So the original tree T models a tree of “states of affairs”,
and the original partition or subset T’ models each node being either under the decision-maker’s control or not.
Then the function f would go from elements of T’ to successors of those elements—nodes of T, to be sure, but there’s a side condition there.
Then the probability measure P is a somewhat more powerful way of attaching probabilities to the non-choice nodes—that is, if you have a distribution over the successors of each non-choice node, then you can obtain a probability measure, but a probability measure over paths would allow some additional correlations.
The function u can be understood (in a finite tree) as labelling leaves with utilities, because paths of maximum length in a finite tree are isomorphic with leaves—but by describing it the way you did, you leave the door open to applying this formalism to an infinite tree.
UP(f) would be the utility of a particular strategy (f), and U(f|\sigma) would be… the utility given a certain initial sequence of events? So \sigma is a finite path segment?
I don’t understand the grammar of “Now either …. and …”—should it be “Now either … or …”? Or is it really “Now assume … and …”?
When you use U(f) later, I am guessing that’s either UP(f) with the P elided, or U(f | the empty path segment) - regardless, we’re going to have to fix a P in order to get a utility, right?
Then the phrase “f(sigma)=f(tau) if the set of x such that \sigma(x) is not equal to tau(x) is a subset of S”—If I understand correctly, sigma and tau are finite path segments, which are isomorphic to nodes in the tree—but are they functions from nodes? If they are functions, wouldn’t they be from, say, initial sections of the integers 0...n TO nodes? If I understand correctly, they’re going to diverge at at most one point—once diverged, since it’s a tree, they’re not going to be able to rejoin. Or were you saying tree and thinking ‘dag’?
I worry about the types of these things; coding it up in something like Haskell or Ocaml might make everything sharper and perhaps suggest simplifications. I’m sure that you can carry through the basic intuition.
Interesting..
So the original tree T models a tree of “states of affairs”, and the original partition or subset T’ models each node being either under the decision-maker’s control or not. Then the function f would go from elements of T’ to successors of those elements—nodes of T, to be sure, but there’s a side condition there. Then the probability measure P is a somewhat more powerful way of attaching probabilities to the non-choice nodes—that is, if you have a distribution over the successors of each non-choice node, then you can obtain a probability measure, but a probability measure over paths would allow some additional correlations. The function u can be understood (in a finite tree) as labelling leaves with utilities, because paths of maximum length in a finite tree are isomorphic with leaves—but by describing it the way you did, you leave the door open to applying this formalism to an infinite tree. UP(f) would be the utility of a particular strategy (f), and U(f|\sigma) would be… the utility given a certain initial sequence of events? So \sigma is a finite path segment?
I don’t understand the grammar of “Now either …. and …”—should it be “Now either … or …”? Or is it really “Now assume … and …”?
When you use U(f) later, I am guessing that’s either UP(f) with the P elided, or U(f | the empty path segment) - regardless, we’re going to have to fix a P in order to get a utility, right?
Then the phrase “f(sigma)=f(tau) if the set of x such that \sigma(x) is not equal to tau(x) is a subset of S”—If I understand correctly, sigma and tau are finite path segments, which are isomorphic to nodes in the tree—but are they functions from nodes? If they are functions, wouldn’t they be from, say, initial sections of the integers 0...n TO nodes? If I understand correctly, they’re going to diverge at at most one point—once diverged, since it’s a tree, they’re not going to be able to rejoin. Or were you saying tree and thinking ‘dag’?
I worry about the types of these things; coding it up in something like Haskell or Ocaml might make everything sharper and perhaps suggest simplifications. I’m sure that you can carry through the basic intuition.
Thanks for thinking about this.