Game Theory Quick Intro to Game Theory Analysis

Game Theory Quick Intro to Game Theory Analysis of Games Design of Games (Mechanism Design) Some References 1

John von Neumann The Genius who created two intellectual currents in the 1930 s, 1940 s Founded Game Theory with Oskar Morgenstern (1928 -44) Pioneered the Concept of a Digital Computer and Algorithms (1930 s) 2

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 Thomas Schelling Nobel 2005 Eric Maskin Nobel 2007 The prize was awarded to three mechanism designers in 2007 Roger Myerson Nobel 2007 3

Game Theory Mathematical framework for rigorous study of conflict and cooperation among rational, intelligent agents Market Buying Agents (rational and intelligent) Social Planner Selling Agents (rational and intelligent) In the Internet Era, Game Theory has become a valuable tool for analysis and design 4

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. 5

A Familiar Game Mumbai Indians 1 Kolkata Knight Riders 2 3 Bangalore Royal. Challengers Punjab Lions Sachin Tendulkar 4 IPL Franchisees IPL CRICKET AUCTION

Sponsored Search Auction Advertisers CPC Major money spinner for all search engines and web portals 7

DARPA Red Balloon Contest Mechanism Design Meets Computer Science, Communications of the ACM, August 2010 8

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

KEY OBSERVATIONS Players are rational, Intelligent, strategic Both conflict and cooperation are “issues” 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 10

Strategic Form Games (Normal Form Games) S 1 Sn N = {1, …, n} Players S 1, … , Sn Strategy Sets S = S 1 X … X Sn 11 U 1 : S R Un : S R Payoff functions (Utility functions)

Example 1: Coordination Game B RVCE MG Road RVCE 100, 100 0, 0 MG Road 0, 0 10, 10 A Models the strategic conflict when two players have to choose their priorities 12

Example 2: Prisoner’s Dilemma No Confess NC C No Confess NC - 2, - 2 - 10, - 1 Confess C 13 -1, - 10 - 5, - 5

Pure Strategy Nash Equilibrium A profile of strategies a pure strategy Nash Equilibrium if response strategy against is said to be is a best A Nash equilibrium profile is robust to unilateral deviations and captures a stable, self-enforcing agreement among the players 14

Nash Equilibria in Coordination Game B College Movie College 100, 100 0, 0 Movie 0, 0 10, 10 A Two pure strategy Nash equilibria: (College, College) and (Movie, Movie); one mixed strategy Nash equilibrium 15

Nash Equilibrium in Prisoner’s Dilemma No Confess NC C No Confess NC - 2, - 2 - 10, - 1 Confess C -1, - 10 - 5, - 5 (C, C) is a Nash equilibrium 16

Relevance/Implications of Nash Equilibrium 17 Players are happy the way they are; Do not want to deviate unilaterally Stable, self-enforcing, self-sustaining agreement Provides a principled way of predicting a steady-state outcome of a dynamic Adjustment process Need not correspond to a socially optimal or Pareto optimal solution

Example 3: Traffic Routing Game x/100 C 45 B A 2 Destination Source 45 D x/100 N = {1, …, n}; S 1 = S 2 = … = Sn = {C, D}

Traffic Routing Game: Nash Equilibrium x/100 C 45 B A 2 Destination Source 45 D Assume n = 4000 U 1 (C, C, …, C) = - (40 + 45) = - 85 U 1 (D, D, …, D) = - (45 + 40) = - 85 U 1 (D, C, …, C) = - (45 + 0. 01) = - 45. 01 U 1 (C, …, C; D, …, D) = - (20 + 45) = - 65 x/100 Any Strategy Profile with 2000 C’s and 2000 D’s is a Nash Equilibrium

Traffic Routing Game: Braess’ Paradox x/100 C 45 B 0 A 2 Destination Source 45 D Assume n = 4000 S 1 = S 2 = … = Sn = {C, CD, D} U 1 (CD, …, CD) = - (40+0+40) = - 80 U 1 (C, CD, …, CD) = - (40+45) = - 85 U 1 (D, CD, …, CD) = - (45+40) = - 85 x/100 Strategy Profile with 4000 CD’s is the unique Nash Equilibrium

Nash’s 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 21

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 22

Example 1: Mechanism Design Fair Division of a Cake Mother Social Planner Mechanism Designer Kid 1 Rational and Intelligent Kid 2 Rational and Intelligent

Example 2: Mechanism Design Truth Elicitation through an Indirect Mechanism Tenali Rama (Birbal) Mechanism Designer Mother 1 Rational and Intelligent Player Baby Mother 2 Rational and Intelligent Player

Mechanism Design: Example 3 Vickrey Auction 1 1 40 2 45 3 60 4 80 Buyers William Vickrey (1914 – 1996 ) Nobel Prize: 1996 25 Winner = 4 Price = 60

Four Basic Types of Auctions Dutch Auction English Auction 1 0, 10, 20, 30, 45, 50, 55, 58, 60, stop. n Buyers 100, 90, 85, 70, 65, 60, stop. Seller Vickrey Auction First Price Auction 40 2 50 3 55 4 60 Buyers 26 40 1 Winner = 4 Price = 60 n Buyers Auctioneer or seller 1 1 2 45 3 60 4 80 Buyers Winner = 4 Price = 60

Vickrey-Clarke-Groves (VCG) Mechanisms Vickrey Clarke Groves Only mechanisms under a quasi-linear setting satisfying Allocative Efficiency Dominant Strategy Incentive Compatibility 27

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 28

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 29

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 30

Cooperative Game with Transferable Utilities

Divide the Dollar Game There are three players who have to share 300 dollars. Each one proposes a particular allocation of dollars to players.

Divide the Dollar : Version 1 § The allocation is decided by what is proposed by player 0 § Characteristic Function

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:

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

Divide the Dollar : Version 4 § It is enough any pair of players has the same proposal § Also called the Majority Voting Game § Characteristic Function

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 Lloyd Shapley 37 Unique solution concept that satisfies symmetry, preservation of carrier, additivity, and Pareto optimality

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)

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