PowerPoint-Prsentation
Strong Implementation of Social Choice Functions in Dominant StrategiesClemens ThielenSven O. Krumke
3rd International Workshop on Computational Social Choice
15 September 2010
TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA1Problem DefinitionSocial choice setting with private information:
Strong Implementability
In this talk: The objective function will always be weighted sum of completion times.2MechanismsTypesBidsSocial Choices...Mechanism:
...
strategy 1strategy ngStrong Implementability
Agent 1
Agent nWork_i = total size of the jobs assigned to I
Hence: work_i/s_i is the total time machine i will be busy
3Utilities and Equilibria Definition:Strong Implementability
valuation of the outputpayment obtained
Intuitively: Small changes in the speeds induce only small changes in the objective value
Technique used in the proof: Geometric rounding(all speeds are rounded to powers of 1+epsilon)4Utilities and EquilibriaStrong Implementability
Definition:
Definition:
Intuitively: Small changes in the speeds induce only small changes in the objective value
Technique used in the proof: Geometric rounding(all speeds are rounded to powers of 1+epsilon)5Strong ImplementationDefinition:
Strong Implementability
Intuitively: Small changes in the speeds induce only small changes in the objective value
Technique used in the proof: Geometric rounding(all speeds are rounded to powers of 1+epsilon)6 Strong Implementability ProblemThe Strong Implementability Problem:
Strong ImplementabilityEncoding length:
Intuitively: Small changes in the speeds induce only small changes in the objective value
Technique used in the proof: Geometric rounding(all speeds are rounded to powers of 1+epsilon)7 Augmented Revelation MechanismsStrong ImplementabilityDefinition:
Augmented Revelation Principle:[Mookherjee, Reichelstein 1990]incentive compatibility8Previous Results
Strong Implementability9Previous Results (2)
Strong Implementability10Our Results
Strong Implementability
11Augmented Revelation PrincipleStrong ImplementabilityAugmented Revelation Principle:[Mookherjee, Reichelstein 1990]
Augmented Revelation Principle for Dominant Strategies:[this paper]
12General Idea (I)Strong ImplementabilityTo obtain an augmented revelation mechanism:
Definition:
see definition to follow soon13Selective EliminationStrong Implementability
agent i
14Selective EliminationStrong Implementability
15Bad Pairs and Elimination Definition:
Definition:
Definition:Strong ImplementabilityTwo Important Steps
Theorem 2 (selective elimination is necessary):
Theorem 3 (selective elimination is sufficient):Strong ImplementabilityStructure of the Algorithm
guess
guess
verifyTheorem 3 + close look at the proofDefinition of selective eliminationStrong Implementability The VerificationStrong ImplementabilityGeneral Approach:
Main Observation:
19 The Payment Polyhedron
Strong Implementability20 The Payment Polyhedron (I)Strong Implementability
Inequalities encode which bids are dominant bids.Incentive compatibility&dominant bids21 The Payment Polyhedron (II)Strong ImplementabilityInequalities encode conditions of selective elimination
22 The Payment Polyhedron (II)Strong ImplementabilityInequalities encode conditions of selective elimination
23Verification Issues
Strong Implementability
Here I am!Verification IssuesWe have to handle strict inequalities.To do so, we must find a point in the relative interior of the polyhedron.This can be done by means of the Ellipsoid Method (directly) or by solving a sequence of LPs.
Byproduct: Payments are of polynomial encoding length.Strong ImplementabilityConclusionStrong Implementability in dominant strategiesNP
Characterization result generalizes to infinite type spaces
Open: Is the problem in P?
Useful(?) results:Augmented Revelation PrincipleSelective elimination procedure with polynomially many stepsPayments of polynomial encoding sizeStrong Implementability
NP-complete!
Thank you!Strong Implementability27
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