Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Paolo Penna and Carmine Ventre

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Concept of mechanisms with verification Construction of optimal

mechanisms w/ verification A class of social choice functions admitting CRMs

w/ verification for any bounded domain Construction of optimal truthful mechanisms

w/ verification for any bounded domain and any cost function of a certain form Shown the technique only for finite domains

collusion-resistant

Routing in Networkss

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Internet

Change over time (link load)

Private Cost

No Input Knowledge

Selfishness

Mechanisms: Dealing w/ Selfishness

Augment an algorithm with a payment function

The payment function should incentive in telling the truth

Design a truthful mechanism

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Truthful Mechanisms

M = (A, P)

s

Utility (true, , .... , ) ≥ Utility (bid, , .... , ) for all true, bid, and , ...,

M truthful if:

Utility = Payment – cost = – true

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

CRMs are “impossible” to achieve Posted price

[Goldberg & Hartline, 05]

Fixed output [Schummer, 02] Unbounded apx

ratios

Coalition C

+

–

∑ Utility (true, true, , .... , ) ≥ ∑ Utility (bid, bid, , .... , ) for all true, bid, C and , ...,

in C in C

Describing Real World: Collusions

“Accused of bribery” 1,030,000 results on Google 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

31TCP

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

(The general) Verification Setting Give the payment if the results are given “in

time” Agent is selected when reporting bid

1. true bid just wait and get the payment

2. true > bid no payment (punish agent )

Utility = Payment – cost = – true

Comparison with [NR99] verification setting Two different declarations

Type Execution time

verification (reported exe time ≥ true one)

Allocation depends on Payments depend on ,

[NR99] verification: agent not caught (i.e., bid ≥ cost)

“Physical” assumptione.g., not usable for TCP example

cost = 10 mins

bid = 3 hrs

,Utility = Payment – reported cost = – bid NR

[NR99] verification: agent caught (i.e., bid < cost)

,Utility = Payment – true cost = – cost NR

Mechanism truthful (resp. CR) in our verification model

Mechanism truthful (resp. CR) in [NR99] verification model

(Easy) Thm.

CRMs w/verification for single-parameter bounded domains Agents aka as “binary” (in/out outcomes)

e.g., controls edges

any number between two known constants bidmin & bidmax

CRMs w/verification for single-parameter bounded domains: ideas Sufficient Properties

Pay all agents(!!!) Algorithm 2-resistants

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e

e’

Truthfulness • e’ has no way to enter the

solution by unilaterally lying• In coalition they can make the

cut really expensive

UtilityC(true)= Pe – 2true

10+Pe

true

11+Pe

true

truePe’ = 0

UtilityC(bid)=Pe’ – 10bid ≥ 10 + Pe – 10 > UtilityC(true)true

Truthful Mechanisms w/ Verification: the threshold

bid < in

bid > out

bid

A(bid, )

(A,P) truthful with verification

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

ths

in

out

ths

ths

2-resistant Algorithms

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

ths

b’

ths

t’≥

b’ =

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

t’ =

in

out

thsb’

thst’

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

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

bid ≥ true (Verification doesn’t work)

Exploiting Verification: CRMs w/verification

At least one agent is caught by verification

Usage of the constant h for bounded domains

Payment (b) =

h - if outths

b’

h if in

Thm. Algorithm A 2-resistant (A,Payment) is a CRM w/ verification

Proof Idea.

Proof (continued)

in

out

thsb’

thst’

No agent is caught by verification Each is not worse by truthtelling

bt

in in

in

in

out

out out

out

Utility (t) = = Utility (b)h - true

true

Utility (t) = h - ≥ h - true ths

t’ = Utility (b)

Payment (b) = h - if out

h if in

thsb’

h - ≥ h -ths

t’

ths

b’ h - true ≥ h -ths

b’

true

Simplifying Resistance Conditiont=(true, true, , .... , )

ths

b’

ths

t’≥

b’ =

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

t’ =

in

out

thsb’

thst’

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

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

bid ≥ true (Verification doesn’t work)

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

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

bid ≥ trueb’ = b-

t’ = t- in

out

thsb’

thst’

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

Optimal CRMs

Applications

Optimal CRMs for: MST k-items auctions Cheaper payments wrt mechanisms of previous

“episode” Optimal truthful mechanisms for

multidimensional agents bidding from bounded domains and non-decreasing cost functions of the form

Cost(bid , ..., bid )

Multidimensional AgentsOutcomes = {X1, ..., Xm}

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

b=(bid , ..., bid )

B(b) optimal algorithm with fixed tie breaking rule

A(bid ) m optimal single-player functions

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

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 polytime if algorithm

is Can we deal with unbounded domains? Threshold-monotone vs. utilitarian algorithms