Issues on the border of economics and computation

נושאים בגבול כלכלה וחישוב

Speaker: Dr. Michael SchapiraTopic: Combinatorial Auctions III

Combinatorial Auctions• Set M of m indivisible items• Set N of n bidders• Preferences are on subsets S – bundles – of

items • Valuation function vi: 2M R

– vi(S) – bidder i’s value for bundle S

– monotone: vi(S) not decreasing in S

– normalized: vi() = 0

Allocation: mutually-disjoint subsets S1, S2, … Sn

Social welfare of allocation: i vi(Si)

What Do We Want?

1. “Good” (w.r.t. efficiency) outcomes (preferably optimal)

2. Incentive compatibility (preferably in dominant strategies)

3. Low running time (in the “natural parameters”: n and m)

Cannot Simply Use VCG!

• Finding optimal allocation is computationally (=NP) hard!

• Cannot compute “approximate” VCG payments.

• The “clash” between Econ and CS. What can we do?

Natural Restrictions on Bidders

• Defn: A valuation v is subadditive (complement-free) if for all S,TM, v(ST) ≤ v(S) + v(T).

• Defn: A valuation v is submodular if for all S,TM, v(ST) + v(ST) ≤ v(S) + v(T).

• Equivalent definition of submodularity: for all STM, and j not in T,

v(T{j})-v(T) ≤ v(S{j})-v(S)

(decreasing marginal utilites)

• Fact: Submodularity implies subadditivity.

Computational Perspective

• Thm: Finding an optimal allocation in combinatorial auctions with submodular bidders is NP-hard.

• Thm: A 2-approximation to the optimal allocation in combinatorial auctions with submodular bidders can be computed in a computationally-efficient manner.

• The 2-approximation algorithm is not truthful. What’s next?

Computational Perspective• Thm: There exists a computationally-

efficient and incentive compatible 2m½-approximation mechanism for auctions with subadditive bidders.

• Thm: No computationally-efficient and incentive compatible mechanism can obtain an approximation ratio of m½-e for auctions with submodular bidders.

• An inherent clash between efficient computation and incentive compatibility.

Incentive Compatibility via VCG?

• We want an algorithm that is incentive compatible in dominant strategies.

• VCG is the only general technique known for making auctions incentive compatible

– 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 i’th bidder.

• Problem: VCG requires finding optimal allocations!

• This is computationally intractable.

• Approximations do not suffice…

• But, that does not mean we cannot use VCG in a more creative way…

Incentive Compatibility via VCG?

• A mechanism M is MIR (= VCG-based) if:– There’s a fixed subset RM of the possible

outcomes (allocations of the m items between the n bidders) = “M’s range”.

– For every valuation profile (v1,…vn) M outputs the optimal partition in RM.

• Fact: MIR mechanisms are truthful (Why?).

RM

allpartitions

Maximal-In-Range Mechanisms

MIR for Subadditive Auctions

• Key idea: limit the set of possible allocations.– either each bidder gets at most one item– or all items are allocated to a single bidder.

• Optimal solution in the set can be found in a computationally efficient manner VCG prices can be computed incentive compatibility.

• We still need to prove that we achieve an approximation.

The Algorithm• Ask each bidder i for vi(M), and for vi(j), for

each item j.

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

• can be done in polynomial time.

1

2

3

A

B

ItemsBidders

v1(A)

v3(B)

The Algorithm (Cont.)

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

• If vi(M)>Val(P)– Allocate all items to i.

• else– Allocate according to P.

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

Proof of Approximation Ratio

Theorem: The algorithm provides an(2m1/2)-approximation for subadditive bidders.

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

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

Sivi(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: Sivi(Qi) ≥ Sivi(Ti)(“small” bundles contribute most of the

social welfare)

Sivi(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| ≥ Sivi(ci) ≥ |OPT|/2m1/2.