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Optimal Collusion-Resistant Mechanisms with Verification
Carmine Ventre
Joint work with Paolo Penna
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Routing in Networkss
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Internet
Change over time (link load)
Private Cost
No Input Knowledge
Selfishness
d
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Mechanisms: Dealing w/ Selfishness
Augment an algorithm with a payment function
The payment function should provide incentives for telling the truth
Design a truthful mechanism
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d
Truthful Mechanisms
M = (A, P)
s
Utility (true, , .... , ) ≥ Utility (false, , .... , ) for all true, false, and , ...,
M truthful if:
Utility = Payment – cost = – true
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VCG Mechanisms
M = (A, P)
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Pe = Ae=∞ – Ae=0 if e is selected
(0 otherwise)
M is truthful iff A is optimal
Pe’ = Ae’=∞ – Ae’=0 = 7
e’Ae’=∞ = 14
Ae’=0 = 10 – 3 = 7
s
Utilitye’ = Pe’ – coste’ = 7 – 3
d
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Inside VCG Payments
Pe = Ae=∞ – Ae=0
Cost of best solution w/o e
Independent from e
h(b–e)
Cost of computed solution w/ e = 0
Mimimum (A is OPT)
A(true) A(false)b–e all but e
Cost nondecreasing in the agents’ bids
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Describing Real World: Collusions
Accused of bribery ~900,000 results on Google 6,463 results on Google news
Are VCGs collusion-resistant mechanisms?
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Collusion-Resistant Mechanisms
Coalition C
+
–
∑ Utility (true, true, , .... , ) ≥ ∑ Utility (false,false, , .... , ) for all true, false, C and , ...,
in C in C
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VCGs and Collusions
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6e1
e2
e3
Pe1(true) = 6 – 1 = 5
e3 reported value
“Promise 10% of my new payment” (briber)
11
Pe1(false) = 11 – 1 – 1 = 9
“Pe3(false)” = 1
bribe
h( ) must be a constantb–e
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Preventing Collusions is expensive Pay all the agents(!!!)
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e
e’
Truthfulness
• e’ to enter the solution by unilaterally lying must underbid (competition, i.e., non-cooperative behaviour)
• In coalition they can make the cut really expensive (cooperative behaviour)UtilityC(true)= Pe – 2
true
10+Petrue
11+Petrue
truePe’ = 0
UtilityC(false)=Pe’ – 10false ≥ 10 + Pe – 10 > UtilityC(true)true
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Constructing Collusion-Resistant Mechanisms (CRMs)
h is a constant function Pay all the agents A(true) A(false)
Coalition C
(A, VCG payments) is a CRM
How to ensure it? “Impossible” for classical mechanisms ([GH05]&[S00])
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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
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The Verification Setting
Give the payment if the results are given “in time”
Agent is selected when reporting false1. true false just wait and get the payment 2. true > false no payment (punish agent )
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Exploiting Verification: Optimal CRMs
No agent is caught by verification
At least one agent is caught by verification
A(true) = A(true, (t1, …, tn))
A(false, (t1, …, tn))
A(false, (b1, …, bn))
= A(false)
A is OPT
For any i ti bi
Cost is monotone
VCG hypotheses
Usage of the constant h for bounded domains
VCGs with verification are collusion-resistant
Any value between bmin e bmax
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Approximate CRMs
Extending technique above: Optimize MinMax + AVCG
MinMax extensively studied in AMD E.g., Interdomain routing and Scheduling
Unrelated Machines Many lower bounds even for two players and
exponential running time mechanisms E.g., [NR99], [AT01], [GP06], [CKV07], [MS07], [G07],
[PSS08], [MPSS09]
MinMax objective functions admit a (1+ε)-apx CRM
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Applications
* = FPTAS for a constant number of machines# = PTAS for a constant number of machines
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Conclusions
Collusion-Resistant mechanisms with verification for arbitrary bounded domains optimizing generalization of utilitarian (VCG) cost functions
Overcome many impossibility results by using a real-world hypothesis (verification)
Efficient Mechanisms Mechanism is polytime if algorithm is
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Further Research
Frugality of payment scheme? Can we deal with unbounded domains? What is the real power of verification? Explore different definitions for the verification
paradigm [Nisan&Ronen, 1999] [Green & Laffont, 1986]...
... for which we can also look for untruthful mechanisms Apply verification to CAs