Redistribution of VCG Payments in Assignment of ...
Transcript of Redistribution of VCG Payments in Assignment of ...
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution of VCG Payments inAssignment of Heterogeneous Objects
Sujit Prakash Gujar
Supervisor : Y [email protected]
E-Commerce LabDepartment of Computer Science and Automation
Indian Institute of Science, Bangalore-12
December 18, 2008
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 1 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Outline of the Talk
Introduction
State of the Art and Research Gaps
Proposed Solution
Experimental Analysis
Summary and Future Work
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 2 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
MotivationGreen Laffont Impossibility Theorem
Motivation
government body wants to allot p land properties among n of itsdifferent subdivisions
an university wants to allot spaces to departments
assignment of p resources among n of its users
assignment should be such that social welfare is maximized
we need true valuations of the agents for these objects
mechanism design comes into picture
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 3 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
MotivationGreen Laffont Impossibility Theorem
Acronyms
DSIC Dominant Strategy Incentive CompatibleAE Allocative Efficiency (Allocatively Efficient)BB Budget BalanceIR Individual RationalityVCG Vickrey-Clarke-Groves Mechanisms
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 4 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
MotivationGreen Laffont Impossibility Theorem
Green-Laffont Impossibility Theorem
Green-Laffont ImpossibilityTheorem [1]: AE + SBB +DSIC is not possible.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 5 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
Redistribution Mechanism
Laffont and Maskin [2] : redistribute the surplus among participatingagents
redistribute the surplus among the participating agents preservingallocative efficiency and DSIC, (Groves mechanism)
refer to it as redistribution mechanism
design an appropriate rebate function
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 6 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
State of the Art and Research Gaps
Cavallo [3]1 : rebate function that depends only on (p + 2) highestbids
Guo and Conitzer [5] : performance ratio of a mechanism as,
minθ∈Θ
Surplus redistributedVCG Surplus
Herve Moulin [6] : notion of efficiency loss,
L(n, p) = maxθ∈Θ
Budget SurplusEfficient Surplus
Guo and Conitzer [7] : designed mechanism which is optimal inexpected sense
1This scheme can be viewed as Bailey [4] scheme applied in the settingSujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 7 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
State of the Art and Research Gaps
Cavallo [3]1 : rebate function that depends only on (p + 2) highestbids
Guo and Conitzer [5] : performance ratio of a mechanism as,
minθ∈Θ
Surplus redistributedVCG Surplus
Herve Moulin [6] : notion of efficiency loss,
L(n, p) = maxθ∈Θ
Budget SurplusEfficient Surplus
Guo and Conitzer [7] : designed mechanism which is optimal inexpected sense
1This scheme can be viewed as Bailey [4] scheme applied in the settingSujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 7 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
State of the Art and Research Gaps
Cavallo [3]1 : rebate function that depends only on (p + 2) highestbids
Guo and Conitzer [5] : performance ratio of a mechanism as,
minθ∈Θ
Surplus redistributedVCG Surplus
Herve Moulin [6] : notion of efficiency loss,
L(n, p) = maxθ∈Θ
Budget SurplusEfficient Surplus
Guo and Conitzer [7] : designed mechanism which is optimal inexpected sense
1This scheme can be viewed as Bailey [4] scheme applied in the settingSujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 7 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
State of the Art and Research Gaps
Cavallo [3]1 : rebate function that depends only on (p + 2) highestbids
Guo and Conitzer [5] : performance ratio of a mechanism as,
minθ∈Θ
Surplus redistributedVCG Surplus
Herve Moulin [6] : notion of efficiency loss,
L(n, p) = maxθ∈Θ
Budget SurplusEfficient Surplus
Guo and Conitzer [7] : designed mechanism which is optimal inexpected sense
1This scheme can be viewed as Bailey [4] scheme applied in the settingSujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 7 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
Notation
θi j The valuation of the agent i for object j
θi = (θi 1, θi 2, . . . , θi p). The vector of valuations of the agent i
Θi = Rp+. Space of valuation of agent i
Θ =∏
i∈N Θi
bi = (bi 1, bi 2, . . . , bi p) ∈ Θi . Bid submitted by agent i
b = (b1, b2, . . . , bn). The bid vector
K The set of all allocations of p objects to n agents,each getting at most one object
k An allocation, k ∈ K
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 8 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
vi (k(b)) The valuation of the allocation to the agent i
ti = vi (k∗(b))−
(v(k∗(b))− v(k∗−i (b))
).
Payment made by agent i in VCG mechanism
t∑
i∈N ti . VCG payment, total payment receivedfrom all the agents
t−i VCG payment received in absence of the agent i
ri Rebate to agent i
pi = ti − ri . Net payment made by agent i in new mechanism
∆ =∑
i∈N pi . Budget imbalance in the system
e The efficiency of the mechanism. = infθ:t 6=0∆t
Table: Notation
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 9 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
WCO Mechanism
Moulin [6] and Guo and Conitzer [5] : Worst Case Optimal (WCO)Mechanism,
ri = f (θ1, θ2, . . . , θi−1, θi+1, . . . , θn) (1)
where,
f (x1, x2, . . . , xn−1) =n−1∑
j=p+1
cjxj
ci =
(−1)i+p−1 (n − p)
(n − 1p − 1
)i
(n − 1
i
) ∑n−1j=p
(n − 1
j
) {n−1∑j=i
(n − 1
j
)}; i = p + 1, . . . , n − 1
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 10 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
WCO Mechanism
Moulin [6] and Guo and Conitzer [5] : Worst Case Optimal (WCO)Mechanism,
ri = f (θ1, θ2, . . . , θi−1, θi+1, . . . , θn) (1)
where,
f (x1, x2, . . . , xn−1) =n−1∑
j=p+1
cjxj
ci =
(−1)i+p−1 (n − p)
(n − 1p − 1
)i
(n − 1
i
) ∑n−1j=p
(n − 1
j
) {n−1∑j=i
(n − 1
j
)}; i = p + 1, . . . , n − 1
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 10 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
WCO Mechanism
Moulin [6] and Guo and Conitzer [5] : Worst Case Optimal (WCO)Mechanism,
ri = f (θ1, θ2, . . . , θi−1, θi+1, . . . , θn) (1)
where,
f (x1, x2, . . . , xn−1) =n−1∑
j=p+1
cjxj
ci =
(−1)i+p−1 (n − p)
(n − 1p − 1
)i
(n − 1
i
) ∑n−1j=p
(n − 1
j
) {n−1∑j=i
(n − 1
j
)}; i = p + 1, . . . , n − 1
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 10 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
Problem We Are Addressing
p heterogeneous objects to assigned among n competing agents,where n ≥ p and agents have unit demand
all the previous work assumes objects are homogeneous
Goal
Design a redistribution mechanism which is individually rational, feasibleand worst case optimal for assignment of p heterogeneous objects amongn agents with unit demand.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 11 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Redistribution MechanismState of the ArtWorst Case Optimal Redistribution MechanismResearch Gaps
Problem We Are Addressing
p heterogeneous objects to assigned among n competing agents,where n ≥ p and agents have unit demand
all the previous work assumes objects are homogeneous
Goal
Design a redistribution mechanism which is individually rational, feasibleand worst case optimal for assignment of p heterogeneous objects amongn agents with unit demand.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 11 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
HETERO
t−i,k : average payment received when agent i is absent along withk other agents
we propose [8],
rHi = α1t−i +
k=L∑k=2
αkt−i,k−1 (2)
where, L = n − p − 1 and for i = p + 1→ n − 1
ci =n−i−1∑k=0
αL−k ×
(i − 1
p
) (n − i − 1
k
)(
n − 1p + 1 + k
) (3)
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 12 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
HETERO
t−i,k : average payment received when agent i is absent along withk other agents
we propose [8],
rHi = α1t−i +
k=L∑k=2
αkt−i,k−1 (2)
where, L = n − p − 1 and for i = p + 1 → n − 1
ci =n−i−1∑k=0
αL−k ×
(i − 1
p
) (n − i − 1
k
)(
n − 1p + 1 + k
) (3)
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 12 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
HETERO
t−i,k : average payment received when agent i is absent along withk other agents
we propose [8],
rHi = α1t−i +
k=L∑k=2
αkt−i,k−1 (2)
where, L = n − p − 1 and for i = p + 1 → n − 1
ci =n−i−1∑k=0
αL−k ×
(i − 1
p
) (n − i − 1
k
)(
n − 1p + 1 + k
) (3)
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 12 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
α’s are given by,
αi =(−1)(i+1)(L− i)!p!
(n − i)!χ
L−i∑j=0
(
i + j − 1j
) n−1∑l=p+i+j
(n − 1
l
) ;
i = 1, 2, . . . , L (4)
where, χ is given by, χ =
(n−p)
n − 1p − 1
∑n−1
j=p
n − 1j
HETERO agrees with WCO mechanism when objects arehomogeneous
advantage : applicable even when objects are heterogeneous
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 13 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
Why it Works?
Conjecture
The proposed scheme, HETERO, is individually rational.
Guo and Conitzer [5] :
Theorem 1
For any x1 ≥ x2 ≥ . . . xn ≥ 0,
a1x1 + a2x2 + . . . anxn ≥ 0 iff
j∑i=1
ai ≥ 0 ∀j = 1, 2 . . . , n.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 14 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
1 define, Γ1 = t−i , Γj = t−i,j−1, j = 2, . . . , L
2 rebate function for agent i ,
r =∑
j
αjΓj
3 note, Γ1 ≥ Γ2 ≥ . . . ≥ ΓL ≥ 0
4 for p = 2, n = 4, 5, 6; p = 3, n = 5, 6, 7; individual rationality followsfrom Theorem 1
5 if∑j
i=1 αi ≥ 0 ∀ j = 1 → L, individual rationality would follow fromTheorem 1
6 Γj ’s are related
7 αj ’s give appropriate weights to the combinations when a particularagent is absent in the system along with j − 1 agents
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 15 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
Experiments and Empirical Evidence
Setup 1
p = 2, n = 5, 6, . . . , 14, # Experiments 200, 000
p = 3, n = 7, 8, . . . , 14, # Experiments 40, 000
p = 4, n = 9, 10, . . . , 14, # Experiments 40, 000
HETERO is individually rational, feasible and performs at least as goodas a worst case optimal mechanism
Setup 2
Assume all the agents have binary valuations on each of these objects
p = 2, n = 5, 6, . . . , 12
Enumerate all possible bids.
HETERO is individually rational, feasible, and worst case optimal
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 16 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
Proposed SchemeExperimental Analysis
Experiments and Empirical Evidence
Setup 1
p = 2, n = 5, 6, . . . , 14, # Experiments 200, 000
p = 3, n = 7, 8, . . . , 14, # Experiments 40, 000
p = 4, n = 9, 10, . . . , 14, # Experiments 40, 000
HETERO is individually rational, feasible and performs at least as goodas a worst case optimal mechanism
Setup 2
Assume all the agents have binary valuations on each of these objects
p = 2, n = 5, 6, . . . , 12
Enumerate all possible bids.
HETERO is individually rational, feasible, and worst case optimal
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 16 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Summary
a redistribution mechanism (HETERO): assigns p heterogeneousobjects among n agents with unit demand
it is individually rational for particular combinations of n and p
experimental analysis : it is individually rational, feasible andperforms equally good as a worst case optimal mechanism
the agents with binary valuations : for some combinations of n andp, it is individually rational, feasible and worst case optimal
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 17 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Directions for Future Work
Ongoing work
prove HETERO is individually rational when p = 2
feasibility
worst case analysis and design of worst case optimal mechanism
all of the above when p > 2
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 18 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Future work
extensions to multi-unit demands
linear rebate function : linear in received bids
no redistribution mechanism with linear rebate function thatredistributes non-zero fraction of VCG surplus in the worst case
characterize situations under which linear rebate functions thatredistribute non-zero fraction of the VCG surplus even in the worstcase
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 19 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
J. R. Green and J. J. Laffont.Incentives in Public Decision Making.North-Holland Publishing Company, Amsterdam, 1979.
J.J. Laffont and E. Maskin.A differential approach to expected utility maximizing mechanisms.In J. J Laffont, editor, Aggregation and Revelation of Preferences.1979.
Ruggiero Cavallo.Optimal decision-making with minimal waste: strategyproofredistribution of vcg payments.In AAMAS ’06: Proceedings of the fifth international jointconference on Autonomous agents and multiagent systems, pages882–889, New York, NY, USA, 2006. ACM.
Martin J Bailey.The demand revealing process: To distribute the surplus.Public Choice, 91(2):107–26, April 1997.
Mingyu Guo and Vincent Conitzer.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 19 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Worst-case optimal redistribution of VCG payments.In EC ’07: Proceedings of the 8th ACM conference on ElectronicCommerce, pages 30–39, New York, NY, USA, 2007. ACM.
H. Moulin.Efficient, strategy-proof and almost budget-balanced assignment.Technical report, Northwestern University, Center for MathematicalStudies in Economics and Management Science, 2007.
Mingyu Guo and Vincent Conitzer.Optimal-in-expectation redistribution mechanisms.In Proceedings of the 7th International Joint Conference onAutonomous Agents and Multiagent Systems (AAMAS-08), 2008.
Sujit Gujar and Yadati Narahari.Redistribution of VCG payments in assignment of heterogeneousobjects.In Proceedings of Internet and Network Economics, 4th InternationalWorkshop, WINE, volume 5385 of Lecture Notes in ComputerScience, pages 438–445. Springer, 2008.
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 20 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Questions?
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 20 / 21
IntroductionState of the Art and Research Gaps
Proposed SolutionSummary and Future Work
SummaryFuture Work
Thank You!!!
Sujit Prakash Gujar (CSA, IISc) Redistribution of VCG Payments December 18, 2008 21 / 21