algorithms for incentive-based computing
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Algorithms for Incentive-Based Computing
Carmine Ventre
Università degli Studi di Salerno
… or Merging Research of Different Fields
Economics
Computer Science
“Worst-case equilibria” by E. Koutsoupias, C. H. Papadimitriou in STACS ‘99
Auctions
7
10
6
First price sealed bid auction
6
Problems? It is not truthful (e.g., auctioneer can not maximize his own revenue)
A
B
Vickrey Auctions10
Second price sealed bid auction
Bid 811
Utility is 0 in place of 1 (= 10 – 9)
Bid 12
Utility is -1 (= 10 – 11) in place of 0
9
This is truthful and generalizes to the concept of mechanism
A
B
Mechanisms
Augment an algorithm with a payment function i.e., design a truthful
mechanism The payment function
should incentive in telling the truth
s1
2
310
2
1
1
4
37
7
1
VCG Mechanisms
s
M = (A, P)
12
310
2
1
1
4
37
7
1
Pe= be if e is selected
(0 otherwise)
9
Utility(3) = payment(3) – cost(3) = 3 – 3 = 0
Utility(9) = payment(9) – cost(9) = 9 – 3 = 6
valuation
Pe = Ae=∞ – Ae=0 if e is selected
(0 otherwise)M is truthful iff A is optimal
Algorithmic mechanism design by N. Nisan and A. Ronen in STOC ’99 (GEB ‘01)
Ae=0 = Ae – be
Vickrey Auction (& VCG Mechanism) Weakness (or Cui Prodest?) It works only for utilitarian problems: i.e., maximizes
the social welfare (e.g., it does not maximize seller revenue) Adaptation to non-utilitarian problems Verification Model
It is not budget balanced Cost-Sharing Budget Balance Mechanisms
It is vulnerable to collusion Cost-Sharing Budget Balance Mechanisms Verification model
… (not here)
Utilitarian problems: objective is to maximize the social welfare (i valuationi(X))BB mechanisms: sum of payments equals the cost of the solution
(skip)
Cost-Sharing Mechanisms
Multicast and Cost-Sharing
A service provider s Selfish customers U Who is getting the service? How to share the cost?
real worth is 7
is worth 5 ( 7)
Pi
Accept or reject the service?
Selfish Agents
Each customer/agent has a private valuation vi for the service
declares a (potentially different) valuation bi
pays Pi for the service
Agents’ goal is to maximize their own utility: ui(bi) := vi – Pi(bi)
Accept iff my
utility ≥ 0!
Coping with Selfishness: Mechanism Design
Algorithm A Who gets serviced (Q(b)) How to reach Q(b)
(Construct tree T) Payment P
How much each user pay
M = (A, P)
bi
bj
P1
P4
P3
P2
M’s Truthfulness (or Strategyproofness)
For all others players’ declarations b-i it holds
ui = ui(vi, b-i) ≥ ui(bi, b-i) = ui
for all bi (ie, truthtelling is a dominant strategy)
M = (A, P)vi
M’s Group Strategyproofness
U
Coalition C
No one gainsAt least one looses (ie, ui > ui)
C is uselessBreaks off C
Does this definition fit our intuition of collusion-resistant mechanisms?
Mechanism’s Requirements
Budget Balance (BB)
i T Pi(b) = COST(T) … (natural “economic” requirements)
Cost-Sharing Budget-Balance Mechanisms
[Penna & V, WAOA ’04]
[Penna & V, SIROCCO ’05]
[Penna & V, STACS ’06]
How to build BB, GSP Mechanisms
Idea: associate prices to service set
U
Q
(Q,i) = COST(Q)
Cost-sharing methods: distribute COST(Q) among users in Q
(Q,i) 0
(Q,i) = 0, i Q
How to build BB, GSP MechanismsCost-sharing method (•,•) Mechanism M( )
(Q,i)U
Drop iQ > bi
UQ1=U
How to build BB, GSP Mechanisms
Q3
Qk
…
Q2
(Qk,i)
(Q2,i)
(Q3,i)
Pi = (Qk,i)
… Monotonicity…
[Moulin & Shenker ’97] & [PV04]
Cost-sharing method (•,•) Mechanism M( )
… for all Q subsets of U … for all Q (possibly) outputted by M
Cross… Self Cross…
(Q1,i)
Prices do not decrease
Group Strategyproof
Changes
Self cross monotonicity: an example
Q
COST(Q)
s
50%50%
s
Pay less than before This is not a cross monotonic cost sharing method!
Self cross monotonicity: an example (2)
Q
COST(Q)
s
100%
s
Pay less than before
This guy pays 0
M() cannot drop him
Idea: some subsets do not “appear”. We need monotone only for possible subsets generated by M()
This is not a cross monotonic cost sharing method!
Sequential Algorithms
A is sequential if for some bid vectors reaches a chain of sets Q1, …, Q|U|, Q|U|+1=Ø
Sequential algorithms admits a self cross-monotonic cost-sharing method
UQ1=U
Q3
Q|U|…
Q2
.
.
.
… Q|U|+1= Ø BB & GSP Mechanisms
Optimal Sequential Algorithm for Steiner Tree Game
s
prune
Q MST(Q)
opt Steiner tree
T + = opt
T + +
U
s
MST
v is the last node added by Prim’s MST algorithm
s
u
Q
v
s
u
Q
v
s
T*>
Qu
v
payv
Adding Fairness to Our Mechanisms
Payment is still self cross-monotonic
Is it possible to have no free rider? No! Unless P=NP
s
prune
Q MST(Q)
U
s
MST
pay
opt Steiner tree
Can we do better without Sequential Algorithms?
M = (A, P)
M for 2 users A is sequential
“Natural” GSP Mechanisms A is sequential
M is SP, BB, …
Mechanisms with Verification
[Ferrante, Parlato, Sorrentino & V, WAOA 2005] [Auletta, De Prisco, Penna, Persiano & V, ICALP 2006] [V, WINE 2006][Penna & V. , 2007]
Motivating Verification Model
Used Car market: The Kelley Blue Book – the Trusted Resource (www.kbb.com)
The Trusted Resource
Can we engage a trusted resource within a mechanism allowing (somehow) bids verification?
Time is trusted…
… unless a time machine will be created
Selfish Task Scheduling
M1 M2 M4M3 M5
t1 t2 t3 t4 t5b1 b2 b3 b4 b5
ti = 1 / si (ie, the inverse of the speed)
Optimal Makespan:
minx maxi ti(X)
Mechanism design: payments
utility = payment - cost
no VCG!
Allocation X
cost = ti(X) = ti • loadi(X)
Awarded independently from the execution!
Verifiable Selfish Agents
ti = 1
i underbids 1/2
1
3
i’s release time should be 2 but…
… i’s finishing time is 4
i overbids
2
1
1 i can wait 2 time slots delivering the results in the right time
IDEA ([Nisan & Ronen, 99]): No payment for underbidding agents
Verification is impossible!
ti(X) = loadi(X) • ti
i bids from the set {1/2, 1, 2}
Verification = observe jobs’ release time
The Power of Verification
bi
loadi
Classical mechanisms Mechanisms w/ Verification
algorithms
Payment functions
NO!
TRUTHFUL
bi
loadiNO!
TRUTHFUL
bi
loadi
ti
Pi(bi, b-i)= Wmax / bi (= Wmax • si)
Related to max possible supported cost
Scaling up for general speeds
Unique
Not unique
[Archer & Tardos, ‘01]
[Archer & Tardos, ‘01] [Auletta & al, ‘04]
The Power of Verification: Breaking Lower Bounds
M1 M2 M4M3 M5
b1 b2 b3 b4 b5
p1 p2 p3 p4 p5 p6 p7 p8 p9
weight
priority
t1 t2 t3 t4 t5
Goal: Design a polytime truthful mechanism optimizing the weighted completion time (ie, weighted sum scheduling)
No 1.54-apx truthful mechanism without verification
[Archer & Tardos, 2001]
(1+)-APX truthful mechanism w/ verification for a constant number of machines
Efficient APX truthful mechanisms w/verification: c-APX algorithm A c(1+)-APX mechanism
(Optimal) Mechanisms with Verification
J1 Jj Jn
… …
M1 Mi Mm… …
bijbi1 bin
… …
agent1 agenth agentk… …
There exists truthful mechanism with verificationWe don’t if truthful mechanisms
without verification do exist polytime
Breaking lower bounds for classical mechanisms concerning many natural problems (eg, variants of SPT problem)
Goal: minimizing the makespan
(althougt not polynomial-time)
Given an algorithm c-apx…
a c(1+)-apxan exact
bib1 bm
Optimal Collusion-Resistant Mechanisms w/ Verification
Coalition C
GSP do not consider side payments
U
+
–
Collusion-Resistant mechanisms are impossible unless using posted-price ([Goldberg & Hartline, 2005])
If OPT is truthful via VCG mechanism without verification
Exists a VCG-like payment function such that OPT is collusion-resistant with verification
Conclusions
Cost-Sharing Games Simple techniques…
… lead to polynomial-time cost-sharing mechanisms for NP-Hard problem Steiner Tree
… not so unfair (unless P=NP) … characterize natural class of cost-sharing mechanisms
Mechanisms with Verification More powerful model…
… breaking known lower bounds for natural problems … dealing with a strong notion of agents’ collusion
Further Research
Cost-Sharing Mechanisms Full characterization
What is the power of not “natural” mechanisms? Price of Fairness Tradeoff between budget balance and efficiency
Mechanisms with Verification What is the real power of verification? Does the revelation principle hold in the
verification setting? Different definitions for the verification paradigm
(e.g., Nisan&Ronen 99)
Questions?
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