Pseudonyms in Cost Sharing Paolo Penna • Riccardo Silvestri • Peter Widmayer • Florian Schoppmann
Pseudonyms “What makes mechanisms immune to fake identities ?” fschopp@stanford.edu fschopp@uni-paderborn.de 1
Pseudonyms “What makes mechanisms immune to fake identities ?” fschopp@stanford.edu fschopp@uni-paderborn.de ‣ Virtual identities are cheap 1
Pseudonyms “What makes mechanisms immune to fake identities ?” fschopp@stanford.edu fschopp@uni-paderborn.de ‣ Virtual identities are cheap ‣ Similar in spirit to falsename-proofness (Yokoo et al., GEB’04) 1
Cost Sharing “ Who should participate in a joint project and at what price ?” Infrastructure for broadband internet access Automated Negotiations in Car Sharing logistics 2
Cost-Sharing Mechanisms ‣ Who should participate and at what price? Q ⊆ { 1, . . . , n } b x = ( x 1 , . . . , x n ) where x i ∈ [ 0, b i ] ‣ Typical requirements • Approximate budget balance: • Economic efficiency (relaxed here: consumer sovereignty) • Strategy-proofness, strategic players optimize net utility = 3
Example: Car Sharing Identical prices, iteratively drop all underbidders? 6 6 v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 4 4 4 ‣ For non-submodular costs: • SP and BB mutually exclusive with identical prices 4
Example: Car Sharing Identical prices, iteratively drop all underbidders? 6 b 1 = 4 6 v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 4 4 4 ‣ For non-submodular costs: • SP and BB mutually exclusive with identical prices 4
Example: Car Sharing Identical prices, iteratively drop all underbidders? 6 b 1 = 4 6 v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 4 4 4 ‣ For non-submodular costs: • SP and BB mutually exclusive with identical prices 4
Example: Car Sharing Serve first two players i bidding b i ≥ 3 for price 3, all others for price 6 6 Alice 6 Bob Cindy v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 3 3 6 ‣ Want Pseudonym-proofness • No multiple bids 5
Example: Car Sharing Serve first two players i bidding b i ≥ 3 for price 3, all others for price 6 6 Alice 6 Adam Bob Cindy v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 3 3 6 ‣ Want Pseudonym-proofness • No multiple bids 5
Example: Car Sharing Serve first two players i bidding b i ≥ 3 for price 3, all others for price 6 6 Alice 6 Adam Bob Cindy v 1 = 3.5 3 3 v 2 = 3.5 v 3 = 6 3 3 6 ‣ Want Pseudonym-proofness • No multiple bids 5
Randomization? ‣ Non-trivial if also collusion resistance required • Randomization over collusion-resistant mechanisms ⇒ collusion-resistant in expectation • And vice versa (Goldberg, Hartline, SODA’05) ‣ not very common in cost-sharing literature • should only be means, but not an end 6
Previous Techniques ‣ Separability: x = ξ ( Q ) ξ b Q x ‣ Moulin mechanisms (Moulin, Soc Choice Welf’99) • Cross-monotonic cost shares: ξ i ( S ∪ j ) ≤ ξ i ( S ) Q := {1, . . . , n } while ∃ i : b i < ξ i ( Q ) • Choose largest b -feasible set, i.e., ∀ i ∈ Q : b i ≥ ξ i ( Q ) Q := Q \ i ‣ Acyclic mechanisms (Mehta et al., EC’07) ‣ Two-price mechanisms (Bleischwitz et al., MFCS’07/09) 7
Name independence ‣ A (separable) reputationproof cost-sharing mechanism satisfies ξ i ( S ∪ i ) = ξ j ( S ∪ j ) for all S and i , j ∉ S Names i j • Follows from S consumer sovereignty 8
Example ‣ Serving three players • Assign a weight to all edges • Exact budget balance: For all triangles, sum of edge weights must be 12 6 Adam Alice 4 3 Bob Cindy 9
Hypergraphs 10
Hypergraphs ‣ Cost shares for s -player sets: • Consider complete ( s – 1)-uniform hypergraph • Assign weight to each hyperedge so that for all s -subsets the sum of all its hyperedges’ weights is ∈ [ 1, β ] 10
Hypergraphs ‣ Cost shares for s -player sets: • Consider complete ( s – 1)-uniform hypergraph • Assign weight to each hyperedge so that for all s -subsets the sum of all its hyperedges’ weights is ∈ [ 1, β ] ‣ System of linear inequalities: (1, …, 1) ≤ A · x ≤ ( β , …, β ) • Gottlieb (Proc. of AMS’66): Incidence matrix A has full rank 10
Hypergraphs ‣ Cost shares for s -player sets: • Consider complete ( s – 1)-uniform hypergraph • Assign weight to each hyperedge so that for all s -subsets the sum of all its hyperedges’ weights is ∈ [ 1, β ] ‣ System of linear inequalities: Poly := { x | (1, …, 1) ≤ A · x ≤ ( β , …, β ) } • Gottlieb (Proc. of AMS’66): Incidence matrix A has full rank 10
How much can cost shares differ? ‣ Suppose x and Q are such that x Q is minimal • W.l.o.g. assume x R = x R’ for all R , R ’ with | Q ∩ R | = | Q ∩ R ’ | • Let p k unique value with x R = p k for all R with | Q ∩ R | = k • Then x Q = p s – 1 and for evey s -subset S with k = | S ∩ Q | • Thus, � monotone in every b i ‣ With a short calculation: 11
But... ‣ For any δ > 0 , given a large enough name space, for each cardinality s there is an s -set S with • When finite number of prices: Coloring • Ramsey’s Theorem (Proc. London Math. Soc’30): Let c , r, s ∈ N with s ≥ r . Then ∃ n : If the r -subsets of any n - set are colored with c colors: ∃ s -set all of whose r -subsets have the same color. 12
Implications ‣ Characterizations of identical prices • New: Separable + 1-budget balance + reputationproof (when at most half the names in use) For excludable public good (i.e., C ( Q ) = 1 ⇔ Q nonempty) • previous characterizations due to, e.g., Dobzinski et al. (SAGT’08) and Deb and Razzolini (Math. Soc. Sciences’99) ‣ Impossibility • Separable, strategyproof, reputationproof, and 1-budget balanced w.r.t. non-submodular costs 13
Ralax Rename-proofness Names Names i j i j S S fschopp@uni-paderborn.de fschopp@stanford.edu 14
Ralax Rename-proofness Names Names i j i j S S fschopp@uni-paderborn.de fschopp@stanford.edu 1 year ago 2 min ago Feedback: 107 positives Feedback: 1 positive ‣ Use reputation for ranking players! 14
Reputationproof Names 1 i j n Reputation high low ‣ No player i can increase her utility unilateraly by bidding with a pseudonym j > i 15
Example ‣ Serve set Q that lexicographically maximizes the vector of net utilities • This mechanism is 6 reputationproof 6 • This mechanism is also group-strategyproof 3 3 (Bleischwitz et al. MFCS’07/09) 3 3 6 16
Conclusion ‣ Renameproof • Identical prices or randomized mechanisms ‣ Reputationproof better reputation ⇒ better price • • In some sense a reasonable derandomization • Most known mechanisms not reputationproof in general 17
Thanks!
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