Combinatorial Auctions without Money

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