More Powerful and Simpler Cost-Sharing Methods

1. More Powerful and Simpler Cost-Sharing Methods Carmine Ventre Joint work with Paolo Penna University of Salerno

2. 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

3. 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

4. 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?

5. Mechanism design Mechanism: M=(A, P) How much each user pay P1(b), …, Pn(b) Who gets the service Q(b) How to serve Q(b) s s CA(Q(b)) Q(b) A = MST A = OPT

6. 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.

7. Mechanism’s desired properties • Budget Balance (BB) • Cost recovery i2Q(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

8. Approximation concepts • -apxBudget 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

9. 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() • Initialize Q Ã U • While 9 i 2 Q s.t. (Q,i) > bi drop i: Q Ã Q n {i} • Return Q, Pi = (Q, i) is cross monotonic if 8 Q’ ½ Q µ U: (Q, i) ·(Q’, i) for every i 2 Q’

10. Extant approach (2) • 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 • Mechanism M() • Initialize Q Ã U • While 9 i 2 Q s.t. (Q,i) > bi drop i: Q Ã Q n {i} • Return Q, Pi = (Q, i) is cross monotonic if 8 Q’ ½ Q µ U: (Q, i) ·(Q’, i) for every i 2 Q’

11. 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

12. Self cross monotonicity: an example CA(Q) 50% 50% s Q s Pay less than before This is not a cross monotonic cost sharing method!

13. Self cross monotonicity: an example (2) CA(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 Q µ U do not “appear”. We need  monotone only for possible subsets generated by M()

14. 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() P0 = U Pj = {Qj-1n {i} | (Qj-1,i) > 0, Qj-12Pj-1} P = [j=0nPj • is self cross monotonic if it is cross monotonic for every pair of sets in P

15. 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 ij i2 i1 … 100 %

16. 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 an = i1 … … a1 = in A drops users in this order

17. 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

18. Open problems • When is cost sharing possible? • Other problems • Steiner forest • Connected facility location • … • Distributed mechanisms? • What is the cost of fairness?