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

**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) 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 • The prize was awarded to three mechanism designers in 2007 Thomas Schelling Nobel 2005 Eric Maskin Nobel 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) Selling Agents (rational and intelligent) Social Planner • 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 Bangalore RoyalChallengers 3 Punjab Lions 4 Sachin Tendulkar IPL Franchisees IPL CRICKET AUCTION

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

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

**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 ** 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

**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) 11

**Example 1: Coordination Game** Models the strategic conflict when two players have to choose their priorities 12

**Example 2: Prisoner’s Dilemma** 13

**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 14

**Nash Equilibria in Coordination Game** Two pure strategy Nash equilibria: (College,College) and (Movie, Movie); one mixed strategy Nash equilibrium 15

**Nash Equilibrium in Prisoner’s Dilemma** (C,C) is a Nash equilibrium 16

**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 17

**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}

**Traffic Routing Game: Nash Equilibrium** C 45 x/100 B A 2 Destination Source x/100 45 D Any Strategy Profile with 2000 C’s and 2000 D’s 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, …,C;D, …,D) = - (20 + 45) = - 65

**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) = - (40+0+40) = - 80 U1 (C,CD, …, CD) = - (40+45) = - 85 U1 (D,CD, …, CD) = - (45+40) = - 85 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

**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 Baby Mother 1 Rational and Intelligent Player Mother 2 Rational and Intelligent Player

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

**1** 1 English Auction n n Auctioneerorseller Buyers 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 First Price Auction Vickrey Auction 1 40 40 1 2 50 Winner = 4 Price = 60 Winner = 4 Price = 60 45 2 55 3 60 3 4 80 4 60 Buyers Buyers

**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

**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** Lloyd Shapley 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 37

**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-DollarShapley Value Version 1 Version 2 Version 3 Version 4 (300, 0, 0) (150, 150, 0) (200, 50, 50) (100, 100, 100)