e-commerce lab, csa, iisc 1 incentive compatible mechanisms for supply chain formation y. narahari...

E-Commerce Lab, CSA, IISc1

Incentive Compatible Mechanisms for Supply

Chain Formation

Y. Narahari

[email protected] http://lcm.csa.iisc.ernet.in/hari

Co-Researchers: N. Hemachandra, Dinesh Garg,

Nikesh Kumar

September 2007

E-Commerce LabComputer Science and Automation,

Indian Institute of Science, Bangalore


OUTLINE

1. Supply Chain Formation Problem

2. Supply Chain Formation Game

3. Incentive Compatible Mechanisms for Network Formation

SCF-DSIC

SCF-BIC

4. Future Work


Talk Based on

1. Y. Narahari, Dinesh Garg, Rama Suri, and Hastagiri. Game Theoretic Problems in Network Economics and Mechanism Design Solutions. Research Monograph to be published by Springer, London, 2008

2. Dinesh Garg, Y. Narahari, Earnest Foster, Devadatta Kulkarni, and Jeffrey D. Tew. A Groves Mechanism Approach to Supply Chain Formation. Proceedings of IEEE CEC 2005.

3. Y. Narahari, N. Hemachandra, and Nikesh Srivastava. Incentive Compatible Mechanisms for Decentralized Supply Chain Formation. Proceedings of IEEE CEC 2007.


The Supply Chain Network Formation Problem

1X 2X3X 4X

n

iiXY

1

Supply Chain Planner

Echelon Manager


Cold Rolling Pickling Slitting Stamping

MasterCoil

1

2

6

2

3

7

3

4

4

5

6

Suppliers

7

Forming a Supply Network for Automotive Stampings


Some Observations

Playersare rational and intelligent

Some of the information is common knowledge

Conflict and cooperationare both relevant

Some information is is private and distributed(incomplete information)

Our Objective: Design an “optimal”Network of supply chain partners, given that the

players are rational, intelligent, and strategic


Simple Example: The Supply Chain Partner Selection Problem

SCP

EM2EM1

A C A CB B

Let us say it is required to select the same partner at the two stages


1. Let us say SCP wants to implement the social choice function: f (x1, x2) = B; f (x1, y2) = A 2. If its type is x2, manager 2 is happy to reveal true type3. If its type is y2, manager 2 would wish to lie4. How do we make the managers report their true types?

Preference Elicitation ProblemSupply Chain

Planner

Echelon Manager 2

x1: A>B>C

x2: C>B>A

Echelon Manager 1

y2: B>C>A


W.E. Walsh and M.P. Wellman. Decentralized Supply Chain Formation: A Market Protocol and Competitive Equilibrium Analysis. Journal of Artificial Intelligence, 2003

M. Babaioff and N. Nisan. Concurrent Auctions Across the Supply Chain. Journal of Artificial Intelligence, 2004

Ming Fan, Jan Stallert, Andrew B Whinston. Decentralized Mechanism Design for Supply Chain Organizations using Auction Markets. Information Systems Research, 2003.

T. S. Chandrashekar and Y. Narahari. Procurement Network Formation: A Cooperative Game Approach. WINE 2005

Current Art


Complete Information Version

• Choose means and standard deviations of individual stages so as to :

subject to

A standard optimization problem (NLP)


1. How to transform individual preferences into social decision (SCF)?2. How to elicit truthful individual preferences (Incentive Compatibility) ?3. How to ensure the participation of an individual (Individual Rationality)?4. Which social choice functions are realizable?

Incomplete Information VersionSupply Chain

Planner

Echelon Manager 2

Type Set 1 Type Set 2

Echelon Manager 1


Strategic form Games

S1

Sn

U1 : S R

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


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}

(1,-1) (-1,1)

(-1,1) (1,-1)


Example 2: Prisoners’ Dilemma


Example 3: Hawk - Dove

2

1

H

Hawk

D

Dove

H

Hawk 0,0 20,5

D

Dove 5,20 10,10

Models the strategic conflict when two players are fighting over a company/territory/property, etc.


Example 4: Indo-Pak Conflict

Pak

India

Healthcare Defence

Healthcare

10,10 -10, 20

Defence

20, -10 0,0

Models the strategic conflict when two players have to choose their priorities


Example 5: Coordination

• In the event of multiple equilibria, a certain equilibrium becomes a focal equilibrium based on certain environmental factors

College MG Road

College

100,100 0,0MG Road

0,0 5,5


Nash Equilibrium

• (s1*,s2

*, … , sn*) is a Nash equilibrium if si

* is a best response for player ‘i’ against the other players’ equilibrium strategies

(C,C) is a Nash Equilibrium. In fact, it is a strongly dominant strategy equilibrium

Prisoner’s Dilemma


Mixed strategy of a player ‘i’ is a probability distribution on Si

is a mixed strategy Nash equilibrium if

is a best response against ,

Nash’s Theorem

Every finite strategic form game has at least one mixed strategy Nash equilibrium

*i

*i

**2

*1 ,...,, n

ni ,...,2,1


John von Neumann(1903-1957)

Founder of Game theory with Oskar Morgenstern


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

Nobel Prize : 1994

John F Nash Jr. (1928 - )


Defined and formalized Bayesian GamesNobel Prize : 1994

John Harsanyi (1920 - 2000)


Reinhard Selten(1930 - )

Founding father of experimental economics and bounded rationalityNobel Prize : 1994


Pioneered the study of bargaining and strategic behaviorNobel Prize : 2005

Thomas Schelling(1921 - )


Robert J. Aumann(1930 - )

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

Nobel Prize : 2005


Lloyd S. Shapley(1923 - )

Originator of “Shapley Value” and Stochastic Games


Inventor of the celebrated Vickrey auction Nobel Prize : 1996

William Vickrey(1914 – 1996 )


Roger Myerson(1951 - )

Fundamental contributions to game theory, auctions, mechanism design


MECHANISM DESIGN


Underlying Bayesian Game

N = {0,1,..,n}0 : Planner1,…,n: Partners

Type setsPrivate Info: Costs

S0,S1,…,Sn

Strategy SetsAnnouncements

Payoff functions

N = {0,1,..,n}0 : Planner1,…,n: Partners

A Natural Setting for Mechanism Design


Mechanism Design Problem

1. How to transform individual preferences into social decision?2. How to elicit truthful individual preferences ?

O: OpenerM: Middle-orderL: Late-orderGreg

Yuvraj Dravid Laxman

O<M<L L<O<M M<L<O


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

i s'iiX

Xn

i si '

s'ii

n ,,1(.)

ii Xu :

(.),(.),,,,(.), nn uu 11


Social Choice Function and Mechanism

f(f(θθ11, …,, …,θθnn))

θθ11 θθnn

XXЄЄ

SS11

g(sg(s11(.), …,s(.), …,snn()()

SSnn

XXЄЄ

x = (yx = (y11((θθ), …, y), …, ynn((θθ), t), t11((θθ), …, ), …, ttnn((θθ))))

(S(S11, …, S, …, Snn, g(.)), g(.))

A mechanism induces a Bayesian game and is A mechanism induces a Bayesian game and is designed to implement a social choice function in an designed to implement a social choice function in an equilibrium of the game.equilibrium of the game.

Outcome Outcome SetSet

Outcome SetOutcome Set


Two Fundamental Problems in Designing a Mechanism

Preference Aggregation Problem

Information Revelation (Elicitation) Problem

For a given type profile of the agents, what outcome

should be chosen ?

n ,,1 Xx

How do we elicit the true type of each agent , which is his

private information ? i i


1 2 n

2 n

1̂ 2̂ n̂

1

Xf n 1:

11 Xu :

11 ,xu

22 Xu : nn Xu :

Xx nfx ˆ,,ˆ 1

22 ,xu nn xu ,

Information Elicitation Problem


1 2 n

2 n

1 2 n

1

Xf n 1:

Xx nfx ,,1

Preference Aggregation Problem (SCF)


1 2 n

2 n

1c 2c nc

1

XCCg n 1:

11 Xu : 22 Xu : nn Xu :

1C 2C nC

Xx nccgx ,,1

11 ,xu 22 ,xu nn xu ,

Indirect Mechanism


Equilibrium of Induced Bayesian Game

iiiiii

iiiiiiiiiidii

SsSsNi

ssgussgu

,,,

))),(),((())),(),(((

A pure strategy profile is said to be dominant strategy equilibrium if

(.)(.),1dn

d ss

iiii

iiiiiiiiiiiiii

SsNi

ssguEssguEii

,,

]|))),(),((([]|))),(),((([ ***

)()(

A pure strategy profile is said to be Bayesian Nash equilibrium

(.)(.), **1 nss

Dominant Strategy-equilibrium Bayesian Nash- equilibrium

Dominant Strategy Equilibrium (DSE)

Bayesian Nash Equilibrium (BNE)

Observation


Implementing an SCF

We say that mechanism implements SCF in dominant strategy equilibrium if

),,( ),,()(),( 1111 nnndn

d fssg

NiiCgM )((.), Xf :

),,( ),,()(),( 11*

1*1 nnnn fssg

We say that mechanism implements SCF in Bayesian Nash equilibrium if

NiiCgM )((.), Xf :

Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic

Theory”, Oxford University Press, New York, 1995.

Dominant Strategy-implementation Bayesian Nash- implementation Observation

Bayesian Nash Implementation

Dominant Strategy Implementation


Properties of an SCF Ex Post Efficiency

For no profile of agents’ type does there exist an

such that and for some

n ,,1 Xx

ifuxu iiii ),(, iiii fuxu ),(, i

Dominant Strategy Incentive Compatibility (DSIC)

Bayesian Incentive Compatibility (BIC)

Nis iiiidi ,,)(

If the direct revelation mechanism has a dominant

strategy equilibrium in which

NiifD )((.),

(.))(.),( 1dn

d ss

Nis iiiii ,,)(*

If the direct revelation mechanism has a Bayesian

Nash equilibrium in which

NiifD )((.),(.))(.),( **

1 nss


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


Social Choice Function

where,


Values and Payoffs

Quasi-linear Utilities


11 Xu : 22 Xu : nn Xu :

Xx

1 2 n

2 n1(.)1 (.)2 (.)n

Policy Maker

Quasi-Linear Environment

011

iiin tnitKkttkX ,,, ,|),,,(

11111 tkvxu ),(),(

project choice Monetary transfer to agent 1

Valuation function of agent 1


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)

SCF is AE if for each , satisfies(.)),(.),(.),((.) nttkf 1 )(k

n

iii

Kkkvk

1

),(maxarg)(

SCF is BB if for each , we have(.)),(.),(.),((.) nttkf 1

01

n

iit )(

Lemma 1An SCF is ex post efficient in quasi-linear

environment iff it is AE + BB

(.)),(.),(.),((.) nttkf 1


A Dominant Strategy Incentive Compatible Mechanism

1. Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.

2. Let the payments be :

Groves Mechanism


VCG Mechanisms (Vickrey-Clarke-Groves)

Vickrey Vickrey AuctionAuction

Generalized Vickrey Generalized Vickrey AuctionAuction

Clarke MechanismsClarke Mechanisms

Groves MechanismsGroves Mechanisms

• Allocatively efficient, Allocatively efficient, individual rational, individual rational, and and dominant strategy incentive compatible with quasi-dominant strategy incentive compatible with quasi-linear utilities.linear utilities.


A Bayesian Incentive Compatible Mechanism

1. Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.

2. Let types of the agents be statistically independent of one another

3.

dAGVA Mechanism


BIC

AE

WBB

IR

SBB

dAGVA

DSIC

EPE

GROVES

MOULIN


CASE STUDY


Casting stage Machining Stage

Transportation stage

C1 C4 M1 M5 T1 T6

… … …

X1 X2 X3

Manager 1

Manager 2 Manager 3

SCP

X=X1+X2+X3

Mechanism Design and Optimization

111 ,

222 , 333 ,

)),,(( *1

*1

*1 )),,(( *

2*2

*2

)),,(( *3

*3

*3

)),,(( 111111 c )),,(( 141414 c )),,(( 212121 c )),,(( 252525 c)),,(( 313131 c )),,(( 363636 c


Information Provided by Service Providers

Partner Id Mean Standard Deviation

Cost

P11 3 1.0 105

P12 3 1.5 70

P13 2 0.5 55

P14 2 1.0 40


Cost

P21 3 0.75 35

P22 2 1.00 40

P23 2 1.25 35

P24 2 0.75 50

P25 1 1.00 70

Information for Manager 1 Information for Manager 2


Contd..


Cost

P31 1 0.25 20

P32 1 0.5 15

P33 2 0.75 12

P34 2 1.00 10

P35 2 1.25 9

P36 2 1.50 8

Information for Manager 3


Centralized Framework

Echelon Payment

1 2 0.3 88.2

2 1 0.2161 101.00

3 1 0.1481 42.0

*i

*i

*i

Solution of the Mean Variance Allocation Optimization problem in a centralized setting


Solutions in the Mechanism Design Setting

Echelon Payment

1 257.00

2 269.80

3 210.80

Echelon Payment

1 14.30

2 13.02

3 19.00

Echelon Payment

1 71.5

2 65.10

3 94.60

SCF-DSIC

Echelon Payment

1 128.70

2 117.18

3 170.28

Echelon Payment

1 143.00

2 130.20

3 189.20

SCF- BIC with belief Probability 0.5SCF- BIC with belief Probability 0.5

SCF- BIC with belief Probability 0.9 SCF- BIC with belief Probability 1.0


Future Work…

• Non-Linear Supply Chains• Deeper Mechanism Design Solutions• Cooperative Game Approach


To probe further…

• Y. Narahari, N. Hemachandra, Nikesh Srivastava. Incentive Compatible Mechanisms for Decentralized Supply Chain Formation. IEEE CEC 2007.

• Y. Narahari, Dinesh Garg, Rama Suri, and Hastagiri Prakash. Emerging Game Theoretic Problems in Network Economics: Mechanism Design Solutions, Springer , To appear: 2007

• Andreu Mascolell, Michael Whinston, and Jerry Green. Microeconomic Theory. Oxford University Press, 1995

• Roger B. Myerson. Game Theory: Analysis of Conflict. Harvard University Press, 1997.


Questions and Answers …

Thank You …


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 1 incentive compatible mechanisms for supply chain formation y. narahari...

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