In the 1930s, John von Neumann and Oscar Morgenstern became the founders of a new interesting area of mathematics, which was called "game theory." In the 1950s, the young mathematician John Nash became interested in this area. The theory of equilibrium became the subject of his dissertation, which he wrote when he was 21 years old. Thus was born a new strategy for games called Nash Equilibrium, which earned the Nobel Prize many years later, in 1994.
The long gap between writing a thesis and universal acceptance was a test for the mathematician. Genius without recognition resulted in serious mental violations, but John Nash was able to solve this problem thanks to his excellent logical mind. His theory of "Nash equilibrium" was awarded the Nobel Prize, and his film adaptation in the film "Beautiful mind" ("Mind Games").
Game Theory Briefly
Since the Nash equilibrium theory explains the behavior of people in terms of interaction, it is therefore worth considering the basic concepts of game theory.
Game theory studies the behavior of participants (agents) under conditions of interaction with each other according to the type of game, when the outcome depends on the decision and behavior of several people. The participant makes decisions, guided by his forecasts regarding the behavior of the others, which is called the game strategy.
There is also a dominant strategy in which the participant gets the optimal result for any behavior of other participants. This is the player's best win-win strategy.
Prisoner's Dilemma and Scientific Breakthrough
The prisoner's dilemma is a case with a game when participants are forced to make rational decisions, reaching a common goal in the context of a conflict of alternatives. The question is which of these options he will choose, recognizing his personal and common interest, as well as the inability to get both. Players seem to be enclosed in harsh gaming conditions, which sometimes makes them think very productively.
This dilemma was explored by American mathematician John Nash. The equilibrium that he brought out became revolutionary of its kind. Particularly vividly, this new thought influenced the opinion of economists on how market players make choices, taking into account the interests of others, with close interaction and intersection of interests.
It is best to study game theory with specific examples, since this mathematical discipline itself is not a dry theoretical one.
Prisoner Dilemma Example
Example, two people robbed, fell into the hands of the police and are being interrogated in separate cells. At the same time, police officers offer each participant favorable conditions under which he will be released if he testifies against his partner. Each of the criminals has the following set of strategies that he will consider:
- Both simultaneously testify and receive 2.5 years in prison.
- Both are silent at the same time and receive 1 year each, since in this case the evidence base of their guilt will be small.
- One gives evidence and gets freedom, while the other is silent and gets 5 years in prison.
Obviously, the outcome of the case depends on the decision of both participants, but they cannot come to an agreement because they are sitting in different cells. The conflict of their personal interests in the struggle for a common interest is also clearly visible. Each prisoner has two options for action and 4 options for outcomes.
Inference Chain
So, criminal A is considering the following options:
- I am silent and my partner is silent - we both will receive 1 year in prison.
- I give my partner and he gives me - we both get 2.5 years in prison.
- I am silent, and my partner is handing me over - I will receive 5 years in prison, and he will be free.
- I rent my partner, and he is silent - I get freedom, and he is 5 years in prison.
We give a matrix of possible solutions and outcomes for clarity.
The table of probable outcomes of the prisoner's dilemma.
The question is what will each participant choose?
"Silence, you cannot speak" or "Silence you cannot speak"
To understand the choice of the participant, you need to go through the chain of his thoughts. Following the reasoning of the criminal A: if I keep silent and keep silent my partner, we will receive a minimum term (1 year), but I can’t find out how he will behave. If he testifies against me, then it’s also better for me to testify, otherwise I can sit down for 5 years. I'd rather sit 2.5 years than 5 years. If he does not say anything, then all the more I need to testify, because this way I will get freedom. Member B also argues in the same way.
It is easy to understand that the dominant strategy for each of the criminals is to testify. The optimal point of this game occurs when both criminals give evidence and receive their "prize" - 2.5 years in prison. Nash's game theory calls it equilibrium.
Nash Optimal Optimal Solution
The revolution of the Nashev view is that such a balance is not optimal if we consider the individual participant and his personal interest. After all, the best option is to keep silent and go free.
Nash equilibrium is a point of contact, where each participant chooses an option that is optimal for him only if the other participants choose a specific strategy.
Considering the option when both criminals are silent and receive only 1 year each, we can call it the Pareto-optimal option. However, it is possible only if the criminals could have agreed in advance. But even this would not guarantee this outcome, since the temptation to backtrack from persuasion and avoid punishment is great. The lack of full trust in each other and the danger of being 5 years old forces one to choose the option with recognition. To reflect on the fact that participants will adhere to the option with silence, acting in concert, is simply irrational. Such a conclusion can be made if we study the Nash equilibrium. Examples only prove that.
Selfish or rational
The theory of Nash equilibrium has yielded stunning conclusions, refuting the principles that existed before. For example, Adam Smith considered the behavior of each of the participants as absolutely selfish, which brought the system into equilibrium. This theory was called the "invisible hand of the market."
John Nash saw that if all the participants act in pursuit of their own interests, then this will never lead to an optimal group result. Considering that rational thinking is inherent in each participant, the choice that the Nash equilibrium strategy offers is more likely.
Purely male experiment
A vivid example is the “blonde paradox” game, which, although it seems inappropriate, is a vivid illustration showing how the Nash game theory works.
In this game you need to imagine that the company of free guys came to the bar. Next is a company of girls, one of which is preferable to the others, say a blonde. How do guys behave to get the best girlfriend for themselves?
So, the guys’s reasoning: if everyone starts to get acquainted with the blonde, then most likely she will not get to anyone, then her friends will not want to meet. Nobody wants to be the second fallback. But if the guys choose to avoid the blonde, then the probability for each of the guys to find a good girlfriend among the girls is high.
The situation of Nash equilibrium is not optimal for guys, because, pursuing only their selfish interests, everyone would choose a blonde. It is evident that the pursuit of only selfish interests will be tantamount to the collapse of group interests. Nash equilibrium will mean that each guy acts in his own personal interests, which are in contact with the interests of the whole group. This is not an optimal option for everyone personally, but optimal for everyone, based on the overall success strategy.
Our whole life is a game
Making decisions in real conditions is very similar to a game when you expect a certain rational behavior from other participants. In business, in work, in a team, in a company, and even in relationships with the opposite sex. From large transactions to ordinary life situations, everything obeys one law or another.
Of course, the considered game situations with criminals and the bar are just excellent illustrations demonstrating Nash's balance. Examples of such dilemmas very often arise in the real market, and this especially works in cases with two monopolists who control the market.
Mixed strategies
Often we are involved not in one but several games at once. Choosing one of the options for one game, guided by a rational strategy, but you get into another game. After several rational decisions, you may find that your result does not suit you. What to do?
Consider two types of strategy:
- A pure strategy is a participant’s behavior that comes from thinking about the possible behavior of other participants.
- A mixed strategy or a random strategy is the alternation of pure strategies randomly or the choice of a pure strategy with a certain probability. This strategy is also called randomized.
Considering this behavior, we get a new look at Nash equilibrium. If earlier it was said that the player chooses a strategy once, then another behavior can be imagined. We can admit the option that players choose a strategy randomly with a certain probability. Games in which Nash equilibria cannot be found in pure strategies always have them in mixed ones.
Nash equilibrium in mixed strategies is called mixed equilibrium. This is such a balance, where each participant chooses the optimal frequency for choosing their strategies, provided that other participants choose their strategies with a given frequency.
