E-Commerce Lab, CSA, IISc 1 Incentive Compatible Mechanisms for Supply Chain Formation Y. Narahari...
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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
E-Commerce Lab, CSA, IISc2
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
E-Commerce Lab, CSA, IISc3
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.
E-Commerce Lab, CSA, IISc4
The Supply Chain Network Formation Problem
1X 2X3X 4X
n
iiXY
1
Supply Chain Planner
Echelon Manager
E-Commerce Lab, CSA, IISc5
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
E-Commerce Lab, CSA, IISc6
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
E-Commerce Lab, CSA, IISc7
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
E-Commerce Lab, CSA, IISc8
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
E-Commerce Lab, CSA, IISc9
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
E-Commerce Lab, CSA, IISc10
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, IISc11
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, IISc12
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
E-Commerce Lab, CSA, IISc13
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)
E-Commerce Lab, CSA, IISc14
Example 2: Prisoners’ Dilemma
E-Commerce Lab, CSA, IISc15
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.
E-Commerce Lab, CSA, IISc16
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
E-Commerce Lab, CSA, IISc17
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
E-Commerce Lab, CSA, IISc18
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
E-Commerce Lab, CSA, IISc19
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
E-Commerce Lab, CSA, IISc20
John von Neumann(1903-1957)
Founder of Game theory with Oskar Morgenstern
E-Commerce Lab, CSA, IISc21
Landmark contributions to Game theory: notions of Nash Equilibrium and Nash Bargaining
Nobel Prize : 1994
John F Nash Jr. (1928 - )
E-Commerce Lab, CSA, IISc22
Defined and formalized Bayesian GamesNobel Prize : 1994
John Harsanyi (1920 - 2000)
E-Commerce Lab, CSA, IISc23
Reinhard Selten(1930 - )
Founding father of experimental economics and bounded rationalityNobel Prize : 1994
E-Commerce Lab, CSA, IISc24
Pioneered the study of bargaining and strategic behaviorNobel Prize : 2005
Thomas Schelling(1921 - )
E-Commerce Lab, CSA, IISc25
Robert J. Aumann(1930 - )
Pioneer of the notions of common knowledge, correlated equilibrium, and repeated games
Nobel Prize : 2005
E-Commerce Lab, CSA, IISc26
Lloyd S. Shapley(1923 - )
Originator of “Shapley Value” and Stochastic Games
E-Commerce Lab, CSA, IISc27
Inventor of the celebrated Vickrey auction Nobel Prize : 1996
William Vickrey(1914 – 1996 )
E-Commerce Lab, CSA, IISc28
Roger Myerson(1951 - )
Fundamental contributions to game theory, auctions, mechanism design
E-Commerce Lab, CSA, IISc29
MECHANISM DESIGN
E-Commerce Lab, CSA, IISc30
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
E-Commerce Lab, CSA, IISc31
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
E-Commerce Lab, CSA, IISc32
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
E-Commerce Lab, CSA, IISc33
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
E-Commerce Lab, CSA, IISc34
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
E-Commerce Lab, CSA, IISc35
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
E-Commerce Lab, CSA, IISc36
1 2 n
2 n
1 2 n
1
Xf n 1:
Xx nfx ,,1
Preference Aggregation Problem (SCF)
E-Commerce Lab, CSA, IISc37
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
E-Commerce Lab, CSA, IISc38
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
E-Commerce Lab, CSA, IISc39
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
E-Commerce Lab, CSA, IISc40
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
E-Commerce Lab, CSA, IISc41
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
E-Commerce Lab, CSA, IISc42
Social Choice Function
where,
E-Commerce Lab, CSA, IISc43
Values and Payoffs
Quasi-linear Utilities
E-Commerce Lab, CSA, IISc44
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
E-Commerce Lab, CSA, IISc45
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
E-Commerce Lab, CSA, IISc46
A Dominant Strategy Incentive Compatible Mechanism
1. Let f(.) = (k(.),I0(.), I1(.),…, In(.)) be allocatively efficient.
2. Let the payments be :
Groves Mechanism
E-Commerce Lab, CSA, IISc47
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.
E-Commerce Lab, CSA, IISc48
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
E-Commerce Lab, CSA, IISc49
BIC
AE
WBB
IR
SBB
dAGVA
DSIC
EPE
GROVES
MOULIN
E-Commerce Lab, CSA, IISc50
CASE STUDY
E-Commerce Lab, CSA, IISc51
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
E-Commerce Lab, CSA, IISc52
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
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
Information for Manager 1 Information for Manager 2
E-Commerce Lab, CSA, IISc53
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
E-Commerce Lab, CSA, IISc54
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
E-Commerce Lab, CSA, IISc55
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
E-Commerce Lab, CSA, IISc56
Future Work…
• Non-Linear Supply Chains• Deeper Mechanism Design Solutions• Cooperative Game Approach
E-Commerce Lab, CSA, IISc57
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.
E-Commerce Lab, CSA, IISc58
Questions and Answers …
Thank You …
E-Commerce Lab, CSA, IISc59
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)