Game Theory

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Game Theory
Quick Intro to Game Theory
Analysis of Games
Design of Games (Mechanism Design)
Some References
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John von Neumann The Genius who created two
intellectual currents in the 1930s, 1940s

• Founded Game Theory with Oskar Morgenstern
  (1928-44)
• Pioneered the Concept of a Digital Computer and
  Algorithms (1930s)

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Robert Aumann Nobel 2005
Leonid Hurwicz Nobel 2007

• Recent Excitement Nobel Prizes for Game Theory
  and Mechanism Design
• The Nobel Prize was awarded to two Game Theorists
  in 2005
• The prize was awarded to three mechanism
  designers in 2007

Thomas Schelling Nobel 2005
Eric Maskin Nobel 2007
Roger Myerson Nobel 2007
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Game Theory

• Mathematical framework for rigorous study of
  conflict and cooperation among rational,
  intelligent agents

Market
Buying Agents (rational and intelligent)
Selling Agents (rational and intelligent)
Social Planner

• In the Internet Era, Game Theory has become a
  valuable tool for analysis and design

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Applications of Game Theory
Microeconomics, Sociology, Evolutionary Biology
Auctions and Market Design Spectrum Auctions,
Procurement Markets, Double Auctions
Industrial Engineering, Supply Chain Management,
E-Commerce, Procurement, Logistics
Computer Science Algorithmic Game Theory,
Internet and Network Economics, Protocol
Design, Resource Allocation, etc.
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A Familiar Game
Mumbai Indians
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Kolkata Knight Riders
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Bangalore RoyalChallengers
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Punjab Lions
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Sachin Tendulkar
IPL Franchisees
IPL CRICKET AUCTION
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Sponsored Search Auction
Advertisers
CPC
Major money spinner for all search engines and
web portals
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DARPA Red Balloon Contest
Mechanism Design Meets Computer Science,
Communications of the ACM, August 2010
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Procurement Auctions
SUPPLIER 1
SUPPLIER 2
Buyer
SUPPLIER n
Supply (cost) Curves
Budget Constraints, Lead Time Constraints,
Learning by Suppliers, Learning by Buyer,
Logistics constraints, Combinatorial Auctions,
Cost Minimization, Multiple Attributes
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KEY OBSERVATIONS
Both conflict and cooperation are issues
Players are rational, Intelligent, strategic
Some information is common knowledge
Other information is private,
incomplete, distributed
Our Goal To implement a system wide solution
(social choice function) with desirable
properties
Game theory is a natural choice for modeling such
problems
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Strategic Form Games (Normal Form Games)
S1
U1 S R
Un S R
Sn
N 1,,n Players
S1, , Sn Strategy Sets S S1 X X Sn
Payoff functions (Utility functions)
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Example 1 Coordination Game
B A RVCE MG Road
RVCE 100,100 0,0
MG Road 0,0 10,10
Models the strategic conflict when two players
have to choose their priorities
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Example 2 Prisoners Dilemma
No Confess NC Confess C
No Confess NC - 2, - 2 - 10, - 1
Confess C -1, - 10 - 5, - 5

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Pure Strategy Nash Equilibrium

• A profile of strategies
  is said to be
• a pure strategy Nash Equilibrium if is
  a best
• response strategy against

A Nash equilibrium profile is robust to
unilateral deviations and captures a stable,
self-enforcing agreement among the players
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Nash Equilibria in Coordination Game
B A College Movie
College 100,100 0,0
Movie 0,0 10,10
Two pure strategy Nash equilibria
(College,College) and (Movie, Movie) one mixed
strategy Nash equilibrium
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Nash Equilibrium in Prisoners Dilemma
No Confess NC Confess C
No Confess NC - 2, - 2 - 10, - 1
Confess C -1, - 10 - 5, - 5
(C,C) is a Nash equilibrium
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Relevance/Implications of Nash
Equilibrium
Players are happy the way they are Do not want
to deviate unilaterally
Stable, self-enforcing, self-sustaining agreement

Need not correspond to a socially optimal
or Pareto optimal solution
Provides a principled way of predicting
a steady-state outcome of a dynamic Adjustment
process
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Example 3 Traffic Routing Game
C
45
x/100

B
A
2
Destination
Source
x/100
45
D


N 1,,n S1 S2 Sn C,D
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Traffic Routing Game Nash Equilibrium
C
45
x/100
B
A

2
Destination
Source
x/100
45
D
Any Strategy Profile with 2000 Cs and 2000 Ds
is a Nash Equilibrium
Assume n 4000 U1 (C,C, , C) - (40 45)
- 85 U1 (D,D, , D) - (45 40) - 85 U1 (D,C,
, C) - (45 0.01) - 45.01 U1 (C, ,CD,
,D) - (20 45) - 65


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Traffic Routing Game Braess Paradox
C
45
x/100
B
A
0

2
Destination
Source
x/100
45
D
Assume n 4000 S1 S2 Sn C,CD, D U1
(CD,CD, , CD) - (40040) - 80 U1 (C,CD, ,
CD) - (4045) - 85 U1 (D,CD, , CD) -
(4540) - 85
Strategy Profile with 4000 CDs is the
unique Nash Equilibrium


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Nashs Beautiful Theorem

• Every finite strategic form game has at least
  one mixed strategy Nash equilibrium
• Computing NE is one of the
• grand challenge problems in CS

Game theory is all about analyzing games
through such solution concepts and predicting the
behaviour of the players Non-cooperative game
theory and cooperative game theory are the major
categories
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MECHANISM DESIGN
Game Theory involves analysis of games
computing NE, DSE, MSNE, etc and analyzing
equilibrium behaviour
Mechanism Design is the design of games
or reverse engineering of games could be
called Game Engineering
Involves inducing a game among the players such
that in some equilibrium of the game, a desired
social choice function is implemented
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Example 1 Mechanism Design Fair Division of a
Cake
Mother Social Planner Mechanism Designer
Kid 1 Rational and Intelligent
Kid 2 Rational and Intelligent
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Example 2 Mechanism Design Truth Elicitation
through an Indirect Mechanism
Tenali Rama (Birbal) Mechanism Designer
Baby
Mother 1 Rational and Intelligent Player
Mother 2 Rational and Intelligent Player
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Mechanism Design Example 3 Vickrey
Auction
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1
1
Winner 4 Price 60
45
2
60
3
4
80
Buyers
William Vickrey (1914 1996 ) Nobel Prize 1996
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Four Basic Types of Auctions
Dutch Auction
100, 90, 85, 75, 70, 65, 60, stop.
0, 10, 20, 30, 40, 45, 50, 55, 58, 60, stop.
Seller
Buyers
Vickrey Auction
First Price Auction
1
40
40
1
Winner 4 Price 60
2
50
Winner 4 Price 60
45
2
55
3
60
3
4
80
4
60
Buyers
Buyers
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Vickrey-Clarke-Groves (VCG) Mechanisms
Vickrey
Clarke
Groves
Only mechanisms under a quasi-linear setting
satisfying Allocative Efficiency Dominant
Strategy Incentive Compatibility
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Concluding Remarks
Game Theory and Mechanism Design have numerous,
high impact applications in the Internet era
Game Theory, Machine Learning, Optimization, and
Statistics have emerged as the most important
mathematical tools for engineers
Algorithmic Game Theory is now one of the
most active areas of research in CS, ECE,
Telecom, etc. Mechanism Design is extensively
being used in IEM
It is a wonderful idea to introduce game theory
and mechanism design at the BE level for CS, IS,
EC, IEM to be done with care
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REFERENCES
Martin Osborne. Introduction to Game
Theory. Oxford University Press, 2003
Roger Myerson. Game Theory and Analysis of
Conflict. Harvard University Press, 1997
A, Mas-Colell, M.D. Whinston, and J.R.
Green. Microeconomic Theory, Oxford University
Press, 1995
N. Nisan, T. Roughgarden, E. Tardos, V.
Vazirani Algorithmic Game Theory, Cambridge
Univ. Press, 2007
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REFERENCES (contd.)
Y. Narahari, Essentials of Game Theory and
Mechanism Design IISc Press, 2012 (forthcoming)
http//www.gametheory.net A rich source of
material on game theory and game theory courses
http//lcm.csa.iisc.ernet.in/hari Course material
and several survey articles can be downloaded
Y. Narahari, Dinesh Garg, Ramasuri, and
Hastagiri Game Theoretic Problems in Network
Economics and Mechanism Design Solutions,
Springer, 2009
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Cooperative Game with Transferable Utilities
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Divide the Dollar Game There are three players
who have to share 300 dollars. Each one proposes
a particular allocation of dollars to players.
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Divide the Dollar Version 1

• The allocation is decided by what is proposed by
  player 0
• Characteristic Function

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Divide the Dollar Version 2

• It is enough 1 and 2 propose the same allocation
• Players 1 and 2 are equally powerful
  Characteristic Function is

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Divide the Dollar Version 3

• Either 1 and 2 should propose the same allocation
  or 1 and 3 should propose the same allocation
• Characteristic Function

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Divide the Dollar Version 4

• It is enough any pair of players has the same
  proposal
• Also called the Majority Voting Game
• Characteristic Function

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Shapley Value of a Cooperative Game

• Captures how competitive forces influence the
  outcomes of a game
• Describes a reasonable and fair way of dividing
  the gains from cooperation given the strategic
  realities
• Shapley value of a player finds its average
  marginal contribution across all permutation
  orderings
• Unique solution concept that satisfies symmetry,
  preservation of carrier, additivity, and Pareto
  optimality

Lloyd Shapley
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Shapley Value A Fair Allocation Scheme

• Provides a unique payoff allocation that
  describes a fair way of dividing the gains of
  cooperation in a game (N, v)

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Shapley Value Examples
Version of Divide-the-Dollar Shapley
Value Version 1 Version 2
Version 3 Version 4
(300, 0, 0)
(150, 150, 0)
(200, 50, 50)
(100, 100, 100)