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WHAT IS GAME THEORY?

Simple principles and concepts for understanding Game Theory and how to exploit it in a variety of situations: complex organizations, existential experiences, repetitive challanges... by understanding why emotions, irrationality, psycho-somatic disorders are natural "equipments"... for healthy winners!

[this text is an abstract from the book, in Italian, "Dalle balle alle bolle: la Finanza sull'Orlo del Caos", by Nicola Antonucci - Hudsucker Ed., for online purchasing: http://www.hudsucker.it/dalle-balle-alle-bolle/ ]

[ for the effective interactions between game theory and complexity theory, let me recommend my article in Italian: Cos’è la complessità… semplicemente? ].

[ for simple, possible applications of "mixed strategies & complexity" in various existential contexts (Sustainable Happiness - Unhappiness, Game - Work, Love - Marriage, Leadership - Mobbing, Youth - Elderness, etc...), I suggest my articles of the series "Dream Wellness & Therapy" ].

In the most different ways, we all are alike -  Làszlò Mérö, mathematician

First of all, we need to do a rapid, but essential, incursion into the history and principles of Game Theory.

Bernard de Mandeville published in 1729 the final version of…- yes, I know, I’m approaching the topic in the roundabout way, but trust me: in this way, I can reach earlier and more effectively the sharing of the fundamental concepts of the game theory - …where was I?  In that remote year it was published the short poem “The Fable of The Bees: or, Private Vices, Public Benefits”. Bernard spread, with it, the thesis that vices are necessary for the balanced development of a nation and of its virtues. As for to say: all that exists (for example, the vice) has a meaning and has to be consciously and attentively integrated also with its opposite (for example, the virtue).

Adam Smith published in 1776 – yes, yes… I’m getting to the point! – “The Wealth of Nations” at the basis of the classic economic doctrine until 1950, when John Nash (See? I’m reaching the point…!) reversed it. Adam spread, over more than a century and a half of economists’ brains, the idea that, in the free market, the selfish research of people’s own interest is useful to the economy of a nation. He invented also the metaphor of the “invisible hand” that providentially accompanies the selfish choices towards a major social wellness – bah, always delegating to others the most difficult tasks…

You now well understand that, to get out from cognitive distortions of so virulent memes, we needed  real scientists, even mathematicians, to reset the issue on new foundations and come to less simple conclusions.

The first scientist was that absolute genius called John von Neumann (János Neumann, Hungarian-American, 1903 – 1957, mathematician, IT expert, eclectic genius) that, beyond some anticipator works of 1928 (“minimax theorem”), founded the modern game theory with the publishing of the book “Theory of Games and Economic Behavior”, written with the economist Oskar Morgenstern in 1944.

He applied a mathematic approach to human decisions, quantifying the value attributed to selfish or altruistic choices, based on virtues or vices, otherwise on competition or collaboration (see the example of the “Dilemma of the Prisoner” in the Learn More Box).

He didn’t have, though, the support of a Providence that, with an invisible hand, could direct such choices towards an Upper Goal.

LEARN-MORE BOX:  The Prisoner’s Dilemma.

An example of some principles and methodological criteria of Game Theory is furnished by the famous, but often scarcely understood, “Prisoner’s Dilemma”.

It is defined as a “complete information game” (meaning that a player knows all the rules of the game), studied by Merrill Flood and Melvin Dresher in 1950 and successively formalized by Albert W. Tucker, to whom we also owe  the name of the dilemma.

The dilemma itself, even if it uses the example of two prisoners to explain the phenomenon, can describe as well various other cases, from the arms race to the competition among companies, from political strategies to sentimental relationships.

The dilemma can be described as follows. Two well-known criminals, that have always “worked” together in robberies, are separately stopped in two different cars for violation of the speed limit, near a jewelry that has just been robbed. The detectives accuse them both and put them in two different cells, preventing them from communicating. To each of them is given a choice: they can either report the crime as eyewitnesses of the robbery performed by the other, or choose not to report it. They are also told that:

a)      if only one of the two criminals denounces the other, he will avoid the penalty, while the other – the accused – will be condemned to 7 years of detention. It’s the “single traitor” situation;

 

b)       if both denounce the other, they will be both condemned to 4 years of detention (3 years remitted for having collaborated with justice, in any case).  It’s the “two traitors” situation;

 

c)       if none of the two denounces the other, they will be both condemned to one year of detention (with the excuse of  rebellion to authorities after the arrest for speed excess). It is the “two faithful friends” situation.

It is, therefore, a “non-cooperative” game, because it isn’t possible, for the players, to previously agree and adopt the more convenient strategy for both. The dilemma that each criminal has to face is: “Shall I cooperate with my friend or shall I compete against him? Do I have to denounce him (betraying him), or not?  Should I betray him, I could totally avoid jail…”

Game Theory models these situations with schemes, more or less articulated and complex. Here follows the simple case of the Prisoner Dilemma just described, where the numbers in brackets represent:

  • the first digit: the years of detention for A,  according to the different combinations between the behavior of A and the behavior of B;
  • the second digit: the years of detention for B,  according to the different combinations between the behavior of A and the behavior of B;

 

 

 

 

 

 

Criminal B

Denounces

(betrays)

 

Doesn’t Denounce

(doesn’t betray)

 

 

 

Criminal A

 

Denounces

(betrays)

 

(4,4)

(0,7)

Doesn’t Denounce

(doesn’t betray)

 

(7,0)

(1,1)

If the aim of the criminals is to minimize the risk of going to jail, the best strategy of this game, then, is (denounce, denounce), i.e. every criminal betrays the other reporting him, because:

denouncing

he risks 0 years  or, at most, 4 years

not denouncing

he risks 1 year or, even, 7 years

It is said then that the strategy “not denouncing  (not betray)” is strictly dominated by the strategy “denouncing (betray)” and, therefore, for every behavior of B, the best result of A is always obtained by betraying the other.  The same works for B.

The result is that, if both choose the most rational option, they come to the so-called “Nash equilibrium” where each of the two prisoners reports the other and they both are convicted for 4 years.

It is a situation of “equilibrium” because, from there, none of the two is interested in changing their own decision since they would increase the numbers of years of detention – given the other’s specific decision!

This is a result of a rational, immoral calculation.

“In hindsight…”, instead, the two criminals regret they didn’t trust more the other, avoiding to report (betray) each other, because in this case they would have been detained for one year only. This would have been the result corresponding to the “Golden Rule”: do to the other what you would like to receive in turn (i.e.: do not denounce, do not betray!).

In this case, it would have been more convenient for the two criminals (in hindsight…) to collaborate together (and not with police) instead of competing one against the other hoping to avoid jail.

But it is not always like this: case by case, in other types of dilemmas, the table here described leads to results where also the Golden Rule produces losing results, and there is need of more articulated strategies (for example “mixed” ones).

END OF THE LEARN-MORE BOX

The aim of John was practical and earthly: to understand when and how two “players” (generally meant as operators or concurrent agents…) could find a final situation of  “balance”, or acceptation of the final result of the “game” (generally meant as negotiation or competitive exchange), in a way that they wouldn’t want to modify it on either sides.

The “von Neumann Theorem” guarantees such balance for finite (limited number of options and moves), “zero-sum” (a player wins exactly what the other loses) games with complete information (the players share the same exhaustive information without any “informational asymmetry”).

Another John – John Nash – took care of bringing prestige to scientific economy and to the newborn Game Theory thanks to his works of 1950 ("Equilibrium Points in N-person Games") and of 1951 (“Non-cooperative Games”).

John demolished with one shoot 180 years of classic economy founded on the dogma, instituted by Adam Smith, of the optimal social result obtained by selfish competitions.

The limit of the zero-sum games, analyzed for the first time by von Neumann, has been solved by the “Nash Theorem” that guarantees, under certain conditions, the existence of a balance, also in competitive situations more complex than those analyzed by von Neumann, like games with numerous participants that can also operate a choice from which everybody is advantaged (or, limit the disadvantage as much as possible): an epochal difference, if compared to the case of the zero-sum games studied previously, at the basis of almost two centuries of economical theory, where the victory of one of the two participants was total and exclusive, or necessarily accompanied by the defeat of the other.

So, as well exemplified by the movie “A beautiful mind”, it is possible to obtain much better (“optimal”) results, by consciously conjugating competition with cooperation – none of the two strategies is by itself the winning one.

The winning strategy is a mix of the two, and of others more – if you have them.

Let’s get back to our point, and to financial efficiency, with the astonishing concept of “mixed strategy”, dissonant with our practical common sense!

Let’s try to understand how, when and how much to compete or cooperate with the Beast (the stock exchange one…) wearing again our diving suit for a new short…

Immersion in “Mixed Strategies”.

Do you want to win in repetitive games (like trading and investments, with numerous operations of selling and buying)? Or in elevate complexity contexts (on this theme, see – in Italian: Cos’è la complessità… semplicemente?).

The Game Theory has demonstrated that every predefined “strategy” (or: sequence of behaviors), even if very articulated and complex, is meant to lose, if confronted with a non-predefined strategy.

Practically, if I define exactly what I will do when x, y, z happens (“pure strategy”), I’ll never gain as much as I could gain with a mixed strategy, which is equivalent to saying: when situations x, y or z will happen, my behavior will be established by…dices – that’s it!

Mixed strategies, therefore, are constituted by possible known behaviors, but also by an unknown sequence. From the childish game “rock-paper-scissors” to the audacious poker, up to  complex finance, the mathematic Game Theory has demonstrated that, to win repetitive games (not one match only!) the decision of the behavioral sequence is to be left to dices – hard to believe, right?!

I didn’t believe it either when I learned the Mixed Strategies, but I learnt to observe them in reality and to live them, because mixed strategies finally explained me the role –for better or worse… - of the human, too human, passions, irrationality, psycho-somatic illnesses, madness…  from which animals are immune, being rigidly programmed by instincts.

John Nash has mathematically demonstrated that, if my sequence of behaviors changes in function of the events x, y or z in a probabilistic way, instead of changing in a predefined way, I then realize a mixed strategy that wins against every pure strategy.

Weird, bizarre….?! Mathematic!

In our everyday and operative life, though, we can’t afford to calculate, for every specific situation, the optimal frequency of the behaviors useful to win and gain – once upon a time there were no calculators, nor a Game Theory…

So, where can we purchase these special dices (with a different number of faces for each problem), or something that attributes a certain percentage P(x) of cases to my behavior associated to x, another percentage P(y) of cases to my behavior associated to y, and so on…?

We can find them inside ourselves: Nature has already equipped us with these instruments, necessary to win and evolve up to our current levels of progress and wellness.

Our special dices are: emotions, irrationality, psycho-somatic disorders, and all that interrupts the irrational domain of the reason, alternating it with other behaviors in a probabilistic way, or rather with a human, very human, mixed strategy.

This is the best vaccine to prevent the well-known and dangerous, especially in finance, “Macbeth Effect” and “Boiled Frog Syndrome”!

(CLICK HERE for a LINK to a funny VIDEO, in Italian, that explains the Macbeth Effect through the paradoxical “Banknote auction”!).

What is traditionally seen as a human limit (i.e., emotionality, irrationality and so on…) is, biologically and evolutionary, an absolutely privileged and effective equipment to compete in an optimal way. Emotions, psycho-somatic disorders and irrationality exist for a reason – let’s use them and don’t waste them, as Game Theory teaches us from a bunch of years.

Humanity has always, almost unconsciously, applied mixed strategies using its own special dices included in hormones, in neurotransmitters and in everything that contributes to elaborate the complex signals coming from the environment.

When our inner calculator elaborates results that produce pleasure or suffering, i.e. stimulations or stress, these latter get bio-chemically involved for conditioning the next apparently accidental, or emotional, or even irrational operations of our internal Value Algebra – actually aimed to enact mixed strategies (except for natural mistakes and omissions).

Here it becomes even more evident that “the unconscious processes are the fully rational ones, while it is the conscious thought that is not completely rational – this one is at most almost-rational (Làszlò Mérö, mathematician), i.e. when it is able to enact a combination of rationality, emotionality and irrationality (with “mixed strategies”).

There are several problems, especially for extreme activities such as financial operations:

  • mixed strategies are typically unaware, and especially unknown, at least until the reading of these lines…
  • our special dices (emotions, irrationality, psycho-somatic disorders…) are unfortunately tricked by cultural gender (female and male) and memetics conditionings: some of our possible expressions are inhibited, atrophied or, on the contrary, stimulated and exalted by external familiar, scholastic, mass-media, ideological, productive interests; the perverse consequences on our Value Algebra, on the Macbeth Effect and… on the purchase of an ordinary banknote, we know them well by now!
  • our special incorporated dices activate, sequentially, different behaviors that can be represented through the Multiple Personalities that we are gifted with, but traditionally repressed in favor of one socially and morally acceptable Personality (the dangerous “upright guy”…).  Even just  two Personalities are seen as pure schizophrenia (do you know “the strange case of dr. Jekyll and Mr. Hyde”…?);
  • the supreme human aim is to become a chorus of proper harmonic, free, creative Multiple Personalities!

I well know that sometimes you have felt like Another, that you executed actions you had excluded a few minutes or days before, or that you refused to correct what you did before, even though all the signals advised you to review your previous decision – the clash and the inner competition among your own Multiple Personalities are the worst and the most dangerous ones.

For further information: nicola.antonucci@libero.it

Nicola Antonucci

P.S.:  For a (“utopian…”) novel (in English,  on Kindle!) that combines Game Theory with the Complexity Theory and with Cognitive Science: Etopia - a nearby Utopia.

Nicola Antonucci

Nicola Antonucci

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