If we draw a tree of all possible timelines (and there is an end to the tree) the older choices will always have more branches that will sprout out because of them. If we are purely looking at the possible endings then the 1 in the first image has a range of 4 possible endings, but 2 only has 2 possible endings. If we’re looking at branches then the 1 has a range of 6 possible branches, while 2 only has 2 possible branches. If we’re looking at ending utility then 1 has a range of [0-8] while 2 only has [7-8]. If we’re looking at the range of possible utility you can experience then 1 has a range from 1->3->0 = 4 utility all the way to 1->2->8 = 11 utility, while 2 only has 2->7 = 9 to 2->8 = 10.
When we talk about the utility of endings it is possible that the range doesn’t change. For example:
Here the “range of utility in endings” tick 1 has (the first 10) is [0-10] and the range of endings the first 0 has (tick 2) is [0-10] which is the same. Of course the probability has changed (getting an ending of 1 utility is not even an option anymore), but the minimum and maximum stay the same.
Now the width of the range of the total amount of utility you could potentially experience can also stay the same. For example the lowest utility tick 1 can experience is 10->0->0 = 10 utility and the highest is 10-0-10 = 20 utility. The difference between the lowest and highest is 10 utility. The lowest total utility that the 0 on tick 2 can experience is 0->0 = 0 utility and the highest is 0->10 = 10 utility, which is once again a difference of 10 utility. The probability has changed (ending with a weird number like 19 is impossible for tick 2). The range has also shifted downwards from [10-20] to [0-10], but the width stays the same.
It just occurred to me that some people may find the shift in range also important for hingeyness. Maybe call that ‘hinge shift’?
Crucially, in none of these definitions is it possible to end up with a wider range later down the line than when you started.
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?
Can you expand on this because it isn’t obvious to me that this is true.
If we draw a tree of all possible timelines (and there is an end to the tree) the older choices will always have more branches that will sprout out because of them. If we are purely looking at the possible endings then the 1 in the first image has a range of 4 possible endings, but 2 only has 2 possible endings. If we’re looking at branches then the 1 has a range of 6 possible branches, while 2 only has 2 possible branches. If we’re looking at ending utility then 1 has a range of [0-8] while 2 only has [7-8]. If we’re looking at the range of possible utility you can experience then 1 has a range from 1->3->0 = 4 utility all the way to 1->2->8 = 11 utility, while 2 only has 2->7 = 9 to 2->8 = 10.
When we talk about the utility of endings it is possible that the range doesn’t change. For example:
(I can’t post images in comments so here is a link to the image I will use to illustrate this point)
Here the “range of utility in endings” tick 1 has (the first 10) is [0-10] and the range of endings the first 0 has (tick 2) is [0-10] which is the same. Of course the probability has changed (getting an ending of 1 utility is not even an option anymore), but the minimum and maximum stay the same.
Now the width of the range of the total amount of utility you could potentially experience can also stay the same. For example the lowest utility tick 1 can experience is 10->0->0 = 10 utility and the highest is 10-0-10 = 20 utility. The difference between the lowest and highest is 10 utility. The lowest total utility that the 0 on tick 2 can experience is 0->0 = 0 utility and the highest is 0->10 = 10 utility, which is once again a difference of 10 utility. The probability has changed (ending with a weird number like 19 is impossible for tick 2). The range has also shifted downwards from [10-20] to [0-10], but the width stays the same.
It just occurred to me that some people may find the shift in range also important for hingeyness. Maybe call that ‘hinge shift’?
Crucially, in none of these definitions is it possible to end up with a wider range later down the line than when you started.
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?