You can only pick one of the choices per unit of time
So I can choose c1∈{a,b},c2∈{a,b} then?
One of the choices will last two units of time, the other will last one
What do we mean by ‘last’? Do you mean that the choice in period 1, c1 , yields benefits (or costs) in periods 1 and 2, while the choice in period 2, c2 , only affects outcomes in period 2?
The choices have different scale,
Can you define this a bit? Which ‘choices’ have different scale, and what does that mean?
Maybe you want to define the sum of benefits U(c1,c2)=...
E.g.,
U(a,b)=a+b,
U(a,a)=a,
U(b,b)=b+βb,
U(b,a)=b,
where a and b are positive numbers, and β<1 is a diminishing returns parameter?
For ‘different scale’ do you just mean something like b>a?
but this difference has nothing to do with lockin,
Not sure what this means.
and you should pick the choice with less scale because it only is available for one unit of time
So this is like, above, U(a,b)=a+b>U(b,b)=(1+β)b if β<a+bb−1
But that’s not just ‘because a has no value in period 2’ but also because of the diminishing returns on b (otherwise I might just choose b in both periods.
Yes. but I think to be very specific, we should call the problems A and B (for instance, the quiz is problem A and the exam is problem B), and a choice to work on problem A equates to spending your resource [1]on problem A in a certain time frame. We can represent this as ai,j where {i} is the period in which we chose a and {j} is the number of times we have picked a before. j is sorta irrelevant for problem A since we only can use one resource max to study but relevant for problem B to represent the diminishing returns via βj .
What do we mean by ‘last’? Do you mean that the choice in period 1, c1 , yields benefits (or costs) in periods 1 and 2, while the choice in period 2, c2 , only affects outcomes in period 2?
Neither if I’m understanding you correctly. I mean that the Scale of problem A in period 2, U(A2), is 0. This also implies that the marginal utility of working on problem A in period 2 is 0. For instance, if I study for my quiz after it happens this is worthless. This is different from the diminishing returns that are at play when repeatedly studying for the same exam.
This is the extreme end of the spectrum though. We can generalize this by acknowledging that the marginal utility of a certain problem is a function of time. For instance, it’s better to knock on doors for an election the day before than 3 years before but probably not infinitely better.
Can you define this a bit? Which ‘choices’ have different scale, and what does that mean?
I think I maybe actually used scale as both meaning MU/resource and as meaning: if we solve the entire problem, how much is that worth? Basically, importance, as described in the ITN framework, except maybe I didn’t mean it as a function of the percent of work done and rather the total. Generally though, I think people consider this to be a constant (which I’m not sure they should...) but this being the case, we are basically talking about the same thing but they are dividing by a factor of 100, which again doesn’t matter for this discussion.
I think what Eliot meant is importance, so that’s what I’m going to define it as, but I think you picked up on this confusion which is my bad.
By choices, I meant the problems, like the quiz or the exam. I think I used the incorrect wording here though since choices also denote a specific decision to spend a resource on a problem. My fault for the confusion.
Maybe you want to define the sum of benefits U(c1,c2)=...
are better notations. although it doesn’t really matter, I got what you were saying.
where a and b are positive numbers, and β<1 is a diminishing returns parameter?
essentially yes but with my notation.
For ‘different scale’ do you just mean something like b>a?
No. taking b to mean U(b1,1), b is the marginal utility of spending a resource in period 1 on problem B, not the total utility to be gained by solving problem b. Using the test example the scale of B is either U(b1,1)1−βsince this is the maximum grade I can achieve based on the convergent geometric sum described or 20% since this is the maximum grade total although maybe it’s literally impossible for me to reach this. I’m not actually sure which to use, but I guess let’s go with 20%, and denote a convergent sum as meaning Tractabilityresources→∞=0.
What I meant was U(B1)>U(A1) or 20% > 10% in the test example
So this is like, above, U(a,b)=a+b>U(b,b)=(1+β)b if β<a+bb−1
I think this was the point I was trying to make with the examples I gave to you. Basically that the decision at t = 1 in a sequence of decisions that maximizes utility over multiple periods of time is not the same as the decision that maximizes utility at t= 1, which is what I believe you are pointing out here. In effect
U(b1,1)>U(a1,1)↛U(b1,1,x2,j)>U(x1,1,x2,j),∀x
But actually, I think the claim I originally made in response to him was actually a lot simpler than that, more along the lines of “A problem being urgent does not mean that its current scale is higher than if it was not urgent”. taking U(Ai) to be the Scale of problem A in the ith period, and taking problem A to be urgent to mean (U(A1)>U(A2)), which I’m getting from the op saying
Some areas can be waited for a longer time for humans to work on, name it, animal welfare, transhumanism.
my original claim in response to Elliot is something like
(U(A1)>U(A2)) and (U(B1)==U(B2)) does not imply U(A1)>U(B1)
where U(A1)==U(B1))
The fact that I get no value out of studying for a Monday quiz on Tuesday doesn’t mean the quiz is now worth more than 10% of my grade. On the flip side if the quiz was moved to Wednesday It would still be worth 10% of my grade.
I think it was maybe not what Eliot meant. That being said, taking his words literally I do think this is what he implied. I’m not really sure honestly haha.
But that’s not just ‘because a has no value in period 2’ but also because of the diminishing returns on b (otherwise I might just choose b in both periods.
Correct. I think there are further specifications that might make my point less niche, but I’m not sure.
As an aside, I’m not sure I’m correct about any of this but I do wish the forum was a little more logic and math-heavy so that we could communicate better.
we could model a situation where you have multiple resources in every period but here I choose to model as if you have a single resource to spend in each period
So I can choose c1∈{a,b},c2∈{a,b} then?
What do we mean by ‘last’? Do you mean that the choice in period 1, c1 , yields benefits (or costs) in periods 1 and 2, while the choice in period 2, c2 , only affects outcomes in period 2?
Can you define this a bit? Which ‘choices’ have different scale, and what does that mean?
Maybe you want to define the sum of benefits U(c1,c2)=...
E.g.,
U(a,b)=a+b,
U(a,a)=a,
U(b,b)=b+βb,
U(b,a)=b,
where a and b are positive numbers, and β<1 is a diminishing returns parameter?
For ‘different scale’ do you just mean something like b>a?
Not sure what this means.
So this is like, above, U(a,b)=a+b>U(b,b)=(1+β)b if β<a+bb−1
But that’s not just ‘because a has no value in period 2’ but also because of the diminishing returns on b (otherwise I might just choose b in both periods.
Does this characterize your case of interest?
Yes. but I think to be very specific, we should call the problems A and B (for instance, the quiz is problem A and the exam is problem B), and a choice to work on problem A equates to spending your resource [1]on problem A in a certain time frame. We can represent this as ai,j where {i} is the period in which we chose a and {j} is the number of times we have picked a before. j is sorta irrelevant for problem A since we only can use one resource max to study but relevant for problem B to represent the diminishing returns via βj .
Neither if I’m understanding you correctly. I mean that the Scale of problem A in period 2, U(A2), is 0. This also implies that the marginal utility of working on problem A in period 2 is 0. For instance, if I study for my quiz after it happens this is worthless. This is different from the diminishing returns that are at play when repeatedly studying for the same exam.
This is the extreme end of the spectrum though. We can generalize this by acknowledging that the marginal utility of a certain problem is a function of time. For instance, it’s better to knock on doors for an election the day before than 3 years before but probably not infinitely better.
I think I maybe actually used scale as both meaning MU/resource and as meaning: if we solve the entire problem, how much is that worth? Basically, importance, as described in the ITN framework, except maybe I didn’t mean it as a function of the percent of work done and rather the total. Generally though, I think people consider this to be a constant (which I’m not sure they should...) but this being the case, we are basically talking about the same thing but they are dividing by a factor of 100, which again doesn’t matter for this discussion.
I think what Eliot meant is importance, so that’s what I’m going to define it as, but I think you picked up on this confusion which is my bad.
By choices, I meant the problems, like the quiz or the exam. I think I used the incorrect wording here though since choices also denote a specific decision to spend a resource on a problem. My fault for the confusion.
Yes basically but I think that
U(a1,1,b2,1)=U(a1,1)+U(b2,1)=U(a1,1)+U(b1,1)
and
U(a1,1,a2,2)=U(a1,1)+U(a2,2)=U(a1,1)+0
and
(U(b1,1,b2,2)=U(b1,1)+U(b2,2)=U(b1,1)+βU(b2,1)=U(b1,1)+βU(b1,1)
are better notations. although it doesn’t really matter, I got what you were saying.
essentially yes but with my notation.
No. taking b to mean U(b1,1), b is the marginal utility of spending a resource in period 1 on problem B, not the total utility to be gained by solving problem b. Using the test example the scale of B is either U(b1,1)1−βsince this is the maximum grade I can achieve based on the convergent geometric sum described or 20% since this is the maximum grade total although maybe it’s literally impossible for me to reach this. I’m not actually sure which to use, but I guess let’s go with 20%, and denote a convergent sum as meaning Tractabilityresources→∞=0.
What I meant was U(B1)>U(A1) or 20% > 10% in the test example
I think this was the point I was trying to make with the examples I gave to you. Basically that the decision at t = 1 in a sequence of decisions that maximizes utility over multiple periods of time is not the same as the decision that maximizes utility at t= 1, which is what I believe you are pointing out here. In effect
U(b1,1)>U(a1,1)↛U(b1,1,x2,j)>U(x1,1,x2,j),∀x
But actually, I think the claim I originally made in response to him was actually a lot simpler than that, more along the lines of “A problem being urgent does not mean that its current scale is higher than if it was not urgent”. taking U(Ai) to be the Scale of problem A in the ith period, and taking problem A to be urgent to mean (U(A1)>U(A2)), which I’m getting from the op saying
my original claim in response to Elliot is something like
(U(A1)>U(A2)) and (U(B1)==U(B2)) does not imply U(A1)>U(B1)
where U(A1)==U(B1))
The fact that I get no value out of studying for a Monday quiz on Tuesday doesn’t mean the quiz is now worth more than 10% of my grade. On the flip side if the quiz was moved to Wednesday It would still be worth 10% of my grade.
I think it was maybe not what Eliot meant. That being said, taking his words literally I do think this is what he implied. I’m not really sure honestly haha.
Correct. I think there are further specifications that might make my point less niche, but I’m not sure.
As an aside, I’m not sure I’m correct about any of this but I do wish the forum was a little more logic and math-heavy so that we could communicate better.
we could model a situation where you have multiple resources in every period but here I choose to model as if you have a single resource to spend in each period
The more I think about this the more confused I get… Going to formalize and answer your questions but it might not be done till tomorrow.