Blog Archives - Ivan's island

Бизнес модели

8/20/2012

Безизходицата е съдба на хората без въображение. Житейските проблеми са решими, понякога по няколко различни начина. Днес ще го демонстрираме, сравнявайки бизнес модела на "Жилет" с този на "Уилкинсън".

"Жилет" залагат на "нарко-модела", масово използван от лидерите, без значение дали те произвеждат автомобили или копирни машини: зарибяваш жертвата с ниски цени и след това я доиш до болка с консумативите. Самобръсначката е сравнително евтина, но се налага честа смяна на ножчетата, които са скъпи. Моята "балканска подсилена" брада убива ножчето на "Жилет" за 2 седмици.

Бизнес моделът на "Уилкинсън" е с човешко лице: продават ти по-евтина самобръсначка с много по-евтини ножчета. Къде е "врътката"? Ножчетата траят не по 2 седмици, а по 3 месеца. Продадеш ли на един мъж самобръсначка "Уилкинсън", той (ако не е рекламно лице на "Жилет" или малоумен) никога няма да премине към конкурента (освен ако конкурентът не подобри и поевтини продукцията си драстично). Казано накратко, за да победиш в конкурентната война няма нужда да продаваш милиони ножчета на един мъж; трябва само да продадеш милиони самобръсначки - по една на всеки мъж. Елементарно, Уотсън, разумният мъж има нужда не от милиони ножчета, а просто от една самобръсначка.

The Garden of Forking Games

8/19/2012

I am afraid of bad drivers. When I am in the car of a bad driver I always think he could do something stupid. The only good thing about bad drivers is that they think the rest drive like them and are therefore careful.

I am afraid of good drivers too. They think the rest drive like them, therefore they are not too careful. A good driver tends not to keep a safe distance from the vehicle in front and is not troubled if the driver of the vehicle behind does the same. This is based on the wrong assumption that should something happen they (the good one and the one behind him) will be able to stop in a moment.

I am also afraid of some good thinkers, who think that all people think like them. Many of those thinkers specialize in game theory, so let me take for example a popular story with a game-theoretic flavor.

There is a faraway island inhabited by 10 people, 2 of whom are blue-eyed and 8 of whom brown-eyed. There is a taboo so they do not use mirrors and do not discuss eye colors. Any islander who by chance discovers his true eye color must commit suicide the same day.

One day an old sailor arrives on the island and oblivious to the local custom greets the islanders by saying how glad he is to see another blue-eyed person like him for the first time in years. What happens then?

Based on those assumptions only, many smart game theorists imagine the following. Nothing happens on the first day as any islander knows there is another one who is blue-eyed. On the second day one of the blue-eyed islanders (let us call him A) is surprised that the only blue-eyed islander he knows (B) did not commit suicide on the first day. To A this says there is another blue-eyed one, whose existence prevents B from realizing he (B) is the only blue-eyed one and kill himself. As A knows that all other islanders are not blue-eyed, he concludes he himself is the blue-eyed one and commits suicide. B thinks the same way and kills himself too. On the next day the rest realize they are brown-eyed (as all blue-eyed ones committed suicide on the second day) and commit mass suicide, leaving the island to the sailor.

I claim that the above assumptions are not enough to justify this particular end of the story. There are other (unstated or even unrealized) assumptions:

1. All islanders are as smart as game theorists. I claim that there is evolutionary explanation why people are not that smart, i.e. why game theorists are not that many. If islander A were smart but B (and all others) were not, then A would have reached the right conclusion but B (and the others) would have not. Therefore, A would have killed himself and B (and all others) would have not. Ask yourself whose grandchildren will inhabit the island.

2. All islanders are as honest as game theorists. I claim that there is evolutionary explanation why people are not that honest, i.e. why game theorists are not that many. If islander A were honest but B (and all others) were not then all would have reached the right conclusion but only A would have killed himself. Ask yourself whose grandchildren will inhabit the island.

3. All islanders are traditionalists. I claim that there is evolutionary explanation why people are not such traditionalists and break with tradition whenever possible. What if A realizes there are better ways to risk his life than commit suicide and commits a successful coup d’etat, resulting in abandoning the stupid taboo among other things? Ask yourself whose children will live in the ruler’s palace?

4. The sailor speaks the language of the islanders (or they speak his). This is a common assumption game theorists, logicians, politicians and businesspeople make. In the real world the islanders would have eaten the sailor long before he had learnt their language. If the sailor was American or English, they do not even need to eat him. He will never speak their language. This is how English became the language of the world, English speakers travelled more than anyone else and never cared to learn foreign languages.

5. The islanders and the sailor share common color perception. Had the sailor been color-blind and called blue-eyed those 2 islanders who in fact were green-eyed, all islanders would not have believed him and would have stayed alive (having him punished for lying). Had they (instead of the sailor) been color-blind, they would have done the same, as they would not have known they are color-blind, i.e. they will have perceived themselves as normal and him as color-blind.

Conclusion
What if the sailor was (or was not) color-blind and some of the blue-eyed and brown-eyed islanders were (or were not): color blind and/or honest and/or smart and/or English speakers? We are entering the Garden of Forking Games. Let us stop and ask the following questions.

Can game theorists control their assumptions or is it that they just do not care? As long as a work based on assumptions like those (that all people are smart, honest and English speaking and are not color-blind or just blind) can earn one a living or even a Nobel Prize in Economics, why care if those assumptions are wrong or insufficient?

What does ABSTRACT ART mean?

8/19/2012

Is abstract art an abstract representation of real objects/processes or a "realistic" representation of abstract objects/processes? Almost no one will consider the latter type of art abstract. Botticelli painted The Birth of Venus realistically and everybody looking at it forgets that Venus is something abstract.

The blue thing below is a painting of mine which I named "The Birth of Square Numbers from the Triangular Numbers". If you have problems seeing the square numbers (1, 4, 9 and 16), then count the white triangles (each one of which is marked by a red dot) between any three blue circles. If you have problems seeing the triangular numbers (3, 6, 10 and 15), you are probably blind.

The painting makes easy to see some abstract things.
4=1+1+2 and 9=1+1+2+2+3.
In fact, for any natural number n the corresponding square number
n^2=1+1+2+2+...+(n-1)+(n-1)+n
Therefore, you can easily see that the n-th square number is the sum of the n-th and (n-1)-th triangular numbers. Does this make my painting abstract or not?

minus-Goldbach Calculator

8/17/2012

While Goldbach’s conjecture is that every even number equal to or greater than 4 can be expressed as the sum of two primes, minus-Goldbach’s conjecture (I call it this way because it’s unnamed but has to be referred to) is that every even number can be expressed as the difference between two primes.

While for every even number the number of corresponding Goldbach’s pairs is finite, the number of minus-Goldbach’s pairs is not (see how easy one can time-jump 107 years ahead and reach the Polignac's conjecture). In order to have the algorithm stop we have to choose an interval, in which a finite number of pairs is calculated (see Step 3 below).

My algorithm for calculating minus-Goldbach’s pairs is following:
Step 1. Choose 2n (any even number that you want expressed as the difference between two primes).
Step 2. Find │2n│ (absolute value of 2n) and its half │n│.
Step 3. To find the primes p1 и p2, the difference of which is 2n, you have to search for them at a distance of │n│ from the natural number x (x=│2n│+i). The values of i are natural numbers between 1 and z. You must also choose the value of z, which is the maximum remoteness of x from │2n│. Thanks to those limitations we are able to calculate a finite number of pairs of primes, otherwise the algorithm wouldn’t be able to stop (see above).
Step 4. For every i=1 .. z, calculate x (x=│2n│+i), p2 (p2=x-│n│) and p1 (p1=x+│n│).
Step 5. If p2 and p1 are primes, i.e. if every time when each is divided by 2 or odd numbers lesser than it (excluding 1) there are remainders existing, then output them.

Thanks to Plamen Antonov, you can see the only 'minus-Golbach' calculator on the web here.

How I proved the Pythagorean theorem, 1

8/16/2012

Everybody can prove the Pythagorean theorem, but have you seen a proof as short as mine is?

Всеки може да докаже Питагоровата теорема, но виждали ли сте доказателство, кратко като моето?

Goldbach calculator, 1

8/15/2012

Insight can emerge from failure to acccomplish something else. Gerd Gigerenzer

Following my naive attempt to prove Goldbach’s conjecture, I found an algorithm for calculating the Goldbach’s partitions. The calculator you can see HERE is created (based on the same algorithm) by the talented programmer, engineer, photographer and aviation historian Plamen Antonov.

They say a picture is worth a thousand words. See on the left Plamen's picture of my words.

Here's my algorithm for the Goldbach calculator:
Step 1. Take Х (an even number greater than 4) and divide it by 2.
Step 2. If the halves X/2 are even go to Step 3, otherwise go to Step 4.
Step 3. Add 1 to the first half, and subtract 1 from the second half.
Step 4. If X1 and X2 (the numbers resulting from Steps 2, 3 and 5) are prime, print X1 and X2.
Step 5. If X2=3 stop, otherwise add 2 to X1, subtract 2 from X2 and go to Step 4.

P.S. 1
In the language of Wolfram Mathematica my approach

gold[x_]:=Module[{lst={},h=x/2},While[h>1,If[PrimeQ[h]&&PrimeQ[x-h],AppendTo[lst,{h,x-h}]];h--];lst];
gold[1787834]//AbsoluteTiming

works much faster than the approaches based on IntegerPartitions or FrobeniusSolve functions (see the musings of some experts here).

P.S. 2
f[n_]:=PrimePi[n/2];
g[n_]:=Module[{lst={}}, Do[If[PrimeQ[n-Prime[i]],AppendTo[lst,{Prime[i],n-Prime[i]}]],{i,1,f[n]}];lst];
g[1787834]//AbsoluteTiming

works much faster than its "older sibling" mentioned in P.S. 1.

P.S. 3
June 22, 2020
Who could have known that one day this would be called Ianakiev's Algorithm?

On Game Theory

8/15/2012

Do game theorists theorize enough when they theorize about games? Do they play the games they theorize about? Not all the time, I think. Let me take for example a well-known game desribed in John Allen Paulos, A Mathematician Plays The Stock Market, Basic Books, 2003, p. 6.

Several players (who know they are a group of players) are told to pick individually a number between 0 and 100, which has to be close to 80% of the average number chosen by the group. The one who comes closest wins a monetary reward. What are the players supposed to do?

Game theorists say the following should happen. Some players might think the average number is 50 and pick 80% of 50, which is 40. Others might think that because of the above they have to pick 80% of 40, which is 32. Others might think that because of the above they have to pick 80% of 32, which is 25.6. This goes on and on … If the group is allowed to play continuously, the players will become fluent in meta-reasoning, i.e. fluent in thinking about thinking (theirs and other players’), and will all pick 0. In this particular instance, reaching 0 is reaching Nash equilibrium, a situation in which players no longer need to do what they have successfully done so far, i.e. no longer need to adjust their behavior with respect to other players' adjustments of behavior.

I have several questions:

1. What does a number between 0 and 100 mean?
Where exactly is the number: in (0, 100) or in [0, 100] or in (0, 100] or in [0,100)? This is an easy question if you have followed game theorists’ reasoning about reaching the Nash equilibrium. The number must be in [0, 100].

2. What does a number mean?
Are the players allowed to choose any number, φ or e for instance? It is natural to assume that numbers means whole numbers, but the reasoning of the game theorists shows that it is not the case (as 25.6 is not a whole number).

3. Is there only one Nash equilibrium if only whole numbers were allowed?
No, there is not. If only whole numbers were allowed, there are three Nash equilibria depending on the rounding algorithm. It is possible to round a number: a) up (to the higher whole number), b) down (to the lower whole number) or c) to the nearest whole number. If game organizers were true theorists they would give all the players an instruction how to round numbers.

If players were told to round down, then Nash equilibrium would be at 0. If they were told to round up the Nash equilibrium would be at 4. If they were told to round to the nearest whole number the Nash equilibrium would be at 2.

4. Is Nash equilibrium possible to achieve when rounding is not allowed?
The reasoning of the game theorists shows that rounding is not allowed. This is why a result such as 25.6 is admissible. This means that any positive number is an admissible answer. Then, Nash equilibrium is impossible to achieve, as 80% of any positive number, however small it is, is still a positive number (not 0). Zero is possible to achieve only if it is the first choice of all players, which is highly unlikely.

The weakness of Game Theory and Economics for that matter is that:
A. It is unreasonable to expect that ordinary people (i.e. the players of the economic game) will behave reasonably provided that those who are expected to be the most reasonable (i.e. those who set the rules of the economic game) do not behave reasonably (see above).
B. It is unreasonable to think there is always only one reasonable behavior. Any behavior is reasonable in the eye of the behaver, meaning that many different (even contradicting) behaviors are reasonable at the same time in the same part of space based on the individual circumstances.

Оградата на Патриарха

8/12/2012

Монолог а ла Габриел Гарсия Маркес и Стефан д-р Чолаков

Настана време неЧЛЕНоразделно и социалистите са зарязали почитането на паметта на най-личния им ЧЛЕН - бай Гошо Консервата, комуто навремето (и приживе, и посмъртно) всеки техен ЧЛЕН беше длъжен да прави теманета a la turque и челoбитни a la russe, че и това не им беше достатъчно, ами караха и нас да се навеждаме, но и те като нас изневериха на култа към ЧЛЕНа на личността и вече не бръснат бай Гошо за СЛИВА, та са зарязали всякаква поддръжка на историческия му дом, и особено на историческата му ограда, вместо да вземат да съберат по някой лев от ЧЛЕНовете си, на които няма да им се счупят ЧЛЕНовете ако вземат та поправят счупената ограда на дома на бай Гошо на ул. "ОпиЧЛЕНска" за да не ни е срам от капиталистическите СЛИВИ, които може "случайно" да се окажат с гадните си капиталистически камери в тези мръсни ЗАДНИ ЧАСТИ на София за да заснемат срама ни и да го изтъпанят по БиБиСи за срам на всички ЧЛЕНове на прогресивното човечество, които най-накрая ще разберат, че у нас е настанало време неЧЛЕНоразделно, та от ламтеж за катерене по стълбите социалистите са забравили изобщо за мазетата, покривите и (както се вижда) оградите, полегнали като ЧЛЕНовете на ЧЛЕНската им маса, чиято деменция ѝ пречи да си дигне ЗАДНИКА и да си го разходи до дома на този, за когото всички знаехме, че МУ е прав и СЛЕД като съгреши дори, ама на кого му пука дали му е прав или не след като оградата му е полегнала на една страна и ще падне така, както ще падне и родния му дом, така както падна и "вечния" му дом, въпреки че на ония меки сини СЛИВИ им трябваше една седмица за да го бутнат.