Combinatorial Auctions with Complement-Free Bidders – An Overview Speaker: Michael Schapira Based on joint works with Shahar Dobzinski & Noam Nisan.

Combinatorial Auctions with Complement-Free Bidders – An Overview

Speaker: Michael Schapira

Based on joint works with Shahar Dobzinski & Noam Nisan

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Talk Structure

Combinatorial auctions with CF bidders Approximating with value queries

An incentive compatible O(m1/2)-approximation for CF bidders using value queries.

Approximating with demand queries. A combinatorial algorithm that obtains a 2-approximation

for XOS bidders. A randomized-rounding algorithm that obtains a 2-

approximation for XOS bidders using demand queries. Open Questions

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Combinatorial Auctions

A set M={1,…,m} of items for sale. n bidders, each bidder i has a valuation function

vi:2M->R+.Common assumptions:

Normalization: vi()=0 Free disposal: ST vi(T) ≥ vi(S)

Goal: find a partition S1,…,Sn such that social welfare vi(Si) is maximized

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Problem 1: finding an optimal allocation is NP-hard. Therefore, we are interested in the possible approximation ratios.

Problem 2: the valuations’ length is exponential in m, while we wish our algorithms to be polynomial in m and n.

Problem 3: how can we be certain that the bidders do not lie?

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We are interested in algorithms that based on the reported valuations {vi }i output an allocation which is an approximation to the optimal social welfare.

We require the algorithms to be polynomial in m and n. That is, the algorithms must run in sub-linear (polylogarithmic) time.

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Access Models

How can we access the input ?

One possibility: bidding languages.

The “black box” approach: each bidder is represented by an oracle which can answer a certain type of queries.

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Access Models Common types of queries:

Value: given a bundle S, return v(S).

Demand: given a vector of prices (p1,…, pm) return the bundle S that maximizes v(S)-jSpj. (demand queries are strictly more powerful than value queries).

General: any possible type of query (the communication model).

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Known Results Finding an optimal solution requires

exponential communication. Nisan-Segal

Finding an O(m1/2-)-approximation requires exponential communication. Nisan-Segal. (this result holds for every possible type of oracle)

Using demand oracles, a matching upper bound of O(m1/2) exists (Blumrosen-Nisan).

Better results might be obtained by restricting the classes of valuations.

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The Hierarchy of CF Valuations

Complement-Free: v(ST) ≤ v(S) + v(T). XOS Submodular: v(ST) + v(ST) ≤ v(S) + v(T).

Semantic Characterization: Decreasing Marginal Utilities.

GS: (Gross) Substitutes: Solvable in polynomial time.

OXS GS SM XOS CF

Lehmann, Lehmann, Nisan

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Talk Structure






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Value QueriesValuation

ClassUpper Bound Lower Bound

General m/(log1/2m) (Holzman, Kfir-Dahav,

Monderer, Tennenholz)

(Incentive Compatible)

m/(logm) (Nisan-Segal)

CF m1/2


XOS m1/2-

SM 2(Lehmann,Lehmann,Nisan) e/(e-1)-(Khot,

Lipton,Markakis, Mehta)

GS 1(Bertelsen, Lehmann)


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Incentive Compatibility & VCG Prices We want an algorithm that is truthful (incentive

compatible). I.e. we require that the dominant strategy of each of the bidders would be to reveal true information.

VCG is the main general technique known for making auctions incentive compatible (if bidders are not single-minded): Each bidder i pays: k≠ivk(O-i) - k≠ivk(Oi)

Oi is the optimal allocation, O-i the optimal allocation of the auction without the i’th bidder.

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Incentive Compatibility & VCG Prices Problem: VCG requires an optimal allocation!

Finding an optimal allocation requires exponential communication and is computationally intractable.

Approximations do not suffice (Nisan-Ronen).

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VCG on a Subset of the Range Our solution: limit the set of possible

allocations. We will let each bidder to get at most one item, or

we’ll allocate all items to a single bidder. Optimal solution in the set can be found in

polynomial time VCG prices can be computed incentive compatibility.

We still need to prove that we achieve an approximation.

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The Algorithm

Ask each bidder i for vi(M), and for vi(j), for each item j.(We have used only value queries)

Construct a bipartite graph and find the maximum weighted matching P.

can be done in polynomial time (Tarjan).

1

2

3

A

B

Items

Bidders

v1(A)

v3(B)

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The Algorithm (Cont.)

Let i be the bidder that maximizes vi(M).

If vi(M)>|P| Allocate all items to i.

else Allocate according to P.

Let each bidder pay his VCG price (with respect to the restricted set).

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Proof of the Approximation RatioTheorem: If all valuations are CF, the algorithm provides an

O(m1/2)-approximation.

Proof: Let OPT=(T1,..,Tk,Q1,...,Ql), where for each Ti, |Ti|>m1/2, and for each Qi, |Qi|≤m1/2. |OPT|= ivi(Ti) + ivi(Qi)

Case 1: ivi(Ti) > ivi(Qi)(“large” bundles contribute most of the social welfare)

ivi(Ti) > |OPT|/2At most m1/2 bidders get at least m1/2 items in OPT. For the bidder i the bidder i that maximizes vi(M), vi(M) > |OPT|/2m1/2.

Case 2: ivi(Qi) ≥ ivi(Ti)(“small” bundles contribute most of the social welfare)

ivi(Qi) ≥ |OPT|/2For each bidder i, there is an item ci, such that: vi(ci) > vi(Qi) / m1/2.(The CF property ensures that the sum of the values is larger than the value of the whole bundle)

{ci}i is an allocation which assigns at most one item to each bidder: |P| ≥ ivi(ci) ≥ |OPT|/2m1/2.

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Talk Structure






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Demand QueriesValuation

ClassUpper Bound Lower Bound

General m1/2

(Blumrosen-Nisan)

m1/2-

(Nisan-Segal)

CF 2(Feige)

2-

XOS e/(e-1)(Dobzinski-Schapira)

e/(e-1)-

SM e/(e-1)-(Feige)

<1 (Feige)

GS 1(Bertelsen, Lehmann)


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XOS

The maximum over additive valuations:

(a:1 b:2 c:3) (a:2)

v({a}) = 2

v({a,b}) = 3

v({a,b,c}) = 6

Examples:

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Algorithm I -ExampleItems: {A, B, C, D, E}. 3 bidders.

• Price vector: p0=(0,0,0,0,0) v1: (A:1 OR B:1 OR C:1) XOR (C:2)Bidder 1 gets his demand: {A,B,C}.

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• Price vector: p1=(1,1,1,0,0) v2: (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2)Bidder 2 gets his demand: {C}

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• Price vector: p1=(1,1,1,0,0) v2: (A:1 OR B:1 OR C:9) XOR (D:2 OR E:2)Bidder 2 gets his demand: {C}

• Price vector: p2=(1,1,9,0,0) v3: (C:10 OR D:1 OR E:2)Bidder 3 gets his demand: {C,D,E}

Final allocation: {A,B} to bidder 1, {C,D,E} to bidder 3.

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Algorithm I Input: n bidders, for each we are given a

demand oracle and an XOS oracle Init: p1=…=pm=0. For each bidder i=1..n

Let Si be the demand of the i’th bidder at prices p1,…,pm.

For all i’ < i take away from Si’ any items from Si.

Let q1,…,qm be the item values in the maximizing clause for Si in vi.

For all j Si update pj = qj.

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Proof

To prove the approximation ratio, we will need these two simple lemmas:

Lemma: The total social welfare generated by the algorithm is at least pj.

Lemma: The optimal social welfare is at most 2pj.

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Proof – Lemma 1

Lemma: The total social welfare generated by the algorithm is at least pj.

Proof: Each bidder i got a bundle Ti at stage i.

At the end of the algorithm, he holds Ai Ti. The definition of the prices guarantees that:

vi(Ai) ≥ jAipj

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Proof – Lemma 2Lemma: The optimal social welfare is at most 2pj.

Proof: Let O1,...,On be the optimal allocation. Let pi,j be the price of the j’th

item at the i’th stage. Each bidder i asks for the bundle that maximizes his

demand at the i’th stage:

vi(Oi)-jOi pi,j ≤ j pi,j – j p(i-1),j

Since the prices are non-decreasing:

vi (Oi )-jOi pn,j ≤ j pi,j – j p(i-1),j

Summing up on both sides:

i vi(Oi )-ijOi pn,j ≤ i (j pi,j –jp(i-1),j)

i vi(Oi )-j pn,j ≤ j pn,j

i vi(Oi ) ≤ 2j pn,j

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Algorithm II (Feige) – Step 1

Solve the linear relaxation of the problem:Maximize: i,Sxi,Svi(S)Subject To: For each item j: i,S|jSxi,S ≤ 1 For each bidder i: Sxi,S ≤ 1 For each i,S: xi,S ≥ 0

Despite the exponential number of variables, the LP relaxation may still be solved in polynomial time using demand oracles.(Nisan-Segal).

OPT*=i,Sxi,Svi(S) is an upper bound for the value of the optimal integral allocation.

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Use randomized rounding to build a “pre-allocation” S1,..,Sn:

• Randomized Rounding: For each bidder i, let Si be the bundle S with probability xi,S, and the empty set with probability 1-Sxi,S.• The expected value of vi(Si) is Sxi,Svi(S)

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Assign every item j, uniformly at random, to one of the bidders i, such that j is in Si.

• Consider a bidder i such that j is in Si. Bidder i gets j with probability 1/(nj +1), where nj is the number of all bidders who got j in Step 2. Since E(nj)=1 we have reached a 2 approximation.

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Talk Structure






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The Approximability of Submodular Bidders

Type of Queries

Upper Bound Lower Bound

Value 2(Lehmann,Lehmann,Nisan) e/(e-1)- (Khot,

Lipton,Markakis, Mehta)

Demand e/(e-1)-(Feige)

<1 (Feige)

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Incentive Compatibility

Valuation Class

Upper Bound Lower Bound

CF O((log2m)/e3) 2-

XOS e/(e-1)-

SM <1 (Feige)

Combinatorial Auctions with Complement-Free Bidders – An Overview Speaker: Michael Schapira Based on joint works with Shahar Dobzinski & Noam Nisan.

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