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 Lab
Computer Science and Automation,
Indian Institute of Science, Bangalore
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E-Commerce Lab, CSA, IISc
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
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E-Commerce Lab, CSA, IISc
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.
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E-Commerce Lab, CSA, IISc
The Supply Chain Network Formation Problem
Supply Chain
Planner
X2
X1
X3
X4
n
Y   Xi
Echelon
Manager
4
i 1
E-Commerce Lab, CSA, IISc
Forming a 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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Some Observations
Players
are rational and intelligent
Some of the information is
common knowledge
Conflict and cooperation
are 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
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E-Commerce Lab, CSA, IISc
Simple Example: The Supply Chain Partner
Selection Problem
SCP
EM1
A
B
EM2
C
A
B
C
Let us say it is required to select the same partner at the
two stages
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Preference Elicitation Problem
Supply Chain
Planner
x2: C>B>A
x1: A>B>C
y2: B>C>A
Echelon
Manager 1
Echelon
Manager 2
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 type
3. If its type is y2, manager 2 would wish to lie
4. How do we make the managers report their true types?
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E-Commerce Lab, CSA, IISc
Current Art
9

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
E-Commerce Lab, CSA, IISc
Complete Information Version
• Choose means and standard deviations of
individual stages so as to :
subject to
A standard optimization problem (NLP)
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Incomplete Information Version
Supply Chain
Planner
1.
2.
3.
4.
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Type Set 1
Type Set 2
Echelon
Manager 1
Echelon
Manager 2
How to transform individual preferences into social decision (SCF)?
How to elicit truthful individual preferences (Incentive Compatibility) ?
How to ensure the participation of an individual (Individual Rationality)?
Which social choice functions are realizable?
E-Commerce Lab, CSA, IISc
Strategic form Games
S1
Sn
N = {1,…,n}
Players
S1, … , Sn
Strategy Sets
U1 : S
R
Un : S
R
Payoff functions
(Utility functions)
S = S1 X … X Sn
• Players are rational : they always strive to maximize their individual payoffs
• Players are intelligent : they can compute their best responsive strategies
• Common knowledge
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Example 1: Matching Pennies
(1,-1)
(-1,1)
(-1,1)
(1,-1)
• Two players simultaneously put down a coin, heads up or tails up.
Two-Player zero-sum game
S1 = S2 = {H,T}
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Example 2: Prisoners’ Dilemma
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E-Commerce Lab, CSA, IISc
Example 3: Hawk - Dove
H
Hawk
D
Dove
H
Hawk
0,0
20,5
D
Dove
5,20
10,10
2
1
Models the strategic conflict when two players are fighting over a
company/territory/property, etc.
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Example 4: Indo-Pak Conflict
Pak Healthcare
Defence
India
Healthcare
10,10
-10, 20
20, -10
0,0
Defence
Models the strategic conflict when two players
have to choose their priorities
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Example 5: Coordination
• In the event of multiple equilibria, a certain
equilibrium becomes a focal equilibrium based
on certain environmental factors
College
MG Road
100,100
0,0
0,0
5,5
College
MG Road
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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
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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

*
1
,  2* ,...,  n*

is a mixed strategy Nash equilibrium if
 i* is a best response against  *i
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,
i  1,2,..., n
E-Commerce Lab, CSA, IISc
John von Neumann
(1903-1957)
Founder of Game theory with Oskar Morgenstern
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John F Nash Jr.
(1928 - )
Landmark contributions to Game theory: notions of Nash
Equilibrium and Nash Bargaining
Nobel Prize : 1994
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John Harsanyi
(1920 - 2000)
Defined and formalized Bayesian Games
Nobel Prize : 1994
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Reinhard Selten
(1930 - )
Founding father of experimental economics and
bounded rationality
Nobel Prize : 1994
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Thomas Schelling
(1921 - )
Pioneered the study of bargaining and strategic behavior
Nobel Prize : 2005
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Robert J. Aumann
(1930 - )
Pioneer of the notions of common knowledge,
correlated equilibrium, and repeated games
Nobel Prize : 2005
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Lloyd S. Shapley
(1923 - )
Originator of “Shapley Value” and Stochastic Games
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William Vickrey
(1914 – 1996 )
Inventor of the celebrated Vickrey auction
Nobel Prize : 1996
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Roger Myerson
(1951 - )
Fundamental contributions to game theory,
auctions, mechanism design
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MECHANISM DESIGN
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Underlying Bayesian Game
N = {0,1,..,n}
0 : Planner
1,…,n: Partners
Type sets
Private Info: Costs
S0,S1,…,Sn
Strategy Sets
Announcements
Payoff functions
A Natural Setting for Mechanism Design
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E-Commerce Lab, CSA, IISc
Mechanism Design Problem
O<M<L
L<O<M
M<L<O
Yuvraj
Dravid
Laxman
Greg
O: Opener
M: Middle-order
L: Late-order
1. How to transform individual preferences into social decision?
2. How to elicit truthful individual preferences ?
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E-Commerce Lab, CSA, IISc
The Mechanism Design Problem
 n agents who need to make a collective choice from outcome set X

Each agent
i privately observes a signal  i which determines i ' s preferences
over the set X

Signal  i is known as agent

The set of agent i ' s possible types is denoted by 
i

The agents types,   1,,n  are drawn according to a probability
distribution function (.)

Each agent is rational, intelligent, and tries to maximize its utility function
i' s
type.
ui : X  i  

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(.), 1,, n ,u1(.), ,un (.) are common knowledge among the agents
E-Commerce Lab, CSA, IISc
Social Choice Function and
Mechanism
θ1
S1
θn
Sn
Outcome Set
Outcome Set
f(θ1, …,θn)
Є
X
x = (y1(θ), …, yn(θ), t1(θ), …,
tn(θ))
g(s1(.), …,sn()
Є
X
(S1, …, Sn, g(.))
A mechanism induces a Bayesian game and is designed to implement
a social choice function in an equilibrium of the game.
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Two Fundamental Problems in
Designing a Mechanism
 Preference Aggregation Problem
For a given type profile
 1,,n of the agents, what outcome x  X
should be chosen ?
 Information Revelation (Elicitation) Problem
How do we elicit the true type
i
of each agent i , which is his
private information ?
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Information Elicitation Problem
n
2
1
2
1
ˆ1
n
ˆ2
ˆn
f : 1   n  X
xX
35

x  f ˆ1,,ˆn

u1 : X  1  
u2 : X  2  
un : X  n  
u1x,1 
u2 x, 2 
un x,n 
E-Commerce Lab, CSA, IISc
Preference Aggregation Problem (SCF)
n
2
1
2
1
1
n
2
n
f : 1  n  X
xX
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x  f 1,,n 
E-Commerce Lab, CSA, IISc
Indirect Mechanism
1
C1
2
C2
2
1
c1
n
Cn
n
c2
cn
g : C1  Cn  X
xX
37
x  g c1,, cn 
u1 : X  1  
u2 : X  2  
un : X  n  
u1x,1 
u2 x, 2 
un x,n 
E-Commerce Lab, CSA, IISc
Equilibrium of Induced Bayesian Game
 Dominant Strategy Equilibrium (DSE)
A pure strategy profile s1d (.),  snd (.)  is said to be dominant strategy
equilibrium if
ui ( g ( sid (i ), si ( i )), i )  ui ( g ( si (i ), si ( i )), i )
i  N ,i  i , si  Si , si  S i
 Bayesian Nash Equilibrium (BNE)
A pure strategy profile s1* (.),  sn* (.)  is said to be Bayesian Nash
equilibrium
E (  i ) [ui ( g ( si* ( i ), s*i ( i )),  i ) |  i ]  E (  i ) [ui ( g ( si ( i ), s*i ( i )),  i ) |  i ]
i  N ,  i  i , si  Si
 Observation
Dominant Strategy-equilibrium
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Bayesian Nash- equilibrium
E-Commerce Lab, CSA, IISc
Implementing an SCF
 Dominant Strategy Implementation
We say that mechanism M  g (.), (Ci )iN implements SCF f :   X
in dominant strategy equilibrium if


g s1d (1 ),  snd ( n )  f (1 , , n )
(1 ,, n )
 Bayesian Nash Implementation
We say that mechanism M  g (.), (Ci )iN  implements SCF f :   X
in Bayesian Nash equilibrium if


g s1* (1 ),  sn* ( n )  f (1 ,, n )
 Observation
Dominant Strategy-implementation
(1 ,, n )
Bayesian Nash- implementation
Andreu Mas Colell, Michael D. Whinston, and Jerry R. Green, “Microeconomic
Theory”, Oxford University Press, New York, 1995.
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E-Commerce Lab, CSA, IISc
Properties of an SCF
 Ex Post Efficiency
For no profile of agents’ type   1,,n  does there exist an
xX
such that ui x,i   ui f ( ),i   i and ui x,i   ui f ( ),i  for some
i
 Dominant Strategy Incentive Compatibility (DSIC)
If the direct revelation mechanism D   f (.), (i )iN  has a dominant
strategy equilibrium
( s1d (.),  snd (.)) in which
sid ( i )   i ,  i  i , i  N
 Bayesian Incentive Compatibility (BIC)
If the direct revelation mechanism D   f (.), (i )iN  has a Bayesian
Nash equilibrium ( s1* (.),  sn* (.)) in which
si* ( i )   i ,  i  i , i  N
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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
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E-Commerce Lab, CSA, IISc
Social Choice Function
where,
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Values and Payoffs
Quasi-linear Utilities
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Quasi-Linear Environment
u1( x,1 )  v1(k,1 )  t1
u1 : X  1  
1(.)
1
1
Valuation function of agent 1
u2 : X  2  
 2 (.)
2
2
un : X  n  
n (.)
n
n
xX
Policy Maker


X  (k, t1,, t n ) | k  K , t i   i  1,, n,  t i  0
i


project choice
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Monetary transfer to agent 1
E-Commerce Lab, CSA, IISc
Properties of an SCF in Quasi-Linear
Environment
 Ex Post Efficiency
 Dominant Strategy Incentive Compatibility (DSIC)
 Bayesian Incentive Compatibility (BIC)
 Allocative Efficiency (AE)
SCF f (.)  (k (.), t1(.), , t n (.)) is AE if for each    , k ( ) satisfies
n
k ( )  arg max  v i (k, i )
kK
i 1
 Budget Balance (BB)
SCF f (.)  (k (.), t1(.), , t n (.)) is BB if for each    , we have
n
 t ( )  0
i
 Lemma 1
i 1
An SCF f (.)  (k (.), t1(.), , t n (.)) is ex post efficient in quasi-linear
environment iff it is AE + BB
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E-Commerce Lab, CSA, IISc
A Dominant Strategy Incentive Compatible
Mechanism
1. Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be
allocatively efficient.
2. Let the payments be :
Groves Mechanism
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VCG Mechanisms (Vickrey-Clarke-Groves)
Groves Mechanisms
Clarke Mechanisms
Generalized Vickrey Auction
Vickrey Auction
• Allocatively efficient, individual rational, and dominant
incentive compatible with quasi-linear utilities.
47
strategy
E-Commerce Lab, CSA, IISc
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
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WBB
SBB
EPE
dAGVA
MOULIN
GROVES
AE
DSIC
BIC
IR
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CASE STUDY
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SCP
Mechanism Design
and Optimization
Manager 1
 1 1 , 1 
(( 1* ,  1* ),  1* )
Manager 2
Casting stage
C4
C1
 2 2 , 2 
((  3* ,  3* ),  3* )




,

3
3
3
Manager 3 Transportation
((  2* ,  2* ),  2* )
…
(( 14 ,  14 ), c14 )
X1
M5
M1
…
(( 11 , 11 ), c11 )
stage
Machining Stage
((  21 ,  21 ), c21 )
T1
T6
…
((  25 ,  25 ), c25 )
X2
((  31 ,  31 ), c31 )
((  36 ,  36 ), c36 )
X3
X=X1+X2+X3
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Information Provided by Service Providers
Partner Id
Mean
Standard
Deviation
P11
3
1.0
105
P12
3
1.5
70
P13
2
0.5
55
P14
2
1.0
40
Information for Manager 1
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Partner Id
Mean
Standard
Deviation
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
Cost
Information for Manager 2
E-Commerce Lab, CSA, IISc
Contd..
Partner Id
Mean
Standard
Deviation
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
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E-Commerce Lab, CSA, IISc
 i*
Centralized Framework

Echelon

*
i
*
i
Payment
1
2
0.3
88.2
2
1
0.2161
101.00
3
1
0.1481
42.0
Solution of the Mean Variance Allocation
Optimization problem in a centralized setting
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E-Commerce Lab, CSA, IISc
Solutions in the Mechanism Design Setting
Echelon
Payment
Echelon
Payment
1
257.00
1
14.30
2
269.80
2
13.02
3
210.80
3
19.00
SCF- BIC with belief Probability 0.5
SCF-DSIC
Echelon
Payment
Echelon
Payment
1
128.70
1
143.00
2
117.18
2
130.20
3
170.28
3
189.20
SCF- BIC with belief Probability 0.9
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Echelon
Payment
1
71.5
2
65.10
3
94.60
SCF- BIC with belief Probability 0.5
SCF- BIC with belief Probability 1.0
E-Commerce Lab, CSA, IISc
Future Work…
• Non-Linear Supply Chains
• Deeper Mechanism Design Solutions
• Cooperative Game Approach
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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.
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Questions and Answers …
Thank You …
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Game Theory
• Mathematical framework for rigorous study of conflict
and cooperation among rational, intelligent agents
Market
Buying Agents
(rational and
intelligent)
59
Selling Agents
(rational and
intelligent)
E-Commerce Lab, CSA, IISc