(optimal) collusion-resistant mechanisms with verification paolo penna & carmine ventre...
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(Optimal) Collusion-Resistant Mechanisms with Verification
Paolo Penna & Carmine Ventre
Università degli Studi di Salerno
Italy
Routing in Networkss
12
3
10
2
1
1
4
37
7
1
d
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
s
12
3
10
2
1
1
4
37
7
1
d
VCG Mechanisms
s
M = (A, P)
12
310
2
1
1
4
37
7
1
Pe = Ae=∞ – Ae=0 if e is selected
(0 otherwise)
M is truthful iff A is optimal
Pe’ = Ae’=∞ – Ae’=0 = 5
e’Ae’=∞ = 10 + 3 + 1
Ae’=0 = 3 + 1 + 2 + 3 + 1 - 3 = 9
s
d
Utilitye’ = Pe’ – coste’ = 5 – 3
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
Describing Real World: Collusions
Accused of bribery 1,030,000 results on Google 1,635 results on Google news
Are VCG mechanisms resistant to collusions?
VCGs and Collusions
s
d
3
1
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
Constructing Collusion-Resistant Mechanisms (CRMs)
h is a constant function A(true) A(false)
Coalition C
(A, VCG payments) is a CRM
How to ensure it? “Impossible” for classical mechanisms ([GH05]&[S00])
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
Verification Setting
Give the payment if the results are given “in time”
Agent i is selected when reporting bi
1. ti bi just wait and get the payment
2. ti > bi no payment (punish agent i)
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
Problem has a truthful VCG Problem has an optimal CRM
Any value between bmin e bmax
Approximating CRMs
Extending technique above: Optimize MinMax + AVCG
Example of MinMax objective functions Interdomain routing Scheduling Unrelated Machines
MinMax objective functions admit a (1+ε)-apx CRM
Lower bound of 2.7… for truthful mechanisms w/o verification
General Monotone Cost Functions Optimizing monotone nondecreasing cost
functions always admits a truthful mechanism with verification (for bounded domain)
Breaking several lower bounds for natural problems Variants of the SPT [Gualà&Proietti, 06] Minimizing weighted sum scheduling
[Archer&Tardos, 01] Scheduling Unrelated Machines [Nisan&Ronen,
99]