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

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

E-Commerce Lab, CSA, IISc Indirect Materials Procurement Suppliers with Volume Contracts Purchase Reqs Vendor identified IISc PReqs CSA PROC. MARKET 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

Ticket Allocation in Software Maintenance Customer . 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

E-Commerce Lab, CSA, IISc Incentive Compatible Broadcast in Ad hoc Wireless Networks ? E-Commerce Lab, CSA, IISc

Internet Routing Tier 3 Tier 2 Tier 1 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 E-Commerce Lab, CSA, IISc 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 E-Commerce Lab, CSA, IISc 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 E-Commerce Lab, CSA, IISc QoS SLA Cost Penalties Web Services Market Web Service Providers Web Service Requestors (rational and intelligent) (rational and intelligent) E-Commerce Lab, CSA, IISc

Sponsored Search Auction E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Sequence of Queries User 1 User 2 User N Google Q1 Q2 Q1 Q3 Q2 Q1 Q2 Q3 E-Commerce Lab, CSA, IISc

Sponsored Search Auction Game E-Commerce Lab, CSA, IISc CPC Advertisers 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

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

E-Commerce Lab, CSA, IISc Strategic form Games S1 U1 : S R Sn Un : S R 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 (1,-1) (-1,1) 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 2 1 0,0 20,5 5,20 10,10 H Hawk D 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 India Healthcare Defence 10,10 -10, 20 20, -10 0,0 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 College MG Road 100,100 0,0 5,5 E-Commerce Lab, CSA, IISc

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

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 a best response against , E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc John von Neumann (1903-1957) Founder of Game theory with Oskar Morgenstern E-Commerce Lab, CSA, IISc

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

E-Commerce Lab, CSA, IISc John Harsanyi (1920 - 2000) Defined and formalized Bayesian Games Nobel Prize : 1994 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Reinhard Selten (1930 - ) Founding father of experimental economics and bounded rationality Nobel Prize : 1994 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Thomas Schelling (1921 - ) Pioneered the study of bargaining and strategic behavior Nobel Prize : 2005 E-Commerce Lab, CSA, IISc

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

E-Commerce Lab, CSA, IISc Lloyd S. Shapley (1923 - ) Originator of “Shapley Value” and Stochastic Games E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc William Vickrey (1914 – 1996 ) Inventor of the celebrated Vickrey auction Nobel Prize : 1996 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Roger Myerson (1951 - ) Fundamental contributions to game theory, auctions, mechanism design E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc MECHANISM DESIGN E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Mechanism Design Problem O<M<L L<O<M M<L<O Yuvraj Dravid Laxman 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 E-Commerce Lab, CSA, IISc 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

E-Commerce Lab, CSA, IISc Indirect Mechanism E-Commerce Lab, CSA, IISc

Social Choice Function and Mechanism Sn θ1 θn Outcome Set Outcome Set f(θ1, …,θn) Є X g(s1(.), …,sn() Є X (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 E-Commerce Lab, CSA, IISc Dominant Strategy Equilibrium (DSE) A pure strategy profile is said to be dominant strategy equilibrium if Bayesian Nash Equilibrium (BNE) A pure strategy profile is said to be Bayesian Nash equilibrium Observation Dominant Strategy-equilibrium Bayesian Nash- equilibrium E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Implementing an SCF Dominant Strategy Implementation We say that mechanism implements SCF in dominant strategy equilibrium if Bayesian Nash Implementation We say that mechanism implements SCF in Bayesian Nash equilibrium if 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

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

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

E-Commerce Lab, CSA, IISc Values and Payoffs Quasi-linear Utilities E-Commerce Lab, CSA, IISc

Quasi-Linear Environment E-Commerce Lab, CSA, IISc Valuation function of agent 1 Policy Maker project choice Monetary transfer to agent 1 E-Commerce Lab, CSA, IISc

Properties of an SCF in Quasi-Linear Environment Ex Post Efficiency Dominant Strategy Incentive Compatibility (DSIC) Bayesian Incentive Compatibility (BIC) Allocative Efficiency (AE) SCF is AE if for each , satisfies Budget Balance (BB) SCF is BB if for each , we have Lemma 1 An SCF is ex post efficient in quasi-linear environment iff it is AE + BB E-Commerce Lab, CSA, IISc

A Dominant Strategy Incentive Compatible Mechanism Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient. Let the payments be : Groves Mechanism E-Commerce Lab, CSA, IISc

VCG Mechanisms (Vickrey-Clarke-Groves) Groves Mechanisms Clarke Mechanisms Generalized Vickrey Auction Vickrey Auction Allocatively efficient, individual rational, and dominant strategy incentive compatible with quasi-linear utilities. E-Commerce Lab, CSA, IISc

A Bayesian Incentive Compatible Mechanism Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient. Let types of the agents be statistically independent of one another dAGVA Mechanism E-Commerce Lab, CSA, IISc

Basic Types of Procurement Auctions Reverse Dutch Auction 1 n 100, 95, 90, 85, 80, 75, 70, 65, 60, stop. Auctioneer or Buyer Reverse English Auction Sellers 1 n 0, 10, 20, 30, 40, 45, 50, 55, 58, 60, stop. Buyer Sellers Reverse Second Price Auction (Reverse Vickrey Auction) Reverse First Price Auction 1 80 75 1 2 75 Winner = 4 Price = 60 2 65 Winner = 4 Price = 60 70 3 3 60 4 60 4 50 Sellers Sellers E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc WBB SBB AE EPE dAGVA BIC IR GROVES DSIC MOULIN E-Commerce Lab, CSA, IISc

Sponsored Search Auctions E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc OUTLINE Sponsored Search Auctions SSA as a Mechanism Design Problem Three Different Auction Mechanisms: GFP, GSP, VCG A New Mechanism: OPT Comparison of Different Mechanisms Ongoing Work E-Commerce Lab, CSA, IISc

Sponsored Search Auction Game E-Commerce Lab, CSA, IISc CPC Advertisers 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

Sponsored Search Auction as a Mechanism Design Problem (Allocation Rule, Payment Rule) E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc X E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc X E-Commerce Lab, CSA, IISc

Bayesian Game Induced by the Auction Mechanism Induces a Bayesian game among advertisers where Set of advertisers Valuation set of advertiser Set of bids for advertiser A pure strategy of advertiser Prior distribution of advertiser valuations Utility payoff of advertiser E-Commerce Lab, CSA, IISc

Strategic Bidding Behavior of Advertisers E-Commerce Lab, CSA, IISc If all the advertisers are rational and intelligent and this fact is common knowledge then each advertiser’s expected bidding behavior is given by Dominant Strategy Equilibrium Strategy profile is said to be Dominant Strategy equilibrium iff Bayesian Nash Equilibrium Strategy profile is said to be Bayesian Nash equilibrium iff E-Commerce Lab, CSA, IISc

Advertisers’ Bidding Strategy E-Commerce Lab, CSA, IISc VCG: Follow irrespective of what the others are doing OPT: Follow if all rivals are also doing so GSP: Never follow strategy . Use the following E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Properties of a Sponsored Search Auction Mechanism E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Google’s Objectives Short Term Long Term Q1 Q2 Q3 Q1 Q2 Q3 Revenue Maximization Click Fraud Resistance Individual Rationality Incentive Compatibility E-Commerce Lab, CSA, IISc

Click Fraud Resistance E-Commerce Lab, CSA, IISc Google’s Objectives Revenue Maximization Choose auction mechanism such that despite strategic bidding behavior of advertisers, expected revenue is maximum Click Fraudulence increase the spending of rival advertisers without increasing its own. is click fraudulent if an advertiser finds a way to Click Fraud Resistance is click fraud resistant If it is not click fraudulent E-Commerce Lab, CSA, IISc

Individual Rationality E-Commerce Lab, CSA, IISc Google’s Objectives Individual Rationality Advertiser’s participation is voluntary Will bid only if the participation constraint is satisfied Why should bother about it ? Advertisers may decide to quit !! What can do about it ? Choose an auction mechanism which is IR E-Commerce Lab, CSA, IISc

Incentive Compatibility E-Commerce Lab, CSA, IISc Google’s Objectives Incentive Compatibility Difficulties faced by an Advertiser In practice, the assumptions like rationality, intelligence, and common knowledge are hardly true Need to invoke sophisticated but impractical software agents to compute the optimal Why should bother about it ? Low ROI switch to other search engines !! E-Commerce Lab, CSA, IISc

Incentive Compatibility E-Commerce Lab, CSA, IISc What can do about it ? Choose an auction mechanism which is IC Dominant Strategy Incentive Compatibility Incentive compatible if truth telling is a dominant strategy equilibrium Auction mechanism is said to be dominant strategy Bayesian Incentive Compatibility Auction mechanism is said to be Bayesian incentive compatible if truth telling is a Bayesian Nash equilibrium E-Commerce Lab, CSA, IISc

Properties of Auction Mechanisms E-Commerce Lab, CSA, IISc Bayesian IC Dominant Individual Strategy Rationality IC VCG GFP OPT GSP E-Commerce Lab, CSA, IISc

Four Different Auction Mechanisms E-Commerce Lab, CSA, IISc GFP GSP VCG OPT (Overture, 1997) (Google, 2002) Notation Feasibility Condition: E-Commerce Lab, CSA, IISc

Generalized First Price (GFP) E-Commerce Lab, CSA, IISc 1 2 m Allocation Rule Allocated the slots in decreasing order of bids Payment Rule Every time a user clicks on the Ad, the advertiser’s account is automatically billed the amount of the advertiser’s bid E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: GFP Q Search Results Sponsored Links 1 2 E-Commerce Lab, CSA, IISc

Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc Allocation Rules Rule: 1 2 m Allocate the slots in decreasing order of bids Greedy Rule: Allocate 1st slot to advertiser Allocate 2nd slot to advertiser Rule: Allocate the slots in decreasing order of Ranking Score Ranking Score = E-Commerce Lab, CSA, IISc

Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc Greedy Observation 1: Greedy Click probability is independent of the identity of advertisers Greedy E-Commerce Lab, CSA, IISc

Generalized Second Price (GSP) E-Commerce Lab, CSA, IISc Payment Rule For every click, charge next highest bid + $0.01 The bottom most advertiser is charged highest disqualified bid +$0.01 charge 0 if no such bid E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: GSP Q Search Results Sponsored Links 1 2 E-Commerce Lab, CSA, IISc

Vickrey-Clarke-Groves (VCG) E-Commerce Lab, CSA, IISc Allocation Rule 1 2 m In decreasing order of bids Payment Rule Case 1 Case 2 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: VCG Q Search Results Sponsored Links 1 2 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Optimal (OPT) 1 2 m Allocation Rule Where is the highest value among (Assumption: is non decreasing: True for Uniform, Exponential) Observation 4: Advertisers are symmetric i.e. Allocation Rule OPT E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Optimal (OPT) Payment Rule Assumptions: Advertisers are symmetric, i.e. Whenever an advertiser bids charge him for every query irrespective of whether his Ad is displayed or not Where is the probability that advertiser will receive a click if he bids and rest of the advertisers bid their true values E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: OPT Q Search Results Sponsored Links 1 2 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: OPT E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Example: OPT E-Commerce Lab, CSA, IISc

Expected Revenue of Seller E-Commerce Lab, CSA, IISc Case 1 Observation E-Commerce Lab, CSA, IISc

Expected Revenue of Seller E-Commerce Lab, CSA, IISc Case 1 E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Conclusions Allocation Payment DSIC BIC IR CFR GSP Decreasing order of the bids Next Highest bid (PPC) X VCG Marginal Contribution OPT (PPP) E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Ongoing Work Deeper Mechanism Design Repeated Games Model Learning Bidding Strategies Cooperative Bidding E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Questions and Answers … Thank You … E-Commerce Lab, CSA, IISc

Vickrey Auction for Ticket Allocation effort, time effort, time effort, time Project lead Ticket Allocator Maintenance Engineers E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc Incentive Compatible Broadcast Problem: Successful broadcast requires appropriate forwarding of the packets by individual selfish wireless nodes. Reimbursing the forwarding costs incurred by the nodes is a way to make them forward the packets. For this, we need to know the exact transit costs of the nodes. We can design an incentive compatible broadcast protocol by embedding appropriate incentive schemes into the broadcast protocol. We shall refer to the problem of designing such robust broadcast protocols as the incentive compatible broadcast (ICB) problem. ? Line Network Bi-connected ad hoc network Source Rooted Broadcast Tree E-Commerce Lab, CSA, IISc

Vickrey Auction for Ticket Allocation E-Commerce Lab, CSA, IISc Maintenance Engineers Bid 1 – Rs. 1000 Bid 2 – Rs. 1500 Bid 3 – Rs. 1200 Allocation Engineer 1 is selected as winner (lowest bid) Payment Engineer 1 is paid 1000 + (1200 – 1000) = 1200 Vickrey Auction is Dominant Strategy Incentive Compatible -- Truth revelation is a best response for each agent Irrespective of what is reported by the other agents E-Commerce Lab, CSA, IISc

Vickrey Auction as a Strategic Form Game E-Commerce Lab, CSA, IISc

GVA for Web Services Composition E-Commerce Lab, CSA, IISc A, B, AB 1 A, B, C 2 A, C, AC Web Service Requestor (client) 3 A, B, C, ABC 4 Web Service Providers E-Commerce Lab, CSA, IISc

GVA for Web Services Composition AB AC ABC 1 30 20 - 40 2 25 3 35 50 4 70 Optimal Allocation: 1AB; 4 C Optimal Cost: 40 + 20 = 60 Optimal Cost without 1 = 70 Optimal Cost without 4 = 65 Payment to provider 1 = 40 + 70 – 60 = 50 Payment to provider 4 = 20 + 65 - 60 = 25 E-Commerce Lab, CSA, IISc