More Powerful and Simpler Cost-Sharing Methods

Carmine VentreJoint work with Paolo Penna

University of Salerno

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 don’t collaborate

saving € 3 millions? How to share the cost?

Town A

Town B

Multicast vs cost-sharing

Service provider s Customers U Who gets serviced? How to share the cost?

Accept or reject the service?

We are selfish

Selfish agents

Each customer/agent has a private valuation for the service (vi) (how

much would pay for the service) declares a (potentially different) valuation (bi)

pays something for the service (Pi)

Agents’ goal is to maximize their own utility: ui(b) := vi – Pi(b)

Is my utility ¸ 0?

Mechanism design

Mechanism: M=(A, P)

Who gets the service Q(b)

How much each user pay

P1(b), …, Pn(b)How to serve Q(b)

CA(Q(b))

s s

A = MST A = OPT

Q(b)

Mechanism’s desired properties No positive transfer (NPT)

Payments are nonnegative: Pi 0

Voluntary Participation (VP) User i is charged less then his reported valuation

bi (i.e. bi ≥ Pi)

Consumer Sovereignty (CS) Each user can receive the transmission if he is

willing to pay a high price.

Mechanism’s desired properties Budget Balance (BB)

Cost recoveryi2Q(b) Pi(b) ¸ CA(Q(b))

Competitiveness: i2Q(b) Pi(b) ¦ CA(Q(b)) Cost Optimality (CO)

CA(Q(b)) = COPT(Q(b)) Group-strategyproof

No coalition of agents has an incentive to jointly misreport their true vi

Approximation concepts

-apx Budget Balance: CA(Q(b)) · Pi(b) · COPT(Q(b))

surplus mechanism Pi · (1+) CA(Q(b))

If A is an -apx algorithm and M is 0 surplus then M is -apx BB The converse is not true

Extant approach

MS provide the mechanism M() is a cost-sharing

method (Q, i) = 0 if i Q i2Q (Q, i) = CA(Q)

If is cross monotonic then M() is GSP, NPT, VP, CS and BB ([MS97])

When is cross monotonic?

Mechanism M()

1. Initialize Q Ã U

2. While 9 i 2 Q s.t. (Q,i) > bi drop i: Q Ã Q n {i}

3. Return Q, Pi = (Q, i)

is cross monotonic if 8 Q’ ½ Q µ U:

Q, i) · (Q’, i)

for every i 2 Q’

Extant approach (2)

Mechanism M()

1. Initialize Q Ã U

2. While 9 i 2 Q s.t. (Q,i) > bi drop i: Q Ã Q n {i}

3. Return Q, Pi = (Q, i)

is cross monotonic if 8 Q’ ½ Q µ U:

Q, i) · (Q’, i)

for every i 2 Q’

MS provide also the converse of the previous result: If CA(Q) is submodular

and non decreasing then any M which is BB, NPT, VP, CS and GSP is “equivalent” to some M(), is a cross monotonic cost sharing method

Our Main Results

If is self cross monotonic then M() has the same properties

Self cross monotonicity is a relaxation of the cross monotonicity condition It is much simpler to obtain

Is this more powerful? We provide the first mechanism for Steiner tree game on

the graphs polytime, CO, BB, VP, NPT and CS Not possible to obtain in general with cross monotonicity Best known result was a 2-BB [JV01]

NP hard problem

Self cross monotonicity: an example

Q

CA(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

CA(Q)

s

100%

s

Pay less than before

This guy pays 0

M() cannot drop him

Idea: some Q µ U do not “appear”. We need monotone only for possible subsets generated by M()

This is not a cross monotonic cost sharing method!

Self cross monotonicity

Intuitively a cost sharing method is self cross monotonic if it is cross monotonic w.r.t. M()’s output

We define P as the possible subsets generated by M()

P0 = U

Pj = {Qj-1n {i} | (Qj-1,i) > 0, Qj-1 2 P

j-1}P = [j=0

n Pj

is self cross monotonic if it is cross monotonic for every pair of sets in P

Reasonable algorithm

An algorithm A is reasonable if it can drop user one by one Exists i1, …, in s.t. A can compute a feasible

solution for Qj = U n {i1, …, ij} If A is reasonable then exists a cost sharing

method self cross monotonic for CA

U i1i2

100 %

…ij

The mechanism for the Steiner Tree Game What about if the optimal algorithm is

reasonable? For the Steiner tree game exists A polytime

reasonable which is optimal (only for the sets in P)

What about A? Consider the Prim’s MST algorithm

s, a1, a2, …, an

MST(Qj) is an optimal steiner tree for Qj

A drops users in this order

an = i1

…

…

a1 = in

Our results in wireless networks (3d – 1)-apx BB, no surplus, GSP, NPT, VP, CS

polytime mechanism Characterization of the pair algorithm, wireless

instances for which a cross monotonic mechanism always produce some surplus Surplus increase exponentially with d Definition of A-bad instances G

A is not optimal CA is not submodular (and badness and submodularity are not

equivalent) Our technique can be used to obtain no surplus

mechanisms for wireless instances

Open problems

When is cost sharing possible? Other problems

Steiner forest Connected facility location …

Distributed mechanisms? What is the cost of fairness?