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Combinatorial Auctions without Money
Dimitris Fotakis, NTUAPiotr Krysta, University of LiverpoolCarmine 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 setsUnknown 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 m1/(b+1)-ε is possible in polynomial-time
The min{m,d+1}-apx algorithm
vi(S1)vi(S2)
Exists S s.t. S intersection S1 is nonempty
S
bi(S1) 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
b
1+δ
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
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