Algorithms for Incentive-Based Computing

1
Algorithms for Incentive-Based Computing

• Carmine Ventre
• Università degli Studi di Salerno

2
or Merging Research of Different Fields
Computer Science
Economics
Worst-case equilibria by E. Koutsoupias, C. H.
Papadimitriou in STACS 99
3
Auctions
10
A
7
6
B
6
First price sealed bid auction
Problems?
It is not truthful (e.g., auctioneer can not
maximize his own revenue)
4
Vickrey Auctions
10
A
Bid 8
Bid 12
11
9
Utility is 0 in place of 1 ( 10 9)
B
Utility is -1 ( 10 11) in place of 0
Second price sealed bid auction
This is truthful and generalizes to the concept
of mechanism
5
Mechanisms

• Augment an algorithm with a payment function
• i.e., design a truthful mechanism
• The payment function should incentive in telling
  the truth

3
10
1
1
s
2
2
1
3
7
7
4
1
6
VCG Mechanisms
valuation
Utility(3) payment(3) cost(3) 3 3
0 Utility(9) payment(9) cost(9) 9 3 6
3
9
10
1
1
2
s
2
1
3
7
4
7
Ae0 Ae be
1
Pe be if e is selected (0 otherwise)
Pe Ae8 Ae0 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)
7
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)

(skip)
Utilitarian problems objective is to maximize
the social welfare (?i valuationi(X))
BB mechanisms sum of payments equals the cost of
the solution
8
Cost-Sharing Mechanisms
9
Multicast and Cost-Sharing
Accept or reject the service?

• A service provider s
• Selfish customers U
• Who is getting the service?
• How to share the cost?

is worth 5 (? 7)
Pi
10
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!
11
Coping with Selfishness Mechanism Design
P1
bj
bi

• Algorithm A
• Who gets serviced (Q(b))
• How to reach Q(b) (Construct tree T)
• Payment P
• How much each user pay

P2
P4
P3
12
Ms Truthfulness (or Strategyproofness)
vi

• 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)

13
Ms Group Strategyproofness
U
Coalition C
No one gains
At least one looses (ie, ui gt ui)
C is useless
Breaks off C
Does this definition fit our intuition of
collusion-resistant mechanisms?
14
Mechanisms Requirements

• Budget Balance (BB)
• ?i ?T Pi(b) COST(T)
• Efficiency (NW) maximize
• NET WORTH(T) WORTH(T) - COST(T)
• where WORTH(T) ?i?T vi
• (natural requirements)

BB and efficiency are mutually exclusive!
15
Mechanisms Requirements

• Budget Balance (BB)
• ?i ?T Pi(b) COST(T)
• (natural economic requirements)

16
Cost-Sharing Budget-Balance Mechanisms

• Penna V, WAOA 04
• Penna V, SIROCCO 05
• Penna V, STACS 06

17
How to build BB, GSP Mechanisms
Cost-sharing methods distribute COST(Q) among
users in Q
? (Q,i) ? 0
Q
U
? (Q,i) 0, i ? Q
? ? (Q,i) COST(Q)
Idea associate prices to service set
18
How to build BB, GSP Mechanisms
Cost-sharing method ? (,) ? Mechanism M(? )
U
? (Q,i)
Drop i
Q
gt bi
19
How to build BB, GSP Mechanisms
Cost-sharing method ? (,) ? Mechanism M(? )
Monotonicity
Pi ? (Qk,i)
Moulin Shenker 97 PV04
Self Cross
Cross
for all Q subsets of U
for all Q (possibly) outputted by M
20
Self cross monotonicity an example
COST(Q)
50
50
s
Q
s
Pay less than before
This is not a cross monotonic cost sharing method!
21
Self cross monotonicity an example (2)
COST(Q)
100
s
This is not a cross monotonic cost sharing method!
Q
This guy pays 0
s
M(?) cannot drop him
Pay less than before
Idea some subsets do not appear. We need ?
monotone only for possible subsets generated by
M(?)
22
Sequential Algorithms

• A is sequential if for some bid vectors reaches a
  chain of sets Q1, , QU, QU1Ø
• Sequential algorithms admits a self
  cross-monotonic cost-sharing method

. . .
BB GSP Mechanisms
QU1 Ø
23
Optimal Sequential Algorithm for Steiner Tree Game
T ? opt
s
Q
Q
u
v
v
T
?
?
opt Steiner tree
v is the last node added by Prims MST algorithm
24
Adding Fairness to Our Mechanisms

• Payment is still self cross-monotonic
• Is it possible to have no free rider?
• No! Unless PNP

opt Steiner tree
25
Can we do better without Sequential Algorithms?
M is SP, BB,
M for 2 users
A is sequential
Natural GSP Mechanisms
A is sequential
26
Mechanisms with Verification

• Ferrante, Parlato, Sorrentino V, WAOA 2005
• Auletta, De Prisco, Penna, Persiano V, ICALP
  2006
• V, WINE 2006
• Penna V. , 2007

27
Motivating Verification Model
Used Car market The Kelley Blue Book the
Trusted Resource (www.kbb.com)
28
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?
29
Selfish Task Scheduling
Awarded independently from the execution!
Mechanism design payments ? utility payment
- cost
Optimal Makespan minx maxi ti(X)
Allocation X ? cost ti(X) ti loadi(X)
M1
M2
M4
M3
M5
t1
t2
t3
t4
t5
b1
b2
b3
b4
b5
ti 1 / si (ie, the inverse of the speed)
30
Verifiable Selfish Agents
Verification observe jobs release time
3
Verification is impossible!
ti(X) loadi(X) ti
1
i bids from the set 1/2, 1, 2
1/2
i underbids
is release time should be 2 but
is finishing time is 4
ti 1
i can wait 2 time slots delivering the results in
the right time
1
i overbids
1
2
IDEA (Nisan Ronen, 99) No payment for
underbidding agents
31
Verification Setting

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

32
The Power of Verification
Classical mechanisms
Mechanisms w/ Verification
algorithms
loadi
loadi
NO!
NO!
TRUTHFUL
TRUTHFUL
bi
Archer Tardos, 01
Auletta al, 04
bi
Payment functions
Not unique
loadi
Unique
Pi(bi, b-i) Wmax / bi ( Wmax si)
Related to max possible supported cost
Scaling up for general speeds
bi
ti
Archer Tardos, 01
33
The Power of Verification Breaking Lower Bounds
Efficient APX truthful mechanisms
w/verification c-APX algorithm A ?
c(1?)-APX mechanism
weight
p2
p3
p4
p5
p6
p7
p8
p9
p1
priority
M1
M2
M4
M3
M5
b1
b2
b3
b4
b5
t3
t1
t2
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
34
Generalizing Verification Setting

• Give the payment if the results are given in
  time (ie, consistently with bi)
• For the outcome computed in bi, ie, X
• ti(X) ? bi(X) ? just wait and get the payment
• ti(X) gt bi(X) ? no payment (punish agent i)

35
(Optimal) Mechanisms with Verification
Breaking lower bounds for classical mechanisms
concerning many natural problems (eg, variants of
SPT problem)


Given an algorithm c-apx
Goal minimizing the makespan
a c(1?)-apx
an exact
There exists truthful mechanism
with verification
We dont if truthful mechanisms without
verification do exist
polytime
(althougt not polynomial-time)
36
Optimal Collusion-Resistant Mechanisms w/
Verification
GSP do not consider side payments
U
Coalition C
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

37
Conclusions

• Cost-Sharing Games
• Simple techniques
• lead to polynomial-time cost-sharing mechanisms
  for NP-Hard problem Steiner Tree
• not so unfair (unless PNP)
• 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

38
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., NisanRonen 99)

39
Questions?