For someone who is not a political expert, Bruce Bueno de Mesquita of New York University makes the events surprisingly accurate. He managed to predict, with an accuracy of several months, the departure of Pereverz Musharaf from his posts. He accurately named Ayatollah Khomeini's successor as leader of Iran 5 years before his death. When asked what the secret is, he replies that he doesn’t know the answer - the game knows it. By game we mean mathematical method, which was originally created for the formation and analysis of strategies various games, namely, game theory. In economics it is used most often. Although it was originally developed to build and analyze strategies in games used for entertainment.
Game theory is a numerical apparatus that allows one to calculate a scenario, or more precisely, the probability of various scenarios of behavior of a system or “game” controlled by various factors. These factors, in turn, are determined by a certain number of “players”.
Thus, game theory, which received the main impetus for development in economics, can be applied in the most different areas human activity. It is too early to say that these programs will be used to resolve military conflicts, but in the future this is quite possible.
BELARUSIAN STATE UNIVERSITY
FACULTY OF ECONOMICS
DEPARTMENT…
Game theory and its application in economics
Course project
2nd year student
Department "Management"
Scientific director
Minsk, 2010
1. Introduction. p.3
2. Basic concepts of game theory p.4
3. Presentation of games page 7
4. Types of games p.9
5. Application of game theory in economics p.14
6. Problems of practical application in management p.21
7. Conclusion p.23
List of used literature p.24
1. INTRODUCTION
In practice, there is often a need to coordinate the actions of firms, associations, ministries and other project participants in cases where their interests do not coincide. In such situations, game theory makes it possible to find the best solution for the behavior of participants who are required to coordinate actions in the event of a conflict of interests. Game theory is increasingly penetrating the practice of economic decisions and research. It can be considered as a tool that helps improve the efficiency of planning and management decisions. It has great importance when solving problems in industry, agriculture, transport, trade, especially when concluding agreements with foreign partners at any level. Thus, it is possible to determine scientifically based levels of reduction in retail prices and the optimal level of inventory, solve the problems of excursion services and the selection of new lines of urban transport, the problem of planning the procedure for organizing the exploitation of mineral deposits in the country, etc. The problem of selecting plots of land for agricultural crops has become a classic one. The game theory method can be used in sample surveys of finite populations and in testing statistical hypotheses.
Game theory is a mathematical method for studying optimal strategies in games. A game is a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy that can lead to winning or losing - depending on the behavior of other players. Game theory helps you choose best strategies taking into account ideas about other participants, their resources and their possible actions.
Game theory is a branch of applied mathematics, or more precisely, operations research. Most often, game theory methods are used in economics, a little less often in others. social sciences- sociology, political science, psychology, ethics and others. Since the 1970s, it has been adopted by biologists to study animal behavior and the theory of evolution. It is very important for artificial intelligence and cybernetics, especially with interest in intelligent agents.
Game theory originates from neoclassical economics. The mathematical aspects and applications of the theory were first outlined in the classic 1944 book by John von Neumann and Oscar Morgenstern, The Theory of Games 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 Nash, Nobel laureate in economics and game theory scientist; and in 2001, based on the book, the film “A Beautiful Mind” was made. Some American television shows, such as Friend or Foe, Alias or NUMB3RS, periodically refer to the theory in their episodes.
A non-mathematical version of game theory is presented in the works of Thomas Schelling, Nobel laureate in economics in 2005.
Nobel laureates in economics for their achievements in the field of game theory were: Robert Aumann, Reinhard Selten, John Nash, John Harsanyi, Thomas Schelling.
2. BASIC CONCEPTS OF GAME THEORY
Let's get acquainted with the basic concepts of game theory. Mathematical model conflict situation is called a game, the parties involved in the conflict are called players, and the outcome of the conflict is called a win. For each formalized game, rules are introduced, i.e. a system of conditions that determines: 1) options for players’ actions; 2) the amount of information each player has about the behavior of their partners; 3) the gain that each set of actions leads to. Typically, winning (or losing) can be quantified; for example, you can evaluate a loss as zero, a win as one, and a draw as ½.
A game is called doubles if it involves two players, and multiple if there are more than two players.
A game is called a zero-sum game, or antagonistic, if the gain of one of the players is equal to the loss of the other, i.e., to complete the game, it is enough to indicate the value of one of them. If we denote a as the gain of one of the players, b as the gain of the other, then for a zero-sum game b = -a, so it is enough to consider, for example, a.
The choice and implementation of one of the actions provided for by the rules is called the player’s move. Moves can be personal and random. A personal move is a player’s conscious choice of one of the possible actions (for example, a move in a chess game). A random move is a randomly chosen action (for example, choosing a card from a shuffled deck). In the future, we will consider only the personal moves of the players.
A player’s strategy is a set of rules that determine the choice of his action at each personal move, depending on the current situation. Usually during the game, with each personal move, the player makes a choice depending on the specific situation. However, it is in principle possible that all decisions are made by the player in advance (in response to any given situation). This means that the player has chosen a specific strategy, which can be specified as a list of rules or a program. (This way you can play the game using a computer.) A game is called finite if each player has a finite number of strategies, and infinite otherwise.
In order to solve the game, or find a solution to the game, you should choose a strategy for each player that satisfies the optimality condition, i.e. one of the players should receive the maximum winnings when the other sticks to his strategy. At the same time, the second player should have a minimal loss if the first one sticks to his strategy. Such strategies are called optimal. Optimal strategies must also satisfy the stability condition, i.e., it must be unprofitable for any player to abandon his strategy in this game.
If the game is repeated quite a few times, then players may not be interested in winning and losing in each specific game, but in the average winning (loss) in all games.
The goal of game theory is to determine the optimal strategy for each player. When choosing an optimal strategy, it is natural to assume that both players behave reasonably in terms of their interests. The most important limitation of game theory is the naturalness of winning as an indicator of efficiency, while in most real economic problems there is more than one indicator of efficiency. In addition, in economics, as a rule, problems arise in which the interests of partners are not necessarily antagonistic.
3. Presentation of games
Games are strictly defined mathematical objects. A game is formed by the players, a set of strategies for each player, and the players' payoffs, or payoffs, for each combination of strategies. Most cooperative games are described by a characteristic function, while for other types the normal or extensive form is more often used.
Extensive form
Game "Ultimatum" in extensive form
Games in extensive, or expanded, form are represented in the form of an oriented tree, where each vertex corresponds to the situation when the player chooses his strategy. Each player is assigned a whole level of vertices. Payments are recorded at the bottom of the tree, under each leaf vertex.
The picture on the left is a game for two players. Player 1 goes first and chooses strategy F or U. Player 2 analyzes his position and decides whether to choose strategy A or R. Most likely, the first player will choose U, and the second - A (for each of them these are optimal strategies); then they will receive 8 and 2 points respectively.
The extensive form is very visual and is especially useful for representing games with more than two players and games with sequential moves. If participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line or outlined with a solid line.
Normal form
|
Player 2 |
Player 2 |
|
|
Player 1 |
4 , 3 |
–1 , –1 |
|
Player 1 |
0 , 0 |
3 , 4 |
|
Normal form for a game with 2 players, each with 2 strategies. |
||
The players chose strategies with the maximum result for themselves, but lost due to ignorance of the other player’s move. Typically, normal form represents games in which moves are made simultaneously, or at least in which all players are assumed to be unaware of what the other participants are doing. Such games with incomplete information will be discussed below.
Characteristic formula
In cooperative games with transferable utility, that is, the ability to transfer funds from one player to another, it is impossible to apply the concept of individual payments. Instead, a so-called characteristic function is used, which determines the payoff of each coalition of players. It is assumed that the gain of the empty coalition is zero.
The basis for this approach can be found in the book of von Neumann and Morgenstern. Studying the normal form for coalition games, they reasoned that if a coalition C is formed in a game with two sides, then the coalition N \ C opposes it. A game for two players is formed, as it were. But since there are many options for possible coalitions (namely 2N, where N is the number of players), the gain for C will be some characteristic value depending on the composition of the coalition. Formally, a game in this form (also called a TU game) is represented by a pair (N, v), where N is the set of all players and v: 2N → R is the characteristic function.
This form of representation can be used for all games, including those without transferable utility. There are currently ways to convert any game from normal form to characteristic form, but the reverse transformation is not possible in all cases.
4. Types of games
Cooperative and non-cooperative.
A game is called cooperative or coalition if players can form groups, taking on certain obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone must play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.
It is often assumed that what makes cooperative games different is the ability for players to communicate with each other. IN general case this is not true. There are games where communication is allowed, but the players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative ones describe situations in great detail and produce more accurate results. Cooperatives consider the game process as a whole. Attempts to combine the two approaches have yielded considerable results. The so-called Nash program has already found solutions to some cooperative games as equilibrium situations of non-cooperative games.
Hybrid games include elements of cooperative and non-cooperative games. For example, players can form groups, but the game will be played in a non-cooperative style. This means that each player will pursue the interests of his group, while at the same time trying to achieve personal gain.
A funny example of the application of game theory is in the fantasy book “The Brave Golem” by Anthony Pearce.
Lots of text
“The point of what I’m about to demonstrate to you all,” Grundy began, “is to collect the required number of points.” The scores can be very different - it all depends on the combination of decisions made by the participants in the game. For example, suppose each participant testifies against his fellow player. In this case, each participant can be awarded one point!
- One point! – said the Sea Witch, showing unexpected interest in the game. Obviously, the sorceress wanted to make sure that the golem had no chance of making the demon Xanth happy with him.
– Now let’s assume that each of the participants in the game does not testify against his friend! – Grundy continued. – In this case, each person can be awarded three points. I want to especially note that as long as all participants act in the same way, they are awarded the same number of points. No one has any advantage over another.
- Three points! - said the second witch.
– But now we have the right to suggest that one of the players began to testify against the second, but the second is still silent! - said Grundy. - In this case, the one who gives this testimony receives five points at once, and the one who is silent does not receive a single point!
- Yeah! – both witches exclaimed in one voice, licking their lips predatorily. It was clear that both of them were clearly going to get five points.
– I kept losing my glasses! – the demon exclaimed. – But you have only outlined the situation, and have not yet presented a way to resolve it! So what is your strategy? No need to waste time!
- Wait, now I’ll explain everything! - Grundy exclaimed. “Each of us four—there are two of us golems and two witches—will fight against our opponents. Of course, the witches will try not to yield to anyone in anything...
- Certainly! – both witches exclaimed again in unison. They perfectly understood the golem at a glance!
“And the second golem will follow my tactics,” Grundy continued calmly. He looked at his double. - Of course, you know?
- Yes, sure! I'm your copy! I understand perfectly well what you think!
- That is great! In that case, let's make the first move so that the demon can see everything for himself. There will be several rounds in each fight so that the entire strategy can be fully realized and give the impression of a complete system. Perhaps I should start.
– Now each of us must mark our own pieces of paper! – the golem turned to the witch. – First you should draw a smiling face. This will mean that we will not testify against a fellow prisoner. You can also draw a frowning face, which means that we think only about ourselves and are giving the necessary evidence against our comrade. We both realize that it would be better if no one turned out to be that same frowning face, but, on the other hand, a frowning face receives certain advantages over a smiling one! But the point is that each of us does not know what the other will choose! We won't know until our playing partner reveals his drawing!
- Get started, you bastard! – the witch cursed. She, as always, could not do without abusive epithets!
- Ready! - Grundy exclaimed, drawing a big smiling face on his piece of paper so that the witch could not see what he had drawn there. The witch made her move, also making a face. One must think that she certainly put on an unkind face!
“Well, now all we have to do is show each other our drawings,” Grundy announced. Turning back, he opened the drawing to the public and showed it in all directions so that everyone could see the drawing. Grumbling something displeased, the Sea Witch did the same.
As Grundy had expected, an angry, dissatisfied face looked out from the witch’s drawing.
“Now you, dear spectators,” Grundy said solemnly, “see that the witch chose to testify against me.” I'm not going to do that. Thus, the Sea Witch scores five points. And, accordingly, I don’t get a single point. And here…
A slight noise rang through the rows of spectators again. Everyone clearly sympathized with the golem and passionately wanted the Sea Witch to lose.
But the game has just begun! If only his strategy was correct...
– Now we can move on to the second round! – Grundy announced solemnly. – We must repeat the moves again. Everyone draws the face that is closest to them!
And so they did. Grundy now wore a gloomy, dissatisfied face.
As soon as the players showed their drawings, the audience saw that they were both now making angry faces.
- Two points each! - said Grundy.
- Seven two in my favor! – the witch shouted joyfully. “You won’t get out of here, you bastard!”
- Let's start again! - Grundy exclaimed. They made another drawing and showed them to the public. The same angry faces again.
– Each of us repeated the previous move, behaved selfishly, and therefore, it seems to me, it is better not to award points to anyone! - said the golem.
– But I still lead the game! - said the witch, happily rubbing her hands.
- Okay, don't make noise! - said Grundy. - The game is not over. Let's see what happens! So, dear audience, we are starting the fourth round!
The players made drawings again, showing the audience what they had drawn on their sheets. Both sheets of paper again showed the same evil faces to the audience.
- Eight - three! - the witch screamed, bursting into evil laughter. “You dug your own grave with your stupid strategy, golem!”
- Fifth round! - Grundy shouted. The same thing happened as in the previous rounds - angry faces again, only the score changed - it became nine - four in favor of the sorceress.
– Now the last, sixth round! - Grundy announced. His preliminary calculations showed that this particular round should become fateful. Now the theory had to be confirmed or refuted by practice.
A few quick and nervous movements of the pencil on the paper - and both drawings appeared before the eyes of the public. Again two faces, now even with bared teeth!
– Ten – five in my favor! My game! I won! – the Sea Witch cackled.
“You really won,” Grundy agreed gloomily. The audience was ominously silent.
The demon moved his lips to say something.
- But our competition is not over yet! - Grundy shouted loudly. – This was only the first part of the game.
- Give you an eternity! – the demon Xanth grumbled displeasedly.
- It's right! - Grundy said calmly. – But one round does not solve anything, only methodicality indicates the best result.
The golem now approached the other witch.
– I would like to play this round with another opponent! - he announced. – Each of us will depict faces, as it was the previous time, then we will demonstrate what we have drawn to the public!
So they did. The result was the same as last time - Grundy drew a smiling face, and the witch just a skull. She immediately gained a full five point lead, leaving Grundy behind.
The remaining five rounds ended with the results that could be expected. Once again the score was ten - five in favor of the Sea Witch.
– Golem, I really like your strategy! - the witch laughed.
– So, you have watched two rounds of the game, dear viewers! - Grundy exclaimed. “Thus, I scored ten points, and my rivals scored twenty!”
The audience, who were also counting points, nodded their heads mournfully. Their count matched that of the golem. Only the cloud named Fracto seemed very pleased, although, of course, it did not sympathize with the witch either.
But Rapunzel smiled approvingly at the golem - she continued to believe in him. She might be the only one left who believed him now. Grundy hoped that he would justify this boundless trust.
Now Grundy approached his third opponent - his double. He was to be his final opponent. Quickly scribbling their pencils on the paper, the golems showed the pieces of paper to the public. Everyone saw two laughing faces.
– Please note, dear viewers, each of us chose to be a good cellmate! - Grundy exclaimed. “And therefore none of us received the necessary advantage over our opponents in this game.” So we both get three points and move on to the next round!
The second round has begun. The result was the same as the previous time. Then the remaining rounds. And in each round, both opponents again scored three points! It was simply incredible, but the public was ready to confirm everything that was happening.
Finally, this round came to an end, and Grundy, quickly running his pencil over the paper, began to calculate the result. Finally he announced solemnly:
- Eighteen to eighteen! In total, I scored twenty-eight points, while my opponents scored thirty-eight!
“So you lost,” the Sea Witch announced joyfully. – Thus, one of us will become the winner!
- Maybe! – Grundy responded calmly. Now another one was coming important point. If everything goes as planned...
– We need to bring this matter to an end! – exclaimed the second golem. “I also still need to fight two Sea Witches!” The game is not over yet!
- Yes, of course, go ahead! - said Grundy. – But just be guided by strategy!
- Yes, sure! – assured his double.
This golem approached one of the witches and the tour began. It ended with the same result with which Grundy himself came out of a similar round - the score was ten to five in favor of the sorceress. The witch actually beamed with inexpressible joy, and the audience fell sullenly silent. Demon Xanth looked somewhat tired, which was not a very good omen.
Now it was time for the final round - one witch had to fight against the second. Each had twenty points, which she was able to get by fighting golems.
“And now, if you allow me to score at least a few extra points...” the Sea Witch whispered conspiratorially to her double.
Grundy tried to remain calm, at least outwardly, although a hurricane of conflicting feelings was raging in his soul. His luck now depended on how correctly he predicted the possible behavior of both witches - after all, their character was, in essence, the same!
Now came perhaps the most critical moment. But what if he was wrong?
- Why on earth should I give in to you! – the second witch croaked to the first. – I myself want to score more points and get out of here!
“Well, if you’re acting so impudent,” the applicant screamed, “then I’ll beat you up so that you won’t be like me anymore!”
The witches, giving each other hateful looks, drew their drawings and showed them to the public. Of course, nothing else but two skulls could have been there! Each scored one point.
The witches, showering each other with curses, began the second round. The result is again the same - again two clumsily drawn skulls. The witches thus scored one more point. The public diligently recorded everything.
This continued in the future. When the round was over, the tired witches discovered that they had each scored six points. Draw again!
– Now let’s calculate the results and compare everything! – Grundy said triumphantly. – Each of the witches scored twenty-six points, and the golems scored twenty-eight points. So what do we have? And we have the result that golems have more points!
A sigh of surprise swept through the rows of spectators. Excited spectators began to write columns of numbers on their pieces of paper, checking the accuracy of the count. During this time, many simply did not count the number of points scored, believing that they already knew the result of the game. Both witches began to growl with indignation, it is unclear who exactly they blamed for what had happened. The eyes of the demon Xant again lit up with a wary fire. His trust was justified!
“I ask you, dear audience, to pay attention to the fact,” Grundy raised his hand, demanding that the audience calm down, “that none of the golems won a single round.” But the final victory will still belong to one of us, the golems. The results will be more telling if the competition continues! I want to say, my dear viewers, that in the eternal duel my strategy will invariably turn out to be winning!
The demon Xanth listened with interest to what Grundy was saying. Finally, emitting clouds of steam, he opened his mouth:
– What exactly is your strategy?
– I call it “Be Firm but Fair”! - Grundy explained. – I start the game honestly, but then I start losing because I come across very specific partners. Therefore, in the first round, when it turns out that the Sea Witch begins to testify against me, I automatically remain a loser in the second round - and this continues until the end. The result may be different if the witch changes her tactics of playing the game. But since this couldn’t even occur to her, we continued to play according to the previous pattern. When I started playing with my double, he treated me well, and I treated him well in the next round of the game. Therefore, our game also went differently and somewhat monotonously, since we did not want to change tactics...
– But you haven’t won a single round! – the demon objected in surprise.
– Yes, and these witches haven’t lost a single round! – Grundy confirmed. – But victory does not automatically go to the one who has the remaining rounds. Victory goes to the one who scores the most points, but this is a completely different matter! I managed to score more points when I played with my double than when I played with the witches. Their selfish attitude brought them a momentary victory, but in the longer term, it turned out that it was because of this that they both lost the entire game. This happens often!
The Game Theory section is represented by three online calculators:
- Solving a matrix game. In such problems, a payment matrix is specified. It is required to find pure or mixed strategies of players and, game price. To solve, you must specify the dimension of the matrix and the solution method.
- Bimatrix game. Usually in such a game two matrices of the same size of payoffs of the first and second players are specified. The rows of these matrices correspond to the strategies of the first player, and the columns of the matrices correspond to the strategies of the second player. In this case, the first matrix represents the winnings of the first player, and the second matrix represents the winnings of the second.
- Games with nature. It is used when it is necessary to select a management decision according to the criteria of Maximax, Bayes, Laplace, Wald, Savage, Hurwitz.
In practice, we often encounter problems in which it is necessary to make decisions under conditions of uncertainty, i.e. situations arise in which two parties pursue different goals and the results of the actions of each party depend on the activities of the enemy (or partner).
A situation in which the effectiveness of a decision made by one party depends on the actions of the other party is called conflict. Conflict is always associated with some kind of disagreement (this is not necessarily an antagonistic contradiction).
The conflict situation is called antagonistic, if an increase in the winnings of one of the parties by a certain amount leads to a decrease in the winnings of the other side by the same amount, and vice versa.
In economics, conflict situations occur very often and are of a diverse nature. For example, the relationship between supplier and consumer, buyer and seller, bank and client. Each of them has their own interests and strives to make optimal decisions that help achieve their goals to the greatest extent. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner and take into account the decisions that these partners will make (they may be unknown in advance). In order to make optimal decisions in conflict situations, a mathematical theory of conflict situations has been created, which is called game theory . The emergence of this theory dates back to 1944, when J. von Neumann’s monograph “Game Theory and Economic Behavior” was published.
The game is a mathematical model of a real conflict situation. The parties involved in the conflict are called players. The outcome of the conflict is called a win. The rules of the game are a system of conditions that determine the players’ options for action; the amount of information each player has about the behavior of their partners; the payoff that each set of actions leads to.
The game is called steam room, if it involves two players, and multiple, if the number of players is more than two. We will only consider doubles games. Players are designated A And B.
The game is called antagonistic (zero sum), if the gain of one of the players is equal to the loss of the other.
The choice and implementation of one of the options provided for by the rules is called progress player. Moves can be personal and random.
Personal move- this is a conscious choice by a player of one of the options for action (for example, in chess).
Random move is a randomly selected action (for example, throwing dice). We will only consider personal moves.
Player strategy is a set of rules that determine the player’s behavior during each personal move. Usually during the game at each stage the player chooses a move depending on the specific situation. It is also possible that all decisions were made by the player in advance (i.e. the player chose a certain strategy).
The game is called ultimate, if each player has a finite number of strategies, and endless- otherwise.
Purpose of Game Theory– develop methods to determine the optimal strategy for each player.
The player's strategy is called optimal, if it provides this player with multiple repetitions of the game the maximum possible average win (or the minimum possible average loss regardless of the opponent’s behavior).
Example 1. Each of the players A or B, can write down, independently of the other, the numbers 1, 2 and 3. If the difference between the numbers written down by the players is positive, then A the number of points equal to the difference between the numbers wins. If the difference is less than 0, he wins B. If the difference is 0, it’s a draw.
Player A has three strategies (action options): A 1 = 1 (write 1), A 2 = 2, A 3 = 3, the player also has three strategies: B 1, B 2, B 3.
| B A | B 1 =1 | B2=2 | B 3 =3 |
| A 1 = 1 | 0 | -1 | -2 |
| A 2 = 2 | 1 | 0 | -1 |
| A 3 = 3 | 2 | 1 | 0 |
Player A's task is to maximize his winnings. Player B’s task is to minimize his loss, i.e. minimize the gain A. This zero-sum doubles game.
From the popular American blog Cracked.
Game theory is about studying ways to make the best move and, as a result, get as much of the winning pie as possible by chopping off some of it from other players. It teaches you to analyze many factors and draw logically balanced conclusions. I think it should be studied after numbers and before the alphabet. Simply because too many people make important decisions based on intuition, secret prophecies, the location of the stars and the like. I have thoroughly studied game theory, and now I want to tell you about its basics. Perhaps this will add common sense into your life.
1. Prisoner's dilemma
Berto and Robert were arrested for bank robbery after failing to properly use a stolen car to escape. The police cannot prove that they were the ones who robbed the bank, but they caught them red-handed in a stolen car. They were taken to different rooms and each was offered a deal: to hand over an accomplice and send him to prison for 10 years, and to be released himself. But if they both betray each other, then each will receive 7 years. If no one says anything, then both will go to prison for 2 years just for car theft.
It turns out that if Berto remains silent, but Robert turns him in, Berto goes to prison for 10 years, and Robert goes free. Each prisoner is a player, and everyone's benefit can be expressed as a "formula" (what they both get, what the other gets). For example, if I hit you, my winning pattern would look like this (I get a rude victory, you suffer from severe pain). Since each prisoner has two options, we can present the results in a table.
Practical Application: Identifying Sociopaths
Here we see the main application of game theory: identifying sociopaths who think only about themselves. True game theory is a powerful analytical tool, and amateurism often serves as a red flag that flags up someone who has no sense of honor. People who make intuitive calculations believe that it is better to do something ugly because it will result in a shorter prison sentence no matter what the other player does. Technically this is correct, but only if you are a short-sighted person putting the numbers higher human lives. This is why game theory is so popular in finance.
The real problem with the prisoner's dilemma is that it ignores the data. For example, it does not consider the possibility of you meeting with friends, relatives, or even creditors of the person you sent to prison for 10 years.
The worst part is that everyone involved in the prisoner's dilemma acts as if they have never heard of it.
And the best move is to remain silent, and after two years, together with good friend use common money.
2. Dominant strategy
This is a situation in which your actions give the greatest payoff, regardless of the actions of your opponent. No matter what happens, you did everything right. This is why many people with the Prisoner's Dilemma believe that betrayal leads to the "best" outcome regardless of what the other person does, and the ignorance of reality inherent in this method makes it look super easy.
Most of the games we play don't have strictly dominant strategies because otherwise they'd be terrible. Imagine if you always did the same thing. There is no dominant strategy in the game of rock-paper-scissors. But if you were playing with a person who had oven mitts on and could only show rock or paper, you would have a dominant strategy: paper. Your paper will wrap his stone or result in a draw, and you cannot lose because your opponent cannot show scissors. Now that you have a dominant strategy, you'd be a fool to try something different.
3. Battle of the sexes
Games are more interesting when they don't have a strictly dominant strategy. For example, the battle of the sexes. Anjali and Borislav go on a date, but cannot choose between ballet and boxing. Anjali loves boxing because she enjoys seeing blood flow to the delight of a screaming crowd of spectators who think they are civilized just because they paid for someone's head to be smashed.
Borislav wants to watch ballet because he understands that ballerinas go through a huge number of injuries and difficult training, knowing that one injury can end everything. Ballet dancers are the greatest athletes on Earth. A ballerina can kick you in the head, but she will never do it, because her leg is worth much more than your face.
Each of them wants to go to their favorite event, but they don't want to enjoy it alone, so here's how they win: highest value- do what they like, smallest value- just being with another person, and zero - being alone.
Some people suggest stubborn brinksmanship: if you do what you want no matter what, the other person must conform to your choice or lose everything. As I already said, simplified game theory is great at identifying fools.
Practical application: Avoid sharp corners
Of course, this strategy also has its significant drawbacks. First of all, if you treat your dating as a "battle of the sexes", it won't work. Break up so that each of you can find someone they like. And the second problem is that in this situation the participants are so unsure of themselves that they cannot do this.
The truly winning strategy for everyone is to do what they want. and after, or the next day, when they are free, go to a cafe together. Or alternate between boxing and ballet until a revolution occurs in the entertainment world and boxing ballet is invented.
4. Nash equilibrium
A Nash equilibrium is a set of moves where no one wants to do anything differently after the fact. And if we can make it work, game theory will replace the entire philosophical, religious, and financial system on the planet, because the "will not to go broke" has become more powerful for humanity driving force than fire.
Let's quickly split $100. You and I decide how many of the hundreds we require and at the same time announce the amounts. If our total is less than a hundred, everyone gets what they wanted. If total more than a hundred, the one who asked for the smallest amount receives the desired amount, and more greedy man gets what's left. If we ask for the same amount, everyone gets $50. How much will you ask? How will you split the money? There is only one winning move.
Requiring $51 will give you the maximum amount no matter what your opponent chooses. If he asks for more, you will receive $51. If he asks for $50 or $51, you will receive $50. And if he asks for less than $50, you will receive $51. In any case, there is no other option that will bring you more money than this one. Nash equilibrium - a situation in which we both choose $51.
Practical Application: Think First
This is the whole point of game theory. You don’t have to win, much less harm other players, but you do have to make the best move for yourself, regardless of what those around you have in store for you. And it’s even better if this move is beneficial for other players. This is the kind of mathematics that could change society.
An interesting variation of this idea is drinking, which can be called a time-dependent Nash Equilibrium. When you drink enough, you don't care about other people's actions no matter what they do, but the next day you really regret not doing something differently.
5. Toss game
The toss is played between Player 1 and Player 2. Each player simultaneously chooses heads or tails. If they guess correctly, Player 1 gets Player 2's penny. If not, Player 2 gets Player 1's coin.
The winning matrix is simple...
...optimal strategy: play completely at random. It's harder than you think because the choice has to be completely random. If you have a heads or tails preference, your opponent can use it to take your money.
Of course, the real problem here is that it would be much better if they just threw one penny at each other. As a result, their profits would be the same, and the resulting trauma might help these unfortunate people feel something other than terrible boredom. After all, this worst game ever existing. And this is the ideal model for a penalty shootout.
Practical application: Penalty
In football, hockey and many other games, extra time is a penalty shootout. And they would be more interesting if they were based on how many times the players full form would be able to do a cartwheel because it would at least be an indication of their physical ability and would be fun to watch. Goalkeepers cannot clearly determine the movement of the ball or puck at the very beginning of their movement, because, unfortunately, in our sports competitions robots are still not participating. The goalkeeper must choose the left or right direction and hope that his choice matches the choice of the opponent who is shooting at goal. This has something in common with playing coins.
However, please note that this is not perfect example similarities with the game of heads and tails, because even with making the right choice direction, the goalkeeper may not catch the ball, and the attacker may not hit the goal.
So what is our conclusion according to game theory? Ball games should end in a "multi-ball" manner, where every minute one-on-one players are given an extra ball/puck until one side achieves a certain result, which is an indication of the true skill of the players, and not a spectacular random coincidence.
At the end of the day, game theory should be used to make the game smarter. Which means it's better.