When the theory brings into play the love

Daniel is a Russian physicist ready to conquer the world with his ideas, but in the midst of publications filled with Greek letters and numbers, he has failed to atract a beautiful girl’s love of the lab at the other side of the campus. That girl that walks around with the sweetest smile and when she meets Daniel in the cafeteria, smiles even more broadly, knowing that he is a foreigner so she does not want to say something wrong... that girl who ... does she make him sweat?... perhaps so much that he has even asked her name...

 

Town was suffering the third winter storm of the season that evening, so Daniel went into the library, hiding as it was his habit to expect bad things to pass, it was his refuge for everything and everyone. 

He had been studying game theory during a while, especially from Nash’s perspective, and he was determined to publish an article about this topic, applying a proposal from economic and biological models. Although there is a lot of articles written on the subject, he wanted to innovate, he wanted to find something that no one else had seen, however he had to find  that something, there was no doubt he was going to find it there, or perhaps here... maybe in this journal... maybe...

He began to move from one side to another around 300 magazines on the table when one of them slid...... when he bent down to lift it without paying attention to the environment, he saw a hand taking his magazine. He was ready to shout do not touch my magazine, when his eyes followed a hand, and then an arm and then a shoulder... he came up to the neck and two brown eyes met with his blue eyes.

The smile was expanded as always... but this time she modulated carefully her words:

-Hey!, I found this!... What are reading?, she said.

-I'm reading Nash’s theory... He said as doubting whether those words came from his voice or if he was listening to through his headphones, he took them off to be able to distinguish between his voice and the music he was hearing, and they fell to the floor.

She hurried to help him to lift them up and while she did it put attention into the magazines on the table and reached to read one of them: game theory and their applications in biological models...

When she concluded reading, she gave him the headphones and then she stretched out her hand: 

-My name is Kathleen, Kat... I study nanotechnology applied to biological models, and try to explain evolutionary patterns... I see that you understand math... my advisor says that my proposal has no future... perhaps you can help me?... huh?, what do I say?, I’m sure you are very busy...

When she dropped his hand, he took it back and he said almost in a whisper:

-My name is Daniel... Dan... and yes!, in fact I want to design a model of application with game theory to economic and evolutionary models, although this  has been done before... I want to find a specific area that allows me to innovate a bit.

She moved a chair beside her and then more magazines fell, but this time he rose and offered her his chair,  he began cleaned the table a little and decided to talk about what he knew and loved so much with her... 

- Please,  sit down Kat, I think  I can explain you from mathematics how species have evolvied, however, remember that maths are only models that represent the reality.

-The science is that, isn't it? Kat, said smiling.

-Yes, yes... it is true... just explain the reality...

- What I know about game theory, is that studies the sharing of profits and therefore decision making among the players, if an entity can understand that it can win or lose during an exchange, it makes decisions based on its experience. I believe the same applies to biological models, as if we believe that species should take into account information from the environment and predators to survive, then they must apply decision-making on environmental responses.

Daniel further opened the eyes and arched eyebrows...

 -Exactly!, that's the idea... Game theory is much related to economic theory, where a zero-sum game is a mathematical representation of a situation in which a participant gains or loses profits and utility is exactly balanced by the gain or loss of the participant, as opposed to a non-zero sum game where both profits and losses are added to a situation in which players interact. To sum game zero resolves le with the Minimax theorem that is closely related to the theory of the Nash equilibrium.

-     I couldn’t understand that, could you explain it to me?, but used simple words, please.

-    It is a conceptual solution to the non-cooperative games that include two or more players, in which it is assumed that it is possible to know the balance of strategies of each player  since has something to gain by changing if strategy during the game.

-    It sounds complicated...

-    It is not, in fact it can be ridiculously simple: maybe if we try to understand biological systems... look, we will assume that you have a bee and a butterfly trying to obtain pollen from two flowers... do you work with biological systems at this level?, or perhaps... microbes, bacteria... 

-    Sounds like a cute example with bees and butterflies, please explain this and then maybe we can apply to spores...

-    Daniel looked at her very happy... well... I’ll explain it with butterflies... There are two flowers and one butterfly and one bee are trying to obtain the maximum pollen, if Butterfly makes it first movement, it can take all the pollen and thus the bee lost, and if the bee makes a first movement, butterfly lost, but both want to profit, so it is important to take a decision on the action and you can see it in simple way with a picture like this:

	Bee takes pollen from a flower	Bee tries to take two flower
The butterfly takes pollen from a flower	Maximum common benefit	Bee gains and butterfly loses
Butterfly tries to take the pollen from two flowers	Butterfly gains and bee loses	Maximum common prejudice

- Crap!, it sounds so easy that even I can understand that, but surely there are a lot of numbers and things involved as I can see in your magazines.

-    I don’t understand it if I see it that way, but let me give you another example to see if I understood... Kat and Dan can learn from each other, and they both make decisions ... then if I do a matrix would be this:

	Kat speaks with Dan	Kat does not speak with Dan
Dan speaks with Kat	Maximum common benefit	Kat goes home
Dan does not speak with Kat	Dan finishes his readings	Maximum common prejudice

-    Yes, more or less like that, but both must have the same level of profit, in this case the gain is different, you would have gone home and I would have read 2 or 3 magazines. 

-I want to study chaotic fluctuations in phenotypic frequency, specifically in those cases where it is necessary the adequacy of a heterozygote based on the measure of a recessive parent the next generation, which I think, is the measure of the adequacy of the homozygotes and heterozygotes would therefore have a destructive effect on the homozygotes and themselves, although to a lesser extent about themselves. This because if a behavior is associated with a low to reply to the environment and high fertility fitness gets a chaotic fluctuation.

Daniel's eyes were opened so much that his glasses were on ready to falling, when he could finally talk, began to mambling few words and then coughed a little to clarify his own ideas...

-Kat... I think that no doubt we both can gain from this talk, there are at least two possible explanations, even it would be possible to make predictions... I can apply my mathematical theories to everything you said,  however, the truth is  I did not understand a half of what you just said, but we can sit down and explain every thing with pears and apples, but I believe that we both would receive profits from all this and it is possible to balance our profits no doubt...

-  I want to know if it is possible to find patterns of adaptation and survival in living populations...

-    Have you ever read about the life of Conway game?

-     Yes, I do!, actually… that what gave me this idea, but I don't understand enough about math,  mmmm but I think that what you just explained and Conway model can help me a lot, now all are algorithms to create stable patterns, as the analysis of the databases, everything is math is our everyday lives, everything is about looking for patterns!.

-   Yes it is, yes... of course... I think you just increase our chances of achieving our dreams...Yes... can I invite a coffee?, I only need to ... collect all these magazines... and... I think it’s not snowing anymore... I can take you in my car, or do you have a car?

-    Let me help you with this… No, I do not have a car, only my bike, but I can leave it on campus, and I come tomorrow on the bus. Climatic variables, increases the probability of accidents as drivers change their behavior patterns...

-    Yes, it is true... well... we can put your bike in my car... I have a porta-bike that I use on weekends... well... only when there is no snow, because as you say... There are changes of patterns in the drivers... Yes... ohm... then if you put your bike in my car, can I take you home?... after coffee, of course...

-  Let me think... If I say yes, then we both get gains, then I should not take the bus tomorrow, and you know where to find me... but in such a case, there will be probably other decisions that each player goes, isn't it?

-    Yes, Yes... the pattern can be exponential... sure!

-    Yes, I figured that... can it become a chaotic fluctuation... the pattern could meet different conditions and therefore... change... evolve...

-    Yes, yes... it's true... yes...

-   Cool!, it seems that mathematics are not so complex!.!

-    No, no... they are not... they can explain many things and as you say... we apply them more and more everyday... I worked applying all this to economic models and I an algorithm to Google, so it seems that it can read your mind when you type something...

-    Amazing!... have you explained biological models?

-    No, but... apparently... I'm about to do it... If you allow it to me!.

Alma Dzib Goodin

The evolution from a mathematical perspective

This time I'm going to share a fascinating proposal about a considered a cellular automaton, which is known as Game of Life, or simply Life among experts in Artificial Intelligence and mathematics, which was created by the English mathematician John Horton Conway in 1970.

The game is determined by its initial state, without need of further input.  The game begins with an initial configuration and it is seen how it evolves.

The game is due to the interest of Conway in a problem presented by the mathematician John von Neumann who was trying to find a machine that could build copies of itself. 

Neumann succeeded when he found a mathematician model for a machine of this type with rules applied in a rectangular grid, which is very similar to the so-called Universal Turing machine, which is described as an automaton artifact that was unveiled in 1936 in the Journal Proceedings of the London Mathematical Society and which is basically a device that manipulates symbols on a strip of tape according to a set of rules that can be adapted to simulate the logic of any algorithm which can help to understand the limits of mechanical calculations, which provides advances to complexity theory.

Returning to the Game of Life, Conway found a way to dramatically simplify the notion of von Neumann, using only 4 rules for the implementation of a cellular automaton.

The game was released in October 1970 in the magazine Scientific American column mathematical games of Martin Gardner and opened a line of mathematical research known as the field of cellular automatons because of the analogy of the emergence, transformation and fall of any society of living organisms belonging to well-known simulation games in which the patterns can evolve.

Life allows an example of emergence and self-organization of patterns by what has attracted the interest of multiple fields of science and giving way to studies of emerging complexity or self-organization systems.

Life  is played in an infinite orthogonal network cell square (I recommend widely to visit the link of Wikipedia that I am here referring), each cell is in one of two possible States, alive or dead.

Every cell interacts with its 8 neighbors, which are the horizontal or vertical, diagonal or adjacent cells, with the passage of time, it is possible to observe the following transitions which are the rules of the game:

The initial pattern is considered the seed of the system. The first generation is created through the application of the above rules simultaneously to every cells of seeds that includes births and deaths which can occur at the same time and discreetly called this a tick, which implies that each generation is a pure function of the precedent and the game continues until the last cell dies.

From these simple rules, life has become one of the examples of what is known as emergent complexity and self-organization systems.

Now, let me explain why this made me jump off my seat and go running the stairs to search Google in this respect:

These simple rules allow the understanding of such complex phenomena such as the arrangement of the petals of a rose, or patterns on the skin of a zebra, and it is that in life, science attempts to explain complex patterns, always applying the idea that simple is best, I remember clearly professor Colm Donaldson saying loudly: make it simple, simplicity is more beautiful in science.

So those simple patterns of interaction are the beginning a complex process of interaction that unlike the rest of the games, life part of the standards themselves to make patterns, while in conventional games, developers create multiple game situations that must be met to advance. 

Designed by Paul Rendell 02/April/00, disponible en http://rendell-attic.org/gol/tm.htm

Life has served even to analyze patterns of conduct testing different models and watching their interaction. For example the so-called R-Pentomino was the first pattern observed by Conway, which is very stable and thus easy to predict, although 1 103 steps are required for this purpose.

This is why some of the programs designed for Life at the beginning were limited to describe the fate of a pattern of small and specific, however with the development of computers, it is now possible to run more complex patterns.

One of the questions of Conway was to determine if the initial pattern of life could grow indefinitely, or if any system could, so it offered a prize of $50.00 to anyone who could respond to questions. In 1970 a group of MIT led by RW Gosper won the prize with a pattern known as Glider Gun, which emits a new agent every 30 generations indefinitely, by what the pattern grows forever.

As they were more and more patterns, other aspects of the game are defined, for example the speed of light is defined as maximum sustainable speed by any moving object, or the rate of spread given in one step either horizontally, diagonally or vertically by generation. In this regard for both processes are the maximum rate at which information can travel and thus determines the speed of the pattern.

Based on that, the mathematicians have played and created different theorems that are observing with developed patterns.

This is part of a set of ideas that are integrated into the Game Theory, and just one of them is known as Nash equilibrium, which is a concept of the solution of a game and decision-making, which analyzes the strategies employed by the players from the benefit that can be obtained by changing their strategies which creates a principle of stability in the solution exchange during the game and is known as the theory of Nash equilibrium.

 

Of course, one of the fields that it has enabled more development is known as artificial life that it is defined as a life made by a human mind and not by nature and relates also with the study of non-organic bodies, beyond of the creations of nature, possessing essential properties that allow you to understand it inside an artificial environment created specifically for its development within a programmable machine (usually you can think in) a computer) so that the artificial life (A Life) allows to understand three properties of the nature that are the reproduction, the emergent properties and evolution. 

I am sure you are wondering how does this explain students would ability to change patterns from simple action rules?, well, it seems that many researchers have tried to apply these principles on biology, personally I think that the program of Neuro-modulation environmental assisted which starting with micro-tasks capable of creating patterns of conduct specifies, can be an applied example, but undoubtedly there is much reading to do simple the complexity of learning.

References: 

Ashwani, K.  (2013) Cellular automation: A discrete approach for modeling and simulation of artificial life systems. International Journal of Scientific and Research Publications.  3 (10) Available at: http://www.ijsrp.org/research-paper-1013/ijsrp-p2213.pdf

Gardner, M (1970) Mathemathical Games: The fantastic combinations of John Conway’s new solitarie game “life”. Available at: http://web.archive.org/web/20090603015231/http://ddi.cs.uni-potsdam.de/HyFISCH/Produzieren/lis_projekt/proj_gamelife/ConwayScientificAmerican.htm

Gymerek, M. (2010) Conway’s game of life. Available at: http://web.mit.edu/sp.268/www/2010/lifeSlides.pdf

Wikipedia. Conways’s Game of Life. Available at: http://en.wikipedia.org/wiki/Conway%27s_Game_of_Life

 If you want to see very cool patterns, you can visit here: http://www.frank-buss.de/automaton/golautomaton.html