a introduction to game theory - sjtu · 2014-12-23 · game theory and cs vthere has been a...
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A Introduction to Game Theory
Xiuting Tao
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Outline
v 1st a brief introduction of Game theory
v 2nd Strategic games
v 3rd Extensive games
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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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v1st a brief introduction of Game theory
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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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The bookvGame theory became a
field since the book of John von Neumann and Oskar Morgenstern published in 1944.
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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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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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Rationality
Assumptions: vhumans are rational beings
vhumans always seek the best alternative in a set of possible choices
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Why assume rationality?
vnarrow down the range of possibilities
vpredictability
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Types of Games
vPerfect vs. Imperfect informationvSequential vs. Simultaneous movesvZero vs. non-zero sum vCooperative vs. conflict
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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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Two examples
v 1) Prisoner’s Dilemma
v 2) Boxed Pigs
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Prisoner’s Dilemma
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Prisoner’s Dilemma
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Cost to press button = 2 units
When button is pressed, food given = 10 units
Boxed Pigs
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Decisions, decisions...
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2nd Strategic game
v < N, (Ai), (ui) >
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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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Nash Equilibrium
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Imperfect Information
vPartial or no information concerning the opponent is given in advance to the player’s decision.
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Bayesian game
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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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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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Variant of BoS with imperfect information
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Variant of BoS with imperfect information
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Bayesian Equilibrium
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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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v3rd Extensive games
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Extensive game with perfect information
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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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allocating two identical indivisible objects
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Payoff tree
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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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Nash Equilibrium
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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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Subgame extensive game
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Subgame Perfect Equilibrium
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Extensive game with imperfect information
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Extensive game with imperfect information
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An example
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Nash equilibrium
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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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