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. 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]

Combinatorial Auctions € 1,000 € 1,200 € 350 Winner and price determination rule Lie (if profitable) € 2,200 € 20 Lie (if profitable)

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

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

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

Characterizing truthfulness

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

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

The min{m,d+1}-apx algorithm v i (S 1 )v i (S 2 ) Exists S s.t. S intersection S 1 is nonempty S b i (S 1 ) verified

Lower bound on approximation (any k)

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 a a b b 1+δ 1 1 0

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