1. Collusion-Resistant
Mechanisms with Verification
Yielding Optimal Solutions
Carmine Ventre (University of Liverpool)
Joint work with:
Paolo Penna (University of Salerno)

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]

12. 2-resistant Algorithms
b=(bid, bid, , .... , )
t=(true, true, , .... , )
bid ≥ true(Verification doesn’t work)
b’ = b- =(bid , , .... , )
t’ = t- =(true , , .... , )
b’ t’
≥
ths ths in
out
t’ b’
ths ths

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

15. Simplifying Resistance Condition
b=(bid, bid, , .... , ) b=(bid , , .... , )
t=(true, true, , .... , )
t=(true , , .... , )
bid ≥ true(Verification doesn’t work)≥ true
bid
b’ = b- b’ = b- =(bid , , .... , )
t’ = t- t’ = t- =(true , , .... , )
in
Optimal
b’ t’
CRMs
≥ thsout
ths in
t’ b’
ths ths
Thm. Optimal threshold-monotone algorithms with
out
fixed tie breaking are n-resistant t’ b’
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?
