> Before starting the drive, the driver determines that always turning at the first intersection will be optimal. I didn’t think we disagreed on that.
But the driver does not have to do any calculation before starting the drive. He can do that, yes. He also can simply choose to only think about the decision when arrived at an intersection. It is possible for him to derive the “action optimals” chronologically before deriving the “planning optimal”. As I said earlier, they are two independent processes.
>Yes, it is. You can verify this by finding the explicit expression for action utility as a function of p....
No, it was not found by maximizing the action utility function. In Aumann’s process, the action utility function was not represented by a single variable p, but with multiple variables representing casually disconnected decisions (observation 1). Because the decisions ought to be the same (observations 2) the action optimal ought to be symmetrical Nash equilibriums or “stable points”. You can see an example in Eliezer Yudkowsky’s post. For this particular problem, there are three stable points for the action utility functions. p=0, p=7/30 and p=1/2. Among these three p=1/2 gives the highest action payoff, 7⁄30 the lowest.
I will take your words for it that p=1/2 also maximizes action utility. But that is just a coincidence for this particular problem. Not how action optimals are found per Aumann.
For the sake of clarity let’s take a step back and examine our positions. Everyone agrees p=1/2 is not the right choice. Aumann thinks it is done through 2 steps.
1. Derive all action optimals using by finding the stable point of the action utility function. ( p=1/2 is one of them, as well as p=0)
2. p=1/2 is rejected because it is not possible for the driver at different intersections to coordinate on it due to absentmindedness.
I disagree with both points 1 and 2, reason being the action utility function is fallacious. Are you rejecting both, or point 2 only, or are you agreeing with him?
For the sake of clarity let’s take a step back and examine our positions. Everyone agrees p=1/2 is not the right choice. Aumann thinks it is done through 2 steps.
1. Derive all action optimals using by finding the stable point of the action utility function. ( p=1/2 is one of them, as well as p=0)
2. p=1/2 is rejected because it is not possible for the driver at different intersections to coordinate on it due to absentmindedness.
I disagree with both points 1 and 2, reason being the action utility function is fallacious. Are you rejecting both, or point 2 only, or are you agreeing with him?
Point 1 is wrong. Action utility measures the wrong thing in this scenario, but does measure the correct thing for some superficially similar but actually different scenarios.
Point 2 is also wrong, because it’s perfectly possible to be able to coordinate in this scenario. It’s just that due to point 1 being wrong, they would be coordinating on the wrong strategy.
So we agree that both these points are incorrect, but we disagree on the reasons for them being incorrect.
OK, I think that is clearer now. I assume you think the strategy to coordinate on should be determined by maximizing the planning utility function. Not by maximizing the action utility function nor finding the stable point of the action utility function. I agree with all of this.
The difference is that you think the self-locating probabilities are valid. The action utility function that uses them is valid but can only be used in superficially similar problems such as multiple drivers being randomly assigned to intersections.
While I think self-locating probabilities are not valid, therefore the action utility functions are fallacious. Whereas in problems where multiple drivers are randomly assigned to intersections, the probability for someone assigned to an intersection is not self-locating probabilities.
Pretty close. I do think that self-locating probabilities can be valid, but determining the most relevant one to a given situation can be difficult. There are a lot more subtle opportunities for error than with more familiar externally supplied probabilities.
In particular, the way in which this choice of self-locating probability is used in this scenario does not suit the payoff schedule and incentives. Transforming it into related scenarios with non-self-locating probabilities is just one way to show that the problem exists.
> Before starting the drive, the driver determines that always turning at the first intersection will be optimal. I didn’t think we disagreed on that.
But the driver does not have to do any calculation before starting the drive. He can do that, yes. He also can simply choose to only think about the decision when arrived at an intersection. It is possible for him to derive the “action optimals” chronologically before deriving the “planning optimal”. As I said earlier, they are two independent processes.
>Yes, it is. You can verify this by finding the explicit expression for action utility as a function of p....
No, it was not found by maximizing the action utility function. In Aumann’s process, the action utility function was not represented by a single variable p, but with multiple variables representing casually disconnected decisions (observation 1). Because the decisions ought to be the same (observations 2) the action optimal ought to be symmetrical Nash equilibriums or “stable points”. You can see an example in Eliezer Yudkowsky’s post. For this particular problem, there are three stable points for the action utility functions. p=0, p=7/30 and p=1/2. Among these three p=1/2 gives the highest action payoff, 7⁄30 the lowest.
I will take your words for it that p=1/2 also maximizes action utility. But that is just a coincidence for this particular problem. Not how action optimals are found per Aumann.
For the sake of clarity let’s take a step back and examine our positions. Everyone agrees p=1/2 is not the right choice. Aumann thinks it is done through 2 steps.
1. Derive all action optimals using by finding the stable point of the action utility function. ( p=1/2 is one of them, as well as p=0)
2. p=1/2 is rejected because it is not possible for the driver at different intersections to coordinate on it due to absentmindedness.
I disagree with both points 1 and 2, reason being the action utility function is fallacious. Are you rejecting both, or point 2 only, or are you agreeing with him?
Point 1 is wrong. Action utility measures the wrong thing in this scenario, but does measure the correct thing for some superficially similar but actually different scenarios.
Point 2 is also wrong, because it’s perfectly possible to be able to coordinate in this scenario. It’s just that due to point 1 being wrong, they would be coordinating on the wrong strategy.
So we agree that both these points are incorrect, but we disagree on the reasons for them being incorrect.
OK, I think that is clearer now. I assume you think the strategy to coordinate on should be determined by maximizing the planning utility function. Not by maximizing the action utility function nor finding the stable point of the action utility function. I agree with all of this.
The difference is that you think the self-locating probabilities are valid. The action utility function that uses them is valid but can only be used in superficially similar problems such as multiple drivers being randomly assigned to intersections.
While I think self-locating probabilities are not valid, therefore the action utility functions are fallacious. Whereas in problems where multiple drivers are randomly assigned to intersections, the probability for someone assigned to an intersection is not self-locating probabilities.
Pretty close. I do think that self-locating probabilities can be valid, but determining the most relevant one to a given situation can be difficult. There are a lot more subtle opportunities for error than with more familiar externally supplied probabilities.
In particular, the way in which this choice of self-locating probability is used in this scenario does not suit the payoff schedule and incentives. Transforming it into related scenarios with non-self-locating probabilities is just one way to show that the problem exists.