1. Game theory refers to an individual or organization, facing certain environmental conditions, under certain rules, relying on the information mastered, selection and implementation from their respective behaviors or strategies, and from their respective respective respective behaviors, and from their respective respective respects The process of obtaining the corresponding results or income is a very important theoretical concept in economics.
What is game theory? The old saying is cloudy, and the world is like chess. Everyone in life is like a player, and each behavior is like a child on an invisible chessboard. The savvy and cautious chess players figured out and restrained each other. Chess game. The game theory is a part of the Chinese and logical part of the chess players who "chess", and systematically become a science. In other words, it is the most reasonable strategy to study how individuals can get the most reasonable strategies in intricate interaction. In fact, the theory of gaming is derived from ancient games or games such as chess and poker. Mathematicians abstract specific problems and study their rules and changes by establishing a complete logical framework and system. This is not easy. Take the simplest two -person game as an example. If you think about it a little, you know that there is a mystery: if the two sides precisely remember that each step of themselves and the opponent is the most "rational", the most "rational" For the player, when you go out, in order to win the chess, you must consider B's thoughts carefully, and when you come out of the child, you must also consider A's thoughts, so A must think of B thinking about his thoughts, of course, he knows A knows A. Thinking of his thoughts of thinking ...
The face of the game theory to analyze the problem, how to analyze the problem, how to find the optimal solution to the abstract mathematical problem of reality, so as to be theoretical What about the possibility of guiding practice? Modern game theory was founded by Hungarian mathematician von Norman in the 1920s. In 1944, he and economist Oscar Morgan Master's masterpiece "Game Theory and Economic Behavior" marked the modern system Initial formation of game theory. For non-cooperation and pure competitive games, Nurman has only two people zero-harmony games-like two people playing chess or playing table tennis, one person will lose one, and the other will lose one. The profit is zero. The abstract game problem here is that it is known how to find a theoretical "solution" if you know the collection of participants (two parties), the strategy collection (all chess), and the profit collection (winning the son). Or "balance", that is, the most "reasonable" and optimal strategy for participating parties? What is "reasonable"? The "smallest and largest" criterion in the application of traditional decision theory, that is, each party of the game assumes that the fundamental purpose of the opponent's skills is to maximize the defeat, and to optimize their countermeasures to the greatest extent. Nomann proves from mathematics. Through a certain linear operation, you can find a "smallest maximum solution" for each two -person game. Through a certain linear operation, the competition between the two sides randomly uses the various steps in a set of optimal strategies in the form of probability distribution, which can finally achieve the most profitable and comparable to each other. Of course, its implicit significance is that this optimal strategy does not depend on the operation of the opponent in the game. In popular terms, the basic "rational" idea reflected by this famous minimum and maximum theorem is "the best hope and the worst plan to do."
2. In economics, "Pigs’payoffs" is an example of a famous game theory. The example is:*There are two ends in the circle*, one big*, and one small*. There is a pedal on one side of the circle. Each step on the pedal, a small amount of food will be dropped on the feeding mouth on the other side of the*circle away from the pedal. If there is one*step on the pedal, the other*has the opportunity to eat the food that falls first. When Xiao*stepped on the pedal, Big*will just eat all the food before Xiao*runs to the groove; if the big*steps on the pedal, there is still the opportunity to run to the small*food before eating the food that falls. Trust tanks, compete for the other half of the disabled. So, what strategies will two take? The answer is: Xiao*will choose the strategy of "take a car", that is, comfortably waiting for the grooves; while Big*is a little disabled, and he is busy running between the pedal and the groove. What is the reason? Because the small*pedal will have nothing, you can eat food without stepping on the pedal. For Xiao*, no matter whether you step on the pedal or not, it is always a good choice to not step on the pedal. On the other hand, it is known that Xiao*will not step on the pedal. Going to the pedal yourself is better than stepping on it yourself, so he has to personally do it. "Little*lying on Da*Run" is caused by the game rules in the story. The core indicators of the rules are: the number of things that falls each time and the distance between the pedal and the feed mouth. If the core indicators, will there be the same scene of the same "small*lying big*run" in the circle? Try it. changing scheme 1: Reduction scheme. Eating only half of the original weight. As a result, Xiao*Big*did not step on the pedal. Xiao*goes, big*will eat the food; big*to step on, Xiao*will also eat the food. Whoever stepping on the pedal means to contribute food to the other party, so no one will have the motivation to step the pedal. If the purpose is to let*people go to the pedal more, the design of this game rules is obviously failed. changing scheme 2: Increase plan. Eating is the original weight. As a result, Xiao*and Da*will go to the pedal. Whoever wants to eat, whoever wants to eat will go to the pedal. Anyway, the other party will not eat the food at a time. Xiao*and Big*are equivalent to living in a "communist" society with relatively rich material, so the awareness of competition will not be strong. For designers of the rules of the game, the cost of this rule is quite high (providing double -portal foods at a time); and because the competition is not strong, it is not good to let the*get more pedal on the pedal. The change scheme three: reduction plus displacement scheme. Eating only half of the original weight, but at the same time move the feeding mouth to the pedal. As a result, Xiao*and Dan*are desperately grabbing the pedal. Waitors do not eat, and more work. Each harvest is just finished. For game designers, this is the best solution. The cost is not high, but the harvest is the biggest. The original version of the "Smart*Game" story gives the weak (small*) in the competition as the inspiration of the best strategy. But for society, because Xiao*failed to participate in competition, Xiao*was not the best state when the social resource allocation was configured when taking the car. In order to make resources the most effective configuration, the designers of the rules are unwilling to see someone on the stool. The government is so, as well as the company's boss. Whether it can completely eliminate the phenomenon of "ride a car" depends on whether the core indicators of the game rules are appropriate. For example, the company's incentive system design has too much rewarding, it is also holding shares, and it is an option. All the company staff has become a millionaire. Essence This is equivalent to the situation described in the "smart*game" . However, if the reward is not strong, and the people have a share (the "small*" without labor), the once-very hard-working big*will not be motivated ---- it's like the "smart*game" reduction solution one. Described. The best incentive mechanism design is like changing solutions three-the method of reduced volume and displacement. The rewards are not everyone has a share, but directly targeting individuals (such as business-proportioned business), which saves costs (for the company's company In terms of), the phenomenon of "ride" has been eliminated, which can achieve effective incentives. The many people have not read the story of "smart*game", but they are consciously using small*strategies. The stock market is waiting for the dealer to lift the retail households; waiting for the new products with profitability in the industrial market, and then make great imitation to make money; those who do not create benefits in the company, but share results, and so on. Therefore, for those who formulate game rules of various economic management, we must know the reason why the "smart*game" indicators have changed. 3. Background knowledge: Principles and applications of Nash game theory r bent.sina, March 21, 2002 17:44 Beijing Evening News 1950 and 1951 Nash about non -non -non -non -non -non -non -non -non -non -non -non -non -Nash Important papers in cooperative game theory have completely changed people's views on competition and markets. He proved the non -cooperative game and its balanced solution, and proved the existence of equilibrium solution, the famous Nash equilibrium. This reveals the internal connection between the balance of the game and the economic balance. Nash's research laid the cornerstone of modern non -cooperative game theory, and later game theory research basically followed this main line. However, Nash's discovery was denied by von Normann, and before that, he was also received by Einstein. However, the nature of challenging authority and contempt of authority in the bones made Nash adhere to his own views and eventually became a generation of masters. If it weren't for more than 30 years of severe mental illness, I am afraid that he had already stood on the podium of the Nobel Prize, and would never share this honor with others.
Nash is a very genius mathematician. His main contribution was made when he was a PhD from 1950 to 1951 in Princeton. However, his genius discovered — the balance of non -cooperative game, that is, "Nash Balance" is not smooth sailing.
1948 Dr. Nash from Nash to Plinson University. He was less than 20 years old that year. At that time, Princeton was outstanding, and the master was like a cloud. Einstein, von Normann, Lev Shez (Director of Mathematics), Albert Tuck, Allen Catkic, Harold Kun, Norman Steen Rod, Elff Fox ... and others are all here. The game theory is mainly created by von Normann (1903-1957). He is a genius born in Hungary. He not only founded the theory of economic game, but also invented a computer. As early as the beginning of the 20th century, Zermelo, Borel, and von Norman had already begun to study the accurate mathematical expression of the game. Until 1939, von Norman met an economist Oscar. Oskar (Oskar), and cooperated with it to make gaming theory into the broad field of economics.
1944 The masterpiece "Game Theory and Economic Behavior", the masterpiece "Game Theory and Economic Behavior", which is co -authored by Oscar Morgan Master, marked the initial formation of modern system game theory. Despite research on gaming -in -nature issues dating back to the 19th century or even earlier. For example, in 1838 Cournot's simple double oligopoly monopoly game; in 1883, Berham and Aicvici in 1925 studied the output and price monopoly of two oligarchs; Sun Wu, the descendant of Chinese militaryist Sun Wu more than 2000 years ago, more than 2000 years ago. The use of gaming theory methods to help Tian Ji horse race winning, etc. It is a bud of early game theory. It is characterized by sporadic. The disconnected research has a lot of accidentality and is very unclear. The concepts and analysis methods of von Norotan and Morgan Stan's "Game Theory and Economic Behavior" proposed in the book "Game Theory and Economic Behavior" laid the theoretical foundation of this discipline. The cooperative game reached its peak in the 1950s. However, the limitations of Noonman's game theory are also increasingly exposed. Because it is too abstract, the application scope is greatly limited. Patent, so the influence is very limited. It was at this time that the non -cooperative game -"Nash Balance" came into being, and it marked the beginning of the new era of game theory! Nash is not a step -by -step student, and he often absent from class. According to his classmates, they couldn't think of when they had completed a compulsory course on the whole ground with Nash, but Nash argued that at least he had served as a scholarship for the algebraic study. Stanod Rhoder was the founder of this discipline, but without a few lessons, Nash determined that the course did not conform to his taste. So, I left again. However, after all, Nash is an extraordinary figure in talents. He is widely involved in every branch of the Mathematics Kingdom, such as topology, algebraic geometry, logic, game theory, etc., which is deeply fascinated by it. Nash often shows his unique self -confidence and arrogance, full of aggressive academic ambitions. In 1950, Nash was busy dealing with the tension. His game theory research work was forced to interrupt. He felt that it was a great waste. As everyone knows, this temporary "abandonment" has made the original vague, clutter and no idea of the thoughts, and gradually formed a clear context under the continuous thinking of subconsciousness, and suddenly inspired! In October of this year, he suddenly felt the trend, and the dream was flowing. One of the most dazzling highlights is the concept of non -cooperative game balance that is called "Nash Balance" in the future. Nash's main academic contributions are reflected in the two papers in 1950 and 1951 (including a doctoral dissertation). In 1950, he wrote his research results as a long doctoral dissertation of "non -cooperative game". In November 1950, it was published in the monthly public bulletin of the National Academy of Sciences, which immediately caused a sensation. Speaking of this relying on his brother David Gail, after a few days after being degraded by von Neumanman, he met Gal, telling him that he had already regarded the "smallest principle" of von Normanmann. (Minimax Solution) pushed to the field of non -cooperative game, and found universal methods and equilibrium points. Gall heard it very seriously. He finally realized that Nash's thinking can better reflect the real situation than von Neumanman's cooperation game, and he is extremely admired by his strict and beautiful mathematics proof. Gall suggested that he immediately sorted out and was released to avoid being first set aside by others. Nash, a fledgling boy, did not know the sinister competition at all, and never thought about it. As a result, Gal served as his "agent" and drafted text messages to the Academy of Sciences on behalf of the Sciences. There are not many articles written by Nash, just a few, but it is enough because they are all fine products. This is also worth our thought. A professor in China requires how many articles are required to publish in the "core publication". According to this standard, Nash may not be qualified.
1996 The Nobel Prize in Merris Dang Morris Dadang Oxjin University Achivis Economic Lecturer did not post any articles. Special talents must have a special selection method.
Nash started to work in the game theory of pure mathematics when he went to college. After entering Princeton University in 1948, he was even more watery. At the age of 20, he has become a world -renowned mathematician. Especially in the field of economic gaming theory, he made epoch -making contributions and is one of the greatest gaming theory masters after von Normann. The famous Nash balanced concept he proposed plays a core role in the theory of non -co -operative game. The subsequent researchers' contributions to gaming theory are based on this concept. Because Nash's balanced proposal and continuous improvement laid a solid theoretical foundation for gaming theory in the fields of economics, management, sociology, political science, and military science.
The dilemma of prisoners
The small story in the theory
It to understand Nash's contribution, first of all, we must know what is wrong with cooperation game. Almost all gaming theory textbooks will talk about "the dilemma of the prisoner" in textbooks, and the examples in each book are similar.
It game theory is mathematics, after all, it is a branch of operational science. Talking about the scriptures naturally, the math language is indispensable. The layman seems to be just a lot of mathematical formulas. Fortunately, the game theory is concerned about daily economic life problems, so you must not eat fireworks on earth. In fact, this theory is borrowed from the issues borrowed from the issue of competition, confrontation, and decision -making from chess, poker, and war. It sounds a bit mysterious, but it actually has important practical significance. Masters of the game theory look at economic and social issues like a chess game, and often make profound reasoning in the game. Therefore, we start with the small things in our daily life, and take the story around us as an example. It is not boring. One day, a rich man was killed at home and the property was stolen. During the detection of the case, the police caught two suspects, Scarfels and Nacus, and found the lost property in their homes from their residence. However, they denied that they had killed people, arguing that they first found that the rich man was killed, and then he just stole something. So the police separated the two and kept them in different rooms for interrogation. Local prosecutors talk to each person separately. The prosecutor said, "Because your theft has conclusive evidence, you can sentence you one year's sentence. However, I can deal with you. If you confess to the crime of killing alone, I only sentence you for three months Imprisonment, but your associates will be sentenced to ten years. If you refuse to confess and be reported by your associates, then you will be sentenced to ten years. Frankly explain, then you have to be sentenced to 5 years. "What should Skalfel and Nacus do? They are facing a dilemma -frank or dependent. Obviously the best strategy is that both parties depend, and the result is that everyone has only been sentenced to one year. However, because the two were in isolation, they could not string their confessions. Therefore, according to Adam Smith's theory, everyone starts from the purpose of self -interest, and they choose to confess to the best strategy. Because frankly can expect a short imprisonment — 3 months, but the premise is that the associates deduct, apparently better than depending on yourself. This strategy is a strategy to harm others. Not only that, there are more benefits frankly. If the other party confessed and relied herself, then he had to sit in 10 years. It's not cost -effective! Therefore, in this case, you should still choose to confess. Even if the two are frank at the same time, it is only 5 years, which is better than that of 10 years. Therefore, the reasonable choice of the two is frank, and the strategies (dependent) and ending (one year sentence sentenced) to both parties will not appear. In this way, both of them choose a frank strategy and the ending of the five -year -old ending is called "Nash Balance", also known as non -coordinated equilibrium. Because each party has no "co -conspiracy" when choosing a strategy, they just choose the most favorable strategy for themselves without considering the interests of social benefits or any other opponents. In other words, this strategy combination is composed of the best strategy combination of people in all bureaus (also known as the parties and participants). No one will take the initiative to change their strategy to make themselves more benefits. "Prisoner's dilemma" has extensive and profound significance. The final ending of personal rationality and collective rationality is a "Nash Balance" and an endless ending for everyone. Both of them first thought of themselves in a frank and deleted strategy, so they must be sentenced to a long sentence. Only when they first think about each other, or conspire each other (stringing) can they get the shortest time in prison. "Nash Balance" first challenged Adam Smith's "invisible hand" principle. According to Smith's theory, in the market economy, everyone starts from the purpose of self -interest, and eventually the whole society achieves the effect of altruism. Let's review the famous sayings of this economics saint in "National Wealth Theory": "By pursuing (individual) his own interests, he often promotes social interests more effectively than what he actually wants to do." From "Nash Balance" we lead to a paradox of the principle of "invisible hands": starting from the purpose of self -interest, the result is not good for ourselves, which is not good for ourselves nor him. This is the fate of the two prisoners. In this sense, the paradox proposed by "Nash Balance" actually shakes the cornerstone of Western economics. Therefore, we can also realize a truth from "Nash Balance": cooperation is a favorable "self -interest strategy". But it must be in line with the following golden laws: according to the way you want others to treat you, but only they can act in the same way. That is, the Chinese say "do not do what you do." But the premise is that people do not want to do it. Secondly, "Nash Balance" is a non -co -operating game balance, and the situation of cooperation between China and Africa is widespread in reality. Therefore, "Nash Balance" is a major development of the theory of cooperation between von Norman and Morgan Midn, and it can even be said to be a revolution.
It from the general sense of "Nash Balance", we can deeply understand the phenomenon of the economic, social, political, politics, national defense, management and daily life phenomenon. We will give examples such as "two dilemma of prisoners". Such as price war, military competition, pollution and so on. The general game problem is composed of three elements: the collection of the parties, participants, strategies, etc., also known as the parties, participants, strategies, etc. )gather. The so -called winning means that if a specific strategic relationship is selected, the utility obtained by the people in each game. All game issues will encounter these three elements.
The price battle game:
Now we often encounter various types of home appliances price wars, color TV war, refrigerator war, air -conditioning war, microwave oven war ... These war The beneficiary is first of all consumers. Whenever you see the price war of a home appliance product, the people will "be okay." Here, we can explain that the ending of the manufacturer's price war is also a "Nash Balance", and the result of the price war is that no one has money to make. Because the profit of both parties is exactly zero. The result of competition is stable, that is, a "Nash Balance". This result may be good for consumers, but it is catastrophic for manufacturers. Therefore, the price war means committing suicide for manufacturers. From this case, we can extend two problems. One is the result of competitive prices or "Nash Balance" may cause an efficient zero -profit ending. Second, if the price war is not adopted, what will happen as a hostile game theory? Each enterprise will consider adopting normal price strategies, or adopt a high price strategy to form a monopoly price, and try its best to obtain monopoly profits. If the monopoly can be formed, the common profit of both parties in the game is the largest. This situation is done by monopoly operations, and the price is usually raised. Another extreme situation is the normal price of manufacturers, and both parties can get profits. From this point, we have led to another basic criterion: "Build your own strategy on the basis of assumed that your opponent will act in accordance with its best interests." In fact, the balance of competition is "Nash Balance" or "non -cooperative game equilibrium". In this state, every manufacturer or consumer decide on the price of all others. In this equilibrium, each enterprise must maximize the profit, and consumers must maximize the utility, which leads to zero profits, which means that the price is equal to marginal costs. In the case of complete competition, non -co -operating behavior has led to the economic efficiency state expected by society. If the manufacturer takes cooperation and decides to turn to monopoly, the economic efficiency of society will be destroyed. This is why WTO and governments of various countries have to strengthen antitrust.
The pollution game:
If pollution in the market economy, but the government does not have a controlled environment. In order to pursue the maximum profit of profits, it is better to sacrifice the environment at the cost of the environment. It will never take the initiative to increase investment in environmental protection equipment. According to the principle of invisible hands, all companies will start from the purpose of self -interest and adopt strategies that regardless of the environment to enter the "Nash Balance" state. If an enterprise starts from the purpose of altruis, invests in pollution, and other companies are still regardless of environmental pollution, then the production cost of this company will increase, the price will increase, and its products will not be competitive, and even the company will go bankrupt. This is an example of the failure of "an invisible and effective competitive mechanism of the invisible hand". Until the mid -1990s, the blind development of Chinese township enterprises caused severe pollution. Only when the government strengthens pollution control can enterprises adopt low pollution strategy combinations. In this case, enterprises will obtain the same profits as high pollution, but the environment will be better.
The trade freedom and barriers:
This problem is particularly important for China that has just joined the WTO. In international trade, any country faces the dilemma of maintaining freedom of trade and the implementation of trade protectionism. The issue of freedom of trade and barriers is also a "Nash Balance". This equilibrium is a strategy of adopting a non -cooperative game between the two parties. As a result, the two parties were damaged for the trade war. Country X attempts to carry out import trade restrictions on countries Y. For example, if you increase tariffs, country Y will inevitably counterattack and increase tariffs. As a result, no one can get benefits. Conversely, such as X and Y can reach a cooperative balance, that is, from the principle of mutual benefit, both parties reduce tariff restrictions. As a result, everyone has obtained the maximum benefits from the freedom of trade, and the total revenue of global trade has also increased.
Introduction to the game theory (Keywords: strategy space, cooperation game, non -cooperative game, Nash balance, group rational, entrustment agency relationship, incentive theory)
Countermeasures, originated from the beginning of this century, the "Game Theory and Economic Behavior" co -authored by von Normann and Morgan Stann laid the theoretical basis for gaming theory. Since the 1950s, Nash, Zelten, Haysni and others have finally matured the game theory and entered practicality. In the past 20 years, game theory, as a tool for analysis and resolution of conflict and cooperation, has been widely used in management science, international politics, ecology and other fields.
It simply speaking, game theory is how to decide on how to decide on a given information structure to maximize its utility, and the balance of decision -making between different decision -making subjects. The game theory consists of 3 basic elements: first, the decision -making subject (Player), and can also be translated as participants or people in the bureau; Called a strategy set; the third is the utility, which is a definition or quantitative participant's interests, and it is also something that all participants really care about, also known as preferences or payment functions. Participants, strategy sets and utility constitute a basic game.
The game theory can be divided into cooperative games and non -cooperative games. The difference between the two is whether the participants can reach a binding agreement during the game. If not, non-cooperative games are called non-cooperative games are the focus of modern game theory. For example, the two companies A and B cooperate to build a VCD production line. The agreement provides the technology of producing VCDs by Party A, and Party B provides factory buildings and equipment. A non -cooperative game is formed when the asset assessment of technology and equipment is formed, because each party tries to maximize its own evaluation value. At this time, if the party B can get the real estimates of the A technology or the reference quotation such as the technology, then the party B will You can make yourself gain advantage in the evaluation; the same is true, as well as the same. As for whether your asset assessment will affect the "collective interest" of the overall operation efficiency of the cooperative company, it will not attach great importance to it. This is a non -cooperative game. When participants choose their own actions, they give priority to how to protect their own interests.
The cooperation game emphasizes collectivism, group rationality (), which is efficiency, fairness, and fairness; instead of cooperative games, it emphasizes personal rationality and individual best decisions. No.
The theory of game theory emphasizes the importance of time and information. It is believed that time and information are the main factor affecting the balance of game. During the game, information transmission between participants determines the choice of action space and optimal strategy. At the same time, there is always a successful problem during the game, and the order of action of the participants has a direct impact on the final balance of the game.
The division of games can be carried out from the order of participants and participants' characteristics, strategic space, and payment of knowledge about other participants. The combination of two angles gets 4 games: complete information static game, full information dynamic game, incomplete information static game, incomplete information dynamic game. Its representatives are Nash, Zelten and Haysni. Strictly speaking, game theory is not a branch of economics. It is just a way, which is why many people see it as a branch of mathematics. Gaming theory has been widely used in the field of politics, economy, diplomacy, and sociology, and it provides a valuable method for solving the conflict and cooperation of different entities.
The use of game theory can prove many interesting issues in real life. For example, there are not many workers, excessive use of public resources, and non -collaborators choose to cooperate with bad guys for good deeds for a period of time. Although these conclusions are based on a strong assumption, that is, participants are rational and have a trend of maximizing their utility. But its conclusion has profound philosophical connotation.
The commissioning and incentive theory in economics can be analyzed by gaming theory. There are many situations in modern enterprises in the context of cooperation. For example, the oligopoly A and B of the monopoly market can protocol the oil output of an output such as the Gulf country to maintain their maximum profits. However, in many cases, there are always situations that increase output in order to maintain their local profits, such as Saudi Arabia often increases production without authorization. As a result, prices have decreased and profit loss. Competitive intelligence often plays an important role in this case. If A masters competition information such as B's actual production capacity, you can adjust your output or even breakthrough agreement to form a new balance.
Game theory: Also known as strategic theory and competition theory to analyze the situation of competition. In the competition and struggle of conflict of interests, the result of competition not only depends on the choice, decision and opportunities of a participant, but also depends on the choice of competitors or other participants. Because the results of competition depend on the choice of people in all bureaus, people in each bureau try to predict the possible choices of others to determine their best countermeasures. The game theory is a discipline with mathematical methods to analyze the form of struggle, also known as conflict analysis. [Example] "Since the Nobel Prize in Economics in 1994 awarded three gaming theory experts, game theory has been widely spread around the world.
You see that the buns in the entire street sell for 5 cents, and you can make 1 dime. So you also do buns and sell them. You plan to sell 4 cents and 8 cents, only 8 cents. This will definitely come to you to buy buns here, so you send it. However, you later discovered that the buns in other shops did not sell 4 cents and 8. This is called: Game theory n00:00 / 00: 5870% shortcut keys to describe space: Play / suspend ESC: exit full screen ↑: increase volume 10% ↓: reduced volume decrease by 10% →: single fast forward 5 seconds studio Here you can drag no longer appear in the player settings to reopen the small window shortcut key description
If you lose or lose, there may be games without win -win.
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