Mechanisms with Verification

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Mechanisms with Verification

• Carmine Ventre
• Penna V, 2007
• V, WINE 06

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Routing in Networks
s
Change over time (link load)
No Input Knowledge
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Selfishness
Private Cost
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Internet
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Mechanisms Dealing w/ Selfishness
s

• Augment an algorithm with a payment function
• The payment function should incentive in telling
  the truth
• Design a truthful mechanism

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VCG Mechanisms
Pe Ae8 Ae0 7
Ae8 10 3 1
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e
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Ae0 3 1 2 3 1 - 3 7
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2
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Pe Ae8 Ae0 if e is selected (0 otherwise)
d
M is truthful iff A is optimal
Utilitye Pe coste 7 3
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Inside VCG Payments
Cost nondecreasing in the agents bids
Pe Ae8 Ae0
Cost of computed solution w/ e 0
Cost of best solution w/o e
Mimimum (A is OPT)
Independent from e
h(be)
A(true) ? A(false)
be all but e
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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?

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VCGs and Collusions
e3 reported value
Pe1(true) 6 1 5
s
Pe1(false) 11 1 1 9
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e3
6
e1
11
Pe3(false) 1
bribe
1
e2
d
h( ) must be a constant
be
Promise 10 of my new payment (briber)
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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
(GH05S00)
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VCG weaknesses

• It is vulnerable to collusion
• Collusion-resistant Mechanisms
• (stay tuned)

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Describing Real World The Trusted Resource
Used Car market The Kelley Blue Book the
Trusted Resource (www.kbb.com)
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The Trusted Resource
Time is trusted
unless a time machine will be created
Can we engage a trusted resource within a
mechanism allowing (somehow) bids verification?
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Time is Trusted

• 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

TCP
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IDEA (Nisan Ronen, 99) No payment for agents
caught by verification
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Exploiting Verification Optimal CRMs
For any i ti ? bi
No agent is caught by verification
A(true) A(true, (t1, , tn))
A is OPT
? A(false, (t1, , tn))
Cost is monotone
? A(false, (b1, , bn))
VCG hypotheses
A(false)
At least one agent is caught by verification
Usage of the constant h for bounded domains
Any value between bmin e bmax
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Approximating CRMs

• Extending technique above Optimize MinMax
  AVCG
• Example of MinMax objective functions
• Interdomain routing
• Scheduling Unrelated Machines

Lower bound of 2.61 for truthful mechanisms w/o
verification KV07
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Summarizing
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VCG weaknesses

• It is vulnerable to collusion
• Collusion-resistant Mechanisms for VCG problems
• It works only for utilitarian problems i.e.,
  minimize the sum of the costs
• Mechanisms minimizing any non-decreasing Cost
  function

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General Monotone Cost Functions

• Optimizing monotone nondecreasing cost functions
  always admits a truthful mechanism with
  verification (for bounded domain)
• Will show technique for Finite Domains
• Breaking several lower bounds for natural
  problems
• Variants of the SPT BilòGualàProietti, 06
• Scheduling Unrelated Machines NisanRonen, 99,
  MS07, CKV07, G07, KV07
• Interdomain Routing MS07, G07

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Task Scheduling NisanRonen99
tasks
Mechanism design payments ? utility payment
- cost
Selfish
machines

• Optimal Makespan
• minx maxi costi(X)

• Verification
• (observe machine behavior)

no VCG!
Allocation X ? costi(X)
ti,n
ti,j
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Verification

• Give the payment if the results are given in
  time
• Machine i gets job j when reporting bi,j
• ti,j ? bi,j ? just wait and get the payment
• ti,j bi,j ? no payment (punish agent i)

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Setup

• Agent i holds a resource of type ti
• X1,, Xk feasible solutions
• (how we use resources)
• costi(X) ti(X) time
• utility payment cost
• Goal minimize m(X,t)

Truthful mechanism running an optimal algorithm
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Existence of the Payments
A(?) ?A(?, b-i) P(?) ? P(?, b-i)
Truthfulness (single player)
P(a) - a(A(a)) ? P(b) - a(A(b))
P(a) ?(a,b) ? P(b)
P(b) - b(A(b)) ? P(a) - b(A(a))
P(b) ?(b,a) ? P(a)
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Existence of the Payments
Truthful mechanism (A, P)
Can satisfy all P(a) ?(a,b) ? P(b)
MalkhovVohra04MV05SaksYu05 Bikhchandan
iChatterjiLaviMu'alemNisanSen06
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Why Verification Helps
Some edges may disappear
X
Y

• True type is a but report b
• a(Y) ? b(Y) ? can simulate b and get P(b)
• a(Y) b(Y) ? no payment (verification helps)

P(a) - a(X) ? P(b) - a(Y)
P(a) - a(X) ? - a(Y)
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Why Verification Helps
Only these edges remain
X
Y
a(Y) ? b(Y)
Negative cycles may desappear
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Optimal Mechanisms

• Algorithm OPT
• Fix lexicographic order
• X1 ? X2 ? ? Xk
• Return the lexicographically minimal
• Xj minimizing m(b,Xj)

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Optimal Mechanisms
a(Y) ? b(Y)
b(Z) ? c(Z)
X
Y
Z
c(X) ? a(X)
m(a(X),b-i(X)) ? m(a(Y),b-i(Y))
? m(b(Y),b-i(Y))
? m(b(Z),b-i(Z)) ?
m(c(Z),b-i(Z))
? m(c(X),b-i(X)) ?
m(a(X),b-i(X))
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Optimal Mechanisms
a(Y) ? b(Y)
b(Z) ? c(Z)
X
Y
Z
c(X) ? a(X)
m(a(X),b-i(X)) m(a(Y),b-i(Y))
m(b(Y),b-i(Y))
m(b(Z),b-i(Z))
m(c(Z),b-i(Z))
m(c(X),b-i(X))
m(a(X),b-i(X))
? Z
? X
X ? Y
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Finite Domains
All vertices in a cycle lead to the same outcome
Theorem Truthful OPT mechanism with verification
for any finite domain and any m(X,b)m(b1(X),,bm(
X)) non decreasing in the agents costs bi(X)
Different proof of existence of exact truthful
mechanism w/ verification for makespan on
unrelated machines NisanRonen99
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Compound Agents
Each agent declares more than a type
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Verification for Compound Agents

• Punish agent i whenever uncovered lying over one
  of its dimensions (e.g., machines)
• Collusion-Resistant mechanisms w/ verification
  w.r.t. known coalitions

a (a1, a2)
b (b1, b2)
Edge (a,b) exists iff a1(Y) ? b1(Y) and a2(Y) ?
b2(Y)
OPT is implementable w/verification
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Compound Agents

• Collusion-Resistant for known coalitions
  mechanisms w/ verification for
• makespan on unrelated machines
• makespan on related machines

Exponential time Exact mechanisms
Polynomial time c (1?) - APX
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Truthful Grids?
Auction
Doughnuts.exe
Can grid nodes declare a completion time before
actually executing Homers task?
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Conclusions

• Mechanisms with Verification a more powerful
  model
• breaking known lower bounds for natural
  problems
• dealing with the strongest notion of agents
  collusion
• describing real-life applications

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Further Research

• What is the real power of verification?
• Does the revelation principle hold in the
  verification setting?
• Different definitions for the verification
  paradigm (e.g., NisanRonen 99)