In 1930 John von Neumann and Oskar Morgenstern became the founders of a new interesting direction of mathematics, which was called "Game Theory".In the 1950s, this trend became interested in the young mathematician John Nash.Equilibrium theory became the subject of his thesis, which he wrote at the age of 21 years old.Thus was born a new strategy game called "Nash equilibrium", deserved the Nobel Prize many years later - in 1994.
long gap between the writing of a dissertation and became universally accepted test for math.Genius without recognition resulted in serious mental disorders, but also the task John Nash was able to solve thanks to the excellent logicheskumu mind.His theory of the "Nash equilibrium" awarded the Nobel and his life in the film adaptation of «Beautiful mind» («A Beautiful Mind").
briefly on game theory
Since Nash equilibrium theory explains human behavior in terms of interaction, so you should consider the basic concepts of game theory.
Game theory studies the behavior of the participants (agents) in the environment interact with each other by the type of game where the outcome depends on the decisions and behavior of a few people.Member takes decisions based on their expectations about the behavior of others, which is called a gaming strategy.
There is also a dominant strategy, in which the participant gets the best results when the behavior of any other participants.This is the best strategy bezproigryshnaya player.
Prisoner's Dilemma and the breakthrough
Prisoner's Dilemma - it is the case with the game, where participants have to make rational decisions, achieving the overall objective of alternatives in situations of conflict.The question is, which of these options it will choose, aware of personal and general interest, and the inability to get something, and more.Players like enclosed in a tough playing conditions, which sometimes makes them think very productive.
this dilemma explored the American mathematician John Nash.The balance, which he led, was revolutionary in its own way.The strong new idea influenced the opinion of economists about how to make a variety of players in the market, taking into account the interests of others, with a dense interaction and intersection of interests.
It is best to study game theory with concrete examples, because this mathematical discipline itself is not dry and theoretical.
Example prisoner's dilemma
example, two men committed the robbery, fell into the hands of the police and are interrogated in separate cells.At the same time the ministers of police offer each participant favorable conditions under which it will be released in the case testify against his partner.Each of criminals there next set of strategies that it will consider:
- Both simultaneously give testimony and receive 2.5 years in prison.
- Both silent and at the same time get to 1 year, as in this case the evidence base of their guilt will be small.
- One testifies and gets free and the other is silent and gets 5 years in prison.
Obviously, the outcome of the case depends on the decision of both parties, but they can not come to an agreement, because sitting in different cells.Also clearly visible to the conflict of their personal interests in the struggle for a common interest.Each prisoner has two options and 4 version outcomes.
chain of logical reasoning
So the offender but to consider the following options:
- I'm quiet and silent is my partner - we both get 1 year in prison.
- I rent a partner and he takes me - we both get 2.5 years in prison.
- I am silent, and my buddy passes - I'll get five years in prison, and he was released.
- I give mate, but he is silent - I get freedom, he is 5 years in prison.
reduce the matrix of possible solutions and outcomes for clarity.
Table probable outcomes prisoner's dilemma.
question is that each participant will choose?
«Shut up, you can not say" or "can not remain silent, to speak»
To understand the choice of the participant, you must go through his chain of thought.Following the arguments offender And if I say nothing and say nothing my partner, we will get a minimum period of time (1 year), but I can not see how he will behave.If he gives evidence against me, I also better to give evidence, otherwise I can sit down for 5 years.I would rather sit down to 2.5 years than 5 years.If he is silent, then the more I need to give evidence, as well, I'll get freedom.Similarly, talks and participant B.
is easy to understand that the dominant strategy for each of the criminals - a testimony.The optimal point of the game occurs when both the offender testify and receive your "prize" - 2.5 years in prison.Game Theory Nash calls this equilibrium.
optimal solution Non-optimal Nash
neshevskogo The revolutionary view that such a balance is not optimal, if we consider the individual participant and his personal interest.After all, the best option - is to remain silent and to be released.
Nash equilibrium - is the point of contact of interest, where each participant chooses this option, which is optimal for him only on condition that other participants choose a particular strategy.
considering an option when both the offender and receive all silent for 1 year, can be called a Pareto-optimal variant.However, it is only possible if the criminals were able to come to an agreement in advance.But even that would not guarantee this outcome, because the temptation to fall back on persuasion and avoid large penalties.The lack of complete trust in each other and get 5 years, the risk of forced confession to choose.Meditate on the fact that participants will stick to the version with silent, acting in concert, simply irrational.Such a conclusion can be made, if we study the Nash equilibrium.Examples only prove right.
Selfishly or manage
theory Nash equilibrium given the stunning conclusions oprovergnuvshie available to this principle.For example, Adam Smith, considered the behavior of each of the participants as being absolutely selfish, and that brings the system into balance.This theory was called the "invisible hand of the market."
John Nash saw that if all parties to act in pursuit of their own interests only, it will never lead to the optimal group result.Given that rational thinking is inherent to each party, a more likely choice, which offers strategy Nash equilibrium.
purely male experiment
striking example is the game "a paradox blonde", which, although it seems out of place, but is a vivid illustration that shows how the game theory Nash.
In this game you need to imagine that the company is free of guys came to the bar.Nearby is the company of girls, one of which is preferable to the other, say the blonde.As the boys to behave, to get the best girlfriend for yourself?
So guys reasoning: if everyone will get acquainted with the blonde, then most likely, it will not get anyone, then her friends will not want to explore.Nobody wants to be the second fall-back option.But if you guys choose to avoid the blonde, then the probability of each of the guys to find a good friend among girls is high.
Nash equilibrium is not optimal for the guys because, pursuing only their own selfish interests, each would choose it blonde.It is seen that only the pursuit of self-interest would be tantamount to the collapse of the group interests.Nash equilibrium would mean that every man acts in their own interests, which are in contact with the interests of the whole group.This sub-optimal option for each person, but the best for everyone, based on the overall strategy for success.
Our whole life is a game
Decision-making in the real world is very similar to the game when you expect a certain rational behavior from the other participants.In business, work in a team, in a company, even in a relationship with the opposite sex.From large transactions and to ordinary life situations everything is subject to a particular law.
course considered for game situations with criminals and the bar - it is just excellent illustration showing the Nash equilibrium.Examples of such dilemmas often arise in the real market, and especially in case of the two monopolists who control the market.
Mixed strategies
vovlekaemy Often we are not in one but in several games.By choosing one of the options one game, guided by a rational strategy, but get a different game.After a few rational decisions you may find that your results are not.What to take?
consider two types of strategies:
- Net strategy - is the behavior of the participant that comes from reflection on the possible behavior of other participants.
- mixed strategy or casual strategy - this alternation of pure strategies random selection or pure strategy with a certain probability.This strategy is called rendomizirovannoy.
Considering this behavior, we get a new perspective on the balance of Nashua.If you previously said that the player chooses the strategy once, and one can imagine a different behavior.You can avoid the possibility that players choose strategies randomly with a certain probability.Games in which you can not find a Nash equilibrium in pure strategies, they are always mixed.
Nash equilibrium in mixed strategies is called a mixed balance. It is a balance, where each participant chooses the optimal frequency selecting their strategies, provided that the other members choose their strategies with a predetermined frequency.
Penalties and mixed strategy
example of a mixed strategy can result in a game of football.The best illustration of the mixed strategy - this is probably the penalties.So, we have a goalkeeper who can jump only in one corner, and the player who is going to beat a penalty.
So, if the first time the player chooses a strategy to make an impact in the left corner and the goalkeeper also fall into this corner and catch the ball, how events can develop a second time?If a player is hit to the opposite corner, it's probably too obvious, but hit the same angle of not less than obvious.Therefore, the goalkeeper, and the batter has no choice but to rely on random selection.
So, alternating random selection of a certain pure strategy, player and goalkeeper pytajutsja get maximum results.
