Crm Esa08 1234869680124198 3
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Transcript of Crm Esa08 1234869680124198 3
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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)
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Routing in Networkss
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3
10
2
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Internet
Change over time (link load)
Private Cost
No Input Knowledge
Selfishness
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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
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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
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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
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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
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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?
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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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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 )
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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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2
1
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10
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
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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
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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)
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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.
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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
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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
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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 )
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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
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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?