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## Second Banker’s Problem – Part 2 – Interest Income and Recovery Time

In Part 1, we defined the Second Banker’s problem and gave a formula for the optimal time to make a purchase.  All of the proofs are given in this PDF.

### Interest Income immediately before the optimal purchase time for the Second Panker’s problem

As with the first banker’s problem, the income from interest just before the optimal purchase time is
$$\mathrm{interest\ income\ during\ purchase} = \frac{c \;r_1 r_2 }{r_2-r_1}.$$

Notice that this income does not depend on the initial amount in the bank account $B_0$.

For example, if

• you initially have \$1000 in the account, • the cost of increasing the interest rate is \$1,100,
• $r_1=0.05=5\%$, and
• $r_2=0.08=8\%$,

then \begin{aligned}\mathrm{interest\ income\ during\ purchase} &= \frac{c \;r_1 r_2 }{r_2-r_1}\\&= \frac{ \1100\cdot 0.05 \cdot 0.08}{0.08-0.05}\\&= \frac{ \55 \cdot 0.08}{0.03} \approx \146.67\mathrm{\ per\ year.}\end{aligned}

### REMARK

Note that the interest income can also be expressed by
\begin{equation}
\label{eqone}
\mathrm{interest\ income\ during\ purchase} = c/t_{\mathrm{pay}}
\end{equation}
where
\begin{equation}
\label{eqtwo}
t_\mathrm{pay} = \frac1{r_1} – \frac1{r_2}= \frac{c}{B_0 r_1 \exp(r_1 t_\mathrm{buy})}
\end{equation}
is the amount of time it takes to pay $c$ dollars from interest starting at the optimal time $t_\mathrm{buy}$.

For the previous example, we get
$$t_\mathrm{pay} = \frac1{r_1} – \frac1{r_2} = \frac1{0.05} – \frac1{0.08} = 20 – 12.5 = 7.5\mathrm{\ years,\ and}$$
$$\mathrm{interest\ income\ during\ purchase} = c/t_{\mathrm{pay}}= 1100/7.5 \approx \146.67/\mathrm{year}.$$

### Recovery Time

If you do purchase the interest rate hike at the optimal time, how many years will you need to wait until the optimal strategy surpasses the never buy strategy. The recovery time for the second banker’s problem is almost, but not quite the same as the first banker’s problem. For the second banker’s problem,
$$t_{\mathrm{surpass}} = t_{buy} + 1/r_1.$$
For the example,
$$t_{\mathrm{surpass}} \approx 21.5228+ \frac{1}{0.05} = 41.5228.$$ The black line shows the results of buying the interest rate increase at the optimal time. The blue line shows what happens if you never buy the interest rate hike and just continue to get 5% interest.
If you buy the interest rate hike at the optimal time, then you will maximize the account balance at all times $t>1/r_1$ years later. (Mathematically, for every $t> t_{\mathrm{buy}} + 1/r_1$, the strategy of purchasing the interest rate upgrade at time $t_{\mathrm{buy}}$ results in an account balance at time $t$ that exceeds the account balance at time $t$ using any other strategy.)

## Ideas That Changed My Life

Morgan Housel posted an article titled “Ideas that Changed My Life”.  See

https://collabfund.com/blog/ideas-that-changed-my-life/

and

https://news.ycombinator.com/item?id=33907245

I really like the idea that there are these great ideas that change your life.  I decided to sit down and try to write up my own list of the ideas that influenced me.  Here is my first draft (ordered from most influential to least):

1) Practice makes you better.

2) You can make decisions that affect the rest of your life.

3) Programming (the idea that many procedures can be broken down into precisely defined steps which can be done by a computer.)

4) People are mostly kind.  women are somewhat more likely to be kind than men.

5) Your conscious mind can make changes to your brain and your body.  You can exercise.  You can teach yourself like or dislike things more.  You can change your habits.

6) You can learn skills from a book or other written materials.

7) The idea and laws of probability (including Baysian)

8) The ability to estimate (Fermi Questions).

9) Information Theory and  Kolmogorov Complexity (Algorithmic Complexity)

10) Polynomial and Exponential Growth  (related to Black Swans)

11) You can use algebra to solve for one variable given a few equations

12) Calculus (Esp Fundamental Theorem of Calculus)

13) Object oriented Programming (Inheritance, Polymorphism)

14) Mathematical proof – the idea that you can take a bunch of axioms and use logic to reach many amazing and useful conclusions.  Furthermore, you can have a high degree of certainty that the result (theorem) is true especially if it is verified by another mathematician or a computer.

15) Game Theory (The Nash Equilibrium often exists and it (or they) can often be found.)

16) Energy and Momentum are both conserved.  That fact arises from the symmetries of the universe (invariance under translation by space time, and rotation (see Emmy Noether)).

17) Optimization ( many problems have an optimal solution and often calculus or math can be used to find it.)

18) You can guess many of the formulas of physics

19) Thermodynamics (Laws of Thermodynamics)

20) Recursion

21) Extending the idea of the Pythagorean theorem   (Application of Hilbert Spaces)

## Optimal Putting in Disc Golf – Revised

About a year ago, I posted a short article about optimal putting in disc golf.  I gave an approximate rule of thumb to use at the end of the article, but it turns out that there is a better thumb rule, so I’m revising the post accordingly below.

I often have to make a decision when putting at the edge of the green during disc golf.  Should I try to put the disc in the basket (choice 1) or should I just try to put the disc near the basket (choice 2)? If I try to put it in the basket and I miss, sometimes the disc will fly or roll so far way that I miss the next shot.

In order to answer this question, I created a simplified model.  Assume:

• If I do try to put it in the basket (choice 1), I will succeed with probability $p_1$.
• If I do try to put it in the basket (choice 1), fail to get it in the basket, and then fail again on my second try, then I will always succeed on the third try.
• If I don’t try to put the disc in the basket (choice 2), I will land near the basket and get it in the basket on the next throw.

Using these assumptions, I can compute the average number of throws for each choice.

For choice 2, I will always use two throws to get the disc in the basket.

For choice 1, there are three possible outcomes:

• outcome 1.1: With probability $p_1$, I get it the basket on the first throw!
• outcome 1.2: With probability $p_2$, I miss the basket, but get the disc in the basket on the second throw.
• outcome 1.3: With probability $p_3$, I miss twice, and get it on the third throw.

I am assuming that if I miss twice, I will always get it on the third try, so

$$p_1 + p_2 + p_3=1.$$

Let $a$ be the average number of throws for choice 1. Then $$a = p_1\cdot 1 +p_2\cdot 2 +p_3\cdot 3.$$

I should choose choice 1 (go for it) if I average fewer throws than choice 2, i.e. if $a<2$.  This occurs when

\begin{aligned}2 >& p_1\cdot 1 +p_2\cdot 2 +p_3\cdot 3 \\2 (p_1+p_2+p_3) >& p_1\cdot 1 +p_2\cdot 2 +p_3\cdot 3\\p_1\cdot 2+p_2\cdot 2+p_3\cdot 2 >& p_1\cdot 1 +p_2\cdot 2 +p_3\cdot 3\\ p_1 >& p_3.\end{aligned}

So, I should choose choice 1 if $p_1> p_3$.  In words,

## The Big O writes on Scientific Error, Bias, and Self-(peer)-review.

I have slightly edited the post below by my friend the big O.

After all my complaining I have reached an actual argument about scientists or, better put, the science class.  They lack rigorous checks and balances.
To think that science can effectively check itself for error and bias just because they’re supposed to is analogous to thinking that the house of representatives can effectively check itself for error and bias just because they’re supposed to. But we know that the job of congressman frequently attracts a type of personality. We know there is an attractive glory to accomplishment in government as there is an attractive glory to accomplishing something in science.  We know that money provides an incentivizing role in even the noblest of endeavors.
And so in government we have branches whose explicit role is to check and balance one another.  And we have a press whose job is to check and seek error with the government itself. And we have a population who feel no embarrassment at checking everyone: we scrutinize the press, we critique the government in general, and we attack specific branches of government, all the while recognizing that government by the people (etc.) is prone to errors and biases specifically because it is a government run by people.
This is not the attitude we have with the class of people who conduct science.  The scientist class has reached such a rarefied status that it lacks equivalent checks and balances. We expect scientists to nobly check themselves.  But i argue they cannot because they’re people.  We need more rigorous outside skepticism than we currently have.
Science has the hardest arguments on earth to develop, prove, justify, and explain because the arguments of science are targeted at revealing something close to objective truth. there are more obstacles and unseen variables between scientific theory and proof than in any other field.  I think we would be better off to consider non-scientist (a “scientist” today being someone who is sanctioned by a university to be labeled as such) checking and balancing as part of – not apart from – the scientific process.
I like the idea of retaining a unembarrassed and reasoned skepticism of the “truth” offered by scientists – particularly in the weak sciences – and instead accept effectiveness (e.g. when science becomes technology) as truth.  When something – a theory or an experiment – works in our daily lives we can label that something as “true enough to be effective” and realize that as an auspicious label.  The rest should invite continued checking and balancing from both in and outside the scientific class.
$$\$$
And then in another email, I added this as a response to my critics:
The one thing I do not attack is the scientific method or reason.
I say the scientific method is not currently being employed to an extent that it could be and we’re worse off for it. As evidence of this I see the very frequent conflation of science – which is an effective process – with scientists – whom I consider to be as flawed and for the same reasons as any other profession.  (That npr study conforms with my personal experience that scientists my be more justifying of their bias than others).  This conflation leads to a citizenry reluctant to be skeptical of scientists and scientific work because they fear they are questioning science.  They are not. they are a part (apparently unknowingly) of science and their reluctance to analyze, question, investigate, criticize scientists and scientist’s work leads to a weakening of science.

The peer review you cite is precisely what I am labeling inadequate.  What if a congressman suggested his law was good because it had been peer reviewed?  What if he said he had special training as a lawyer or an economist or a historian?  Would that be a satisfying rational not to have separate gov’t branches, press, or citizenry actively challenge his work?  Scientific peer review is equivalent to gov’t peer review – it is necessary but less adequate than broadening that peer review to others currently only limitedly involved or allowed in the process.

Again:  re: the scientific method, I don’t believe that a science that includes only an academia certified science class does or even can adequately follow the scientific method to its fullest rigor anymore than congress can adequately run the government to its fullest rigor minus the critical analysis of outside agencies such as other branches of gov’t, the press, and citizenry.

Imitation is the greatest form of flattery.

1. The just plain fun retro turbo Pascal editor.
2. The Rush movie reminds me of the “Who is your arch-enemy?” post (see also 37 signals) and “The smackdown learning model“.
3. The rule of three for re-usable code.
4. The PCP Theorem and Zero Knowledge Proofs suggest avenues of attack on P vs NP
5. God’s number is 20 (assuming God invented Rubik’s cube)
6. Why Your Brain Needs More Downtime (Scientific American, Link from Carl)
7. Keith’s Photos of Caves, Critters, and Nature

I wonder if there is some way to identify which internet queries are the most frequently asked and least frequently successfully answered.

I bring this up because my three most popular blog posts are:

The 100 theorems post and the deep support post are similar.  Both are fairly short simple answers to potential internet queries. The 100 theorems post is an answer to “What are the most useful theorems in mathematics?” and the “deep support” article answers the very technical AI question “What is a deep support vector machine?”

Cheers, Hein

## Time Off

The last week of classes and finals week were kind of brutal both for me and my students.  (It wasn’t that bad, but it was a lot of work.)  So, I’m taking some time off blogging.  Cheers, Hein

This is just too cute.  Type

data:text/html, <html contenteditable>

into the URL tab and then type in the blank page.

## Schwarzenegger Bandit Success Formula

In “Does Luck Matter More Than Skill?“, Cal Newport writes

<success of a project> = <project potential> x <serendipitous factors>,

where <project potential> is a measure of the rareness and value of your relevant skills, and the value of the serendipitous factors is drawn from something like an exponential distribution.

and

If you believe that something like this equation is true, then this approach of becoming as good as possible while trying many different projects, maximizes your expected success.

Indeed, we can call this the Schwarzenegger Strategy, as it does a good job of describing his path to stardom. Looking back at his story, notice that he tried to maximize the potential in every project he pursued (always “putting in the reps”). But he also pursued a lot of projects, maximizing the chances that he would occasionally complete one with high serendipity. His breaks, as described above, all required both rare and valuable skills, and luck. And each such project was surrounded in his life by other projects in which things did not turn out so well.

If success is measured in dollars, then I bet the distributions of <serendipitous factors> have fat 1/polynomial tails because there are a lot of people with great skills, but the wealth distribution among self-made billionaires is something like C/earnings^1.7.  For many skills, like probability of hitting a baseball, the amount of skill seems to be proportional to log(practice time) plus a constant.  For other skills, like memorized vocabulary, the amount of skill seems proportional to (study time)^0.8 or the Logarithmic Integral Function.  Mr Newport emphasizes the “rareness” of skill also.  Air is important, but ubiquitous, so no one charges for it despite it’s value.  In baseball, I imagine that increasing your batting average a little bit can increase your value a lot.  I wonder what the formulas for <project potential> are for various skills.  If we could correctly model Newport’s success equation, we could figure out the correct multi-armed bandit strategy for maximizing success.  (Maybe we could call it the Schwarzenegger Bandit Success Formula.) You may even be able to add happiness into the success formula and still get a good bandit strategy for achieving it.

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