 4⁄9. So what’s wrong?
My position is that AB+CD is meaningless. It is a fallacious payoff because self-locating probabilities are invalid. This also resolves double-halfers problems. But let’s leave that aside for the time being.
If I understand correctly, your position is maximizing AB+CD is not correct. Because when deciding we should be maximizing the payoff of runs instead of this decision. Here I just want to point out the payoff for runs (planning utility function) does not use self-locating probabilites. You didn’t say if AB+CD is something meaningful or not.
Aumann thinks AB+CD is meaningful. However, maximizing it is wrong. He pointed out that the decision at X and at Y are causally disconnected, yet the two decisions ought to be the same. A symmetrical Nash equilibrium. So the correct decision is a stable point of AB+CD. The problem is when there are multiple stable points, which one is the optimal decision, the one with the highest AB+CD? Aumann says no. It should be the point that maximizes the planning payoff function (the function not using self-locating probabilities).
I am not convinced by this. First of all, it lacks compelling reason. The explanation is ad-hoc “due to the absentmindedness”. Second, by his reasoning, AB+CD effectively plays no part in the decision-making process. The decision maximizing planning utility function is always going to be a stable point of AB+CD, and it is always going to be chosen regardless of what value it gives for AB+CD. So the whole argument about AB+CD being meaningful lacks substantial support.
Each of A,B,C,D are measurable, and so AB+CD is measurable as well, but since that combination isn’t the expected value we want to be maximizing, I would certainly say it’s not useful and at that point I wouldn’t really care whether it’s meaningful (if it does “mean” anything, it would be something convoluted and not really relevant to the decision).
I think the interesting thing is what does AB+CD actually means? If we treat the fraction of decisions at X as the probability here is X, same for Y, then AB+CD should be the expected payoff of this decision. Typically the best decision should be derived by maximizing it. But clearly, that leads to wrong results such as 4⁄9. So what’s wrong?
My position is that AB+CD is meaningless. It is a fallacious payoff because self-locating probabilities are invalid. This also resolves double-halfers problems. But let’s leave that aside for the time being.
If I understand correctly, your position is maximizing AB+CD is not correct. Because when deciding we should be maximizing the payoff of runs instead of this decision. Here I just want to point out the payoff for runs (planning utility function) does not use self-locating probabilites. You didn’t say if AB+CD is something meaningful or not.
Aumann thinks AB+CD is meaningful. However, maximizing it is wrong. He pointed out that the decision at X and at Y are causally disconnected, yet the two decisions ought to be the same. A symmetrical Nash equilibrium. So the correct decision is a stable point of AB+CD. The problem is when there are multiple stable points, which one is the optimal decision, the one with the highest AB+CD? Aumann says no. It should be the point that maximizes the planning payoff function (the function not using self-locating probabilities).
I am not convinced by this. First of all, it lacks compelling reason. The explanation is ad-hoc “due to the absentmindedness”. Second, by his reasoning, AB+CD effectively plays no part in the decision-making process. The decision maximizing planning utility function is always going to be a stable point of AB+CD, and it is always going to be chosen regardless of what value it gives for AB+CD. So the whole argument about AB+CD being meaningful lacks substantial support.
Each of A,B,C,D are measurable, and so AB+CD is measurable as well, but since that combination isn’t the expected value we want to be maximizing, I would certainly say it’s not useful and at that point I wouldn’t really care whether it’s meaningful (if it does “mean” anything, it would be something convoluted and not really relevant to the decision).