Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

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Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions. Carmine Ventre (University of Liverpool) Joint work with: Paolo Penna (University of Salerno). Routing in Networks. s. Change over time (link load). No Input Knowledge. 3. 10. 1. 1. 2. Selfishness.

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Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Carmine Ventre (University of Liverpool)

Joint work with:

Paolo Penna (University of Salerno)

Routing in Networks

s

No Input Knowledge

3

10

1

1

2

Selfishness

Private Cost

2

1

3

7

7

4

1

Internet

Mechanisms: Dealing w/ Selfishness

s

• Augment an algorithm with a payment function
• The payment function should incentive in telling the truth
• Design a truthful mechanism

3

10

1

1

2

2

1

3

7

7

4

1

M = (A, P)

Truthful Mechanisms

s

Utility = Payment – cost = – true

M truthful if:

Utility (true, , .... , ) ≥ Utility (bid, , .... , )

for all true, bid, and , ...,

Optimization & Truthful Mechanisms
• Objectives in contrast
• Many lower bounds (even for two players and exponential running time mechanisms)
• Variants of the SPT [Gualà&Proietti, 06]
• Minimizing weighted sum scheduling [Archer&Tardos, 01]
• Scheduling Unrelated Machines [Nisan&Ronen, 99], [Christodoulou & Koutsoupias & Vidali 07], …
• Workload minimization in interdomain routing [Mu’alem & Schapira, 07], [Gamzu, 07]
• & a brand new computational lower bound
• CPPP [Papadimitriou &Schapira & Singer, 08]
• Study of optimal truthful mechanisms
Collusion-Resistant Mechanisms

∑ Utility (true, true, , .... , ) ≥ ∑ Utility (bid, bid, , .... , )

• CRMs are “impossible” to achieve
• Posted price [Goldberg & Hartline, 05]
• Fixed output [Schummer, 02]
• Unbounded apx ratios

for all true, bid, C and , ...,

in C

in C

Coalition C

+

Describing Real World: Collusions
• “Accused of bribery”
• 1,635 results on Google news
• Can we design CRMs using real-world information?
Describing Real World: Verification
• TCP datagram starts at time t
• Expected delivery is time t + 1…
• … but true delivery time is t + 3
• It is possible to partially verify declarations by observing delivery time
• Other examples:
• Distance
• Amount of traffic
• Routes availability

TCP

3

1

IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

Verification Setting
• Give the payment if the results are given “in time”
• Agent is selected when reporting bid
• truebid just wait and get the payment
• true>bid no payment (punish agent )
CRMs w/verification for single-parameter bounded domains

s

• Agents aka as “binary” (in/out outcomes)
• e.g., controls edges
• Sufficient Properties
• Pay all agents(!!!)
• Algorithm 2-resistant

3

10

1

1

2

2

true

true

10+Pe

11+Pe

1

3

7

Truthfulness

true

Pe’ = 0

e

7

• e’ has no way to enter the solution by unilaterally lying
• In coalition they can make the cut really expensive

2

4

1

10

e’

bid

true

UtilityC(bid)=Pe’ – 10

≥ 10 + Pe– 10 > UtilityC(true)

true

UtilityC(true)= Pe – 2

Truthful Mechanisms w/ Verification: the threshold

bid < in

bid > out

(A,P) truthful with verification

A(bid, )

ths

ths

ths

in

out

bid

[Auletta&De Prisco&Penna&Persiano,04]

2-resistant Algorithms

t=(true, true, , .... , )

t-=(true , , .... , )

b-=(bid , , .... , )

b=(bid, bid, , .... , )

bid ≥ true

(Verification doesn’t work)

b’ =

t’=

t’

b’

t’

b’

in

ths

ths

ths

ths

out

Exploiting Verification: CRMs w/verification

h - if out

Payment (b) =

h if in

b’

ths

• (A,Payment) is a CRM w/ verification

Thm. Algorithm A 2-resistant

Proof Idea.

At least one agent is caught by verification

Usage of the constant h for bounded domains

any number between bidmin & bidmax

Proof (continued)
• Each is not worse by truthtelling
• No agent is caught by verification

h - if out

Payment (b) =

t

b

h if in

in

in

out

in

in

out

out

in

out

out

t’

t’

b’

b’

b’

t’

true

true

ths

ths

ths

ths

ths

ths

• h - true

= Utility (b)

= Utility (b)

• h - true ≥ h -

b’

Utility (t) =

Utility (t) =

ths

• h - ≥ h - true
• h - ≥ h -
Simplifying Resistance Condition

t-

b-

b=(bid , , .... , )

b-=(bid , , .... , )

b=(bid, bid, , .... , )

t-=(true , , .... , )

t=(true, true, , .... , )

t=(true , , .... , )

bid ≥ true

(Verification doesn’t work)

bid ≥ true

b’ =

b’ =

t’=

t’=

in

Optimal CRMs

t’

t’

b’

b’

b’

t’

out

in

ths

ths

ths

ths

ths

ths

Thm. Optimal threshold-monotone algorithms with fixed tie breaking are n-resistant

out

Applications
• Optimal CRMs for:
• MST
• k-items auctions
• Cheaper payments wrt [Penna&V,08]
• Optimal truthful mechanisms for multidimensional agents bidding from bounded domains and non-decreasing cost functions of the form

Cost(bid , ..., bid )

Multidimensional Agents

Outcomes = {X1, ..., Xm}

View bid as a virtual coalition C of m single-parameter agents

bid =(bid(X1), .... ,bid(Xm))

b=(bid , ..., bid )

B(b) optimal algorithm with fixed tie breaking rule

A(bid ) m single-player functions

P (b) = ∑ payment (bid )

in C

Lemma. If every A is m-resistant then (B,P) is truthful

Thm. For non-decreasing cost function of the form

Cost(bid , ..., bid )

every A is threshold-monotone

Every A is m-resistant

(B,P) is truthful

Conclusions
• Optimal CRMs with verification for single-parameter bounded domains
• Optimal truthful mechanisms for multidimensional bounded domains
• Construction tight (removing any of the hypothesis we get an impossibility result)
• Overcome many impossibility results by using a real-world hypothesis (verification)
• For finite domains: Mechanisms polytimeif algorithm is
• Can we deal with unbounded domains?