Game Theory—Introduction

曽 道 智Zeng, Dao-Zhi

Tohoku University

April 21, 2019

曽 道 智 Zeng, Dao-Zhi (Tohoku University) Game Theory—Introduction April 21, 2019 1 / 24

Welcome to Game Theory

Time: 1st class on TuesdayGoogle Classroom (Codes anuzdcp (main), x2xiooo)Lecturer

Name: 曽道智 (ゼン・ダオズ) Zeng, Dao-ZhiMail: zeng@se.is.tohoku.ac.jp (Anytime is OK for questions!)Url: http://www.se.is.tohoku.ac.jp/̃ zeng/index.htmlMajor: Regional Science, Urban Economics, SpatialEconomics, Conflict ResolutionAspiration: Teaching knowledge worldwide applicable andfostering regional specialists with international viewpoints

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Lecture plan

Date Content1st 4/21 Introduction2nd 4/28 Strategic form game3rd 5/12 Extensive form game with perfect information4th 5/19 Application examples5th 5/26 Application in imperfect competition markets6th 6/2 Mixed strategy7th 6/9 General extensive-form games　8th 6/19 Repeated games　9th 6/23 Application examples10th 6/30 Strategic form games with imperfect information11th 7/7 Bargaining game12th 7/14 Application13th 7/21 Application14th 7/28 Cooperative game15th 8/4 Test

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References

Textbook渡辺隆裕, 2008: ゼミナールゲーム理論入門,日本経済新聞出版社。

Lecture 1: Chapters 1&13

References岡田章, 2011: ゲーム理論(新版),有斐閣.武藤滋夫, 2011: ゲーム理論,オーム社.

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Game Theory

Game?Gambling machinehow to win?how to avoid a loss?

Game TheoryA science to explore howmultiple agents makedecisions in situations withconflicts and cooperationdescribed by games.

Nobel Laureates1994: John Nash, ReinhardSelten, and John Hasanyi2005: Thomas Schelling andRobert Aumann2007: Leonid Hurwicz, EricMaskin, and Roger Myerson2012: Alvin E. Roth andLloyd S. Shapley2014: Jean Tirole.

John Maynard Smith: theCrafoord Prize for his applicationof game theory to biology

曽 道 智 Zeng, Dao-Zhi (Tohoku University) Game Theory—Introduction April 21, 2019 5 / 24

Three components

PlayerDM（e.g.individuals, firms,governments）agentsrepresentinguncertain factors(e.g., weather) arenot DMs.rationalintelligent

StrategyAction plans made by playersIt may depend on the states．Astrategy describes how a playerchooses a specific action in everypossible state．

Payoffnumerical expression of a player’spreferencedefined for every possible outcomeof players’ strategiesanother name—utility

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Game forms

matrix (prisoner’s dilemma)

Prisoner 1Prisoner 2

Silent Betray

Silent −1,−1 −10,−0.5Betray −0.5,−10 −8,−8

Graph

Teacher

Student

Student’s payoff −2 1

Teacher’s payoff −2 1

tough easy

stop continue stop Continue

−2 0−1 0

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Equilibrium concept

A theoretical concept to predict plausible results of gamesdominant strategies, Nash equilibrium· · · perfect equilibrium

曽 道 智 Zeng, Dao-Zhi (Tohoku University) Game Theory—Introduction April 21, 2019 8 / 24

Rules of Games

Game rule＝ {Player，Strategy，Payoff}complete information

All players know the rule of the game completelycommon knowledgecomplete , perfect

Games with incomplete informationSome different Japanese terms for players,プレーヤー,プレイヤー

limited to a small number of playersrationallity: players maximize their own payoffsmechanism design

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Classification of Games I

2-player game, 3-player game, n-player gamenoncooperative and cooperative games

Noncooperative games Players do not cooperate. Toinvestigate economic behavior when individual players maketheir decisions based on their own interests

zero-sum and nonzero-sum

Cooperative games Players cooperate with each other. Toanalyze how players divide the total value of a result fromcooperation in a fair wayThe forming of a coalition depends on the related players’decisionsNG theory is more basic than CG theory. Some studies focuson how s coalition is formed based on NG theory.

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Classification of Games II

Games in strategic form, games in extensive form, games incoalitional form

Representative models to study games by mathematics

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The History of Game Theory I

John von Neumann（1928）On the theory of games of strategyto prove the minimax theory (Brouwer’s fixed-point theorem)to formulate n-persion games in strategic form，to propose theconcept of coalitions, characteristic function based on themaximin value，to analyze zero-sum 3-person cooperativegames.

von Neumann and Morgenstern（1944）Theory of Games and Economic Behavior

John Nash（1950,1953）Nash equilibriumNash bargaining solution，Nash program

Harold W. Khun（1953）develop extensive games

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The History of Game Theory II

complete information, perfect recall, behavioral strategy, andetc.the existence of a pure-strategy equilibrium in any game withperfect recallthe equivalence between behavioral strategy and mixedstrategy

Other research in 1950scooperative gameShapley value (Shapley, 1953)Core (Shapley, 1953; Gillies, 1959)The existence of a general equilibrium by Arrow and Debreu

Research in 1960sCooperative game

cooperative games without side payments（Auman and Peleg,1960; Auman, 1961）

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The History of Game Theory III

Bargaining set（Auman and Maschler, 1964）Kernel（Davis and Maschler, 1965）Nucleolus（Schmeidler, 1969）The existence of a core（Bondareva, 1963; Shapley, 1967）

Noncooperative gamethe core of market games（Schubik, 1959; Debreu and Scarf,1963）subgame perfect equilibrium（Selten, 1965）Bayesian equilibrium and games with incomplete information（Harsanyi, 1967-68）

Research in 1970s and early 1980sNoncooperative game (refinement of equilibria)perfect equilibrium（Selten, 1975）sequential equilibrium（Kreps and Wilson, 1982）Repeated gamesFolk theorem（Rubinstein, 1979）

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The History of Game Theory IV

Bargaining gamesnoncooperative bargaining games（Rubinstein, 1982）noncooperative bargaining model for n-player coalition games（Selten, 1981）

Research in the middle and late 1980sbounded rationalityevolutionary game theory（Maynard Smiths and Price, 1973）learning theoryadaptation, rational learning, adaptive learning

Research after 1990sBehavioral game theory (Camerer, 2003)

Social preferences, guessing game

game experimentsto explore rational behavior

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von Neumann

https://commons.

wikimedia.org/wiki/File:

JohnvonNeumann-LosAlamos.gif

1903–1957. Born inHungary. Move to U.S. in1930. Died in 1957.started learning calculus atthe age of 8awarded Ph.D. inmathematics, physics,chemistry when he was 23great contribution inmathematics, physics,meteorology, economics,computer science, nucleardevelopment · · ·

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Oskar Morgenstern

https://mises.org/profile/

oskar-morgenstern-0

1902–1977. Born inGermany. Passed away atPrinceton.his mother: an illegitimatedaughter of AustrianEmperor Emperor FrederickIIIAdolf Hitler took over Viennain 1938. Fired by theUniversity of Vienna andstayed in Princeton as alecturer.

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John Nash

https://commons.

wikimedia.org/wiki/File:

John_Forbes_Nash,_Jr._by_

Peter_Badge.jpg

1928–2015. Born in Virginia.Taxi accident.Entered Carnegie MellonUniversity to learn Chemistrywhen he was 17. Turn toInternational EconomicsMaster degree when he was 20.Moved to Princeton for a Ph.D.Ph.D. when he was 22. Advisor:Prof. Albert William TuckerAwarded Nobel prize by 3papersA beautiful mind

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A mathematical genius

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Reinhard Selten

https://commons.

wikimedia.org/wiki/File:

Reinhard_Selten2.jpg

1930-2016. Born inGermany.Bachlor: majored inmathematics.Ph.D: experimentaleconomicsStarted game theoryresearch in 1965Contribution: extensivegames and refinement ofNash equilibriaGood at Esperanto

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A photo taken at Kagawa University

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John Charles Harsanyi

https://upload.wikimedia.org/

wikipedia/en/6/6b/John_

Harsanyi.jpg

1920–2000. Born inBudapest and died atBerkeley.entered chemical eng. at theU. of Lyon when he was 19.Due to the 2nd WW,returned to Hungary to studypharmacology at the U. ofBudapestBachelor when he was 24Awarded a Ph.D. when hewas 27Moved to Australia in 1950and moved to the U.S. in1960s

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New Trend: Bounded rationality

Evolutionary gametheory

re-import from biologyshort-sighted action,imitationmutation, selectionreplicator dynamics,migration of skilledworkers—

Behavioral Game TheoryExperimental economicsSubjects： college studentsObserve human behaviorExamination of theory.Revise theory if necessaryFairness and altruisticbehavior

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Homework

Give example(s) in which more than one people makedecisions in the real world.What is your motivation to take this course? What do youexpect from it?

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