A Bit of a Math/Logic Problem

A phenomenon called ‘Hyperbolic Discounting’ is a well-known phenomenon in both human and, I think, animal psychology (definitely human). The idea is that we treat events in the near future as more important than events in the far future. It’s been discovered that the weighting we give events is inversely proportional to how far away they are in time.

Another explanation of the phenomenon:

Again, the way humans currently discount is proportional to the distance in time. So, something immediate is not discounted at all (divide by 1), something that is (for example, not an accurate value) a day away, divide by 2, something that is 2 days away, divide by 3, etc.

Some people consider this a highly irrational way to calculate values. The reason is that the relative weighting of, for example, two consecutive days changes depending on how close you are to those two days.

For example, if we’re comparing the weightings of March 30 and March 31, and we’re comparing it on Jan 01, the two weightings will be quite similar, but if we’re comparing them on March 29, March 30 has a very higher rating.

So, the assumptions being made in this puzzle are:

  1. That discounting is rational, potentially, but that
  2. To be rational, relative weightings of days in the future shouldn’t change based on how close you are to those days.

Assuming that, these are the questions. Their difficulty depends on your ability to abstractly reason and your familiarity with types of mathematical equations:

  1. Why is the inversely proportional weighting system irrational, given the 2 assumptions?
  2. What system of discounting would be rational, given the 2 assumptions?

Warning: If you look it up on the internet yourself, prior to solving the problem, you will find spoilers. If you want to solve it yourself, don’t look it up. If you need a better explanation of anything, ask, as I’m sure it’s not perfectly clear what I’m talking about to some people.
Please: If you create an answer, put it in [ tab ] tags so that people who want to figure it out themselves don’t read the answer.

I have a high psychological interest in this topic and can explain extensive details as to why it occurs, but to ask for “the” math model seems to be an irrational question. You provide two premises that have a true/false nature and then ask for math models presuming the premises are true. The true problem is that the math is both over simplified and very misleading (another psychological issue arising in economics and quantum physics due to presumptive focus on pure math).

So I don’t know how to even address your intentions.
My opinion is that your premises are not both true.

It’s potentially the case that they’re not both true, I’m not committed to them being true. I just encountered the position that they are both true, and myself figured out 1 and 2 and thought it was an interesting problem (if a little bit simple for the mathematically initiated).

If you’d like, feel free to go into detail about which one isn’t true and why (both, if you like).
Though I can’t personally see a justification for the belief that 1 is true but 2 not.
As far as I can see, they go together – they’re both false or they’re both true.

Premise (1) is true.
Premise (2) is false.

You cannot make a credible decision making model based merely upon the certainty of today and the uncertainty of tomorrow merely because tomorrow becomes today and the certainty factor switches, so of course it isn’t going to be consistent and it really shouldn’t be.

But then you ask;

The inversely proportional weighting system of (1) isn’t irrational.
And how do you provide a rational solution given both a true and a false premise?

You don’t know how to accept false premises?
“Accept the premise that Jesus Christ was a homosexual and was married to John the Baptist. What would that imply about the common Christian abhorrence of homosexuality?”
You wouldn’t know how to figure out what it would imply, simply because the premise isn’t true?

Anyway, in my example of March 30 and 31, I don’t see why the (weighting of March 30) / (weighting of March 31) should be vastly higher when it’s the 29th than when it’s January 1st.

Not mathematically.

“What is 28/3 given that 2+2=3?”
Can you figure that out?

In reality a decision is changed because a decision is Rationally made based upon PRESENT information, not merely upon previous presumptive calculations.

Psychologically, imagine that Congress decided that the next week, starting Monday, everyone in America is to drive on the left side of the road versus the right. They made this decision for very sound reasons that only the super elite such as themselves could comprehend.

Comes next Monday. What do you seriously think would happen?

Your disassociated cognitive calculations are not as elite as you imagine, but more importantly, they are not in charge. So what if Congress decided that something “is to occur”. Reality rules, not presumptuous would-be Gods.

What you are doing right now is not based upon some decision you made yesterday, although that might play into the total, but rather upon what your subconscious re-evaluated as a better choice for today. Imagine that you decided that you are to be in your office on your PC at 10:00 AM tomorrow. But during the night your entire house got destroyed by a tornado. Are you going to stand where your office used to be and pretend that you are on the Internet?

What most people do not realize is that most of their decisions are not being made by their conscious mind in the first place. But further, they are not cognitively aware of what information has been relevant in making them. So they make cognitive decisions based on irrelevant information as far as the truer decision maker is concerned. They lose discipline and the respect of their own subconscious mind, all because they presumed to know more than those who have to suffer the decisions. If Congress doesn’t understand the people, do you seriously think Congress is going to get what it really wanted? Or is it going to have a serious and perhaps endless problem on its hands?

The calculations for tomorrow have to be based up the Bayesian type of understanding, which inherently changes upon the immediate information at hand because probabilities are always about the available information versus the real truth. As new information comes into the picture, which includes information that the cognitive mind didn’t have yesterday and probably still doesn’t, the true decision maker assesses the situation regardless of whatever the “Congress” decided.

You have a situation in every mind of multiple decision makers, “multiprocessors”. Each has a different paradigm of information and sources. When any one ignores any other, the decisions become weak if not totally futile. Buddhism is largely about the effort to dispense with such dissonance. But until such time occurs, decisions will always, as they should, be based upon CURRENT information versus prior prediction.

Or would you prefer sticking with 2000 year old action plans?

I think you know that that’s not relevant. I didn’t say anything mathematically impossible.
The question is about the mathematical type of formula that would be discounting in such a way that the relative weightings of the 30th and the 31st don’t change depending on how close you get to them. There IS an answer to the question, there IS a formula type that doesn’t change relative weightings of the 30th and 31st depending on the current date (given that the date of evaluation if before the 30th, of course).

You’re misunderstanding a whole lot.

There’s nothing in the problem about new information changing the evaluations of the different days. That’s not part of the problem. That’s not an issue. Obviously if new information arises to change how you value potential actions, nobody’s saying to ignore it. That’s a ridiculous straw man.

Let me explain it in simpler terms:
If someone said to you that you can choose to get an apple in an hour or an apple in a day, most people would choose to get it in an hour. The belief that this choice is rational is premise 1.

Now, similarly, if you were given the option of getting an apple in 30 days, versus 31 days, you would also choose 30 days. ‘This is rational’ is again premise 1.

Now, to get into the meat of what the issue is (and it’s NOT new information, that’s nothing to do with it – it’s ONLY about time-proximity, so no more of that silly stuff), we’re going to start comparing apples and oranges.
Let’s say a person values oranges a little bit more than apples, such that if they were offered to choose from an apple in 30 days, or an orange in 31 days, they would choose an orange in 31 days.

Now, he values oranges only a little more than apples, and though he’d choose the orange when it’s 31 days away as opposed to the apple 30 days away, if you reduce the time distance, it changes. Let’s say now, you offer an apple in 15 days, or an orange in 16 days. The relative weightings of 15 days versus 16 days is higher than the relative weightings of 30 days versus 31 days, and it’s higher enough that now he’d choose the apple in 15 days. “The relative cost of a day’s wait is considered differently whether that day’s wait is near or far.”

This is what it’s about. Not new information – there’s no new information. It’s just about how soon the dates are.

Now, it’s some peoples’ idea that, as long as the interval between the apple offer and the orange offer remains the same, the decision should remain the same. If he chooses the orange when it’s one day later, in a month’s time, he should choose the orange when it’s one day later in half a month’s time, in a week’s time, etc etc.

The problem in the OP is, what formula type do you use, that’s different from the Inversely Proportional formula type, to make sure that, as long as the interval between the options is the same, the choice is the same? Whether or not it’s true that such a formula would be more rational isn’t required to figure out the answer. It’s nothing like assuming 2+2=3. 2+2=3 is a mathematically incorrect statement, whereas there is a formula type which would account for a weighting system that I described. This formula type is not ‘mathematically incorrect,’ and in fact it’s found in nature. Regardless of whether it’s more rational, the problem is solvable.

Now, SEPARATELY, I’d still like to know why you think it’s rational to change from the orange to the apple when it’s 15 days away as opposed to 30. This is not required for you to solve the problem, please try to understand that. Whether 2 is true is not required to solve the problem assuming 2 is true. There exists a mathematically correct formula type which models real phenomena that answers the question. You don’t have to break any laws of mathematics.

Based upon what you just stated, it is a null equation, “A=A”, decisions don’t change.
But you apparently have seriously little understanding of the actual issue being addressed.

To anyone wishing to find the solution: the above attempt isnt it. PM me if you want the answer James

You are ignoring why any decisions change and claiming as a premise that they rationally should not.
If the premise is that it is rational to not change a decision, then the equation is “decision today = decision tomorrow”, end of story.
There is nothing to discuss.

Ok well if you want the answer and an explanation, PM me.

Well out of mere curiosity, I will. But I get the strong sense that you are thinking something very different from what you are saying.

You seem to be drawing a dichotomy between being a mindless animal versus a cognitive android. Either will get you killed, so I don’t know how you could call either “rational”. Rational decision making requires information updating and since you provide no updating information formula, there is no rational equation that you could support.

From the PM and understanding (I think) what you were aiming for, I think that I would have used a different example, perhaps the procrastination concern;

A person procrastinates so as to avoid a decision with a specific amount of urge to do so. Typically one imagines that if put off, that decision will be more easily made later due to less urge to put it off. The reasons for that aren’t relevant for now and the result turns out to actually be that the decision really is made more easily later. Using a hyperbolic discounting formula, we can calculate about how much easier the decision will become.

But what if we didn’t want it to calculate as any easier as time progressed? We couldn’t use that hyperbolic formula. What if we wanted to calculate that no matter how long we put off a decision, it would be exactly as difficult later as it is currently once we got to that day? What formula could we use?

Procrastination is itself partly because of hyperbolic discounting, or at least exacerbated by it.
It’s not so much about ‘ease of decision’ as it is about ‘Reward now, or reward later?’
With hyperbolic discounting, as opposed to (the method that is the solution to the OP), rewards closer to the present are hyperbolically (if that’s a word) more valuable than rewards in the future. Hence, procrastination – I could study, and ace my exam (in the future), or I could have fun playing XBox now.

If presented with ‘Study, or play XBox’ a week in advance, a lot of people would say ‘Study – Long-term rewards for some short-term effort’. But as the day draws nearer, hyperbolic discounting rapidly makes the short-term rewards of XBox more and more attractive, while the long-term rewards of studying don’t change much. So, while you’d say a week ahead of time you’d rather study, when you’re in the moment you may very well want to play XBox.

Whereas someone who uses a discounting method that is (the solution to this thread), what you say a week ahead of time is also what you say when you’re in the moment. The relative values of the two decisions don’t change depending on how close they are to you, only depending on how close they are to each other.

[tab]1) Because it directly contradicts point 2
2) One that keeps the ratios of the probabilities the same, presumably - I’m not sure off the top of my head, but inverse log?[/tab]

I’d also disagree with assumption 2, unless there are very compelling reasons to believe it.

I’ll PM you the message I sent JamesP, OnlyHumean. Your answer was, I think, on the right track. Either that or it’s correct and I just didn’t understand it.

Yep, got it thanks. Mine was poorly phrased but that’s what I was aiming for.