What I meant was, what point were you trying to make with that statement? According to Aumann’s paper, every planning-optimal solution is also an action-optimal solution, so the decision procedure they endorse will end up picking the planning-optimal solution.
My understanding of their paper has changed somewhat since we began this discussion. I now believe that repeating the planning-optimal analysis at every decision node is only guaranteed to give ideal results in simple cases like this one in which every decision point is in the same information set. In more complicated cases,
I can imagine that the policy of planning-optimal-for-the first-move, then action-optimal-thereafter might do better. I would need to construct an example to assert this with confidence.
(My complaint is just that it goes about it in an unnecessarily round-about way.) If theirs is a correct policy, then the policy of just recomputing the planning-optimal solution must also be correct.
In this simple example, yes. Perhaps not in more complicated cases.
That seems to disprove your “only correct policy” claim. I thought your “sequential equilibrium” line was trying to preempt this argument, but I can’t see how.
And I can’t see how to explain it without an example
While I wait, did you see anything in Aumann’s paper that hints at “the policy of planning-optimal-for-the first-move, then action-optimal-thereafter might do better”? Or is that your original research (to use Wikipedia-speak)? It occurs to me that if you’re correct about that, the authors of the paper should have realized it themselves and mentioned it somewhere, since it greatly strengthens their position.
Answering that is a bit tricky. If I am wrong, it is certainly “original research”. But my belief is based upon readings in game theory (including stuff by Aumann) which are not explicitly contained in that paper.
Please bear with me. I have a multi-player example in mind, but I hope to be able to find a single-player one which makes the reasoning clearer.
Regarding your last sentence, I must point out that the whole reason we are having this discussion is my claim to the effect that you don’t really understand their position, and hence cannot judge what does or does not strengthen it.
Ok, I now have at least a sketch of an example. I haven’t worked it out in detail, so I may be wrong, but here is what I think. In any scenario in which you gain and act on information after the planning stage, you should not use a recalculated planning-stage solution for any decisions after you have acted upon that information. Instead, you need to do the action-optimal analysis.
For example, let us complicate the absent-minded driver scenario that you diagrammed by adding an information-receipt and decision node prior to those two identical intersections. The driver comes in from the west and arrives at a T intersection where he can turn left(north) or right(south). At the intersection is a billboard advertising today’s lunch menu at Casa de Maria, his favorite restaurant. If the billboard promotes chile, he will want to turn right so as to have a good chance of reaching Maria’s for lunch. But if the billboard promotes enchiladas, which he dislikes, he probably wants to turn the other way and try for Marcello’s Pizza. Whether he turns right or left at the billboard, he will face two consecutive identical intersections (four identical intersections total). The day is cloudy, so he cannot tell whether he is traveling north or south.
Working this example in detail will take some work. Let me know if you think the work is necessary.
My understanding of their paper has changed somewhat since we began this discussion. I now believe that repeating the planning-optimal analysis at every decision node is only guaranteed to give ideal results in simple cases like this one in which every decision point is in the same information set. In more complicated cases, I can imagine that the policy of planning-optimal-for-the first-move, then action-optimal-thereafter might do better. I would need to construct an example to assert this with confidence.
In this simple example, yes. Perhaps not in more complicated cases.
And I can’t see how to explain it without an example
While I wait, did you see anything in Aumann’s paper that hints at “the policy of planning-optimal-for-the first-move, then action-optimal-thereafter might do better”? Or is that your original research (to use Wikipedia-speak)? It occurs to me that if you’re correct about that, the authors of the paper should have realized it themselves and mentioned it somewhere, since it greatly strengthens their position.
Answering that is a bit tricky. If I am wrong, it is certainly “original research”. But my belief is based upon readings in game theory (including stuff by Aumann) which are not explicitly contained in that paper.
Please bear with me. I have a multi-player example in mind, but I hope to be able to find a single-player one which makes the reasoning clearer.
Regarding your last sentence, I must point out that the whole reason we are having this discussion is my claim to the effect that you don’t really understand their position, and hence cannot judge what does or does not strengthen it.
Ok, I now have at least a sketch of an example. I haven’t worked it out in detail, so I may be wrong, but here is what I think. In any scenario in which you gain and act on information after the planning stage, you should not use a recalculated planning-stage solution for any decisions after you have acted upon that information. Instead, you need to do the action-optimal analysis.
For example, let us complicate the absent-minded driver scenario that you diagrammed by adding an information-receipt and decision node prior to those two identical intersections. The driver comes in from the west and arrives at a T intersection where he can turn left(north) or right(south). At the intersection is a billboard advertising today’s lunch menu at Casa de Maria, his favorite restaurant. If the billboard promotes chile, he will want to turn right so as to have a good chance of reaching Maria’s for lunch. But if the billboard promotes enchiladas, which he dislikes, he probably wants to turn the other way and try for Marcello’s Pizza. Whether he turns right or left at the billboard, he will face two consecutive identical intersections (four identical intersections total). The day is cloudy, so he cannot tell whether he is traveling north or south.
Working this example in detail will take some work. Let me know if you think the work is necessary.
Ok, I see. I’ll await your example.