Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions
Carmine Ventre (University of Liverpool)
Joint work with:
Paolo Penna (University of Salerno)
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
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 )
CRMs w/verification for single-parameter bounded domains Agents aka as “binary” (in/out outcomes)
e.g., controls edges Sufficient Properties
Pay all agents(!!!) Algorithm 2-resistant
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12
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2
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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
any number between bidmin & bidmax
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 [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 AgentsOutcomes = {X1, ..., Xm}
bid =(bid(X1), .... ,bid(Xm))
b=(bid , ..., bid )
B(b) optimal algorithm with fixed tie breaking rule
A(bid ) m 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?