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Game Theoretic Problems in Network Economics and Mechanism Design Solutions

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## Game Theoretic Problems in Network Economics and Mechanism Design Solutions

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**Game Theoretic Problems in Network Economics and Mechanism**Design Solutions Y. Narahari hari@csa.iisc.ernet.in Co-Researchers: Dinesh Garg, Rama Suri, Hastagiri, Sujit Gujar September 2007 E-Commerce Lab Computer Science and Automation, Indian Institute of Science, Bangalore E-Commerce Lab, CSA, IISc**OUTLINE**Examples of Game Theoretic Problems in Network Economics Mechanism Design Case Study: Sponsored Search Auctions Future Work E-Commerce Lab, CSA, IISc**Talk Based on**Y. Narahari, Dinesh Garg, Rama Suri, Hastagiri Game Theoretic Problems in Network Economics and Mechanism Design Solutions Research Monograph in the AI & KP Series To Be Published by Springer, London, 2008 E-Commerce Lab, CSA, IISc**Supply Chain Network Formation**Supply Chain Network Planner Stage Manager E-Commerce Lab, CSA, IISc**Indirect Materials Procurement**Suppliers with Volume Contracts Purchase Reqs Vendor identified IISc PReqs PROC. MARKET CSA Catalogued Suppliers without Volume Contracts RFQ Reqs PURCHASE SYSTEM EE Quotes PHY Auction Non Catalogued Suppliers Optimized Order(s) recommendations ADM PO’s to Suppliers E-Commerce Lab, CSA, IISc**Customer**. . . Ticket Allocation in Software Maintenance Team of Maintenance Engineers Web Interface Product #1 Queue Product Lead #1 . . . . . . Based on Type of Application Or product, problems are distributed to various Queues . . . Product #100 Queue Product Lead #100 Level 1 Product Maintenance Processes E-Commerce Lab, CSA, IISc**Ticket Allocation Game**effort, time effort, time effort, time Project lead (Ticket Allocator) (rational and intelligent) Maintenance Engineers (rational and intelligent) E-Commerce Lab, CSA, IISc**Resource Allocation in Grid Computing**E-Commerce Lab, CSA, IISc**?**Incentive Compatible Broadcast in Ad hoc Wireless Networks E-Commerce Lab, CSA, IISc**Tier 3**Tier 2 Tier 1 Internet Routing Tier 1: UU Net, Sprint, AT&T, Genuity Tier 2: Regional/National ISPs Tier 3: Residential/Company ISP E-Commerce Lab, CSA, IISc**Web Service Composition**Web Service Web Service Web Service A B C Service Providers1, 2 Service Providers 2,3 Service Providers 3,4 There could be alternate service providers for each web service How do we select the best mix of web service providers so as to execute the end-to-end business process at minimum cost taking into account QOS requirements? E-Commerce Lab, CSA, IISc**Web Services Composition Game**A, B, AB 1 A, B, C 2 A, C, AC Web Service Requestor (client) (rational and intelligent) 3 A, B, C, ABC 4 Web Service Providers (rational and intelligent) E-Commerce Lab, CSA, IISc**Web Services Market Game**QoS SLA Cost Penalties Web Services Market Web Service Requestors Web Service Providers (rational and intelligent) (rational and intelligent) E-Commerce Lab, CSA, IISc**Sponsored Search Auction**E-Commerce Lab, CSA, IISc**User 1**Google User 2 User N Sequence of Queries Q1 Q1 Q3 Q2 Q1 Q3 Q2 Q2 E-Commerce Lab, CSA, IISc**Sponsored Search Auction Game**Advertisers CPC E-Commerce Lab, CSA, IISc**Some Important Observations**Players are rational and intelligent Conflict and cooperation are both relevant issues Some information is common knowledge Some information is is private and distributed (incomplete information) Our Objective: Design a social choice function With desirable properties, given that the players are rational, intelligent, and strategic E-Commerce Lab, CSA, IISc**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) E-Commerce Lab, CSA, IISc**Strategic 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) • Players are rational : they always strive to maximize their individual payoffs • Players are intelligent : they can compute their best responsive strategies • Common knowledge E-Commerce Lab, CSA, IISc**Example 1: Matching Pennies**• Two players simultaneously put down a coin, heads up or tails up. Two-Player zero-sum game S1 = S2 = {H,T} E-Commerce Lab, CSA, IISc**Example 2: Prisoners’ Dilemma**E-Commerce Lab, CSA, IISc**Example 3: Hawk - Dove**Models the strategic conflict when two players are fighting over a company/territory/property, etc. E-Commerce Lab, CSA, IISc**Example 4: Indo-Pak Budget Game**Models the strategic conflict when two players have to choose their priorities E-Commerce Lab, CSA, IISc**Example 5: Coordination**• In the event of multiple equilibria, a certain equilibrium becomes a focal equilibrium based on certain environmental factors E-Commerce Lab, CSA, IISc**Nash Equilibrium**• (s1*,s2*, … , sn*) is a Nash equilibrium if si* is a best response for player ‘i’ against the other players’ equilibrium strategies Prisoner’s Dilemma (C,C) is a Nash Equilibrium. In fact, it is a strongly dominant strategy equilibrium E-Commerce Lab, CSA, IISc**Nash’s Theorem**Every finite strategic form game has at least one mixed strategy Nash equilibrium Mixed strategy of a player ‘i’ is a probability distribution on Si is a mixed strategy Nash equilibrium if is abest response against , E-Commerce Lab, CSA, IISc**John von Neumann**(1903-1957) Founder of Game theory with Oskar Morgenstern E-Commerce Lab, CSA, IISc**John F Nash Jr.(1928 - )**Landmark contributions to Game theory: notions of Nash Equilibrium and Nash Bargaining Nobel Prize : 1994 E-Commerce Lab, CSA, IISc**John Harsanyi**(1920 - 2000) Defined and formalized Bayesian Games Nobel Prize : 1994 E-Commerce Lab, CSA, IISc**Reinhard Selten**(1930 - ) Founding father of experimental economics and bounded rationality Nobel Prize : 1994 E-Commerce Lab, CSA, IISc**Thomas Schelling**(1921 - ) Pioneered the study of bargaining and strategic behavior Nobel Prize : 2005 E-Commerce Lab, CSA, IISc**Robert J. Aumann**(1930 - ) Pioneer of the notions of common knowledge, correlated equilibrium, and repeated games Nobel Prize : 2005 E-Commerce Lab, CSA, IISc**Lloyd S. Shapley**(1923 - ) Originator of “Shapley Value” and Stochastic Games E-Commerce Lab, CSA, IISc**William Vickrey**(1914 – 1996 ) Inventor of the celebrated Vickrey auction Nobel Prize : 1996 E-Commerce Lab, CSA, IISc**Roger Myerson**(1951 - ) Fundamental contributions to game theory, auctions, mechanism design E-Commerce Lab, CSA, IISc**MECHANISM DESIGN**E-Commerce Lab, CSA, IISc**L<O<M**M<L<O O<M<L Mechanism Design Problem Yuvraj Laxman Dravid O: Opener M:Middle-order L: Late-order Greg • How to transform individual preferences into social decision? • How to elicit truthful individual preferences ? E-Commerce Lab, CSA, IISc**The Mechanism Design Problem**• agents who need to make a collective choice from outcome set • Each agent privately observes a signal which determines preferences over the set • Signal is known as agent type. • The set of agent possible types is denoted by • The agents types, are drawn according to a probability distribution function • Each agent is rational, intelligent, and tries to maximize its utility function • are common knowledge among the agents E-Commerce Lab, CSA, IISc**Two Fundamental Problems in Designing a Mechanism**• Preference Aggregation Problem For a given type profile of the agents, what outcome should be chosen ? • Information Revelation (Elicitation) Problem How do we elicit the true type of each agent , which is his private information ? E-Commerce Lab, CSA, IISc**Information Elicitation Problem**E-Commerce Lab, CSA, IISc**Preference Aggregation Problem (SCF)**E-Commerce Lab, CSA, IISc**Indirect Mechanism**E-Commerce Lab, CSA, IISc**Social Choice Function and Mechanism**S1 Sn θ1 θn Outcome Set Outcome Set g(s1(.), …,sn() X f(θ1, …,θn) X Є Є (S1, …, Sn, g(.)) x = (y1(θ), …, yn(θ), t1(θ), …, tn(θ)) A mechanism induces a Bayesian game and is designed to implement a social choice function in an equilibrium of the game. E-Commerce Lab, CSA, IISc**Equilibrium of Induced Bayesian Game**• Dominant Strategy Equilibrium (DSE) A pure strategy profile is said to be dominantstrategy equilibriumif • Bayesian Nash Equilibrium (BNE) A pure strategy profile is said to be BayesianNash equilibrium • Observation Dominant Strategy-equilibrium Bayesian Nash- equilibrium E-Commerce Lab, CSA, IISc**We say that mechanismimplements SCF in**dominant strategy equilibrium if We say that mechanism implements SCF in Bayesian Nash equilibrium if Implementing an SCF • Dominant Strategy Implementation • Bayesian Nash Implementation • Observation Dominant Strategy-implementation Bayesian Nash- implementation Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic Theory”, Oxford University Press, New York, 1995. E-Commerce Lab, CSA, IISc**Properties of an SCF**• Ex Post Efficiency For no profile of agents’ type does there exist an such that and for some • Dominant Strategy Incentive Compatibility (DSIC) If the direct revelation mechanism has a dominant strategy equilibrium in which • Bayesian Incentive Compatibility (BIC) If the direct revelation mechanism has a Bayesian Nash equilibrium in which E-Commerce Lab, CSA, IISc**Outcome Set**Project Choice Allocation I0, I1,…, In : Monetary Transfers x = (k, I0, I1,…, In) K = Set of all k X = Set of all x E-Commerce Lab, CSA, IISc**Social Choice Function**where, E-Commerce Lab, CSA, IISc**Values and Payoffs**Quasi-linear Utilities E-Commerce Lab, CSA, IISc**Policy Maker**Quasi-Linear Environment Valuation function of agent 1 project choice Monetary transfer to agent 1 E-Commerce Lab, CSA, IISc