Shapley Value Analysis of
Cooperative Formation of
Procurement Networks
Y. Narahari
Computer Science and Automation,
Indian Institute of Science, Bangalore
Joint Work with
T.S. Chandrashekar, GM ISL, Bangalore
September 2007
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E-Commerce Lab, CSA, IISc
OUTLINE
1. Supply Chain Formation Problem: Motivation and Approaches
2. Cooperative Games in Characteristic Form

The Shapley Value
3. The Multi-Unit Procurement Network Formation (MPNF) Game
4. Results on Shapley Value Analysis
5. Conclusions and Future Work
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Supply Network for Automotive
Stampings
Master
Coil
Cold
Rolling
Pickling
Slitting
Stamping
1
2
3
4
2
3
4
5
6
7
6
7
Suppliers
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Procurement Network Formation: Why

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.
Global Supply Chains Vendor
with
Management
specialized suppliers Programs
Network Formation
is critical to Supply Chain Planning
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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.


Construction Industry.


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.
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
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game theoretic techniques to “Procurement Network
Formation”
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Cooperative Game with Transferable Utilities
(TU Games)
T  ( N , v)
N  {0,1,..., n} set of players
v : 2   characteristic function ; v( )  0
N
C  N is called a coalition.
There are 2 | N |  1 possible coalitions
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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 !

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Cooperative game theory offers several solution concepts:

The Core

Shapley Value

Kernel

Nucleolus
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Divide the Dollar Game
There are three players who have to share 300 dollars.
Each one proposes a particular allocation of dollars to
players.
N  {0,1,2}
S 0  S1  S 2  {( x0 , x1 , x2 )  3 : x1  0; x2  0; x3  0;
x1  x2  x3  300}
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Divide the Dollar : Version 1

The allocation is decided by what is proposed by player 0
ui ( s0 , s1 , s2 )  xi
0
if s0  ( x0 , x1 , x2 )
otherwise

Apex Game or Monopoly Game

Characteristic Function
v({0})  300
v({1})  v({2})  v({1,2})  0
v({0,1})  v({0,2})  v({0,1,2})  300
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Divide the Dollar : Version 2

It is enough 0 and 1 propose the same allocation
ui ( s0 , s1 , s2 )  xi
0

if s0  s1  ( x0 , x1 , x2 )
otherwise
Players 0 1nd 1 are equally powerful; Characteristic Function is:
v({0})  v({1})  v({2})  0
v({0,1})  300
v({0,2})  v({1,2})  0
v({0,1,2})  300
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Divide the Dollar : Version 3

Either 0 and 1 should propose the same allocation or 0 and 2
should propose the same allocation
ui (s0 , s1 , s2 )  xi
0

if s0  s1  ( x0 , x1 , x2 ) or s0  s2  ( x0 , x1 , x2 )
otherwise
Characteristic Function
v({0})  v({1})  v({2})  v({1,2})  0
v({0,1})  v({0,2})  v({0,1,2})  300
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Divide the Dollar : Version 4

It is enough any pair of players has the same proposal
ui ( s0 , s1 , s2 )  xi
if s0  s1  ( x0 , x1 , x2 )
or s0  s2  ( x0 , x1 , x2 )
or s1  s2  ( x0 , x1 , x2 )
0
otherwise

Also called the Majority Voting Game

Characteristic Function
v({0})  v({1})  v({2})  0
v({0,1})  v({0,2})  v({1,2})  v({0,1,2})  300
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Shapley Value

Shapley (1950)

Gives a unique payoff allocation for a TU game (N, v) that
describes a fair way of dividing the gains from cooperation.

Axiomatically developed
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
Symmetry

Carrier

Linearity
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Shapley Value : Expression

Given : A Coalitional Game - (N , v)

Shapley value
 (v)  ( 0 (v),...,  n (v))
where
 i (v ) 
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| C |!(| N |  | C | 1)!
{v(C  {i})  v(C )}

| N |!
C  N i
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Shapley Value: Examples
Version of Divide-the-Dollar
Shapley Value

Version 1
(300,0,0)

Version 2
(150,150,0)

Version 3
(200,50,50)

Version 4
(100,100,100)
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The MPNF Game
(Multi-Unit Procurement Network
Formation Game)
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Supply Network for Automotive
Stampings
Master
Coil
Cold
Rolling
Pickling
Slitting
Stamping
1
2
3
4
2
3
4
5
6
7
6
7
Suppliers
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Procurement Feasibility Graph
Cold – Rolled
Stage
Post-Picking
Stage
Post Slitting
Stage
Stamped
Stage
Finished
Stage
S
T
Master
Coil
Stage
• Each vertex represents an intermediate state of the stamping
• Each edge represents a value adding operation
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Master Coil
Stage
Supplier-ID,
Per unit-cost,
Capacity
ColdRolled
Stage
• Capacity is specified as an upper-bound on
the flow along the edge.
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MPNF Situation
  (G, N , , b, d )
G  Procuremen t feasibility Graph  (V,E )
N  Set of Suppliers
 : E  N edge ownership mapping
b  Valuation of the buyer for a single unit of the item
d  number of units demanded by the buyer
An MPNF situation leads to an MPNF game in
characteristic form, (N,v)
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MPNF Problem
G  (V , E ) ; N  Set of Suppliers
c(e)  cost per unit flow of edge e
l (e)  minimum flow along edge e
u (e)  maximum flow along edge e
f (e)  flow on edge e
 (e)  Supplier owning edge e
I ( j )  Set of incoming edges at vertex j
O ( j )  Set of outgoing edges at vertex j
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b
 valuation of the buyer for a single unit
C
of the item
 Coalition of the suppliers
EC
 Set of edges owned by Suppliers in
coalition C
FC
 Flow in the netwok from S to T
using only the edges in EC
 ( FC )  Suppliers in C who facilitate the flow FC
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d
 quantity demanded by the buyer
x
 decision variable, actual flow from S to T
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Objective Function
Maximize the surplus v(C) for C = N
where
v(C )  bx 
 f ( e) c ( e)
eEC
bx
 Total value to buyer from a flow x
 f (e)c(e)  Total cost to the suppliers in C to achieve the
eEC
required flow
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Constraints
(1) Flow conservati on constraint s
 f(e)   f(e)
eI ( j )  Ec
j  V \ {S , T }
eO ( j )  Ec
(2) Flow constraint s
 f(e)  x   f(e)
eI (T )  Ec
eO ( S )  Ec
(3) Demand constraint s
0 xd
(4) Lower and upper bounds on flow
l (e)  f (e)  u (e) e  EC
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MPNF Cooperative Game
• (N, v)
• N = Set of all suppliers = {1,2,…,n}
• v(C) = Maximum flow that can be derived using
suppliers in coalition C
CN
Immediate Questions
• What is the core of (N, v) ?
• What is the Shapley value of (N, v)?
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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
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An example to show implications of using Shapley value
Sample procurement networks
(4, [0,2])
T
s
t
(3, [0,2])
b = 5, d=2
a
(2, [0,2])
s
L
(3, [0,2])
a
(2, [0,2])
t
(2, [0,2])
s
(2, [0,2])
R
(3, [0,2])
b
(0, [0,2])
b = 5, d=2
t
b = 5, d=2
Agent 1
Agent 2
27 3
Agent
(4, [0,2])
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Characteristic function and Shapley value allocations
Example cont’d:
Network T
S
Network L
Value
v(S)
S
Value
v(S)
{i}
0
{i}
0
{1,2}
0
{1,2}
0
{1,B}
4
{1,B}
4
{2,B}
2
{2,B}
0
{1,2,B}
4
{1,2,B}
4
1(N,v) = 4/3
1(N,v) = 2
2(N,v) = 2/6
2(N,v) = 0
B(N,v) = 14/6
B(N,v) = 2
Network R
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) = 9/6; 2(N,v) = 1/6
3(N,v) = 5/6; B(N,v) = 9/6
Makes allocations to agents who are not critical to network formation
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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.,
vb ( N )  vb ( N \ {i})  vb ({i}), i  N
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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., vb(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)
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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.
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Implementing the Shapley value of the MPNF game
A high level description of the extensive form game
• Actions in Stage 1:
• Agents make their choices simultaneously.
• Every agent chooses an action from the action set given by
•
A  {(bi | bi  R ; bij  R}
1
|N|
Net bid of agent i
•
Bi 

jN \{ i}
bij 
b
jN \{ i}
Value that agent i is willing to
transfer to agent j in order to
become the proposer
|N| dimensional bid vector of
agent i
ji
• Agent with highest net bid chosen as the proposer 
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Implementing the Shapley value of the MPNF game
A high level description of the extensive form game
• Actions in Stage 2:
• Agent  makes an offer (N, ) to form the coalition N and implement the
outcome 
• Outcome consists of a reassignment of edge capacities and division of
surplus among the agents in the coalition.
• Actions in Stage 3: All agents other than the proposer either accept or reject the
offer made by agent .
• If offer is accepted than edge capacities and surplus is transferred as agreed
otherwise the game is replayed without agent .
• Result: This extensive form game implements the Shapley value of the MPNG
game in sub-game perfect Nash equilibrium.
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Conclusion and Future work

Investigated the interrelation between using the Shapley value as a solution
concept for the MPNF game and the ownership structure of the procurement
network.

Also, developed a protocol to implement the Shapley value of the MPNF
game.

Future work:
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
Examine other solution concepts such as the Kernel and Nucleolus for the
MPNF game.

Investigate the incomplete information versions of this game.
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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
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Questions and Answers …
Thank You …
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