Combinatorial Auctions with Complement-Free Bidders

1
Combinatorial Auctions with Complement-Free
Bidders An Overview

• Speaker Michael Schapira
• Based on joint works with Shahar Dobzinski Noam
  Nisan

2
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

3
Combinatorial Auctions

• A set M1,,m of items for sale.
• n bidders, each bidder i has a valuation function
  vi2M-gtR.
• Common assumptions
• Normalization vi(?)0
• Free disposal S?T ? vi(T) vi(S)
• Goal find a partition S1,,Sn such that social
  welfare Svi(Si) is maximized

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Combinatorial Auctions

• 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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Combinatorial Auctions

• 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.

6
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)-Sj?Spj.
  (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-e)-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
Lehmann, Lehmann, Nisan
OXS ? GS ? SM ? XOS ? CF

• Complement-Free v(S?T) v(S) v(T).
• XOS
• Submodular v(S?T) v(S??T) v(S) v(T).
• Semantic Characterization Decreasing Marginal
  Utilities.
• GS (Gross) Substitutes Solvable in polynomial
  time.

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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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Value Queries
Valuation Class Upper Bound Lower Bound
General m/(log1/2m) (Holzman, Kfir-Dahav, Monderer, Tennenholz) (Incentive Compatible) m/(logm) (Nisan-Segal)
CF m1/2 (Incentive Compatible)
XOS m1/2-e
SM 2(Lehmann,Lehmann,Nisan) e/(e-1)-e (Khot, Lipton,Markakis, Mehta)
GS 1(Bertelsen, Lehmann) (Incentive Compatible)
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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 Sk?ivk(O-i) - Sk?ivk(Oi)
• Oi is the optimal allocation, O-i the optimal
  allocation of the auction without the ith 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 well 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).

Bidders
Items
1
v1(A)
A
2
B
3
v3(B)
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The Algorithm (Cont.)

• Let i be the bidder that maximizes vi(M).
• If vi(M)gtP
• 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 Ratio

• Theorem If all valuations are CF, the algorithm
  provides an O(m1/2)-approximation.
• Proof Let OPT(T1,..,Tk,Q1,...,Ql), where for
  each Ti, Tigtm1/2, and for each Qi, Qim1/2.
  OPT Sivi(Ti) Sivi(Qi)

Case 1 Sivi(Ti) gt Sivi(Qi) (large bundles
contribute most of the social welfare) ? Sivi(Ti)
gt OPT/2 At 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) gt OPT/2m1/2.
Case 2 Sivi(Qi) Sivi(Ti) (small bundles
contribute most of the social welfare) ? Sivi(Qi)
OPT/2 For each bidder i, there is an item
ci, such that vi(ci) gt vi(Qi) / m1/2. (The CF
property ensures that the sum of the values is
larger than the value of the whole bundle) cii
is an allocation which assigns at most one item
to each bidder P Sivi(ci) OPT/2m1/2.
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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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Demand Queries
Valuation Class Upper Bound Lower Bound
General m1/2 (Blumrosen-Nisan) m1/2-e (Nisan-Segal)
CF 2 (Feige) 2-e
XOS e/(e-1) (Dobzinski-Schapira) e/(e-1)-e
SM e/(e-1)-e (Feige) rlt1 (Feige)
GS 1(Bertelsen, Lehmann) (Incentive Compatible)
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XOS

• The maximum over additive valuations

(a1?? b2 ? c3)?? (a2)
v(a) 2
Examples
v(a,b) 3
v(a,b,c) 6
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Algorithm I -Example

• Items A, B, C, D, E. 3 bidders.
• Price vector p0(0,0,0,0,0) v1 (A1 OR B1 OR
  C1) XOR (C2)Bidder 1 gets his demand A,B,C.

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Algorithm I -Example

• Items A, B, C, D, E. 3 bidders.
• Price vector p0(0,0,0,0,0) v1 (A1 OR B1 OR
  C1) XOR (C2)Bidder 1 gets his demand A,B,C.
• Price vector p1(1,1,1,0,0) v2 (A1 OR B1 OR
  C9) XOR (D2 OR E2)Bidder 2 gets his demand
  C

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Algorithm I -Example

• Items A, B, C, D, E. 3 bidders.
• Price vector p0(0,0,0,0,0) v1 (A1 OR B1 OR
  C1) XOR (C2)Bidder 1 gets his demand A,B,C.
• Price vector p1(1,1,1,0,0) v2 (A1 OR B1 OR
  C9) XOR (D2 OR E2)Bidder 2 gets his demand
  C
• Price vector p2(1,1,9,0,0) v3 (C10 OR D1 OR
  E2)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 p1pm0.
• For each bidder i1..n
• Let Si be the demand of the ith bidder at prices
  p1,,pm.
• For all i lt 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 Spj.
• Lemma The optimal social welfare is at most 2Spj.

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Proof Lemma 1

• Lemma The total social welfare generated by the
  algorithm is at least Spj.
• 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) Sj?Aipj

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Proof Lemma 2

• Lemma The optimal social welfare is at most
  2Spj.
• Proof
• Let O1,...,On be the optimal allocation. Let pi,j
  be the price of the jth item at the ith stage.
• Each bidder i asks for the bundle that maximizes
  his demand at the ith stage
• vi(Oi)-Sj?Oi pi,j Sj pi,j Sj p(i-1),j
• Since the prices are non-decreasing
• vi (Oi )-Sj?Oi pn,j Sj pi,j Sj p(i-1),j
• Summing up on both sides
• Si vi(Oi )-SiSj?Oi pn,j Si (Sj pi,j
  Sjp(i-1),j)
• Si vi(Oi )-Sj pn,j Sj pn,j
• Si vi(Oi ) 2Sj pn,j

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Algorithm II (Feige) Step 1

• Solve the linear relaxation of the problem
• Maximize Si,Sxi,Svi(S)
• Subject To
• For each item j Si,Sj?Sxi,S 1
• For each bidder i SSxi,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).
• OPTSi,Sxi,Svi(S) is an upper bound for the
  value of the optimal integral allocation.

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Algorithm II (Feige) Step 2

• 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-SSxi,S.
• The expected value of vi(Si) is SSxi,Svi(S)

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Algorithm II (Feige) Step 3

• 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

• 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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The Approximability of Submodular Bidders
Type of Queries Upper Bound Lower Bound
Value 2(Lehmann,Lehmann,Nisan) e/(e-1)-e (Khot, Lipton,Markakis, Mehta)
Demand e/(e-1)-e (Feige) rlt1 (Feige)
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Incentive Compatibility
Valuation Class Upper Bound Lower Bound
CF O((log2m)/e3) 2-e
XOS e/(e-1)-e
SM rlt1 (Feige)