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### Resource Allocation in Grid Computing

Game Theoretic Problems in Network Economics and Mechanism Design Solutions

Y. Narahari

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

Examples of Game Theoretic Problems in Network Economics

Mechanism Design

Case Study: Sponsored Search Auctions

Future Work

E-Commerce Lab, CSA, IISc

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

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

.

.

.

Ticket Allocation in Software MaintenanceTeam 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

E-Commerce Lab, CSA, IISc

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

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

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

QoS

SLA

Cost

Penalties

Web

Services

Market

Web Service Requestors

Web Service Providers

(rational and intelligent)

(rational and intelligent)

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

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

Landmark contributions to Game theory: notions of Nash Equilibrium and Nash Bargaining

Nobel Prize : 1994

E-Commerce Lab, CSA, IISc

(1920 - 2000)

Defined and formalized Bayesian Games

Nobel Prize : 1994

E-Commerce Lab, CSA, IISc

(1930 - )

Founding father of experimental economics and bounded rationality

Nobel Prize : 1994

E-Commerce Lab, CSA, IISc

(1921 - )

Pioneered the study of bargaining and strategic behavior

Nobel Prize : 2005

E-Commerce Lab, CSA, IISc

(1930 - )

Pioneer of the notions of common knowledge, correlated equilibrium, and repeated games

Nobel Prize : 2005

E-Commerce Lab, CSA, IISc

(1923 - )

Originator of “Shapley Value” and Stochastic Games

E-Commerce Lab, CSA, IISc

(1914 – 1996 )

Inventor of the celebrated Vickrey auction

Nobel Prize : 1996

E-Commerce Lab, CSA, IISc

(1951 - )

Fundamental contributions to game theory, auctions, mechanism design

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

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

- 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

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

- 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

Quasi-Linear Environment

Valuation function of agent 1

project choice

Monetary transfer to agent 1

E-Commerce Lab, CSA, IISc

SCF is AE if for each , satisfies

An SCF is ex post efficient in quasi-linear environment iff it is AE + BB

SCF is BB if for each , we have

Properties of an SCF in Quasi-Linear Environment

- Ex Post Efficiency

- Dominant Strategy Incentive Compatibility (DSIC)

- Bayesian Incentive Compatibility (BIC)

- Allocative Efficiency (AE)

- Budget Balance (BB)

- Lemma 1

E-Commerce Lab, CSA, IISc

A Dominant Strategy Incentive Compatible Mechanism

- Letf(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.
- Let the payments be :

Groves Mechanism

E-Commerce Lab, CSA, IISc

Clarke Mechanisms

Generalized Vickrey Auction

Vickrey Auction

VCG Mechanisms (Vickrey-Clarke-Groves)- Allocatively efficient, individual rational, and dominant strategy incentive compatible with quasi-linear utilities.

E-Commerce Lab, CSA, IISc

A Bayesian Incentive Compatible Mechanism

- Letf(.) = (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

1

Reverse English Auction

100, 95, 90, 85,

80, 75, 70, 65,

60, stop.

n

n

AuctioneerorBuyer

Sellers

Basic Types of Procurement AuctionsReverse Dutch Auction

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

Winner = 4 Price = 60

65

2

70

3

60

3

4

50

4

60

Sellers

Sellers

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

- 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

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

E-Commerce Lab, CSA, IISc

Bayesian Game Induced by the Auction Mechanism

Induces a Bayesian game

Mechanism

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

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

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

Properties of a Sponsored Search Auction Mechanism

E-Commerce Lab, CSA, IISc

Q1

Q1

Q3

Q2

Q3

Q1

Q1

Q3

Q2

Google’s Objectives

Short Term

Long Term

- Revenue Maximization
- Click Fraud Resistance
- Individual Rationality
- Incentive Compatibility

E-Commerce Lab, CSA, IISc

is click fraudulent if an advertiser finds a way to

increase the spending of rival advertisers without increasing its own.

Google’s Objectives

Revenue Maximization

such that despite

Choose auction mechanism

strategic biddingbehavior of advertisers, expected revenue is maximum

Click Fraudulence

Click Fraud Resistance

is click fraud resistant If it is not click fraudulent

E-Commerce Lab, CSA, IISc

- What can do about it ?

Choose an auction mechanism

which is IR

Google’s Objectives

Individual Rationality

- Advertiser’s participation is voluntary
- Will bid only if the participation constraint is satisfied

Advertisers may decide to quit !!

E-Commerce Lab, CSA, IISc

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

is said to be dominant strategy

Auction mechanism

Incentive compatible if truth telling is a dominant strategy equilibrium

Incentive Compatibility

What can do about it ?

which is IC

Choose an auction mechanism

Dominant Strategy Incentive Compatibility

Bayesian Incentive Compatibility

is said to be Bayesian incentive

Auction mechanism

compatible if truth telling is a Bayesian Nash equilibrium

E-Commerce Lab, CSA, IISc

Properties of Auction Mechanisms

Bayesian IC

Individual

Dominant

Strategy

Rationality

- VCG

IC

- GFP

- OPT

- GSP

E-Commerce Lab, CSA, IISc

GFP

VCG

GSP

(Overture, 1997)

(Google, 2002)

Four Different Auction Mechanisms

Notation

Feasibility Condition:

E-Commerce Lab, CSA, IISc

2

m

Generalized First Price (GFP)

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

2

m

Generalized Second Price (GSP)

Allocation Rules

Rule:

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)

Observation 1:

Greedy

Click probability is independent of the

identity of advertisers

Greedy

E-Commerce Lab, CSA, IISc

Generalized Second Price (GSP)

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

2

Case 1

m

Case 2

Vickrey-Clarke-Groves (VCG)

Allocation Rule

In decreasing order of bids

Payment Rule

E-Commerce Lab, CSA, IISc

2

m

Optimal (OPT)

Allocation Rule

Where is the highest value among

(Assumption: is non decreasing: True for Uniform, Exponential)

Observation 4:

Allocation Rule

OPT

Advertisers are symmetrici.e.

E-Commerce Lab, CSA, IISc

Where is the probability that advertiser will receive a click if he bids and rest of the advertisers bid their true values

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

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

E-Commerce Lab, CSA, IISc

- Deeper Mechanism Design
- Repeated Games Model
- Learning Bidding Strategies
- Cooperative Bidding

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

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

N = { 1, 2, 3 }

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

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

- Optimal Allocation: 1AB; 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

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