2. Routing in Networks
s
Change over time No Input
(link load) Knowledge
3
10
1 1
2
Selfishness Private Cost
2
1
3
7
7
4
1
Internet
3. Mechanisms: Dealing w/ Selfishness
s
Augment an algorithm
3
with a payment function
10
1 1
The payment function
2
should incentive in
2
telling the truth
1
3
7
Design a truthful
mechanism
7
4
1
4. Truthful Mechanisms
s
M = (A, P)
– cost = – true
Utility = Payment
M truthful if:
, .... , ) ≥ Utility (bid,
Utility (true, , .... , )
for all true, bid, and , ...,
5. 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
6. Collusion-Resistant Mechanisms
CRMs are
“impossible” to
achieve
Coalition C
Posted price
[Goldberg &
Hartline, 05]
Fixed output
[Schummer, 02]
Unbounded apx
ratios
∑ Utility (true, true, , .... , ) ≥ ∑ Utility (bid, bid, , .... , )
+
in C in C
for all true, bid, C and , ...,
–
7. 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?
8. Describing Real World: Verification
TCP datagram starts at time
t
Expected delivery is time t +
1…
… but true delivery time is t
TCP
1
3 +3
It is possible to partially
verify declarations by
observing delivery time
Other examples:
Distance
Amount of traffic
Routes availability
IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification
9. Verification Setting
Give the payment if the results are given “in
time”
Agent is selected when reporting bid
true bid just wait and get the payment
1.
true > bid no payment (punish agent )
2.
10. CRMs w/verification for single-
parameter bounded domains
Agents aka as “binary” (in/out outcomes)
e.g., controls edges
Sufficient Properties
Pay all agents(!!!)
s
Algorithm 32-resistant
true
Truthfulness Pe’ = 0
10
true
e
1 1
• e’ has no way to enter the 11+Pe
2
2
solution by unilaterally lying
2
true
1
• In coalition they 7can make the 10+Pe
10
3
e’
cut really expensive 7
4
true 1
Pe – 2
UtilityC(true)=
UtilityC(bid)=Pbid – 10 ≥ 10 + Pe – 10 > UtilityC(true)
true
e’
11. Truthful Mechanisms w/ Verification:
the threshold
(A,P) truthful with verification
A(bid, )
bid < in
ths
in bid > out
ths
out
bid
ths
[Auletta&De Prisco&Penna&Persiano,04]
13. Exploiting Verification: CRMs
w/verification
b’
h- if out
ths
Payment (b) =
h if in
(A,Payment) is a CRM
Thm. Algorithm A 2-resistant
w/ verification
Proof Idea.
At least one agent is caught by verification
Usage of the constant h for bounded domains
any number between bidmin & bidmax
14. b’
Proof (continued) h- if out
ths
Payment (b) =
h if in
Each is not worse
No agent is caught by verification
by truthtelling
t b
in in
out in
in
out out
in out
out
t’ b’
true true
ths ths
Utility (t) = h - true = Utility (b)
Utility (t) = h - true ≥ h --trueb’ = Utility (b)
t’
t’ b’
≥h thsths
ths
ths
16. 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 )
17. Multidimensional Agents
Outcomes = {X1, ..., Xm}
View bid as a virtual coalition C
bid =(bid(X1), .... ,bid(Xm))
of m single-parameter agents
b=(bid , ..., bid )
B(b) optimal algorithm with A(bid ) m single-player
fixed tie breaking rule functions
P (b) = ∑ payment (bid )
in C
Lemma. If every A is m-resistant then (B,P) is truthful
Thm. For non-decreasing cost
Every A is (B,P) is
function of the form
m-resistant truthful
Cost(bid , ..., bid )
every A is threshold-monotone
18. 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?