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!