Bayesian Combinatorial
Auctions
Giorgos Christodoulou, Annamaria Kovacs, Michael Schapira
האוניברסיטה העברית בירושליםהאוניברסיטה העברית בירושליםThe Hebrew University of JerusalemThe Hebrew University of Jerusalem
Combinatorial Auctions
Combinatorial Auctions
opt=9
Combinatorial Auctions
Objective: Find a partition of the items
biddersitems
valuations
that maximizes the social welfare
(normalized)
(monotone)
ValuationsSubmodular (SM)
The marginal value of the item decreasesas the number of items increases.
Fractionally-subadditive (FS)
additive
FS Valuations
a b c0 9 00 5 55 5 04 4 4
items
add. valuations
Combinatorial Auctions - Challenges
StrategicWe want bidders to be truthful.VCG implements the opt. (exp. time)
Computationalapproximation algorithms (not
truthful)
Unknown Valuations
Huge Gaps
Submodular (SM)
Fractionally-subadditive (FS)
1-1/e-[Feige-Vondrak]
1-1/e[Dobzinski-Schapira] O(log(m) log log(m))
[Dobzinski]
Solution?
We do not know whether reasonable truthful and polynomial-time approximation algorithms exist.
How can we overcome this problem?
An old/new approach.
Partial Informationis
drawn from D
Complete Information
Auction SettingPlayer i will bidStrategy Profile Algorithm = allocation +
payments
Utility of player i
Bayesian Combinatorial
AuctionsQuestion: Can we design an auction for which any Bayesian
Nash Equilibrium provides good approximation to the
social welfare?
(Pure) Bayesian Nash [Harsanyi]
•Bidding function
•Informal: In a Bayesian Nash (B1,…,Bn), given a probability distribution D, Bi(vi) maximizes the expected utility of player i (for all vi).
( )
Bayesian PoA
Optimal Social Welfare
Expected Social of a B.N.E.
for fixed v
Bayesian PoA = biggest ratio between SW(OPT) and SW(B) (over all D, B)
Bayesian PoA
Price of Anarchy
[Gairing, Monien, Tiemann, Vetta]
Second PricePlayer i will bid
Strategy Profile
Algorithm:Give item j to the player i with the highest
bid. Charge I the second highest bid.
Utility of player i
Second Price
Social Welfare = 1
Second Price
Social Welfare =
Second Price
Social Welfare =
PoA=1/
Supporting Bids
Bidders have only partial info (beliefs)•They want to avoid risks. (ex-post IR)
Supporting Bids:(for all S)
Lower Bound
opt=2
Lower Bound
Nash=1PoA=2
Our ResultsBayesian setting:The Bayesian PoA for FS valuations
(supporting bids, mixed) is 2.
Complete-information setting:FS Valuations: Existence of pure N.E.Myopic procedure for finding one.PoS=1.
•SM Valuations: Algorithm for computing N.E. in poly time.
ValuationsSubmodular (SM)
The marginal value of the item decreasesas the number of items increases.
Fractionally-subadditive (FS)
additive
Upper Bound(full-info case)
Lemma. For any set of items S,
where is the maximizing additive valuation for the set S.
Upper Bound
Let be a fixed valuation profile
Upper Bound
Let be a fixed valuation profile
optimum partition:Nash partition:
Upper Bound
Since b is a N.E
Let be a fixed valuation profile
optimum partition:
maximum additive valuation wrt
Nash partition:
Upper Bound
Since b is a N.E
Let be a fixed valuation profile
optimum partition:
maximum additive valuation wrt
Nash partition:
Upper BoundSince b is a N.E
and so
Upper BoundSince b is a N.E
and so
using lemma we get
Upper BoundSince b is a N.E
and so
using lemma we get
and so
Upper Bound
summing up
But…
Open Question: Does a (pure) BN with supporting bids always exist?
Open Question: Can we find a (mixed) BN in polynomial time?
We consider the full-information setting.
The Potential ProcedureStart with item prices 0,…,0.
Go over the bidders in some order 1,…,n.
In each step, let one bidder i choose his most demanded bundle S of items.
Update the prices of items in S according to i’s maximizing additive valuation for S.
Once no one (strictly) wishes to switch bundle, output the allocation+bids.
Theorem: If all bidders have fractionally-subadditive valuation functions then the Potential Procedure always converges to a pure Nash (with supporting bids).
Proof: The total social welfare is a potential function.
The Potential Procedure
Theorem: After n steps the solution is a 2-approximation to the optimal social welfare (but not necessarily a pure Nash). [Dobzinski-Nisan-Schapira]
Theorem: The Potential Procedure might require exponentially many steps to converge to a Pure Nash.
The Potential Procedure
Open Question: Can we find a pure Nash in polynomial time?
Open Question: Does the Potential Procedure converge in polynomial time for submodular valuations?
The Potential Procedure
The Marginal-Value ProcedureStart with bid-vectors bi=(0,…,0).
Go over the items in some order 1,…,m.
In each step, allocate item j to the bidder i with the highest marginal value for j.
Set bij to be the second highest marginal value.
Theorem: The Marginal-Value Procedure always outputs an allocation that is a 2-approximation to the optimal social-welfare. [Lehmann-Lehmann-Nisan]
Proposition: The bids the Marginal-Value Procedure outputs are supporting bids and are a pure Nash equilibrium.
The Marginal-Value Procedure
Open Questions
Can a (mixed) Bayesian Nash Equilibrium be computed in poly-time?
Algorithm that computes N.E. in poly time for FS valuations.
Second Price
Design an auction that minimizes the PoA for B.N.E.
Thank you!