Ending in negative numbers wouldn’t change anything. The amount of endings will still shrink, the amount of branches will shrink, the range of the possible utility of the endings will still shrink or stay the same length, the range of the total amount of utility you could generate over the future branches will also shrink or stay the same length. Try it! Replace any number in any of my models with a negative number or draw your own model and see what happens.
If in the first image we replace the 0 with a −100 (much wider) what happens? The amount of endings for 1 is still larger than 3. The amount of branches for 1 is still larger than 3. The width of the range of the possible utility of the endings for 1 is [-100 to 8] and for 3 is [-100 to 6] (smaller). The width of the range of the total amount of utility you could generate over the future branches is [1->3->-100 = −96 up to 1->2->8= 11] for 1 and [3->-100= −97 up to 3->6= 9] for 3 (smaller). Is this a good example of what you’re trying to convey? If not could you maybe draw an example tree, to show me what you mean?
Its seems like it’s only impossible because that is how you’ve drawn it. Not that it isn’t actually mathematically impossible.
Why couldnt one of the final branches in your example be −100?
Ending in negative numbers wouldn’t change anything. The amount of endings will still shrink, the amount of branches will shrink, the range of the possible utility of the endings will still shrink or stay the same length, the range of the total amount of utility you could generate over the future branches will also shrink or stay the same length. Try it! Replace any number in any of my models with a negative number or draw your own model and see what happens.
It wasn’t about being negative or not. My question works just as well with a positive number.
I was trying to get at what happens when the range of one of the final branches goes wider than another final branch.
If that is the case, then it is mathematically possible for a more recent hinge to be hingier than a hinge further back in time.
If in the first image we replace the 0 with a −100 (much wider) what happens? The amount of endings for 1 is still larger than 3. The amount of branches for 1 is still larger than 3. The width of the range of the possible utility of the endings for 1 is [-100 to 8] and for 3 is [-100 to 6] (smaller). The width of the range of the total amount of utility you could generate over the future branches is [1->3->-100 = −96 up to 1->2->8= 11] for 1 and [3->-100= −97 up to 3->6= 9] for 3 (smaller). Is this a good example of what you’re trying to convey? If not could you maybe draw an example tree, to show me what you mean?