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I am looking for a keyword for a certain concept that is somewhat related to delayed gratification (psychology) and optimisation but I don't really know where to start or even where should this post go in the forum Embarassed

But anyway, some explanation is in order, I'll give a few examples, then prose the general problem in the end:

[1] Suppose my sister is playing a cute little farm management game (harvest moon, farmville etc.) and she likes how the chicken looks in the game so she said to me that she's going to keep buying all the chicken and fill her farm with it.
One chicken costs some amount of money, and is able to produce eggs periodically which she can then sell to get some gold back and eventually gain a profit.
Consider further that she have lots of other options to gain money such as planting various crops and milking cows for their milk etc. The point is that there are better (and worse) ways to earn money in this game than if we just have chickens laying their eggs.
So in order to fill her chicken farm up to their full potential in the least amount of time, shouldn't she play it optimally first without any chickens until she have sufficient amount of money to buy the exact number of chickens needed in the end?
But if the game is almost limitless, she could be spending almost all the time chickenless trying to reach a chicken goal.

[2] Suppose, for some reason, that my brother hates waiting in the lunch queue. So after a week, he noted the time when people start to diminish and the cafeteria is not as crowded. The problem is that there's always an in-flow of people and there's a line of at least 1 person at any given time. If his goal is to also try to get to eat sooner, what should he do?
He could just go for it right after the lunch break, getting to eat early.
Or he could wait a while until there's less people so he's more comfortable being in the queue.

[3] Suppose my professor is planning to move to a career in business. He wants to donate most of the money he earns from his work to research. Suppose further that he could be very successful in the business world, earning more money than he normally would, and that this would be more likely the case if he starts out with more money.
The problem is that if he donated some portion to research, it would take him longer to get to the same stage in his new career than if he started out with all the money. So again, he could advance his business without apportioning his earnings to scientific research, so that he can later donate a larger sum. Thus able to give more in the end within a shorter amount of time.

Example 1 and 3 are clearly instances of the same problem, one on a trifling scale, while another with a very real application. But on hindsight, I don't think example 2 fits in very well.
In general, I can formulate these problems as follows:
Given some fixed amount of time t that we have, and a resource r that we want to distribute, what is the best course of action to take so as to maximise donation of r within time t. We have the condition that if after some time we have some resource left, we can produce even more resources. How fast we can produce r is proportional to the amount r itself that we have left. So if we donate all r resources from the beginning, we won't be able to produce more to donate.

Mathematically speaking, this is a simple maximisation problem and the solution is to wait until the end at time t, then we can donate an optimal amount of resource.
But it seems like there's some ethical issues involved in the problem, plus the fact that we may not even know what t is. Because if t is very large or unknown, we would be trying to maximise r for a very long time in hope to maximise the contribution in the end, without really donating any portion of it along the way.

I would be really interested to know exactly what topic this is in so that I can look up a formal treatment of this sort of problems Smile
i don't know what you are talking about but anyway best of luck for your venture. Smile
This appears to be a non-scaling problem, in that the actual measurements change dependant on the scale of the time axis. By that I mean that the actual quantity changes depending on how finely you measure it. This is not intuitive for most people. We expect the measurement to stay the same and the scale to change.

For those unfamiliar with this sort of maths, think about the coastline of an island. You measure it from space via satellite an get, say 400 km. You then do it by hand, using a 100 metre baseline and basic surveying techniques. Your answer will be larger. because anything less than 100m now impacts the measurement - you have to measure around it. As you go finer and reduce the scale, the measurement grows. The limit is going around every sub-atomic particle at the boundary and tends to infinity - or at least to bloody big.

Now consider the 'gift'.
A man cannot give more than he takes from the environment - anything else is a notional construct, including money, and depends on a common agreement to have any value at all. Labour is a concrete and has value regardless of individual differences. A man can only give so much labour - and he then needs others to do some of the stuff he should otherwise be doing for himself - gathering food, cooking, etc..

So, considered over a long time-scale it makes sod all difference. Energy in = energy out and the rest is just local eddies in the current.
As time-scale tightens then the measurement change. At small scales we see the bumps in the trend, produced by the notional and the other variables acting on the concrete. So, at a decade scale Bill Gates has definitely had a huge impact. At the millennia scale a small blip and at the millennia scale probably unmeasurable. At larger scales it all smooths down into energy in <->energy out.
The problem with these messy biological apps is that they need arbitrary measurements.

Then we have the measurement problem - how do you measure the gain? Do you include all the negatives in the exploitation chain that enables a westerner to live in the lifestyle they do? If so then good luck with the sums......

The broad problem is that we are crap at this type of thing but we should be good at it. Applying the most powerful tool-kit we have to ourselves is still new and we are learning, but slowly.
As usual, I think religion is mainly to blame. With it all controlled by a Deity who is ultimately inscrutable then there is no point trying to understand anything about the biological squishy bits of the world - and we just mainly stuck to the rocky bits, they didn't complain when we assigned them arbitrary quantities like mass.

Now, I have just realised that this is a philosophical answer to a question straddling maths and philosophy and science, and we are in the science forums. This reply belongs in p&r. I'll formulate a mathematical answer later,
What about use simulation technique to study those?
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