The Power of Two Prices Beyond Cross-Monotonicity Yvonne - PowerPoint PPT Presentation

Introduction Significance The Power of Two Prices Conclusion The Power of Two Prices Beyond Cross-Monotonicity Yvonne Bleischwitz Burkhard Monien Florian Schoppmann Karsten Tiemann Department of Computer Science International Graduate School Dynamic Intelligent Systems University of Paderborn August 30, 2007 Intern. Grad. School, University of Paderborn Florian Schoppmann 1 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Sharing the Outcome-Dependent Cost of a Binary Service Protocol 1. Customers submit binding bids for service 2. Service provider determines set of served customers S and how to distribute the incurred cost C ( S ) among them Fundamental in economics, e.g.: ◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products Mechanism Design Problem ◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed Intern. Grad. School, University of Paderborn Florian Schoppmann 2 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Sharing the Outcome-Dependent Cost of a Binary Service Protocol 1. Customers submit binding bids for service 2. Service provider determines set of served customers S and how to distribute the incurred cost C ( S ) among them Fundamental in economics, e.g.: ◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products Mechanism Design Problem ◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed Intern. Grad. School, University of Paderborn Florian Schoppmann 2 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Sharing the Outcome-Dependent Cost of a Binary Service Protocol 1. Customers submit binding bids for service 2. Service provider determines set of served customers S and how to distribute the incurred cost C ( S ) among them Fundamental in economics, e.g.: ◮ Sharing cost of public infrastructure projects ◮ Distributing volume discounts ◮ Allocating development costs of built-to-order products Mechanism Design Problem ◮ Strategic bidding: Cheating possibilities ◮ Incentives for truthful bidding needed Intern. Grad. School, University of Paderborn Florian Schoppmann 2 / 23 ·

Introduction Significance The Power of Two Prices Conclusion A Cost-Sharing Scenario (1/2) Computing center with large cluster of parallel machines ◮ Offering customers (uninterrupted) processing times ◮ Cost proportional to makespan Machine A Customer 1 Machine B Customer 2 Customer 3 Machine C Customer 4 Customer 5 Makespan({1, 2, 3, 4, 5 }) = 7 Intern. Grad. School, University of Paderborn Florian Schoppmann 3 / 23 ·

Introduction Significance The Power of Two Prices Conclusion A Cost-Sharing Scenario (2/2) Each customer “owns” a job Different valuations for having job processed ◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 23 ·

Introduction Significance The Power of Two Prices Conclusion A Cost-Sharing Scenario (2/2) v 1 = 2 v 2 = 4 Each customer “owns” a job v 5 = 1.5 v 3 = 3 v 4 = 2 Different valuations for having job processed ◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 23 ·

Introduction Significance The Power of Two Prices Conclusion A Cost-Sharing Scenario (2/2) v 1 = 2 Makespan([5]) = 7 v 2 = 4 Each customer “owns” a job v 5 = 1.5 v 3 = 3 v 4 = 2 Different valuations for having job processed ◮ Cost for processing job oneself ◮ Virtual cost for not processing ◮ Costs of competing offers Intern. Grad. School, University of Paderborn Florian Schoppmann 4 / 23 ·

Introduction Significance The Power of Two Prices Conclusion The Model ◮ n ∈ N players: ◮ Have private valuations v i ∈ R for service, v := ( v i ) i ∈ [ n ] ◮ Submit bids b i ∈ R to service provider, b := ( b i ) i ∈ [ n ] ◮ Service provider uses mechanism to determine outcome: Definition (Cost-Sharing Mechanism) Mechanism Q ( b ) (“White Box”) b ( Q × x ) : player R n → 2 [ n ] × R n x ( b ) set [ n ] ◮ Desirable that b = v but this cannot be a priori guaranteed Intern. Grad. School, University of Paderborn Florian Schoppmann 5 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Common Assumptions for Cost-Sharing Mechanisms Consider only mechanisms with the following properties ∀ i ∈ [ n ] : ◮ NPT (No Positive Transfer) = no negative payments: x i ( b ) ≥ 0 ◮ VP (Voluntary Participation) = obey bids: x i ( b ) ≤ b i and ( i / ∈ Q = ⇒ x i ( b ) = 0 ) ◮ CS (Consumer Sovereignty): i ∈ R : ∀ b ∈ R n : ( b i ≥ b + ∃ b + ⇒ i ∈ Q ( b )) i = Note that players may opt not to receive the service: ∀ b ∈ R n : ( b i < 0 = ⇒ i / ∈ Q ( b )) Intern. Grad. School, University of Paderborn Florian Schoppmann 6 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Modeling Assumptions ◮ Players are rational, only maximize � v i − x i ( b ) if i ∈ Q ( b ) u i ( b ) := 0 if i / ∈ Q ( b ) (That is, utilities are quasi-linear) ◮ A coalition K ⊆ [ n ] forms if ∃ v ∈ R n for which it is successful. Successful means: There are bids b K ∈ R K such that ◮ u i ( v − K , b K ) ≥ u i ( v ) for all i ∈ K and ◮ u i ( v − K , b K ) > u i ( v ) for at least one i ∈ K . Players do not sacrifice their own utility to help others! Intern. Grad. School, University of Paderborn Florian Schoppmann 7 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Desirable Properties of Mechanisms ◮ GSP (Group-Strategyproofness): No coalition is ever successful Definition ( n -Player Cost Function) Function C : 2 [ n ] → R ≥ 0 with C ( A ) = 0 ⇐ ⇒ A = ∅ Typically, costs stem from a combinatorial optimization problem Convention: C denotes cost of service provider if he uses an optimal solution (tractable in this work) ◮ β -BB ( β -Budget-Balance, with β ≥ 1): � C ( Q ( b )) ≤ x i ( b ) ≤ β · C ( Q ( b )) i ∈ [ n ] Intern. Grad. School, University of Paderborn Florian Schoppmann 8 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Implications of GSP Theorem (Moulin, 1999) Let ( Q × x ) be a GSP mechanism, b , b ′ ∈ R n , and Q ( b ) = Q ( b ′ ) . Then x ( b ) = x ( b ′ ) . ◮ Payments independent of bids ◮ Bids only determine set of served players Definition ( n -Player Cost-Sharing Method) Function ξ : 2 [ n ] → R n . β -BB defined as before: � ∀ A ⊆ [ n ] : C ( A ) ≤ ξ i ( A ) ≤ β · C ( A ) i ∈ [ n ] Intern. Grad. School, University of Paderborn Florian Schoppmann 9 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Implications of GSP Theorem (Moulin, 1999) Let ( Q × x ) be a GSP mechanism, b , b ′ ∈ R n , and Q ( b ) = Q ( b ′ ) . Then x ( b ) = x ( b ′ ) . ◮ Payments independent of bids ◮ Bids only determine set of served players Definition ( n -Player Cost-Sharing Method) Function ξ : 2 [ n ] → R n . β -BB defined as before: � ∀ A ⊆ [ n ] : C ( A ) ≤ ξ i ( A ) ≤ β · C ( A ) i ∈ [ n ] Intern. Grad. School, University of Paderborn Florian Schoppmann 9 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Moulin Mechanisms ◮ Basically only one general approach to design GSP mechanisms ◮ GSP relies on cross-monotonic cost-sharing method ξ , i.e., ∀ A , B ⊆ [ n ] and ∀ i ∈ A : ξ i ( A ) ≥ ξ i ( A ∪ B ) Algorithm (for computing M ξ : R n → 2 [ n ] × R n ≥ 0 , Moulin, 1999) Input: ξ : 2 [ n ] → R n ≥ 0 , b ∈ R n ; Output: Q ∈ 2 [ n ] , x ∈ R n ≥ 0 1: Q := [ n ] 2: while ∃ i ∈ Q : b i < ξ i ( Q ) do Q := { i ∈ Q | b i ≥ ξ i ( Q ) } 3: x := ξ ( Q ) Theorem (Moulin, 1999) M ξ satisfies GSP and β -BB if ξ is cross-monotonic and β -BB. Intern. Grad. School, University of Paderborn Florian Schoppmann 10 / 23 ·

Introduction Significance The Power of Two Prices Conclusion Previous Research (1/2) Characterizations. ◮ Submodular costs: For all A ⊆ B ⊆ [ n ] and i / ∈ B : C ( A ∪ { i } ) − C ( A ) ≥ C ( B ∪ { i } ) − C ( B ) . Theorem (Moulin, 1999, for submodular costs) Any GSP, 1-BB mechanism has cross-monotonic cost-shares. Conversely, a 1-BB cross-monotonic ξ exists. ◮ Upper-Continuous mechanisms: If i ∈ Q ( b − i , b i ) for every bid b i > b ∗ i , then i ∈ Q ( b − i , b ∗ i ) . Theorem (Immorlica et. al., 2005) Any upper-continuous, GSP, β -BB mechanism has cross-monotonic cost-shares. Trivially, M ξ is upper-continuous. Intern. Grad. School, University of Paderborn Florian Schoppmann 11 / 23 ·