Pseudonyms in Cost Sharing Paolo Penna Riccardo Silvestri Peter - - PowerPoint PPT Presentation

Pseudonyms in Cost Sharing

Paolo Penna • Riccardo Silvestri • Peter Widmayer • Florian Schoppmann

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Pseudonyms

“What makes mechanisms immune to fake identities?”

fschopp@stanford.edu fschopp@uni-paderborn.de

• Virtual identities are cheap

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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)

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Pseudonyms

“What makes mechanisms immune to fake identities?”

fschopp@stanford.edu fschopp@uni-paderborn.de

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“Who should participate in a joint project and at what price?”

Cost Sharing

Car Sharing Infrastructure for broadband internet access Automated Negotiations in logistics

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• Who should participate and at what price?
• Typical requirements
• Approximate budget balance:
• Economic efficiency (relaxed here: consumer sovereignty)
• Strategy-proofness,

strategic players optimize net utility =

Cost-Sharing Mechanisms

b Q ⊆ {1, . . . , n} x = (x1, . . . , xn) where xi ∈ [0, bi]

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Example: Car Sharing

Identical prices, iteratively drop all underbidders?

• For non-submodular costs:
• SP and BB mutually exclusive with identical prices

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 4 4 4 6

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Example: Car Sharing

Identical prices, iteratively drop all underbidders?

• For non-submodular costs:
• SP and BB mutually exclusive with identical prices

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 4 4 4 6 b1 = 4

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Example: Car Sharing

Identical prices, iteratively drop all underbidders?

• For non-submodular costs:
• SP and BB mutually exclusive with identical prices

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 4 4 4 6 b1 = 4

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Example: Car Sharing

Serve first two players i bidding bi ≥ 3 for price 3, all others for price 6

• Want Pseudonym-proofness
• No multiple bids

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 3 3 6 6 Alice Bob Cindy

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Example: Car Sharing

Serve first two players i bidding bi ≥ 3 for price 3, all others for price 6

• Want Pseudonym-proofness
• No multiple bids

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 3 3 6 6 Alice Bob Cindy Adam

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Example: Car Sharing

Serve first two players i bidding bi ≥ 3 for price 3, all others for price 6

• Want Pseudonym-proofness
• No multiple bids

v1 = 3.5 v2 = 3.5 v3 = 6 6 3 3 3 3 6 6 Alice Bob Cindy Adam

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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

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Previous Techniques

• Separability: x = ξ(Q)
• Moulin mechanisms (Moulin, Soc Choice Welf’99)
• Cross-monotonic cost shares:

ξi (S ∪ j) ≤ ξi (S)

• Choose largest b-feasible set,

i.e., ∀i ∈ Q: bi ≥ ξi (Q)

• Acyclic mechanisms (Mehta et al., EC’07)
• Two-price mechanisms (Bleischwitz et al., MFCS’07/09)

Q := {1, . . . , n} while ∃ i: bi < ξi (Q) Q := Q \ i b Q x ξ

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Name independence

• A (separable) reputationproof cost-sharing

mechanism satisfies ξi (S ∪ i) = ξj (S ∪ j) for all S and i,j ∉ S

• Follows from

consumer sovereignty S i j Names

• Serving three players
• Assign a weight to all edges
• Exact budget balance: For all triangles, sum of edge

weights must be 12

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Example

Adam Alice Bob Cindy 6 3 4

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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]

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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

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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

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Hypergraphs

Poly := { x | }

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How much can cost shares differ?

• Suppose x and Q are such that xQ is minimal
• W.l.o.g. assume xR = xR’ for all R, R’ with |Q ∩ R| = |Q ∩ R’|
• Let pk unique value with xR = pk for all R with |Q ∩ R| = k
• Then xQ = ps – 1 and for evey s-subset S with k = |S ∩ Q|
• Thus,

monotone in every bi

• With a short calculation:

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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.

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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

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Ralax Rename-proofness

S i j Names S i j Names fschopp@uni-paderborn.de fschopp@stanford.edu

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Ralax Rename-proofness

• Use reputation for ranking players!

S i j Names S i j Names fschopp@uni-paderborn.de fschopp@stanford.edu 1 year ago 2 min ago Feedback: 107 positives Feedback: 1 positive

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Reputationproof

• No player i can increase her utility unilateraly

by bidding with a pseudonym j > i

Reputation 1 n i high low j Names

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Example

• Serve set Q that lexicographically maximizes the

vector of net utilities

• This mechanism is

reputationproof

• This mechanism is

also group-strategyproof (Bleischwitz et al. MFCS’07/09) 6 3 3 3 3 6 6

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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

Thanks!