When Nash theory brings into play some love

  By Alma Dzib Goodin

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!.