bayesian combinatorial auctions giorgos christodoulou, annamaria kovacs, michael schapira...

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!