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Page 1: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

A Introduction to Game Theory

Xiuting Tao

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Page 2: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Outline

v 1st a brief introduction of Game theory

v 2nd Strategic games

v 3rd Extensive games

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Page 3: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Game theory and CS

vThere has been a remarkable increase in work at the interface of computer science and game theory in the past decade.

----Joseph Y. Halpern《Computer Science and Game Theory: A Brief Survey》

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Page 4: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

v1st a brief introduction of Game theory

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Page 5: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

In the beginningvSome game-theoretic ideas traced back to

the 18-th century.vEmile Borel (1871~1956) and John von

Neumann (1903~1957) began the major development of game theory.

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Page 6: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

The bookvGame theory became a

field since the book of John von Neumann and Oskar Morgenstern published in 1944.

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Page 7: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

John Nash(1928- )

vReceived his Ph.D. from Princeton University with a 28-page thesis on his 22-nd birthday.Invented the notion of Nash

equilibrium.vWrote a seminal paper on

bargain theory.

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Page 8: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

What is Game Theory?

Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome

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Page 9: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Rationality

Assumptions: vhumans are rational beings

vhumans always seek the best alternative in a set of possible choices

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Page 10: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Why assume rationality?

vnarrow down the range of possibilities

vpredictability

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Page 11: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Types of Games

vPerfect vs. Imperfect informationvSequential vs. Simultaneous movesvZero vs. non-zero sum vCooperative vs. conflict

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Page 12: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Types of game and Equilibriumorder

Information Strategic game Extensive game

perfectNash

Equilibrium(John Nash)

subgame perfect equilibrium((Reinhard SeltenReinhard Selten ))

imperfect

Bayesian equilibrium

(John (John HarsanyiHarsanyi ))

perfect Bayesian equilibrium

((Reinhard SeltenReinhard Selten ))

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Page 13: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Two examples

v 1) Prisoner’s Dilemma

v 2) Boxed Pigs

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Page 14: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Prisoner’s Dilemma

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Prisoner’s Dilemma

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Page 16: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Cost to press button = 2 units

When button is pressed, food given = 10 units

Boxed Pigs

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Page 17: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Decisions, decisions...

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Page 18: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

2nd Strategic game

v < N, (Ai), (ui) >

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Page 19: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Nash Equilibrium

“If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium. "

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Page 20: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Nash Equilibrium

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Page 21: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Imperfect Information

vPartial or no information concerning the opponent is given in advance to the player’s decision.

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Page 22: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Bayesian game

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Page 23: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

BoSTwo people wish to go out together to a concert of music by either Bach or Stravinsky. Their main concern is to go out together, but one person prefers Bach and the other person prefers Stravinsky. Representing the individuals' preferences by payoff functions

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Page 24: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Variant of BoS with imperfect information

Consider another variant of the situation modeled by BoS, in which neither player knows whether the other wants to go out with her.

Specifically, suppose that player 1 thinks that with probability ½ player 2 wants to go out with her, and with probability ½ player 2 wants to avoid her, and player 2 thinks that with probability 2/3 player 1 wants to go out with her and with probability 1/3 player 1 wants to avoid her.

Assume that each player knows her own preferences.

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Page 25: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Variant of BoS with imperfect information

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Page 26: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Variant of BoS with imperfect information

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Page 27: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Payoffs

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Page 28: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Bayesian Equilibrium

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Page 29: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Bayesian Equilibrium

v (B, y) (B, y)v (B, y) (S, n)v (B, n) (B, y)v (B, n) (S, n)

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Page 30: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

v3rd Extensive games

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Page 31: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Extensive game with perfect information

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Page 32: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

An example

Two people use the following procedure to share two desirable identical indivisible objects. One of them proposes an allocation, which the other then either accepts or rejects. In the event of rejection, neither person receives either of the objects. Each person cares only about the number of objects he obtains.

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Page 33: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

allocating two identical indivisible objects

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Page 34: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Payoff tree

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Page 35: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Strategy

In the just example:

The strategies of player 1: {(2,0),(1,1),(0,2)}The strategies of player 2:{yyy,yyn,yny,ynn,nyy,nyn,nny,nnn}

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Page 36: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Nash Equilibrium

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Page 37: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

a problem

vThe Nash equilibria of the example game are ((2, 0), yyy), ((2, 0), yyn), ((2, 0), yny), ((2, 0), ynn), ((1, 1),nyy), ((1, 1), nyn), ((0, 2), nny), ((2, 0), nny), and ((2, 0), nnn).

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Page 38: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Subgame extensive game

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Page 39: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Subgame Perfect Equilibrium

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Page 40: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Extensive game with imperfect information

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Page 41: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Extensive game with imperfect information

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Page 42: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

An example

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Page 43: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Nash equilibrium

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Page 44: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

Four parts of the textbook

v Strategic games (Part I)

v Extensive games with perfect information (Part II)

v Extensive games without perfect information (Part III)

v Coalitional games (Part IV)

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Page 45: A Introduction to Game Theory - SJTU · 2014-12-23 · Game theory and CS vThere has been a remarkable increase in work at the interface of computer science and game theory in the

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