e-commerce lab, csa, iisc 1 the core and shapley value analysis of cooperative formation of supply...

E-Commerce Lab, CSA, IISc1

The Core and Shapley Value Analysis of Cooperative

Formation of Supply Chain NetworksY. Narahari

Computer Science and Automation,Indian Institute of Science, Bangalore

Joint Work withT.S. Chandrashekar, GM ISL, Bangalore

September 2007


OUTLINE

1. Supply Chain Formation Problem: Motivation and Approaches

2. Cooperative Games in Characteristic Form

The Core

The Shapley Value

3. The Multi-Unit Procurement Network Formation (MPNF) Game

4. Non-Emptiness of Core and other Results

5. Shapley Value Analysis

6. Conclusions and Future Work


Procure-ment

InboundLogistics

Assembly Outbound LogisticsCustomer

Orders

1

2

6

4

5

7

3

4

4

5

7

Supply Chain Partners

8

Supply Chain Formation Problem


Cold Rolling Pickling Slitting Stamping

MasterCoil

1

2

6

2

3

7

3

4

4

5

6

Suppliers

7

Supply Network for Automotive Stampings


Service 1 Service 2 Service 3 Service 4

Jobs

1

2

6

2

3

7

3

4

4

5

6

Service providers

7

Service Network Formation


Procurement Models Research: A Bird’s Eye View

Pro

du

ct A

ttri

bu

tes

System Attributes

Single item, Single attribute

Multi item, single attribute

Single item, configurableattributes

Businessconstraints

Costcomplementarities

Capacityconstraints

Single item, multi attribute

Multi item, multi attribute

Multi item, configurable attributes

Research and implementation focus on auction based models*

Auction models mostly geared towards scenarios involving adjacent echelons in the supply chain.

What is the focus?

NetworkAspects

Thistalk

* T.S. Chandrashekar, Y. Narahari, Charles H. Rosa, Devadatta Kulkarni, Jeffrey D. Tew and Pankaj Dayama. “Auction Based Mechanisms for Electronic Procurement”. IEEE Transactions on Automation Science and Engineering, 2007.


Procurement Network Formation: Why

Global Supply Chainswith

specialized suppliers

Vendor Management

Programs

Economic activity often involves inter relationships at multiple levels of production.

Supply chains are deep.

Individual entities in the supply chain are rational economic agents.

Implicit involvement in deciding the supplier’s supplier with a focus on quality and inventory.

Going further one can expect explicit involvement in price setting, capacity planning, etc.

Network Formationis critical to Supply Chain Planning


Approach 1: Optimization

• A global optimization approach is used to obtain an optimal mix of supply chain partners

• Mostly ILP and MIP formulations• Heuristic approaches are followed for solving

large scale problems

Does not take into account the rationality of the supply chain partners


William E. Walsh, Michael P. Wellman, and Fredrik Ygge. Combinatorial auctions for supply chain formation. In Second ACM Conference on Electronic Commerce, 2000.

William E Walsh. Market Protocols for Decentralized Supply Chain Formation. PhD Thesis, Michigan, Ann Arbor, 2002.

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

Dinesh Garg, Y. Narahari, Earnest Foster, Devadatta Kulkarni, Jeffrey Tew. Groves Mechanisms for Decentralized Supply Chain Formation. IEEE CEC 2005.

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

This approach is based on a non-cooperative game theory

These models have Vickrey type payments built into them which can be unacceptably high in a network context.

Approach 2: Mechanism Design


Example 2: The Supply Chain Network Formation Problem

1X 2X3X 4X

n

iiXY

1

Supply Chain Planner

Echelon Manager


The Supply Chain Network Formation Problem

1X 2X3X 4X

n

iiXY

1

Supply Chain Planner

Echelon Manager


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


BIC

AE

WBB

IR

SBB

dAGVA

DSIC

EPE

GROVES

MOULIN


The buyer and suppliers cooperate to form a supply network and the surplus generated is shared among the players

T. S. Chandrashekar. Procurement Network Formation: A Cooperative Game Theoretic Approach. Ph D Thesis, CSA, IISc, March 2007

Approach 3: Cooperative Game Approach


Procurement Network Formation: A Cooperative Approach

Anecdotal evidence suggests that negotiation and bargaining are key mechanisms to settle procurement contracts.

Industrial Electronics - IMEC to Texas Instruments.

Automotive Industry - Delphi and Lear to GM, Ford and DaimlerChrysler.

Automation Equipment - Symbol Technologies + Paxar to Home Dept and Walmart.

Nagarajan. M and Sosic. G. Game Theoretic Analysis of Cooperation among Supply Chain Agents: Review and Extensions. Technical Report. Sauder School of Business, Univ. of British Columbia, Vancouver, Canada, August 2005.

Construction Industry.

Bajari, P.L and McMillan, R S and Tadelis, S. Auctions versus Negotiations in Procurement: An Empirical Analysis. NBER Working Paper Series, Department of Economics, Stanford University, June 2003.

Negotiation and bargaining are at the heart of cooperative game theory.

Natural to apply “Negotiation and Bargaining” based mechanisms or cooperative

game theoretic techniques to “Procurement Network Formation”


Cooperative Game with Transferable Utilities (TU Games)

coalitions possible are There

. a called is

12

0)(2:

},...,1,0{

),(

||

N

N

NC

vv

nN

vNT

coalition

; functionsticcharacteri

players of set


Given a TU game, two central questions are:

Which coalition(s) should form ?

How should a coalition that forms divide the surplus among its

members ?

The second question has implications for answering the first question !

Cooperative game theory offers several solution concepts:

The Core

Shapley Value


Divide the Dollar GameThere are three players who have to share 300 dollars. Each one proposes a particular allocation of dollars to

players.

}300

;0;0;0:),,{(

}2,1,0{

210

3213

210210

xxx

xxxxxxSSS

N


Divide the Dollar : Version 1

The allocation is decided by what is proposed by player 0

Apex Game or Monopoly Game

Characteristic Function

300})2,1,0({})2,0({})1,0({

0})2,1({})2({})1({

300})0({

vvv

vvv

v

otherwise

if

0

),,(),,( 2100210

xxxsxsssu ii



It is enough 0 and 1 propose the same allocation

Players 0 1nd 1 are equally powerful; Characteristic Function is:

300})2,1,0({

0})2,1({})2,0({

300})1,0({

0})2({})1({})0({

v

vv

v

vvv

otherwise

if

0

),,(),,( 21010210

xxxssxsssu ii



Either 0 and 1 should propose the same allocation or 0 and 2

should propose the same allocation


300})2,1,0({})2,0({})1,0({

0})2,1({})2({})1({})0({

vvv

vvvv

otherwise

or if

0

),,(),,(),,( 2102021010210

xxxssxxxssxsssu ii



It is enough any pair of players has the same proposal

Also called the Majority Voting Game


300})2,1,0({})2,1({})2,0({})1,0({

0})2({})1({})0({

vvvv

vvv

otherwise

or

or

if

0

),,(

),,(

),,(),,(

21021

21020

21010210

xxxss

xxxss

xxxssxsssu ii


The Core

Core of (N, v) is the collection of all allocations (x0 , x1 ,…, xn) satisfying:

Individual Rationality

Coalitional Rationality

Collective Rationality

Ci

i CvxNC )( ,

},...,1,0{ nN Let

Niivxi })({

)(...10 Nvxxx n


The Core: Examples

Version of Divide-the-Dollar Core

Version 1

Version 2

Version 3

Version 4

)}0,0,300{(

}300;0;0:)0,,{( 101010 xxxxxx

{(300, 0, 0)}

Empty


Some Observations

If a feasible allocation x = ( x0 ,…, xn ) is not in the core, then there

is some coalition C such that the players in C could all do strictly

better than in x.

If an allocation x belongs to the core, then it implies that x is a

Nash equilibrium of an appropriate contract signing game, so

players are reasonably happy.

Empty core is bad news so also a large core!


Shapley Value : Expression

Provides a unique payoff allocation that describes a fair way of

dividing the gains of cooperation in a game (N, v)

iNCi

n

CviCvN

CNCv

vvv

)}(}){({|!|

)!1|||(||!|)(

))(),...,(()( 0

where


Shapley Value: Examples

Version of Divide-the-Dollar Shapley Value

Version 1

Version 2

Version 3

Version 4

(150,150,0)

(300,0,0)

(200,50,50)

(100,100,100)


The MPNF Game

(Multi-Unit Procurement Network Formation Game)


Cold Rolling Pickling Slitting Stamping

MasterCoil

1

2

6

2

3

7

3

4

4

5

6

Suppliers

7

Supply Network for Automotive Stampings


S T

• Each vertex represents an intermediate state of the stamping• Each edge represents a value adding operation

MasterCoil Stage

Cold – Rolled Stage

Post-Picking Stage

Post SlittingStage Stamped

Stage

FinishedStage

Procurement Feasibility Graph


Supplier-ID,Per unit-cost,Capacity

Master CoilStage

Cold-RolledStage

• Capacity is specified as an upper-bound on the flow along the edge.


buyer theby demanded units of number

item the of unit single a for buyer the of Valuation

mapping ownership edge

Suppliers of Set

)( Graphy feasibilit tProcuremen

d

b

NE

N

V,EG

dbNG

:

),,,,(

An MPNF situation leads to an MPNF game in characteristic form, (N,v)

MPNF Situation


jjO

jjI

ee

eef

eeu

eel

eec

NEVG

vertex at edges outgoing of Set

vertex at edges incoming of Set

edge owning Supplier

edge onflow

edge alongflow maximum

edge alongflow minimum

edge offlow unit per cost

Suppliers of Set

)(

)(

)(

)(

)(

)(

)(

;),(

MPNF Problem


TSx

d

FCF

E

TSF

C

E

C

b

CC

C

C

C

to fromflow actual variable,decision

buyer by the demandedquantity

flow thefacilitate who in Suppliers )(

in edges only the using

to fromnetwok thein Flow

coalition

in Suppliersby owned edges ofSet

suppliers theof Coalition

item theof

unit singlea for buyer theof valuation


Maximize the surplus v(C) for C = N

where

flow required

the achieve to in suppliers the to cost Total

flow a from buyer to value Total

Cecef

xbx

ecefbxCv

ECe

ECe

)()(

)()()(

Objective Function


C

ESOeETIe

EjOeEjIe

Eeeuefel

dx

f(e)xf(e)

TSVjf(e)f(e)

cc

cc

flow on bounds upper and Lower (4)

sconstraint Demand (3)

sconstraintFlow (2)

sconstraint onconservatiFlow

)()()(

0

},{\

)1(

)()(

)()(

Constraints


• (N, v)• N = Set of all suppliers = {1,2,…,n}• v(C) = Maximum flow that can be derived using

suppliers in coalition C

Immediate Questions• What is the core of (N, v) ?• What is the Shapley value of (N, v)?

NC

MPNF Cooperative Game


Core and Shapley Value Analysis

Effect of Suppliers’ Profile

Effect of Demand

Effect of buyer’s valuation


Bondereva-Shapley Characterization

players the of

called are LP above the of solutions optimal The

to subject

:LP the Consider

saspiration balanced

NCCvx

x

Cii

Nii

x n

)(

min


balanced. is it iff coreempty -non has gameA

,any for iff balanced be to said is gameA

to subject

:is LP this of dual The

),(

)()()(1)(

),(

1)(

)()(max

}{

}{

12

vN

NvCvCNiC

vN

NiC

CvC

NCiC

iC

NCn

Bondereva-Shapley Characterization


Profitable Flow

flow profitable one

least at is there if are and say We

if profitable be to said is flow A

trivial-non),(

0)(

),(),,,,(

vN

Cv

F

vNdbNG

c


Flow – Veto Supplier (Indispensable Supplier)

S

X

Y

Z

T

A,2 B,6

A,3 C,8

B,4 C,4

• S X T and S Z T are surplus maximizing floors • B is involved in both, hence is an f - veto agent• A and C are not f- veto agents

G

iNi

vNdbNG

inflow maximizing surplusevery in edge one least at

owns supplier the if agent veto -f an is

;

),(),,,,(


Effect of Suppliers’ Profile on the Non-Emptiness of the core

Lemma : In any core allocation, every non-f-veto agent gets zero payoff.

Theorem 1 : Let Nf = Set of f-veto agents. The game (N,v) has non-empty core iff

(1) Nf is non-empty and the game (Nf, vf) is balanced where vf(D) = v( D U (N \Nf))(2) Every profitable flow contains an edge owned by an f- veto agent


Managerial Implications of Theorem1

f- veto agents are the ones with bargaining power. The elements of the core reflect the relative bargaining power of the f-veto suppliers

If the buyer is made a part of the MPNF graph, then the core is always non-empty.

Helps the buyer to map out a strategy for supplier development so as to ensure stability of the network.

Advises the suppliers on what additional capabilities they need to get to move into the f-veto set.


• Nf = {2}

• Core = {(0,12,0)}

• B-Core = {(9+x, 0, 3-x, 0): x <=3}

S B

C

T

1,2 2,6

2,3 3,8

2,4 2,4

A

Z

b = 20d = 1

0,0


• Nf = Φ

• Core = Φ

• B-Core = {(12,0,0,0)}

S B

C

T

1,22,6

1,3 3,5

2,4 3,4

A

Z

b = 20d = 1

0,0


• Nf = {1,2}

• Core = {(x, 12-x): 0<= x <= 12}• B-Core = {(x, y, z) : x,y,z >= 0, x + y + z = 12}

S B

C

T

1,2 2,6

1,3 2,5

2,4 1,4

A

Z

b = 20d = 1

0,0


Effect of Demand

Theorem 2

*empty non is agents veto-f ofset that thesuch

0a exists there,),,,,( Given

dd

),(d* dbNG

ImplicationIf the buyer desires the formulation of a stable procurement network, then the demand has to be carefully chosen to be above some threshold.


Effect of Buyer’s ValuationTheorem 3:

b*b

,b*

,b

db(G,N,ψμ(N,v)

iff balanced is

game that thesuch ]0[a exists or there )2(

]0[ balancedeither (1)

is )1,, from arising game MPNFAn

Implication• The buyer needs to choose the budget carefully to ensure formation of a stable procurement network.• High budgets however could lead to instability due to protracted negotiations among the players


Implementation of the Core through an Extensive form

game• Theorem :

An extensive form game can be constructed that implements in sub-game perfect Nash equilibrium the core allocations of any given MPNF game with non-empty core


Shapley Value of MPNF Games

• Makes a positive allocation of surplus to agents who own edges in any flow that generates positive surplus

• Need to understand conditions under which the Shapley value makes positive allocation to only agents involved in surplus maximizing flows

• Need to understand conditions under which the Shapley value allocation is stable– Convex Games


An example to show implications of using Shapley value

Sample procurement networks

(2, [0,2])

b = 5, d=2

s s

s

t

t

t

a a

b

(2, [0,2])

(3, [0,2])

(3, [0,2])

(4, [0,2])

(2, [0,2]) (2, [0,2])

(3, [0,2]) (0, [0,2])

(4, [0,2])

b = 5, d=2

b = 5, d=2

Agent 1

Agent 2

Agent 3

T

L R


Characteristic function and Shapley value allocations

Example cont’d:

S Value v(S)

{i} 0

{1,2} 0

{1,B} 4

{2,B} 2

{1,2,B} 4

S Value v(S)

{i} 0

{1,2} 0

{1,B} 4

{2,B} 0

{1,2,B} 4

S Value v(S)

{i}, {1,2}, {1,3}, {2,3}, {2,B}, {3,B}, {1,2,3},

{2,3,B}

0

{1,B} 2

{1,2,B} 2

{1,3,B} 4

{1,2,3,B} 4

1(N,v) = 4/3

2(N,v) = 2/6

B(N,v) = 14/6

1(N,v) = 2

2(N,v) = 0

B(N,v) = 2

1(N,v) = 9/6; 2(N,v) = 1/6

3(N,v) = 5/6; B(N,v) = 9/6

Network T Network L Network R

Makes allocations to agents who are not critical to network formation


Characterization of Shapley value allocations and ownership structure

Definition 1: The set of all agents, denoted SM(F) who own edges

in a surplus maximizing flow F of the procurement graph are

called the SM-agents, i.e., SM = {i = ψ(e), i N: e ψ(F)}

associated with the surplus maximizing flow F.

Proposition: The characteristic function of the MPNF game with

the buyer included as an agent is zero-monotonic, i.e.,

NiiviNvNvbbb

}),({}){\()(


Characterization of Shapley value allocations and ownership structure (cont’d)

Theorem: If the Shapley value rule allocates all the surplus value

in the MPNF game only to agents i SM then for every flow FS

provided by a coalition S that includes an agent i SM, either:

1. FS is not profitable, i.e., vb(S) = 0 or

2. If FS is profitable, then we have (FS) SM ≠ 0, and there is a

set

SSM (FS) SM such that v(SSM) = v((FS)


Implementation of the Shapley value of the MPNF game

A non-cooperative way to the cooperative solution

• The Shapley value is an exogenous viewpoint of a cooperative scenario.

• We need a succinct game form that allows us to implement the Shapley value.

• What do we mean by “implement the Shapley value”?

• The Shapley value allocation vector must correspond to the Nash

equilibrium of a non-cooperative (extensive form) game.

• Implementation Theory has investigated this topic rigorously.

• See: J Moore, Implementation, contracts, and renegotiation in

environments with complete information. Chapter in Advances in

Economic Theory, VI World Congress of the Econometric Society.


Future Perspective

• Combinatorial Procurement situations• Multi-Commodity Network Flow Formulation• Non-linear Flows• Comparison with Non-cooperative approaches• Extension to Incomplete Information Scenarios• Other Solution Concepts for Cooperative Games


Key References

• T.S. Chandrashekar. Procurement Network Formation: A Cooperative Game Theoretic Approach. PhD Thesis, February 2007, CSA, IISc.

• Roger B. Myerson. Game Theory: Analysis of Conflict. Harvard University Press. 1998


Questions and Answers …

Thank You …

e-commerce lab, csa, iisc 1 the core and shapley value analysis of cooperative formation of supply...

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