More Powerful and Simpler Cost-Sharing Methods Carmine Ventre Joint 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 (v i ) (how much would pay for the service)  declares a (potentially different) valuation (b i )  pays something for the service (P i ) Agents’ goal is to maximize their own utility: u i (b) := v i – P i (b) Is my utility ¸ 0?

Mechanism design Mechanism: M=(A, P) Who gets the service Q(b) How much each user pay P 1 (b), …, P n (b) How to serve Q(b) C A (Q(b)) ss A = MSTA = OPT Q(b)

Mechanism’s desired properties No positive transfer (NPT)  Payments are nonnegative: P i  0 Voluntary Participation (VP)  User i is charged less then his reported valuation b i (i.e. b i ≥ P i ) 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 recovery  i 2 Q(b) P i (b) ¸ C A (Q(b))  Competitiveness:  i 2 Q(b) P i (b) ¦ C A (Q(b)) Cost Optimality (CO)  C A (Q(b)) = C OPT (Q(b)) Group-strategyproof  No coalition of agents has an incentive to jointly misreport their true v i

Approximation concepts  - apx Budget Balance:  C A (Q(b)) ·  P i (b) ·  C OPT (Q(b))  surplus mechanism   P i · (1+  ) C A (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  i 2 Q  (Q, i) = C A (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) > b i drop i: Q Ã Q n {i} 3.Return Q, P i =  (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) > b i drop i: Q Ã Q n {i} 3.Return Q, P i =  (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 C A (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 C A (Q) s 50% s Pay less than before This is not a cross monotonic cost sharing method!

Self cross monotonicity: an example (2) Q C A (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(  ) P  0 = U P  j = {Q j-1 n {i} |  (Q j-1,i) > 0, Q j-1 2 P  j-1 } P  = [ j=0 n P  j  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 i 1, …, i n s.t. A can compute a feasible solution for Q j = U n {i 1, …, i j } If A is reasonable then exists a cost sharing method self cross monotonic for C A Ui1i1 i2i2 100 % … ijij

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, a 1, a 2, …, a n MST(Q j ) is an optimal steiner tree for Q j A drops users in this order a n = i 1 … a 1 = i n

Our results in wireless networks (3 d – 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 C A 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?