Overview of Game Theory and its Applications

1941 words (8 pages) Essay

8th Feb 2020 Mathematics Reference this

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Introduction

Game theory consists of seven aspects: players, action, strategies, information, utility, outcome and equilibrium. Participants in all game models of game theory are defined as rational participants who do not trust others. For each participant, he or she cannot improve his or her situation as long as others do not change the strategy. Nash Equilibrium is a combination of the best strategies for all participants. Because the purpose of the game is actually to maximise the value of itself and hence there will be factors that tend to Nash Equilibrium. What is more, reaching Nash Equilibrium will produce positive results for all players to some extent (PETROSJAN and GRAUER, 2002). Game theory has a wide range of applications and has become a significant research and analysis tool in the fields of economics, political science, military strategy, and contemporary computer science.

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Game theory has four categories, first, cooperative game and non-cooperative game. The difference between these two is whether there is a binding agreement between the parties that interact with each other. If there is, it is a cooperative game. If not, it is a non-cooperative game. Second, simultaneous game and sequential game. According to Heap and Varoufakis (2004), simultaneous game means that in the game, the participants choose at the same time or not at the same time, but the latter actor does not know what specific actions were taken by the previous actor. It could be related to Prisoner’s Dilemma, which is simultaneous decision-making and belongs to the simultaneous game. Sequential game means that in the game, the actions of the participants are in order, and the latter actors can observe the actions selected by the first actors, in addition, playing chess is a dynamic game. Thirdly, perfect information game and imperfect information game. Perfect information game means that in the process of the game, each participant has accurate information about the characteristics, strategy and payoff of other participants. Imperfect information game refers to the inaccuracy of the participants’ understanding of the characteristics, strategy and payoff of other participants. Fourthly, zero-sum game and non-zero-sum game. Zero-sum game belongs to the non-cooperative game. It refers to the parties involved in the game under strict competition, one party’s income must mean the other party’s loss, the sum of the gains and losses of the parties to the game is zero, there is no possibility of cooperation between the two sides, it could be mentioned that gambling is a zero-sum game. Non-zero-sum game is a game under cooperation, the sum of the gains or losses of the parties in the game is not zero, and the income is not equal to the loss of others. The two sides of the game have the possibility of win-win and then achieve cooperation value.

The common features of the game are: first, the game has certain rules; secondly, the game will have a result that can be converted into numbers; thirdly, the game’s strategy is interdependent; fourthly, the game’s strategy is crucial. Most game in daily life are conducted in an interactive environment that requires two or more participants. Participants are interdependent and cannot continue the game normally if in the absence of one participant. Participants create intricate interactions in the game, and the outcome of the game is determined by the behaviour of the game participants. In a strategic interactive environment, the players are not facing a passive world, but in an interdependent network. The achievement of players’ goals is not entirely controlled by themselves, but also depends on the strategy choices of other players, determined by the results of interaction rather than one-way choice. Strategic interaction logically requires each player not only to consider their own interests and goals, but also to consider the various possible strategies of other players, including other players’ anticipation and response to their own behaviour, the impact of other players’ behaviour on themselves, and so on.

Game Model

Take the stone game: The basis of the game is two players. The form of the game is that the player takes turns to catch the stone. The standard of the player’s victory is the player’s victory that captures the last stone.

Player settings: Player A is the first to take the stone, and player B is the stone.

Specific situation: There are 1 pile containing n stones, and the two players take the items from the pile in turn. It is required to take at least one at a time and take at most m. The person who took the last stone won.

The concept of equilibrium: Introducing a concept, a state of balance, also known as a strange situation. When faced with this situation, it will fail. Any non-equilibrium state can become equilibrium after one operation. Each player will try to balance the situation after he has finished the stone, leaving this balance to the other side. Therefore, Player A can win in an initially unbalanced game, and Player B can win in an initially balanced game.

The last singular situation is n=(0). A strange situation is that n = (m + 1), then no matter how many players take the first player, the other party can take away all the remaining items at a time to win.

The determination of the strange situation: The general singular situation is n=(m+1)*i, where i is a natural number, ie n%(m+1)=0. In the face of this situation, no matter how I take it, the other party can always restore it to n%. (m+1) = 0, up to n = (m + 1) situation.

Player’s strategy: It is to leave the current non-singular situation into a strange situation to the other side. If the current number of stones is (m+1)*i+s, then s stones will be taken away to achieve a strange situation.

Second situation:

There are k piles of n stones, and two people take turns taking as many items as possible from a certain pile. It is stipulated that at least one is taken at a time, and many are not limited. The person who took the last stone won.

 The last singular situation is (0,0…,0). Another singular situation is (n, n, 0…0), as long as the opponent always takes as many items as I take, and finally faces (0,0…,0).

The determination of the strange situation: For an ordinary situation, how to judge whether it is a strange situation? For a situation (s1, s2, … sk), the XOR operation of all stone numbers, s1^s2^s3^…^sk, if the result is 0, then the situation (s1, s2, …sk) is a strange situation (balance), otherwise it is not (unbalanced).From the perspective of binary bits, the number of 1s in each bit is even when the situation is singular.

Player’s strategy: It is to turn the non-singular situation facing the situation into a strange situation and leave it to the other side. That is to say, after taking a few stones from a certain pile, the number of 1s in each bit is changed to an even number, and there is generally only one such method. You can change the number of stones in one pile to the value of the XOR operation of other piles (if this value is smaller than the original number of stones).

Third situation:

There are 2 piles of n stones, and two people take the same number of items from one pile or two piles at the same time. It is stipulated to take at least one at a time, and many are not limited. The person who took the last stone won.

  The last singular situation is (0,0). The strange situation that followed was (1, 2), (3, 5), (4, 7), (6, 10)…

Now think of them as a sequence of singular situations (0,0), (1,2), (3,5), (4,7), (6,10)…

The determination of the strange situation: We will find the law of this sequence, let the kth singular situation element of the sequence be (Ak, Bk), and k be a natural number. Then, when the initial condition k=0, A0=B0=0, the recursive relationship is that the Ak of the next singular situation is the smallest natural number that has not appeared before, and Ak = Bk + k.

The general formula for the sequence of this singular situation can be expressed as:

Ak = [k*(1+sqrt(5.0)/2]

Bk = Ak + k

Where k=0,1,2,…,n, the square brackets represent the int rounding function.

With this general formula, reverse, for a situation, only need to determine whether A is a multiple of a certain k of the golden section, and then confirm whether B is equal to A + k.

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The result of a game ending is called gains and losses. The gains and losses of each player at the end of a game are not only related to the strategy chosen by the players themselves, but also related to a set of strategies determined by the players in the overall situation which usually called the payoff matrix.

Real game with Game Theory

The Evolution of Trust is a game based on game theory. Each time through third person’s vision to control the proportion of each character and the number of games to achieve the most beneficial role in different situations. By letting players choose between different scenarios and changing over time, showing how The Evolution of Trust evolves and letting players know how to choose the most powerful game strategy. The rule of this game is that the player needs to spend one gold coin to participate in the game in each round and can choose to cooperate or deceive the opponent. If both parties choose to cooperate, then each side can get two gold coins. If one party chooses to cheat and the other chooses to cooperate, the deceiver cannot only spend even one gold coin, but also get three gold coins, but the partner spends a gold coin but cannot get anything. Nevertheless, if both parties choose to deceive, neither party will receive gold coins. It turns out that if the game is less than five rounds, cheating will maximize the player’s interests. But if it is more than five rounds, then the better strategy is for the player to collaborate from the start and copy the opponent’s last action.

Conclusion

In a sense, the concept of the game is defined by its rules, and the interactions in the game depend on the shared rules. Every game has its own rules, once the rules are destroyed, the entire game world will collapse (Huizinga, n.d.). In game theory, the game rules include the setting of basic game conditions such as action mode, utility structure and information mastery, which constitute the basis of the game, determine the nature of the game, and affect the outcome of the game.

Bibliography

Introduction

Game theory consists of seven aspects: players, action, strategies, information, utility, outcome and equilibrium. Participants in all game models of game theory are defined as rational participants who do not trust others. For each participant, he or she cannot improve his or her situation as long as others do not change the strategy. Nash Equilibrium is a combination of the best strategies for all participants. Because the purpose of the game is actually to maximise the value of itself and hence there will be factors that tend to Nash Equilibrium. What is more, reaching Nash Equilibrium will produce positive results for all players to some extent (PETROSJAN and GRAUER, 2002). Game theory has a wide range of applications and has become a significant research and analysis tool in the fields of economics, political science, military strategy, and contemporary computer science.

Game theory has four categories, first, cooperative game and non-cooperative game. The difference between these two is whether there is a binding agreement between the parties that interact with each other. If there is, it is a cooperative game. If not, it is a non-cooperative game. Second, simultaneous game and sequential game. According to Heap and Varoufakis (2004), simultaneous game means that in the game, the participants choose at the same time or not at the same time, but the latter actor does not know what specific actions were taken by the previous actor. It could be related to Prisoner’s Dilemma, which is simultaneous decision-making and belongs to the simultaneous game. Sequential game means that in the game, the actions of the participants are in order, and the latter actors can observe the actions selected by the first actors, in addition, playing chess is a dynamic game. Thirdly, perfect information game and imperfect information game. Perfect information game means that in the process of the game, each participant has accurate information about the characteristics, strategy and payoff of other participants. Imperfect information game refers to the inaccuracy of the participants’ understanding of the characteristics, strategy and payoff of other participants. Fourthly, zero-sum game and non-zero-sum game. Zero-sum game belongs to the non-cooperative game. It refers to the parties involved in the game under strict competition, one party’s income must mean the other party’s loss, the sum of the gains and losses of the parties to the game is zero, there is no possibility of cooperation between the two sides, it could be mentioned that gambling is a zero-sum game. Non-zero-sum game is a game under cooperation, the sum of the gains or losses of the parties in the game is not zero, and the income is not equal to the loss of others. The two sides of the game have the possibility of win-win and then achieve cooperation value.

The common features of the game are: first, the game has certain rules; secondly, the game will have a result that can be converted into numbers; thirdly, the game’s strategy is interdependent; fourthly, the game’s strategy is crucial. Most game in daily life are conducted in an interactive environment that requires two or more participants. Participants are interdependent and cannot continue the game normally if in the absence of one participant. Participants create intricate interactions in the game, and the outcome of the game is determined by the behaviour of the game participants. In a strategic interactive environment, the players are not facing a passive world, but in an interdependent network. The achievement of players’ goals is not entirely controlled by themselves, but also depends on the strategy choices of other players, determined by the results of interaction rather than one-way choice. Strategic interaction logically requires each player not only to consider their own interests and goals, but also to consider the various possible strategies of other players, including other players’ anticipation and response to their own behaviour, the impact of other players’ behaviour on themselves, and so on.

Game Model

Take the stone game: The basis of the game is two players. The form of the game is that the player takes turns to catch the stone. The standard of the player’s victory is the player’s victory that captures the last stone.

Player settings: Player A is the first to take the stone, and player B is the stone.

Specific situation: There are 1 pile containing n stones, and the two players take the items from the pile in turn. It is required to take at least one at a time and take at most m. The person who took the last stone won.

The concept of equilibrium: Introducing a concept, a state of balance, also known as a strange situation. When faced with this situation, it will fail. Any non-equilibrium state can become equilibrium after one operation. Each player will try to balance the situation after he has finished the stone, leaving this balance to the other side. Therefore, Player A can win in an initially unbalanced game, and Player B can win in an initially balanced game.

The last singular situation is n=(0). A strange situation is that n = (m + 1), then no matter how many players take the first player, the other party can take away all the remaining items at a time to win.

The determination of the strange situation: The general singular situation is n=(m+1)*i, where i is a natural number, ie n%(m+1)=0. In the face of this situation, no matter how I take it, the other party can always restore it to n%. (m+1) = 0, up to n = (m + 1) situation.

Player’s strategy: It is to leave the current non-singular situation into a strange situation to the other side. If the current number of stones is (m+1)*i+s, then s stones will be taken away to achieve a strange situation.

Second situation:

There are k piles of n stones, and two people take turns taking as many items as possible from a certain pile. It is stipulated that at least one is taken at a time, and many are not limited. The person who took the last stone won.

 The last singular situation is (0,0…,0). Another singular situation is (n, n, 0…0), as long as the opponent always takes as many items as I take, and finally faces (0,0…,0).

The determination of the strange situation: For an ordinary situation, how to judge whether it is a strange situation? For a situation (s1, s2, … sk), the XOR operation of all stone numbers, s1^s2^s3^…^sk, if the result is 0, then the situation (s1, s2, …sk) is a strange situation (balance), otherwise it is not (unbalanced).From the perspective of binary bits, the number of 1s in each bit is even when the situation is singular.

Player’s strategy: It is to turn the non-singular situation facing the situation into a strange situation and leave it to the other side. That is to say, after taking a few stones from a certain pile, the number of 1s in each bit is changed to an even number, and there is generally only one such method. You can change the number of stones in one pile to the value of the XOR operation of other piles (if this value is smaller than the original number of stones).

Third situation:

There are 2 piles of n stones, and two people take the same number of items from one pile or two piles at the same time. It is stipulated to take at least one at a time, and many are not limited. The person who took the last stone won.

  The last singular situation is (0,0). The strange situation that followed was (1, 2), (3, 5), (4, 7), (6, 10)…

Now think of them as a sequence of singular situations (0,0), (1,2), (3,5), (4,7), (6,10)…

The determination of the strange situation: We will find the law of this sequence, let the kth singular situation element of the sequence be (Ak, Bk), and k be a natural number. Then, when the initial condition k=0, A0=B0=0, the recursive relationship is that the Ak of the next singular situation is the smallest natural number that has not appeared before, and Ak = Bk + k.

The general formula for the sequence of this singular situation can be expressed as:

Ak = [k*(1+sqrt(5.0)/2]

Bk = Ak + k

Where k=0,1,2,…,n, the square brackets represent the int rounding function.

With this general formula, reverse, for a situation, only need to determine whether A is a multiple of a certain k of the golden section, and then confirm whether B is equal to A + k.

The result of a game ending is called gains and losses. The gains and losses of each player at the end of a game are not only related to the strategy chosen by the players themselves, but also related to a set of strategies determined by the players in the overall situation which usually called the payoff matrix.

Real game with Game Theory

The Evolution of Trust is a game based on game theory. Each time through third person’s vision to control the proportion of each character and the number of games to achieve the most beneficial role in different situations. By letting players choose between different scenarios and changing over time, showing how The Evolution of Trust evolves and letting players know how to choose the most powerful game strategy. The rule of this game is that the player needs to spend one gold coin to participate in the game in each round and can choose to cooperate or deceive the opponent. If both parties choose to cooperate, then each side can get two gold coins. If one party chooses to cheat and the other chooses to cooperate, the deceiver cannot only spend even one gold coin, but also get three gold coins, but the partner spends a gold coin but cannot get anything. Nevertheless, if both parties choose to deceive, neither party will receive gold coins. It turns out that if the game is less than five rounds, cheating will maximize the player’s interests. But if it is more than five rounds, then the better strategy is for the player to collaborate from the start and copy the opponent’s last action.

Conclusion

In a sense, the concept of the game is defined by its rules, and the interactions in the game depend on the shared rules. Every game has its own rules, once the rules are destroyed, the entire game world will collapse (Huizinga, n.d.). In game theory, the game rules include the setting of basic game conditions such as action mode, utility structure and information mastery, which constitute the basis of the game, determine the nature of the game, and affect the outcome of the game.

Bibliography

  • Heap, S. and Varoufakis, Y. (2004). Game theory. New York: Routledge.
  • Huizinga, J. (n.d.). Homo Iudens.
  • PETROSJAN, L. and GRAUER, L. (2002). STRONG NASH EQUILIBRIUM IN MULTISTAGE GAMES. International Game Theory Review, 04(03), pp.255-264.

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