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

Introduction toIntroduction to Game TheoryGame Theory

Presented by Presented by 蘇柏穎蘇柏穎 2004/12/92004/12/9

OutlineOutline What is Game Theory?What is Game Theory? ApplicationsApplications The formulations of a gameThe formulations of a game The strategies of a gameThe strategies of a game One example: The Prisoner’s One example: The Prisoner’s

DilemmaDilemma

Explanation of Game TheoryExplanation of Game Theory Game Theory can be regarded as a Game Theory can be regarded as a

multi-agentmulti-agent decisiondecision problem problem There are many people contending There are many people contending

for limited for limited rewards/payoffsrewards/payoffs

大老二大老二

Rules that Players FollowRules that Players Follow

1. They have to make certain moves 1. They have to make certain moves on which their payoff dependson which their payoff depends

2. These people have to follow certain 2. These people have to follow certain strategiesstrategies while making these while making these movesmoves3. Each player is supposed to behave 3. Each player is supposed to behave rationallyrationally

SummarySummary Game Theory is the analysis of Game Theory is the analysis of

situations which involve conflict situations which involve conflict and/or cooperationand/or cooperation

OutlineOutline What is Game Theory?What is Game Theory? ApplicationsApplications The formulations of a gameThe formulations of a game The strategies of a gameThe strategies of a game One example: The Prisoner’s One example: The Prisoner’s

DilemmaDilemma

Business and EconomicsBusiness and Economics

Artificial IntelligenceArtificial Intelligence

Everything About CompetitionEverything About Competition

OutlineOutline What is Game Theory?What is Game Theory? ApplicationsApplications The formulations of a gameThe formulations of a game The strategies of a gameThe strategies of a game One example: The Prisoner’s One example: The Prisoner’s

DilemmaDilemma

Two Formulations of a GameTwo Formulations of a Game

Extensive formExtensive form game treegame tree

Strategic formStrategic form matrix gamematrix game

Extensive form: Extensive form: The Game Tree FormulationThe Game Tree Formulation

The Game Tree for The Game Tree for Paper-Scissors-StonePaper-Scissors-Stone

U1

U2

(0,0) (1,-1) (-1,1) (-1,1) (0,0) (1,-1) (0,0)(-1,1)(1,-1)

StoneScissorsPaper

Game in Extensive FormGame in Extensive Form The internal nodes of T are the The internal nodes of T are the decision decision nodesnodes of the players of the players The edges of T are the The edges of T are the actionsactions that the pl that the player can chooseayer can choose Each leaf of T is labeled by an n-tuple (hEach leaf of T is labeled by an n-tuple (h11,,hh22,…,h,…,hnn) of ) of payoffspayoffs

Strategic form: Strategic form: The Matrix FormulationThe Matrix Formulation

The Strategic Form for The Strategic Form for Paper-Scissors-StonePaper-Scissors-Stone

StoneStone ScissorScissorss

PaperPaper

StoneStone 0, 00, 0 1, -11, -1 -1, 1-1, 1

ScissorScissorss

-1, 1-1, 1 0, 00, 0 1, -11, -1

PaperPaper 1, -11, -1 -1, 1-1, 1 0, 00, 0

Row player

Column player

Comparison Between Two FormsComparison Between Two Forms

Tree formulationTree formulation can represent all can represent all possible states that a game can havepossible states that a game can have

Matrix formulationMatrix formulation can only can only represent one interaction represent one interaction

Matrix is easier to store and do Matrix is easier to store and do arithmetic computationsarithmetic computations

OutlineOutline What is Game Theory?What is Game Theory? ApplicationsApplications The formulations of a gameThe formulations of a game The strategies of a gameThe strategies of a game One example: The Prisoner’s One example: The Prisoner’s

DilemmaDilemma

Strategies of a GameStrategies of a Game Deterministic strategiesDeterministic strategies Randomized strategiesRandomized strategies

• Mixed strategiesMixed strategies• Behavioral strategiesBehavioral strategies• General strategiesGeneral strategies

Deterministic StrategiesDeterministic Strategies The The pure strategiespure strategies are the are the

deterministic strategiesdeterministic strategies

Random Strategies:Random Strategies:

Mixed StrategiesMixed Strategies

Mixed StrategiesMixed Strategies The player has a set of The player has a set of pure pure

strategies Sstrategies S He defines a He defines a probability distributionprobability distribution

over the pure strategiesover the pure strategies He can only throw a dice at the He can only throw a dice at the

beginningbeginning of the game of the game After that, he must stick to the After that, he must stick to the

strategy he choosesstrategy he chooses

Random Strategies:Random Strategies:

Behavioral StrategiesBehavioral Strategies

Behavioral StrategiesBehavioral Strategies The player defines a The player defines a probability probability

distributiondistribution among action edges of among action edges of each decision node.each decision node.

When he reaches a When he reaches a decision nodedecision node, he , he throws a dice to decide which throws a dice to decide which strategy should be usedstrategy should be used

Random Strategies:Random Strategies:

General StrategiesGeneral Strategies

General StrategiesGeneral Strategies A general strategy of a player is A general strategy of a player is

probability distributionprobability distribution of all of all behavioral strategiesbehavioral strategies of the player of the player

OutlineOutline What is Game Theory?What is Game Theory? ApplicationsApplications The formulations of a gameThe formulations of a game The strategies of a gameThe strategies of a game One example: The Prisoner’s One example: The Prisoner’s

DilemmaDilemma

The Prisoner’s DilemmaThe Prisoner’s Dilemma Two robbers are arrested by the Two robbers are arrested by the

policepolice They are interrogated respectivelyThey are interrogated respectively Each of them have two choices: Each of them have two choices:

confess or deceiveconfess or deceive

Payoff of the GamePayoff of the Game

ConfessConfess DeceiveDeceive

ConfessConfess 20, 2020, 20 0, 300, 30

DeceiveDeceive 30, 030, 0 10, 1010, 10

Competition over 5 Competition over 5 different strategiesdifferent strategies

http://http://www.gametheory.net/Web/PDilwww.gametheory.net/Web/PDilemma/Pdilemma.htmlemma/Pdilemma.html

If people do not believe that Game If people do not believe that Game Theory is simple, it is only because Theory is simple, it is only because they do not realize how complicated they do not realize how complicated life is.life is.

~Modified ~Modified from from von Neumann, von Neumann, John John

Thanks EveryoneThanks Everyone