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

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) PiPi Accept or reject the service?

Selfish Agents Each customer/agent  has a private valuation v i for the service  declares a (potentially different) valuation b i  pays P i for the service Agents’ goal is to maximize their own utility: u i (b 1, …, b n ) := v i – P i (b 1, …, b n ) 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) bibi bjbj P1P1 P4P4 P3P3 P2P2

M’s Strategyproofness For all others players’ declarations b -i it holds u i = u i (v i, b -i ) ¸ u i (b i, b -i ) = u i for all b i (ie, truthtelling is a dominant strategy) M = (A, P) vivi

M’s Group Strategyproofness U Coalition C No one gains At least one looses (ie, u i < u i ) C is useless Breaks off C

Mechanism’s Requirements Budget Balance (BB)   i 2 Q(b) P i (b) = C A (Q(b)) Cost Optimality (CO)  C A ( ¢ ) is minimum 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

Beyond CS Property M is not upper continuous  E.g., serve i for all bids strictly greater than 1 bibi  i (b -i ) Serviced Not 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 submodularP is cross monotonic [MS99] Sufficient condition too [MS99] M UC & with no free-ridersP is cross monotonic [IMM05] And the algorithm?

Extant Approach & Algorithms A is able to reach any set in 2 U  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 Q 1, …, Q |U|, ; Sequential algorithm leads to GSP, BB, … mechanisms ([PV04], [IMM05])  Steiner tree game BB mechanisms ([PV04, PV05]) NP-hard problem U Q 1 =U Q3Q3 Q |U| … Q2Q … Q |U|+1 = ;

Our Results M = (A, P) M for 2 usersA is sequential M GSP & UCA 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 ) P 2 must be at least Unbounded payments 1 2 b1b1 b2b2 SP, VP ) P 2 (b 1,0)=0 SP, VP ) P 1 (0, b 2 )=0 SP ) P 1 (b 1,b 2 )= P 2 (b 1,b 2 )=0 Users not separable No payments

Three Users: Working Mechanism 123 SetsCost U3 {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 b 1 >1, b 2 >1 and b 3 > 1 Serve {i, j} if b i > 1 and b j > 1 Serve {i, i+1 mod 3} if b i > 1 {1,2,3} ) P 1 =P 2 =P 3 =1 {1,2} ) P 1 =1, P 2 =0 {1,3} ) P 1 =0, P 3 =1 {2,3} ) P 2 =1, P 3 =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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? Games Upper Continuous Mechanism any (non polytime)poly-time With Sequential Algorithms P =2 U 1 [PV04, IMM05]  ·  (2 U ) [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]