The Algorithmic Structure of Group Strategyproof Budget- Balanced Cost-Sharing Mechanisms Paolo...
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The Algorithmic Structure of Group Strategyproof Budget-Balanced Cost-Sharing Mechanisms
Paolo Penna & Carmine Ventre
Università di Salerno
Italy
Why Cost-Sharing Methods?
Town A needs a water distribution system A’s cost is € 11 millions
Town B needs a water distribution system B’s cost is € 7 millions
A and B construct a unique water distribution system for both cities The total cost is € 15
millions Why not collaborate and
save € 3 millions? How to share the cost?
Town A
Town B
11
7
15
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(b1, …, bn) := vi – Pi(b1, …, bn)
Accept iff my
utility ¸ 0!
Coping with Selfishness: Mechanism Design
Algorithm A Who gets serviced (Q(b)) How to reach Q(b)
Payment P How much each user pay
M = (A, P)
bi
bj
P1
P4
P3
P2
M’s 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
Mechanism’s Requirements
Budget Balance (BB) i2Q(b) Pi(b) = CA(Q(b))
Cost Optimality (CO) CA(¢) is minimum
No positive transfer (NPT) Payments are nonnegative: Pi 0
Voluntary Participation (VP) User i is charged less then his reported valuation b i (i.e. bi ≥
Pi) Consumer Sovereignty (CS)
Each user can receive the transmission if he is willing to pay a high price
Beyond CS Property
M is not upper continuous E.g., serve i for all bids strictly greater than 1
bii(b-i)
ServicedNot Serviced
M SP
Fix i, b-i
CS
M is upper continuous E.g., serve i for all bids greater or equal than 1
Characterizing GSP, BB, … Mechanisms
M = (A, P)
Cost function is submodular P is cross monotonic
[MS99]
Sufficient condition too
[MS99]
M UC & with no free-riders P is cross monotonic
[IMM05]
And the algorithm?
Extant Approach & Algorithms
A is able to reach any set in 2U Cost hard to compute for
some subset (e.g. Steiner tree) Polynomial-time
mechanisms Relax BB condition
Switched beam wireless antenna Generalized cost-sharing
games
U
Sequential Algorithms
A is sequential if for some bid vectors reaches a chain of sets Q1, …, Q|U|, ;
Sequential algorithm leads to GSP, BB, … mechanisms ([PV04], [IMM05]) Steiner tree game BB
mechanisms ([PV04, PV05]) NP-hard problem
UQ1=U
Q3
Q|U|…
Q2
.
.
.
… Q|U|+1 = ;
Our Results
M = (A, P)
M for 2 users A is sequential
M GSP & UC A is sequential
M is SP, BB, …
9 M for 3 users SP, BB with A not sequential
The Two Users Case
No singleton is reached by A
A cannot reach U
A is not sequential
M=(A,P) SP, BB, …
Users compete for the resource
SP ) P2 must be at least 7
1 2
510 712
Unbounded payments
1
2
b1
b2
SP, VP ) P2(b1,0)=0
SP, VP ) P1(0, b2)=0
SP ) P1(b1,b2)= P2(b1,b2)=0
Users not separable
No payments
Three Users: Working Mechanism
1 2 3
Sets Cost
U 3
{1,2} 1
{1,3} 1
{2,3} 1
A is not sequential (no singleton in the sets)
Mechanism M = (A,P)
Serve U if b1>1, b2>1 and b3 > 1
Serve {i, j} if bi > 1 and bj > 1
Serve {i, i+1 mod 3} if bi > 1
{1,2,3} ) P1=P2=P3=1
{1,2} ) P1=1, P2=0
{1,3} ) P1=0, P3=1
{2,3} ) P2=1, P3=0
M is not UC nor GSP (user 3 can help user 1)
Hints for the General Case
Full Coverage (U reachable) Weak Separation (a singleton reachable) Using Upper Continuity & GSP Working in P P
P
Bids only in {0,B}
A user bidding B is serviced no matter what b-i is
Conclusions Introduction of generalized cost-sharing games (modeling
many real-life applications) Simple technique of [PV04, IMM05] is not less powerful than
more complex one for UC mechanisms Relaxing BB does not allow to solve more problems
Are sequential algorithms necessary for not UC & GSP mechanisms too?
GamesUpper Continuous Mechanism
any (non polytime) poly-time
With Sequential Algorithms
P=2U 1 [PV04, IMM05]· (2U) [PV04]
¸ ({U}) [this work]
P has a sequence 1 [PV04, IMM05]· () [PV04]
¸ ({U}) [this work]
With No Sequential Algorithm
P has no sequence unbounded [this work]
unbounded [this work]