Author: Sergey Alexeev - Page 2 - Backward induction

Kinda cool phrases

These phrases are kinda cool:

x) Fish don’t know they’re in water until they experienced air;
x) A man can do as he will but not will as he will;
x) Mathematics is a profession where you can not tell if a person is working or sleeping;
x) If you don’t read the newspaper you are uninformed; if you do read the newspaper you are misinformed;
x) All models are wrong but some are useful;
x) In god we trust the rest bring data.

Intermediate Microeconomics (UTS)

Dear students, please use the videos below if you missed a class. Please note, however, that these are, in essence, my “preparation notes”, and are not designed to completely substitute tutorials. I’m using pencasts as a motivation to work through the problems (a credible threat of a sort).

Solutions
Solutions by Jonathon Livermore
Pencasts:
Tutorial 1. Q1 Q2 (notes)
Tutorial 2. Q1 Q2 Q3 Q4
Tutorial 3. Q1 Q2 Q3 Q4 (notes)
Tutorial 4. Q1 Q2 Q3 Q4 (notes)
Tutorial 5. Q1 Q2 Q3 (notes)
Tutorial 6. Q1 Q2 Q3 Q4 (notes)
Tutorial 7. Q1 Q2 (notes)
Tutorial 8. Q1 Q2
Tutorial 9. Q1 Q2 Q3
Tutorial 10. Q1 Q2 Q3
Tutorial 11. Q1
Students’ reviews

About maths in economics…. hmm

Von Neumann: “As a mathematical discipline travels far from its empirical source, or still more, if it is a second or third generation only indirectly inspired by ideas coming from “reality”, it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely l’art pour l’art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much “abstract” inbreeding, a mathematical subject is in danger of degeneration.”(source)

Marshal: “I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules — (1) Use mathematics as a shorthand language, rather than an engine of inquiry. (2) Keep to them until you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can’t succeed in 4, burn 3. This last I did often.” (source)

Some simple probability formulas with examples

A known relationship that is usually given axiomatically:

$P(B|A) = \frac{{P(AB)}}{{P(A)}}$

Upon rearrangement gives the multiplication rule of probability:

$P(AB) = P(A)P(B|A) = P(B)P(A|B)$

Now observe a cool set up that is handy to keep in mind for proving the law of total probability and Bayes’ theorem.

Imagine that $B$ happens with one and only one of $n$ mutually exclusive events $A_1, A_2,..., A_n$ , i.e.:

$B = \sum\limits_{i = 1}^n {B{A_i}}$

By addition rule:

$B = \sum\limits_{i = 1}^n {P(B{A_i})}$ .

Now by multiplication rule:

$B = \sum\limits_{i = 1}^n {P({A_i})P(B|{A_i})}$ .

This is the law of total probability

From the same set up imagine that we want to find the probability of even $A_i$ if $B$ is known to have happened. By the multiplication rule:

$P(A_i B) = P(B)P(A_i|B) = P(A_i)P(B|A_i)$

By neglecting $P(A_i B)$ and dividing the rest through $P(B)$ we get:

$P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{P(B)}}$

And applying the law of total probability to the bottom we have the Bayes’ equation

$P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{\sum\limits_{j = 1}^n {P({A_j})P(B|{A_j})} }}$

Bunch of examples:

Problem: $P_t (k)$ is a known probability of receiving $k$ phone calls during time interval $t$ . Also $k=0,1,2,...$ . Assuming that a number of received calls during two adjeicent time periods are independent find the probability of receiving $s$ calls for the time interval that equal $2t$ .

Solution: Let $A_{b.b + t}^k$ be an event consisted of $k$ call in the interval $b$ till $b+t$ . Then clearly

$A_{0,2t}^s = A_{0,t}^0A_{t,2t}^s + ... + A_{0,t}^sA_{t,2t}^0$

which means that the event $A_{0,2t}^s$ can be seen as sum of $s+1$ mutually exclusive events, such that in the first interval of duration $t$ number of calls received is $i$ and in the second interval of the same duration number of received calls is $s-i$ ( $i=0,1,2,...,s$ ). By rule of addition

$P(A_{0,2t}^s) = \sum\limits_{i = 0}^s {P(A_{0,t}^iA_{t,2t}^{s - i})}$ .

By the rule of multiplication

$P(A_{0,t}^iA_{t,2t}^{s - i}) = P(A_{0,t}^i)P(A_{t,2t}^{s - i})$

If we change the notation so that

${P_{2t}}(s) = P(A_{0,2t}^s)$

then

${P_{2t}}(s) = \sum\limits_{i = 0}^s {{P_t}(s) \cdot P(s - i)}$ .

It is known that under quite general conditions

${P_t}(k) = \frac{{{{(at)}^k}}}{{k!}}\exp \{ - at\} {\rm{ }}(k = 0,1,2...)$

(Recall that the Poisson distribution is an appropriate model if the following assumptions are true. (a) $k$ is the number of times an event occurs in an interval and $k$ can take values $0,1,2,...$ . (b) The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently. (c) The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals (that kinda a lot to take on faith really). (d) Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur. (e) The probability of an event in a small sub-interval is proportional to the length of the sub-interval. Or instead of those assumptions, the actual probability distribution is given by a binomial distribution and the number of trials is sufficiently bigger than the number of successes one is asking about (binomial distribution approaches Poisson).)

Parametrisation then gives

${P_{2t}}(s) = \sum\limits_{i = 0}^s {\frac{{{{(at)}^s}}}{{i!(s - i)!}}\exp \{ - 2at\} } {\rm{ = }}{(at)^s}\exp \{ - 2at\} \sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}}}$

Note that

$\sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}} = \frac{1}{{s!}}\sum\limits_{i = 0}^s {\frac{{s!}}{{i!(s - i)!}}} = \frac{1}{{s!}}{{(1 + 1)}^s} = \frac{{{2^s}}}{{s!}}}$

Then

${P_{2t}}(s) = \frac{{{{(2at)}^s}\exp \{ - 2at\} }}{{s!}}{\rm{ }}(s = 0,1,2,...)$

The key point is that if for time interval $t$ we have that parametrized formula for $2t$ we have the one above. It holds true for any multiples of $t$ as well.

Social dark matter

If you look at the ants you see that they are able to coordinate their actions. They navigate in the uncertainty by sharing some information amongst each other, and they do it by licking another ants’ acid (that’s kinda gross). Ants are able to share important information that can be used to coordinate their actions so that all of them become better off.

People don’t leak acid for this purpose most of the times. However, the most important exchange of information is still done by exchanging fluids. An exchange of genetic information.

But how do people are able to exchange information and coordinate their actions? They do this because of the ideas that they have in their minds. Crazy! You know that there is a table, you know that there is a glass of water. But ideas are not real, they don’t exist. Yet, they do exist and are real, but only for one species on earth.

Imagine that you are an alien that hovers above Earth and observes peoples’ actions. You would be amazed to see that somehow people manage to coordinate with each other. You’ll see someone at one part of the planet makes, idontknow, a windshield for an airplane that is assembled in another part of the planet.

But how is this possible? How is this happening that we people developed something that is real only for us and we use it to do much more than we would have done if we were alone?

The most amazing thing is that we don’t think explicitly that emotions inside us and the ideas that follow actually exist and emerged so we would be able to do more by to coming together.

Just like an and that is licking someone else’s acid doesn’t see the full picture he’s just taking one step at a time following a familiar taste.

Morals. Right. Wrong. Prices. God. Education. Marriage. Patriotism. Love. Anger. All of it a exist in us not for its own sake but for us to be able to become stronger by coordinating actions with each other.

So it means that there is an enormous collection of objects that are not actually real, don’t exist in a physical form, yet they are real for people.

In some sense ideas – from the point of view of an alien that hovers above Earth and doesn’t actually observe ideas but see the result of their existence indirectly – is a dark matter.

This dark matter that nobody directly observes serve as the glue that keeps people together.

A good question is that who are scientists, especially social scientists, especially those who are able to actually say something meaningful. In the universe of ants, they are the ones that for some reason instead of following a familiar smell to make a step forward raise their heads and look around. In some sense, they stop being ants and they become aliens that hovers about above Earth. An advantage that they have over aliens is that they actually see what aliens can only guess exists.

Scientists create ideas and most of the scientific papers don’t do anything good; don’t build bridges, don’t treat patients, don’t grow vegetables. They just collect, create and structure ideas. And this is where an alien that hovers above Earth is the most confused.

Why would someone like an idea of a fixed point in a multidimensional space, or Nash equilibrium or a measure theory? Very little will be ever used to make a physical world any better.

Why is there so many ideas and people care about them? What does it take to come up with an idea that is useless but so beautiful that people allow those freaks who came up with it to exist in a society and even prosper?

Some theories I guess are useful. It seems like people due to their computational limitations are unable to perceive the world in its true complexity. That’s why we need theories that degenerate reality, turns something very complicated into something much more simpler. In some sense, we need theories and they exist because we are unable to process the world as it is. We can focus on one force at a time, one concept at the time. In this sense, the reality is a million theories that happen at the same time but we fail to see it.

Personally, an explanation that makes sense to me is that due to evolution human brain became a very a complicated system. A probability of a glitch in the complicated system is much higher. Most talented scientists are indeed a little bit crazy.

That’s why scientific methods are so time-consuming to learn. We never evolved to use our brains for this. We great at gossiping and calling people “bad” and stuff.

I like this post on it.

A simple fact about sets

Out of $n$ elementary events one can get

$\sum_{m=1}^{n} C_{n}^{m} = 2^n - 1$

possible outcomes. Where $C_{n}^{m}$ is an event that contains $m$ elementary events. Take set

$\{ a,b,c\}$

with the size as the only characteristic $n=3$ . Then it power set

$\{ \{ a\} ,\{ b\} ,\{ c\} ,\{ a,b\} ,\{ a,c\} ,\{ b,c\} ,\{ a,b,c\} ,\{ \emptyset \} \}$

contains ${2^3} = 8$ elements. $3$ event for one element each, $C_{3}^{1}$ . Then $3$ events with two element, $C_{3}^{2}$ . Finally, $1$ event for one with all elements, $C_{3}^{1}$ . A emply set is an impossible event.

I personally think that this simple fact is amazing, but some would say it is kinda boring. Here is an interesting question for those.

A pack of cards that has $36$ cards is randomly split equally into halves. What is the probability that halves have equal amount black and red cards?

This is just another set with $36$ elements of two type.

$p = \frac{{C_{18}^9 \times C_{18}^9}}{{C_{36}^{18}}} = \frac{{{{(18!)}^4}}}{{36!{{(9!)}^4}}}$

The denominator indicates all possible equally likely ways the pack can be split.

Instead of computing that manually one can use this asymptotic equality

$n!\ \approx \sqrt {2\pi n} \cdot {n^n}{e^{ - n}}$

Thus

$18!\ \approx {18^{18}}{e^{ - 18}}\sqrt {2\pi \cdot 18}$

$9!\ \approx {9^9}{e^{ - 9}}\sqrt {2\pi \cdot 9}$

$36!\ \approx {36^{36}} \cdot {e^{ - 36}}\sqrt {2\pi \cdot 36}$

Which means

$p \approx \frac{{{{(\sqrt {2\pi \cdot 18} \cdot {{18}^{18}} \cdot {e^{ - 18}})}^4}}}{{\sqrt {2\pi \cdot 36} \cdot {{36}^{36}} \cdot {e^{ - 36}}{{(\sqrt {2\pi \cdot 9} \cdot {9^9} \cdot {e^{ - 9}})}^4}}}$

Simple algebra yields

$p \approx \frac{2}{{\sqrt {18\pi } }} \approx \frac{4}{{15}} \approx 0.26$

The result fascinates me. The graph visualizes data from a real experiment where a pack is split equally $100$ times and $\mu$ is a cumulated sum if exactly $9$ red cards are observed in on of the halves. What is crazy is that we were able to see the results of this experiments without doing any experiments, by simply reasoning mathematically about things.

More on this topic: Гнеденко-1988

Distribution of a ordered pile of rubble

Imagine a pile of rubble ( $X$ ) where the separated elements of the pile are stones ( $x_i$ ). By picking $n$ stones we form a sample that we can sort by weight. A sequence $x_1,x_2,...,x_n$ becomes $x_{(1)},x_{(2)},...,x_{(m)},...x_{(n)}$ , where $(m)$ is called “rank”.

Pretend that we do the following. Apon picking a sample and sorting it we put stones into $n$ drawers and mark each drawer by rank. Now repeat the procedure again and again (picking a sample, sorting and putting stones into drawers). After several repetitions, we find out that drawer # $1$ contains the lightest stones, whereas drawer # $n$ the heaviest. An interesting observation is that by repeating the procedure indefinitely we would be able to put all parenting set (the whole pile or the whole range of parenting distribution) into drawers and later do the opposite — take all stones (from all drawers) mix them to get back the parenting set. (The fact that distributions (and moments) of stones of particular rank and the parenting distribution are related is probably the most thought-provoking)

Now let us consider the drawers. Obviously, the weight of stones in a given drawer (in a rank) is not the same. Furthermore, they are random and governed by some distribution. In other words, they are, in turn, a random variable, called order statistics. Let us label this random variable $X_{(m)}$ , where $m$ is a rank. Thus a sorted sample looks like this

$X_{(1)},X_{(2)},...,X_{(m)},...,X_{(n)}$

Its elements $X_{(m)}$ (a set of elements (stones) $x$ from the general set $X$ (pile) with rank $m$ (drawer)) are called $m$ order statistics.

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Elements $X_{1}$ and $X_{(n)}$ are called “extreme”. If $n$ is odd, a value with number $m=\frac{(n+1)}{2}$ is central. If $m$ is of order $\frac{n}{2}$ this statistics is called “ $m$ central” A curious question is how define “extreme” elements if $n \to \infty$ . If $n$ increases, then $m$ increases as we.

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Let us derive a density function of $m$ order statistics with the sample size of $n$ . Assume that parenting distribution $F(x)$ and density $f(x)$ are continues everywhere. We’ll be dealing with a random variable $X_{(m)}$ which share the same range as a parenting distribution (if a stone comes from the pile it won’t be bigger than the biggest stone in that pile).

Untitled

The figure has $F(x)$ and $f(x)$ and the function of interest $\varphi_n (\cdot)$ . Index $n$ indicates the size of the sample. The $x$ axis has values $x_{(1)},...,x_{(m)},...,x_{(n)}$ that belong to a particular realization of $X_{(1)},X_{(2)},...,X_{(m)},...,X_{(n)}$

The probability that m-order statistics $X_{(m)}$ is in the neuborhood of $x_{(m)}$ is by definition (recall identity: $dF = F(X + dx) - F(x) = \frac{{F(x + dx) - F(x)}}{{dx}} \cdot dx = f(x) \cdot dx &s=-3$ ):

$dF_{n}(x_{(m)})=p[x_{(m)}<X_{(m)}<x_{(m)}+dx_{(m)}]=\varphi_n (x_{(m)})dx_{m}$

We can express this probability in term of parenting distribution $F(x)$ , thus relating $\varphi_n (x_{(m)})$ and $F(x)$ .

(This bit was a little tricky for me; read it twice with a nap in between) Consider that realization of $x_1,...,x_i,...,x_n$ is a trias (a sequence generated by parenting distribution, rather then the order statistics; remember that range is common) where “success” is when a value $X<x_{(m)}$ is observed, and “failure” is when $X>x_{(m)}$ (if still necessary return to a pile and stone metaphor). Obviously, the probability of success is $F(x_{(m)})$ , and of a failure is $1-F(x_{(m)})$ . The number of successes is equal to $m-1$ , failures is equal to $n-m$ , because $m$ value of $x_m$ in a sample of a size $n$ is such that $m-1$ values are less and $n-m$ values are higher than it.

Clearly, that the process of counting of successes has a binomial distribution. (recall that probability of getting exactly $k &s=-3$ successes in $n &s=-3$ trials is given by pms: $p(k;n,p) = p(X = k) = \left( \begin{array}{l} n\\ k \end{array} \right){p^k}{(1 - p)^{n - k}} &s=-3$ In words, $k &s=-3$ successes occur with $p^k &s=-3$ and $n-k &s=-3$ failures occur with probability $(1-p)^{n-k} &s=-3$ . However, the $k &s=-3$ successes can occur anywhere among the $n &s=-3$ trials, and there are $\left( \begin{array}{l} n\\ k \end{array} \right) &s=-3$ different ways of distributing $k &s=-3$ successes in a sequence of $n &s=-3$ trials. A little more about it)

The probability for the parenting distribution to take the value close to $x_{(m)}$ is an element of $dF(x_{(m)})=f(x_{(m)})dx$ .

The probability of sample to be close to $x_{(m)}$ in such a way that $m-1$ elements are to the left of it and $n-m$ to the rights, and the random variable $X$ to be in the neighborgood of it is equal to:

$C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})dx$

Note that this is exactly $p[x_{(m)}<X_{(m)}<x_{(m)}+dx_{(m)}]$ , thus:

$\varphi_n (x_{(m)})dx_{m}=C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})dx$

Furthermore if from switching from $f(x)$ to $\varphi_n (x_{(m)})$ we maintaine the scale of $x$ axis then

$\varphi_n (x_{(m)})=C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})$

The expression shows that the density of order statistics depends on the parenting distribution, the rank and the samples size. Note the distribution of extreme values, when $m=1$ and $m=n$

The maximum to the right element has the distribution $F^{n}(x)$ and the minimumal $1-[1-F(x)]^n$ . As an example observe order statistics for ranks $m=1,2,3$ with the sample size $n=3$ for uniform distribution on the interval $[0,1]$ . Applying the last formula with $f(x)=1$ (and thus $F(x)=x$ we get the density of the smallest element

$\varphi_3 (x_{(1)})=3(1-2x+x^2)$ ;

the middle element

$\varphi_3 (x_{(2)})=6(x-x^2)$

and the maximal

$\varphi_3 (x_{(3)})=3x^2$ .

With full concordance with the intuition, the density of the middle value is symmetric in regard to the parenting distribution, whereas the density of extreme values is bounded by the range of the parenting distribution and increases to a corresponding bound.

Note another interesting property of order statistics. By summing densities $\varphi_3 (x_{(1)}), \varphi_3 (x_{(2)}), \varphi_3 (x_{(3)})$ and dividing the result over their number:

$\frac{1}{3}\sum\limits_{m = 1}^3 {{\varphi _3}({x_{(m)}}) = \frac{1}{3}(3 - 6x + 3{x^2} + 6x - 6{x^2} + 3{x^2}) = 1 = f(x)}$

on the interval $[0,1]$

The normolized sum of order statistics turned out to equla the parenting distribution $f(x)$ . It means that parenting distibution is combination of order statistics $X_{(m)}$ . Just like above had been mentioned that after sorting the general set by ranks we could mix the sorting back together to get the general set.

Further read: Ефимов-1980; Arnord-balakrishnan-2008.

Math is the extension of common sense

What makes math? Isn’t it just common sense?

Yes. Mathematics is common sense. On some basic level, this is clear. How can you explain to someone why adding seven things to five things yields the same result as adding five things to seven? You can’t: that fact is baked into our way of thinking about combining things together. Mathematicians like to give names to the phenomena our common sense describes: instead of saying, “This thing added to that thing is the same thing as that thing added to this thing,” we say, “Addition is commutative.” Or, because we like our symbols, we write:

For any choice of a and b, a + b = b + a.

Despite the official-looking formula, we are talking about a fact instinctively understood by every child.

Multiplication is a slightly different story. The formula looks pretty similar:

For any choice of a and b, a × b = b × a.

The mind, presented with this statement, does not say “no duh” quite as instantly as it does for addition. Is it “common sense” that two sets of six things amount to the same as six sets of two?

Maybe not; but it can become common sense. Eight groups of six were the same as six groups of eight. Not because it is a rule I’d been told, but because it could not be any other way.

We tend to teach mathematics as a long list of rules. You learn them in order and you have to obey them, because if you don’t obey them you get a C-. This is not mathematics. Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

Now let’s be fair: not everything in mathematics can be made as perfectly transparent to our intuition as addition and multiplication. You can’t do calculus by common sense. But calculus is still derived from our common sense—Newton took our physical intuition about objects moving in straight lines, formalized it, and then built on top of that formal structure a universal mathematical description of motion. Once you have Newton’s theory in hand, you can apply it to problems that would make your head spin if you had no equations to help you. In the same way, we have built-in mental systems for assessing the likelihood of an uncertain outcome. But those systems are pretty weak and unreliable, especially when it comes to events of extreme rarity. That’s when we shore up our intuition with a few sturdy, well-placed theorems and techniques, and make out of it a mathematical theory of probability.

The specialized language in which mathematicians converse with each other is a magnificent tool for conveying complex ideas precisely and swiftly. But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking. That’s exactly wrong.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved is little different from the way we think about more down-to-earth problems. I find it helpful to keep in mind an image of Iron Man punching a hole through a brick wall. On the one hand, the actual wall-breaking force is being supplied, not by Tony Stark’s muscles, but by a series of exquisitely synchronized servomechanisms powered by a compact beta particle generator. On the other hand, from Tony Stark’s point of view, what he is doing is punching a wall, exactly as he would without the armor. Only much, much harder.

To paraphrase Clausewitz: Mathematics is the extension of common sense by other means.

Without the rigorous structure that math provides, common sense can lead you astray. That’s what happened to the officers who wanted to armor the parts of the planes that were already strong enough. But formal mathematics without common sense—without the constant interplay between abstract reasoning and our intuitions about quantity, time, space, motion, behavior, and uncertainty—would just be a sterile exercise in rule-following and bookkeeping. In other words, math would actually be what the peevish calculus student believes it to be.

A citation from Ellenberg’s “How Not To Be Wrong…” book. Kinda liked it.

A conjecture on mating

What is dating and why do we even need it? Here is mine theory. I have not cross-referenced it with existing sciency literature, thus, it could lack originality or could be just nuts (it’s really just some random thoughts). The theory naturally follows from several observations, so I start with those. Medical science has a good understanding of how a perfect, textbook, human body looks like. In reality, a perfect body does not exist. It is just an idea that is useful to understand what is right and what is wrong with a patient. A deviation from this conceptual body can help in classification. Noteworthy is that there has been a considerable change in classifying deviation into the right and wrong. Many diseases that were classified before as a subject for a treatment, today let roll on their own.

Nature never creates perfect bodies, because it is not sure how a perfect body looks like. The process of a human creation by nature can be understood as following. Design a perfect, textbook, body and then introduce disturbances into some system of the body. The disturbances are known as mutations, and the whole process is known as evolution. Put differently, nature randomizes human bodies and then the environment trims the randomization that was not useful. It is conceptually indistinguishable from how a programmer develops a code of a program. There is a core functionality and over time the programmer introduces features and see if they make the program better. The key difference is that the programmer controls the “trimming” process. He’d know very well which feature came through testing and which require further testing because they are promising, but initial tests were not very successful. Well here is an amazingly awesome news. There is a conceptual analog of a programmer. A woman. Natur is agnostic about which features became successful and which were not. Let’s start with counterposition. If there were no women then the progress of medical science (with its moto “no pain is great”) generates this:

Ok, that might be obvious. So, having woman improves the sorting process and trims unsuccessful mutations. But how exactly? This is the best bit. The process of a woman picking a man has exactly the same characteristics as a patient picking a doctor and a firm picking a worker. When you come to see a doctor you would like him to know medical stuff more than you do. When you come to see a surgeon you would like him to make right choices during surgery when you are asleep and unavailable for consultation. The problem is that when you see a doctor you see a head, two legs, and two arms. These observables are not very useful to infer the unobserved characteristics of a person that actually matter for you. That is why you use potentially useless and silly observables as proxies for unobservables that matter.

Let me start with “a flip of a coin in the vacuum”. Imagine all people have perfect, text-book bodies, they are exactly the same. Then people can form a group to archive the economy of scale (to hunt elephants or to produce iphones or to make healthy fat well-nourished kids) with anyone. Then one does not need friends, family or anyone really. There is no need to designate anyone as special. If you feel like having a beer or sex you just talk to the next person next to you ask if he/she don’t mind and just do it. The same happens with kids, you have kids and if you need anyone to babysit you just give a kid to a next person on the street. One does not even have to go home in the same place every night. Just crush to the closest bed. This is a benchmark.

Now imagine nature intentionally introduce noise to every person. Sort of introducing random features and then the environment needs to test the feature by killing the versions that are no good. Now people are different and they possess characteristics that could be useful in the current environment or could be useless. Now it matters who is in your group. To form groups quickly our brain can classify people into a bad person (immoral people) and good person (moral person). There are even the whole institutions that people created to facilitate the sorting, reputation and even… church. The church is kinda like education, it helps to send signals about types. A religious person is an unconditional contributor (speaking a language of public goods provision game). Religious people are usually intense, so for them, it is a computational shortcut (it requires extra commentary, don’t worry about it at this point. In short, usually, people spend tons of time to sort people into bad and good and to come out as good. That’s what peoples’ brain is hardwired to do. Some of us decide not to spend too much time strategizing, but simply contribute all the time (hard working people, like productive scientists), but they expect others to contribute when it is crucial for them).

A family is a special case of a group. A man possesses some properties that are unobserved, thus a woman chooses observable as a proxy. Good physiqueis good, but it is not a sufficient indicator for skills. Money is better. Both already serve as a better proxy, they convey more information. Those observable are more likely to indicate strong providing properties of a person. A woman also wants a man to be responsive to incentives, thus, non-cognitive skill also matter. A woman wants someone who has good social skills. This approach refines the sorting process and makes it very intelligent; you could be narcoleptic, so you probably will lose a fight with a crocodile and won’t survive in a forest a day, but you still do fine as a scientist. In this manner, a narcoleptic gene still persists in the population even though it manifests in a really really weird behavior of passing out randomly during the day. It is not designated for trimming, the opposite it designated as a potential feature that might over year constitute a “perfect” text-book body.

It could be shown that if the world consisted of identical people a woman would interact with whoever is closer, thus it could be stated that the total amount of time a woman interacts with a man (in aggregate) is:

$S=M \times T$

S is given, it could be that interaction is needed due to a physical property of environment (a group is necessary because there are many dinosaurs or food is scarce, thus several people need to search for it to get a healthy fat kid). For the reasoning at hand is it given. M is a number of men in the social fragment. If all men are the same then a woman is indifferent, thus all stock of man is used because time per a man (T) is high. If the environment is too risky, woman are too cautious and they socialize with less and less per man, thus for given stock of men and in a given environment more and more men are designated for being trimmed.

I think that this conjecture naturally follows the idea that is advocated by any famous social scientists that ever existed (e.g. Hayek, Friedman). People have evolutionarily developed to construct social structures and they do work with fantastic efficiency. States, markets, mating all of these examples of social structures existed way before scientists had any say in it. People should interfere as less with those as possible, any interferences have to be very gentle. A woman has to be free in her choices because if she is not, then tons of terrible men are not trimmed.

Some interesting manifestations of it: ban on divorce produces massive suicide rates and violence, ban on abortions produces massive criminalization

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Game theory is ridiculous

Game theory is ridiculous. The first acquaintance with main “solution concepts” usually produces a question “wtf?!” in a man with a good common sense.

Good economics approximates essentials with assumptions to overcome limitations of verbal reasoning. Assumptions in game theory mostly exist to confuse readers without really saying anything that matters.

I believe those are not assumptions but conventions and the only question that a man with a good common sense should be trying to ask is “why so many individually silly things when come together say so many astonishingly amazing stories?!”