🚲 🍦 🦓 Game Theory and its Application in Life 🤾🏼 🎋 🚳

Hello reader!

Some of you have seen the qwerty letter set. Qwerty is a keyboard layout. Look at your keyboard. You will see the letters “q” “w” “e” “r” “t” “y” in the top row. And for what reason are we interested in the keyboard layout?

A long time ago, when people used typewriters, they typed pretty quickly. This created problems: the heads of the typewriter, beating on paper and typing letters on it, clung to each other, which led to breakage. A qwerty layout was created in which the letters in the words next to each other were placed at the greatest possible distance from each other. Thus, the problem was solved.

Nobody has been using typewriters for a long time, and the problem of contact between the printheads has disappeared. The fact that we stopped using the inconvenient keyboard layout is logical. But there is a catch - this fact does not exist, people are used to typing on the qwerty layout and do not want to relearn.

Now, having entered the settings, you can switch the keyboard layout to “dvorak”. Printing will accelerate at times, while training will take only a week. Unfortunately, it is not beneficial for anyone to be the only retrainer, because it will be inconvenient to work on any computer other than a personal one. And also, unfortunately or fortunately, people are too lazy to relearn. Although together, by making efforts and relearning ourselves, we could increase the throughput of typing at times.

To summarize: with the massive use of qwerty, the transition of an individual player to dvorak is not effective, although the transition of society to dvorak is effective.

The concept of "game theory"

Game theory examines the conflicts of two or more parties called games. Under study are the games themselves, the strategies used in games, as well as patterns of behavior in games. The behavior of the players is determined by strategies. The strategies inherent in players are called "behaviors."

Take an example:

There is an automaton that responds to your actions. If you put a coin in it, your opponent will receive three coins - and vice versa, if your opponent puts a coin in the machine, you will get 3 coins.

In this case, there are 2 players in the game - “Naive” and “Strategist”. They can trust the enemy, therefore, put a coin or cheat and not put a coin.

What will happen? If the first player and his opponent trust, the first player will receive 3 coins, giving 1 and his opponent will receive 3 coins by giving 1. If player number 1 trusts, and the opponent deceives, the player will not receive anything by giving 1 coin. If the first player deceives, and the enemy trusts, the player will receive 3 coins without spending a single one. If both participants try to cheat, then they will not get anything.

For the convenience of player 1, denote I1, and player 2, denote I2.

Table:

On the table we clearly see the possible options for the development of games, then we will build many similar tables. What conclusions can we draw from the table?

Let’s try to find the most profitable strategy - a plan, following which we will get the greatest benefit. So which of the strategies is the most profitable?

If the enemy trusts, I1, choosing the strategy "Deceive" will receive the highest gain. If the adversary deceives us, then the Deceive strategy also wins. Although cruel, a deception strategy is always the best.

But what are behaviors? These are strategies that certain players constantly use. Recall the names of our players - “Strategist” and “Naive”. Perhaps their names were given based on the strategies they use? Yes it is. And here are the strategies used by the players: “Strategist” looks at the previous action of the opponent and analyzes it, “Naive” in turn always trusts.

It is also necessary to mention the Nash equilibrium. Nash equilibrium is a situation in which no participant can increase his gain by changing his strategy if other participants do not change their strategies. Remember the introduction? Namely, the game “qwerty". If all users of gadgets retrained on dvorak, it would be better for society, but by no means, retraining only a few players is not profitable - this is Nash equilibrium.

Terms and types of games

Game theory is a branch of mathematical economics. He studies conflicts, their solution.

A game is a conflict of two or more parties, in which each of the parties pursues its own personal interests.

The outcome of the game is a win, a loss or a draw, as well as a reward received.

Strategy - conclusions from which the choice of actions in the game comes from.

A behavior model is a player’s inherent strategy or strategies.

Nash equilibrium - This is the name of a set of strategies in a game for two or more players, in which no participant can increase their winnings by changing their strategy if other participants do not change their strategies. Often in games with equilibrium, a change in the strategy of all participants will lead to an increase in winnings, but it is unprofitable for each individual participant in the game to change the strategy.

Cooperative and non-cooperative. The game is called cooperative, when players can join in groups, make commitments to other players and coordinate their actions. Unlike cooperative games, non-cooperative ones are games where everyone should play only for themselves. Hybrid games include elements of cooperative and non-cooperative games. This means that each player will pursue the interests of his group and at the same time try to make a personal profit.

Symmetrical and asymmetric. The game is symmetrical when the players will have the same rewards accordingly. In other words, if the players switch places, they will receive winnings for the same moves as without changing places. Many of the studied games for two players are symmetrical.

With a zero sum and a non-zero sum. Zero-sum games - games with a constant fund of the game, available game resources cannot become more or less. In this case, the sum of all winnings is equal to the sum of all losers for each move. An example of such a game is poker. In non-zero sum games, winning one player does not necessarily mean losing another player. The result of such a game may be less than or greater than zero.

Parallel and sequential. In parallel games, all players can perform an action in a given period of time. All parties make their move in the given period of time, not knowing the actions of opponents, until the end of the game. In sequential games, participants can make moves in a predetermined or random order, but at the same time they receive some information about the previous actions of others.

With complete or incomplete information. In a game with full information, participants know all the moves made up to the current moment, as well as possible strategies of opponents. Full details are not available in parallel games. In a game with incomplete information, players have only partial information about the opponent.

Games with an infinite number of steps. Games with an infinite number of steps, as the name implies, have no limit on the number of steps. Games with a finite number of steps are the exact opposite; they are limited by their number.

Discrete and continuous games. Discrete games - games with a limited number of steps, events, outcomes. Continuous games - games lasting an infinite amount of time.

Game Analysis

The game "Ultimatum"

They play 1 time. There are 2 players. The first can divide the sum of 200 decillion francs between themselves and the enemy. The adversary may agree with the decision of the first player - to split the win, or refuse. In case of failure, no one gets anything.

Let's classify the game!

This is a non-cooperative game, because You cannot join in groups. This is not a symmetrical game, as 1 and 2 players have different actions in the game. This is a game with a non-zero amount, because all winnings may be lost. This is a sequential game, because decisions are made in turn - 1, and then 2 players. This is a game with full information, as the second player is available information about the actions of the first player. This is a game with not an infinite number of steps - only 2 steps. This is a discrete game, because the number of actions is limited.

We play as 1 player. How to choose a strategy? Imagine the possible development.

n> 0: Any reasonable player will agree to share the winnings, because no one will refuse to become the second or even the first richest person on our planet.

n = 0: The player can both agree or refuse.

Thus, the optimal strategy for 1 player is to offer the enemy 1 decillion francs, taking the remaining 199 to himself.

Game "Deer Hunting"

The essence of the game - a group of hunters of 2 people went hunting for a deer in the region with a very large number of rabbits. The aim of the hunters is to kill the deer. Each player's goal is to kill the prey. Although the highest benefit for all players is a deer, each of the hunters can kill a hare, receiving personal gain, but frightening off a deer.

Classification.

This is a cooperative game - players can join in groups. This is a symmetrical game, because players have the same choice of actions. This is a game with a nonzero amount, because the entire winnings vary. This is a parallel game, because decisions are made at the same time, arbitrarily. This is a game with full information, as Both players have access to information about each other’s actions. This is a game with an infinite number of steps - only 1 step is available. This is a discrete game because the number of actions is limited.

Let's build the scheme: The

reward for the deer is definitely higher, but the chance to stay with nothing is high. Playing with a trusted partner you can trust, you can arrange to kill the deer. Otherwise, it is better to choose the “Hare” strategy.

The game "Bototto"

2 players play. Each of them can write 3 digits, but not in descending order. The sum of the digits must be 6. A player whose 2 digit positions exceed 2 opponent positions wins.

Classification.

This is a non-cooperative game - players cannot join in groups. This is a symmetrical game, because players have the same choice of actions. This is a zero-sum game, because all winnings are fixed. This is a parallel game, because decisions are made at the same time, arbitrarily. This is a game with incomplete information, as Both players do not have access to information about the opponent’s action. This is a game with not an infinite number of steps - only 1 step. This is a discrete game, because the number of actions is limited.

The choice of strategy.

There are 3 options for each player (the game is symmetrical):

(2-2-2) or (1-2-3) or (1-1-4).

(1-1-4) versus (1-2-3) entails a draw.

(1-2-3) versus (2-2-2) entails a draw.

(2-2-2) beats (1-1-4).

Thus (2-2-2) is the optimal strategy.

This game also has Our balance: any combination of strategies (2-2-2) and (1-2-3).

Game "Princess and the Beast"

In a dark, dark cave ... A dark, dark night ... A dark, dark monster ... Looked for a dark, dark princess ... A dark, dark cave had dark, dark borders known to dark, dark players ...

Simply put, the princess and a monster appeared in a cave, the borders of which known to both the princess and the monster. The monster's goal is to catch the princess, and the princess's goal is to hold out as long as possible. The monster can grab the princess at a short distance relative to the size of the cave. Both players have freedom of movement.

Game classification

This is a non-cooperative game - players cannot join in groups. This is not a symmetrical game, as players do not have the same choice of actions. This is a zero-sum game, because all winnings are fixed. This is a parallel game, because decisions are made at the same time, arbitrarily. This is a game with incomplete information, as both players are not available information about each other's actions. This is a game with an infinite number of steps - the steps are unlimited. This is a game with an infinite number of steps, because the number of actions is not limited.

Game solution

This game was not resolved until the late 1970s. But later a strategy was found. The strategy for the princess is this: the princess goes to a random point and waits at this point for a certain amount of time, not too short and not too long. Then the princess moves to another random point and so on.

An optimal search strategy is proposed for the monster, in which the whole room is divided into many small rectangles. The monster randomly selects a rectangle and searches in it, then randomly selects the next rectangle and so on.

By the way, the obvious strategy - to start from a random end and cut off the retreat path in a zigzag fashion - is not optimal.

The game "Guess 2/3 of the average"

In 2005, a Danish newspaper called Politiken invited its readers to play the following game: anyone could send the publisher a real number from 0 to 100, the sender of the closest to 2/3 of the arithmetic average of the numbers sent won 5,000 Danish kroner.

This game demonstrates the difference between completely rational behavior and real actions of players.

Imagine that all participants in the game are rational and know that all other participants are rational. Which number is optimal in this situation?

Obviously, it makes no sense to call a number greater than 66. (6) because two-thirds of the arithmetic mean cannot be greater. However, if all players think this way, all numbers will be no more than 2/3 * 66. (6) = 44. (4). Repeating this argument infinitely many times, we conclude that the number 0 is the only correct move. Therefore, if all players reason rationally, they all must choose the number 0.

However, in real life the situation is different. Even if the player is rational, he knows that many of his opponents are not rational, which means that he will have to take into account that their numbers will be greater than 0. It can be assumed that the majority will send more or less random numbers, then the average will be 50, two-thirds from 50 approximately 33. If we go further and assume that quite a lot of people guess at number 33, then we can choose two-thirds of 33, that is, 22. Further iterations will yield ~ 15, ~ 10, etc., but it seems unlikely that a sufficiently significant number of players will calculate so far.

Game "Volunteer Dilemma"

Playing with a volunteer dilemma simulates a situation in which each player can either make a small sacrifice that benefits everyone, or instead wait in the hope of benefiting from someone else's victim.

One example is the scenario in which the power supply was cut off for the entire area. All residents know that the electricity company will not solve the problem until it calls and notifies at least one person about what happened, paying for the call. If no one wants to call, all participants will receive a negative win. If any person decides to become a volunteer, the rest will benefit, of course, if they do not become volunteers.

In this game, players decide on their own whether to sacrifice themselves for the good of the group. If no one sacrifices something voluntarily, everyone loses.

No matter how hard we try, we cannot find a winning strategy by playing with rational players. But what will happen in life? After all, not all people are rational!

Game Theory History

Already in the 18th century, optimal solutions and strategies for mathematical modeling were proposed. Some problems were considered in the 19th century by Augustine Augustine Kruno and Joseph Louis Francois Bertan.

At the beginning of the 20th century, Emmanuel Lasker, Ernst Firidrich Gemelo and Ferdinand Felix Eduard Justin Emile Borel put forward the idea of a mathematical theory of conflict of interest.

The mathematical theory of games comes from neoclassical economics. For the first time, mathematical aspects and applications of the theory were presented in the classic book of John von Neumann and Oscar Morgenstern in 1944, "Game Theory and Economic Behavior."

This area of mathematics has found some reflection in public culture. In 1998, the American writer and journalist Sylvia Nazar published a book about the fate of John Forbes Nash, and in 2001, based on the book, the film "Mind Games" was shot.

After graduating from the Carnegie Polytechnic Institute with two degrees - bachelor and master - John Nash entered Princeton University, where he attended lectures by John von Neumann. In his writings, Nash developed the principles of "control dynamics." John Nash defended his doctorate in game theory in 1949 and was awarded the Nobel Prize in economics.

The first concepts of game theory were analyzed by antagonistic games, when there are losers and winners at their expense. Nash is developing analysis methods in which all participants either win or lose.

These situations are called "Nash equilibrium" or "non-cooperative equilibrium" when the parties use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will worsen their situation.

These works by Nash made a significant contribution to the development of game theory, and mathematical tools for economic modeling were revised. Nash shows that Adam Smith’s classic approach to competition, when each for himself, is not optimal. Strategies are more beneficial when everyone is trying to benefit for themselves and do better for others.

Although game theory initially considered economic models, it remained a formal theory in mathematics until the 1950s. But already in the 1950s, attempts were made to apply the methods of game theory not only in economics, but also in biology, cybernetics, technology and anthropology.

During the Second World War and immediately after it, the military became seriously interested in the theory of games, who saw in it a powerful apparatus for studying strategic decisions.

In 1960-1970, interest in game theory was weakened, despite significant mathematical results achieved by then. In the mid-1980s, active practical application of game theory began, especially in the field of economics and management.

Over the past 20-30 years, the importance of game theory and interest in it has grown significantly. Some areas of modern economic theory cannot be stated without the application of game theory.

A number of famous scientists became Nobel Prize winners in economics for their contribution to the development of game theory, which describes socio-economic processes. John Nash, thanks to his research in game theory, has become one of the leading experts in the field of the Cold War, which confirms the magnitude of the tasks that game theory deals with.

The winners of the Economics Prize in memory of Alfred Nobel for achievements in the field of game theory and economic theory are: Robert Auman, Reinhard Zelten, John Nash, John Harsanyi, William Wickrey, James Mirrlis, Thomas Schelling, George Akerlof, Michael Spence, Joseph Stiglitz, Leonid Hurwitz , Eric Maskin, Roger Myerson, Lloyd Shapley, Alvin Roth, Jean Tyrol.

Application of Game Theory in Life

The game "Cork"

The cork from a bottle of champagne shot so hard that it reached a telephone with an open navigator.

Imagine the situation that you have a choice: either go along the highway during a traffic jam, or choose an empty circular path, which is 2 times longer than the highway. The maximum permissible speed in traffic congestion is 3 times less than the maximum permissible speed, without it.

Everything is simple here. The path is x, the speed is y.

Traffic jam - 1 x / 1 y
Empty road - 2 x / 3 y
Let's try to substitute the numbers.

Traffic jam - 50/10 = 5
Empty road 100/30 = 3.3
Let's try others, different from the previous numbers.

Traffic jam - 100/320 = 0.3
Empty road - 200/960 = 0.2
According to the results, we can conclude: in any case, an empty road will be faster.

But that is not all, this experience has a continuation. Many people, without knowing it, will use the theory of games and choose an empty road, which in turn will become busy. Taking this into account, perhaps you will choose the first option, having analyzed some factors: the average arrival of cars, the capacity of roads, the time needed to form a traffic jam and the time of approaching a fork in the road.

Game "Mafia Game"

You and your friends play the Mafia. Remains alive: "Peaceful inhabitant", "Mafiosi" and "Maniac". What are the chances to win peace? It would seem - no.

As we see, if: the

Mafia will kill the Maniac, and the Maniac will kill the Mafia - Peaceful will win.

The Mafia will kill the Maniac, and the Maniac will kill the Peaceful - The Mafia will win.

The Mafia will kill Mirny, and the Maniac will kill the Mafia - the Maniac will win.

The mafia will kill Mirny, and the Maniac will kill Mirny - Draw.

If decisions are spontaneous and random, the chances of peace are 25%.

Of course, no one wants to have a chance to either lose or get a draw, because the chance to either lose or win is better. Consequently, the choice to kill the peaceful is ruled out. Consequently, the Mafia will kill the Maniac and the Maniac will kill the Mafia - Peace will win.

Game "Movie"

Imagine - after a long working day you return home, hoping to go to bed immediately after arrival. The trip will last 1 hour 50 minutes. Suddenly, you had a desire to watch a movie, and the last movie coupon was left in the streaming service. You have a choice of 2 films: one of them is “The Matrix”, which lasts 2 hours, the second - “The Disgusting Eight”, which lasts 3 hours. Also, the last you really wanted to see.

So, let's try to understand what we should look at. It’s important to consider that you will only receive the next movie coupons in a week.

Your interest in the Abominable Eight is very large, but, unfortunately, we cannot translate interest and desire to sleep in one size and compare them, because it is very personal and depends on many factors: such as: desire to sleep, time to wake up, the importance of tomorrow's affairs, the ability to watch a movie at another time, the battery level of the phone, etc.

Fortunately, the human brain can process a huge amount of information. But the creation of a universal solution, even such a simple task for us, is very difficult and requires a lot of time and resources.

The game "Adverse Monopoly"

Perhaps this is one of the most common games in the world of economics. Recall that game theory is a branch of mathematical economics.

Microsoft, Sony, Disney ... Guess the common trait of these corporations? Each of them, to one degree or another, is a monopolist in his market. Microsoft, namely Windows in the field of operating systems. Sony, to be more precise - Play Station, in the field of game consoles. Disney in the entertainment industry.

All 3 companies manage most of the market by regulating and setting standards. Once they made a coup, made what became the pinnacle of opportunity. You can recall some Microsoft operating systems, Play Station 2 and the game The Last of Us, Disney cartoons, popular around the world.

But, corporations are primarily interested in profit. Having conquered the market and secured their status, they began to produce fairly mediocre products and services. Windows 8 and problems Windows 10, Play Station Vita, Avengers - mediocre products that do not deserve their status.

Customers, united, can make companies change their strategy - to begin to produce better products. By abandoning the company's services and products, customers could reduce the market by forcing the company to find ways to return the market.

But, unfortunately, people, unlike birds and some other creatures, are not endowed with the ability to unite so productively and harmoniously.

The chances of the situation described above are very scarce. And the players understand this.

Each participant of the game is not profitable to abandon Windows, because most players are used to it and it wakes them not only to understand, and not only install Linux, but also to understand the differences between Linux Kali and Linux Ubuntu.

Each participant in the game is not profitable to refuse one or another product, because he knows that he will not benefit personally.

At the heart of this game is Nash Balance, with which we are already familiar. But let's update our possibly distorted memories!

Nash equilibrium is a set of strategies in a game for two or more players, in which no participant can increase their winnings by changing their strategy if other participants do not change their strategies.

Of course, we can imagine a situation in which the former customers of the above companies abandoned the products of our companies.

In this case, Microsoft, Sony, Disney would create products of such quality and such capabilities, which and what is needed for the return of the market.

Perhaps they would be: “Windows Infinity open source,” “games not only with Keanu Reeves and Norman Reedus, but with the whole of Hollywood, in addition to Quentin Tarantino as a director,” “Avengers with meaning and a good plot.”

Alas, this is not achievable. It is very difficult to solve this Nash equilibrium with the size of 100 million participants.

I would also like to note some details:

Not only "our trinity" has this position. Hundreds and hundreds of companies play this game.

There are different types of this game. Sometimes a corporation does not occupy a monopoly position, but has a circle of “loyal” customers, or only their products provide certain opportunities. An example of this is Apple.

The game "Bertrand Model"

Is it profitable for shops to reduce the price of a product? Obviously not, but not so simple.

Imagine a game - 2 stores sell the same product at a 20% mark-up, buying it from the manufacturer at the same price. The same price = the same demand = the same income.

Suddenly, one of the shops lowers the price. What will happen? He will have more demand and consequently greater earnings. This is why price cuts are sometimes profitable.

The game "Narrow Road"

X and Igrik ride towards each other along a narrow road. In order not to crash into each other, both need to pull over.

The game is to choose the side of rotation. Each of the players must choose a side that does not coincide with the side of the opponent. What to choose? To solve such a game, traffic rules have been created.

Game Theory Application

Why do you need game theory? In the "History" section, you could observe the development of game theory and mention of its application. So let's find out why game theory is needed, where it is used, and even how game theory can come in handy for you!

Biology

To begin with, it should be noted: the behavior of animals is largely determined genetically, also, some types of behavior are more consistent with the situation than others.

A partially erroneous thought “the fittest survive” is widespread, at least the highest criterion of biological fitness is not survival, but reproductive success.

Animals pass on their genes to the next. Then, the more adaptable phenotype becomes relatively larger in the next generation than the less adaptable phenotype. It is this selection process that changes the combination of genotype and phenotype and can ultimately lead to the formation of a stable state.

New genetic mutations occur from time to time, spontaneously. Many of them create a phenotype that does not mix well with the environment and therefore disappears. However, sometimes mutations can lead to new phenotypes, making them more adaptive to the environment.

The number of more adapted animal mutations will increase while the non-adapted mutations may disappear, and mutations that are not currently part of this population may try to capture it.

Similar situations are used in game theory. Behavior can be seen as a strategy for the interaction of animals with other animals. The only difference is that in animals the choice of strategy is not carried out using targeted decisions.

Sociology and Psychology

Game theory is used in sociology to understand, explain and control games with a social component. In turn, in psychology, game theory studies the actions of each individual isolated player. In one form or another, game theory is used by psychologists, sociologists, politicians, marketers, and many other people.

Sociologists are trying to understand the reasons for the actions of groups of players and use the knowledge gained. They simulate games, conduct research to find the most profitable strategy.

Policy

In politics, game theory is used to analyze situations and interactions between players (usually countries), to solve games and to find the best strategies. Countries have a number of conflicts: territories, trade, alliances ... Game theory helps to reach a compromise.

The same game theory is used in the voting - candidates resort to different strategies to increase the chances of winning.

Economy

In economics, game theory is applied universally. Earlier you met the game "Adverse Monopoly", this is a very good example of the game. Economic games - auctions, monopoly and oligopoly models, markets and much more.

In economics, there are models that characterize certain games and are universal - and can be applied in all games that are suitable for the characteristic.

Unconscious application

Often, we apply game theory without even realizing it. We build logical chains, analyze situations and come up with strategies using game theory, but without knowing it. Above are the games "Film", "Cork" and some others in which players play constantly.

Our brain analyzes games, not betraying this value. From this statement the question arises: can knowledge of game theory be useful to an ordinary person?

The benefits of knowing Game Theory

Game theory is useful to many different specialists, but does the game theory need an ordinary person?

There is no practical universal application of game theory for an ordinary person. In life, to analyze the game, standing with a leaf and a pen opposite the counter with cookies, choosing a product is not a good idea, because you can cope with this task without using the methods of game theory.

Game theory is useful when:

Important decisions. There are situations in our lives that require very thoughtful choices that can change a lot of things. In such situations, game theory can be extremely useful and even necessary.
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Game Theory and its Application in Life