PowerPoint-Prsentation

Strong Implementation of Social Choice Functions in Dominant StrategiesClemens ThielenSven O. Krumke

3rd International Workshop on Computational Social Choice

15 September 2010

krumke@mathematik.uni-kl.de

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