e-commerce lab, csa, iisc 1 the core and shapley value analysis of cooperative formation of supply...
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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
E-Commerce Lab, CSA, IISc2
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
E-Commerce Lab, CSA, IISc3
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
E-Commerce Lab, CSA, IISc4
Cold Rolling Pickling Slitting Stamping
MasterCoil
1
2
6
2
3
7
3
4
4
5
6
Suppliers
7
Supply Network for Automotive Stampings
E-Commerce Lab, CSA, IISc5
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
E-Commerce Lab, CSA, IISc6
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.
E-Commerce Lab, CSA, IISc7
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
E-Commerce Lab, CSA, IISc8
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
E-Commerce Lab, CSA, IISc9
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
E-Commerce Lab, CSA, IISc10
Example 2: The Supply Chain Network Formation Problem
1X 2X3X 4X
n
iiXY
1
Supply Chain Planner
Echelon Manager
E-Commerce Lab, CSA, IISc11
The Supply Chain Network Formation Problem
1X 2X3X 4X
n
iiXY
1
Supply Chain Planner
Echelon Manager
E-Commerce Lab, CSA, IISc12
Complete Information Version
• Choose means and standard deviations of individual stages so as to :
subject to
A standard optimization problem (NLP)
E-Commerce Lab, CSA, IISc13
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
E-Commerce Lab, CSA, IISc14
BIC
AE
WBB
IR
SBB
dAGVA
DSIC
EPE
GROVES
MOULIN
E-Commerce Lab, CSA, IISc15
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
E-Commerce Lab, CSA, IISc16
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”
E-Commerce Lab, CSA, IISc17
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
E-Commerce Lab, CSA, IISc18
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
E-Commerce Lab, CSA, IISc19
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
E-Commerce Lab, CSA, IISc20
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
E-Commerce Lab, CSA, IISc21
Divide the Dollar : Version 2
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
E-Commerce Lab, CSA, IISc22
Divide the Dollar : Version 3
Either 0 and 1 should propose the same allocation or 0 and 2
should propose the same allocation
Characteristic Function
300})2,1,0({})2,0({})1,0({
0})2,1({})2({})1({})0({
vvv
vvvv
otherwise
or if
0
),,(),,(),,( 2102021010210
xxxssxxxssxsssu ii
E-Commerce Lab, CSA, IISc23
Divide the Dollar : Version 4
It is enough any pair of players has the same proposal
Also called the Majority Voting Game
Characteristic Function
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
E-Commerce Lab, CSA, IISc24
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
E-Commerce Lab, CSA, IISc25
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
E-Commerce Lab, CSA, IISc26
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!
E-Commerce Lab, CSA, IISc27
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
E-Commerce Lab, CSA, IISc28
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)
E-Commerce Lab, CSA, IISc29
The MPNF Game
(Multi-Unit Procurement Network Formation Game)
E-Commerce Lab, CSA, IISc30
Cold Rolling Pickling Slitting Stamping
MasterCoil
1
2
6
2
3
7
3
4
4
5
6
Suppliers
7
Supply Network for Automotive Stampings
E-Commerce Lab, CSA, IISc31
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
E-Commerce Lab, CSA, IISc32
Supplier-ID,Per unit-cost,Capacity
Master CoilStage
Cold-RolledStage
• Capacity is specified as an upper-bound on the flow along the edge.
E-Commerce Lab, CSA, IISc33
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
E-Commerce Lab, CSA, IISc34
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
E-Commerce Lab, CSA, IISc35
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
E-Commerce Lab, CSA, IISc36
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
E-Commerce Lab, CSA, IISc37
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
E-Commerce Lab, CSA, IISc38
• (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
E-Commerce Lab, CSA, IISc39
Core and Shapley Value Analysis
Effect of Suppliers’ Profile
Effect of Demand
Effect of buyer’s valuation
E-Commerce Lab, CSA, IISc40
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
E-Commerce Lab, CSA, IISc41
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
E-Commerce Lab, CSA, IISc42
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
E-Commerce Lab, CSA, IISc43
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
;
),(),,,,(
E-Commerce Lab, CSA, IISc44
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
E-Commerce Lab, CSA, IISc45
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.
E-Commerce Lab, CSA, IISc46
• 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
E-Commerce Lab, CSA, IISc47
• 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
E-Commerce Lab, CSA, IISc48
• 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
E-Commerce Lab, CSA, IISc49
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.
E-Commerce Lab, CSA, IISc50
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
E-Commerce Lab, CSA, IISc51
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
E-Commerce Lab, CSA, IISc52
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
E-Commerce Lab, CSA, IISc53
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
E-Commerce Lab, CSA, IISc54
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
E-Commerce Lab, CSA, IISc55
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
}),({}){\()(
E-Commerce Lab, CSA, IISc56
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)
E-Commerce Lab, CSA, IISc57
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.
E-Commerce Lab, CSA, IISc58
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
E-Commerce Lab, CSA, IISc59
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
E-Commerce Lab, CSA, IISc60
Questions and Answers …
Thank You …