Introduction to Game Theory Presented by 蘇柏穎 2004/12/9 2004/12/9

Outline What is Game Theory? What is Game Theory? Applications Applications The formulations of a game The formulations of a game The strategies of a game The strategies of a game One example: The Prisoner ’ s Dilemma One example: The Prisoner ’ s Dilemma

Explanation of Game Theory Game Theory can be regarded as a multi-agent decision problem Game Theory can be regarded as a multi-agent decision problem There are many people contending for limited rewards/payoffs There are many people contending for limited rewards/payoffs

大老二

Rules that Players Follow 1. They have to make certain moves on which their payoff depends on which their payoff depends 2. These people have to follow certain strategies while making these strategies while making these moves moves 3. Each player is supposed to behave rationally rationally

Summary Game Theory is the analysis of situations which involve conflict and/or cooperation Game Theory is the analysis of situations which involve conflict and/or cooperation

Outline What is Game Theory? What is Game Theory? Applications Applications The formulations of a game The formulations of a game The strategies of a game The strategies of a game One example: The Prisoner ’ s Dilemma One example: The Prisoner ’ s Dilemma

Business and Economics

Artificial Intelligence

Everything About Competition

Outline What is Game Theory? What is Game Theory? Applications Applications The formulations of a game The formulations of a game The strategies of a game The strategies of a game One example: The Prisoner ’ s Dilemma One example: The Prisoner ’ s Dilemma

Two Formulations of a Game Extensive form Extensive form game tree game tree Strategic form Strategic form matrix game matrix game

Extensive form: The Game Tree Formulation

The Game Tree for Paper-Scissors-Stone U 1 U 2 (0,0)(1,-1)(-1,1) (0,0)(1,-1)(0,0)(-1,1)(1,-1) Stone Scissors Paper

Game in Extensive Form The internal nodes of T are the decision nodes of the players The internal nodes of T are the decision nodes of the players The edges of T are the actions that the player can choose The edges of T are the actions that the player can choose Each leaf of T is labeled by an n- tuple (h 1,h 2, …,h n ) of payoffs Each leaf of T is labeled by an n- tuple (h 1,h 2, …,h n ) of payoffs

Strategic form: The Matrix Formulation

The Strategic Form for Paper-Scissors-Stone Stone Scissor s Paper Stone 0, 0 1, -1 -1, 1 Scissor s -1, 1 0, 0 1, -1 Paper -1, 1 0, 0 R owp l ayer C o l umnp l ayer

Comparison Between Two Forms Tree formulation can represent all possible states that a game can have Tree formulation can represent all possible states that a game can have Matrix formulation can only represent one interaction Matrix formulation can only represent one interaction Matrix is easier to store and do arithmetic computations Matrix is easier to store and do arithmetic computations

Outline What is Game Theory? What is Game Theory? Applications Applications The formulations of a game The formulations of a game The strategies of a game The strategies of a game One example: The Prisoner ’ s Dilemma One example: The Prisoner ’ s Dilemma

Strategies of a Game Deterministic strategies Deterministic strategies Randomized strategies Randomized strategies Mixed strategiesMixed strategies Behavioral strategiesBehavioral strategies General strategiesGeneral strategies

Deterministic Strategies The pure strategies are the deterministic strategies The pure strategies are the deterministic strategies

Random Strategies: Mixed Strategies

The player has a set of pure strategies S The player has a set of pure strategies S He defines a probability distribution over the pure strategies He defines a probability distribution over the pure strategies He can only throw a dice at the beginning of the game He can only throw a dice at the beginning of the game After that, he must stick to the strategy he chooses After that, he must stick to the strategy he chooses

Random Strategies: Behavioral Strategies

The player defines a probability distribution among action edges of each decision node. The player defines a probability distribution among action edges of each decision node. When he reaches a decision node, he throws a dice to decide which strategy should be used When he reaches a decision node, he throws a dice to decide which strategy should be used

Random Strategies: General Strategies

A general strategy of a player is probability distribution of all behavioral strategies of the player A general strategy of a player is probability distribution of all behavioral strategies of the player

Outline What is Game Theory? What is Game Theory? Applications Applications The formulations of a game The formulations of a game The strategies of a game The strategies of a game One example: The Prisoner’s Dilemma One example: The Prisoner’s Dilemma

The Prisoner’s Dilemma Two robbers are arrested by the police Two robbers are arrested by the police They are interrogated respectively They are interrogated respectively Each of them have two choices: confess or deceive Each of them have two choices: confess or deceive

Payoff of the Game ConfessDeceive Confess 20, 20 0, 30 Deceive 30, 0 10, 10

Competition over 5 different strategies Web/PDilemma/Pdilemma.ht ml Web/PDilemma/Pdilemma.ht ml

If people do not believe that Game Theory is simple, it is only because they do not realize how complicated life is. ~Modified from von Neumann, John If people do not believe that Game Theory is simple, it is only because they do not realize how complicated life is. ~Modified from von Neumann, John

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