This post demonstrates that ignoring counterfactuals can cause you to do worse even if you only care about your particular branch. This doesn’t take you all the way to expected utility over branches, but I can’t see any obvious intermediate positions.
I was pointing out a typo in the Original Post. That said, that’s a great summary.
Perhaps an intermediate position could be created as follows:
Given a graph of ‘the tree’ (including the branch you’re on), position E is
expected utility over branches
position B is
you only care about your particular branch.
Position B seems to care about the future tree (because it is ahead), but not the past tree. So it has a weight of 1 on the current node and it’s descendants, but a weight of 0 on past/averted nodes, while Position E has a weight of 1 on the “root node” (whatever that is). (Node weights are inherited, with the exception of the discontinuity in Position B.)
An intermediate position is placing some non-zero weight on ‘past nodes’, going back along the branch, and updating the inherited weights. Aside from a weight of 1⁄2 being placed along all in branch nodes, another series could be used, for example: r, r^2, r^3, … for 0<r<1. (This series might allow for adopting an ‘intermediate position’ even when the branch history is infinitely long.)
There’s probably some technical details to work out, like making all the weights add up to 1, but for a convergent series that’s probably just a matter of applying an appropriate scale factor for normalization. For r=1/2, the infinite sum is 1, so no additional scaling is required. However this might not work (the sum across all node’s rewards times their weight might diverge) on an infinite tree where the rewards grow too fast...
(This was an attempt at outlining an intermediate position, but it wasn’t an argument for it.)
This post demonstrates that ignoring counterfactuals can cause you to do worse even if you only care about your particular branch. This doesn’t take you all the way to expected utility over branches, but I can’t see any obvious intermediate positions.
I was pointing out a typo in the Original Post. That said, that’s a great summary.
Perhaps an intermediate position could be created as follows:
Given a graph of ‘the tree’ (including the branch you’re on), position E is
position B is
Position B seems to care about the future tree (because it is ahead), but not the past tree. So it has a weight of 1 on the current node and it’s descendants, but a weight of 0 on past/averted nodes, while Position E has a weight of 1 on the “root node” (whatever that is). (Node weights are inherited, with the exception of the discontinuity in Position B.)
An intermediate position is placing some non-zero weight on ‘past nodes’, going back along the branch, and updating the inherited weights. Aside from a weight of 1⁄2 being placed along all in branch nodes, another series could be used, for example: r, r^2, r^3, … for 0<r<1. (This series might allow for adopting an ‘intermediate position’ even when the branch history is infinitely long.)
There’s probably some technical details to work out, like making all the weights add up to 1, but for a convergent series that’s probably just a matter of applying an appropriate scale factor for normalization. For r=1/2, the infinite sum is 1, so no additional scaling is required. However this might not work (the sum across all node’s rewards times their weight might diverge) on an infinite tree where the rewards grow too fast...
(This was an attempt at outlining an intermediate position, but it wasn’t an argument for it.)