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Combinatorial Auctions without Money

Combinatorial Auctions without Money . Dimitris Fotakis , NTUA Piotr Krysta , University of Liverpool Carmine Ventre , Teesside University. Main question. Money pervasive in (Algorithmic) Mechanism Design to adjust incentives of algorithms.

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Combinatorial Auctions without Money

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  1. Combinatorial Auctions without Money DimitrisFotakis, NTUA PiotrKrysta, University of Liverpool Carmine Ventre, Teesside University

  2. Main question • Money pervasive in (Algorithmic) Mechanism Design to adjust incentives of algorithms. • Money necessarily evil (Gibbart-Satterthwaite theorem) but… • Unavailable, morally unacceptable and sometimes at odds with the objective of the mechanism • Money vs verification of agents’ behavior (and the punishment of those caught lying) in Combinatorial Auctions (CAs): • What class of algorithms can we use here? [MN02] • What is the best approximation guarantee we can achieve? [PT09]

  3. CombinatorialAuctions € 1,200 € 20 € 2,200 € 1,000 € 350 Lie (if profitable) Lie (if profitable) Winner and price determination rule

  4. What is the objective? • Want to make society better, yet we charge bidders to enforce truthfulness!?! • CAs without money for a really happy society Revenue Social welfare e.g., VCG

  5. What do we know of the bidders? Unknown 3-minded bidder Known 2-minded bidder € 1,200 € 1,000 € 350 € 20 ? 3 sets € 2,200 ?

  6. Verification in CAs [Krysta&V10] • No overbidding on awarded set [Celik06] [Penna&V09] (and references therein) € 1,200 € 1,000 € 350 € 1,300 ? € 50 €900 OK if outcome φ, Caught lying otherwise

  7. Characterizing truthfulness

  8. Backward compatibility for single minded bidders (k=1) • This is [MN02, LOS01] monotonicity, known to characterize CAs with money • Same class of truthful CAs! • Any truthful CA with money can be turned into one without money by implementing verification

  9. Approximation guarantee of monotone algorithms (any k) Recall that no O(d/log d) and no m1/(b+1)-ε is possible in polynomial-time

  10. The min{m,d+1}-apx algorithm S bi(S1) vi(S2) vi(S1) verified Exists S s.t. S intersection S1 is nonempty

  11. Lower bound on approximation (any k)

  12. Lower bound for deterministic mechanisms • B.c. there exists algorithm A better than 2 apx • Then A must assign both {a} and {b} • Wlog, say A gives {a} to the girl and {b} to the boy • Now if the boy says 0 for {b}, A must keep granting him {b} (by truthfulness) • A’s solution has then SW 1+δ, OPT is 2+δ • A is not better than 2-apx 1+δ a 1+δ 1 1 0 b

  13. Conclusions • We have shown the advantages/limitations of trading verification with money in the realm of CAs • Characterization of truthfulness which makes an interesting parallel with CAs with money • Host of bounds obtained mainly via known algorithmic techniques • Close the gaps • Apply framework to different problems/domains

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