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Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב

Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב. Speaker: Dr. Michael Schapira Topic: Combinatorial Auctions III. Combinatorial Auctions. Set M of m indivisible items Set N of n bidders Preferences are on subsets S – bundles – of items

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Issues on the border of economics and computation נושאים בגבול כלכלה וחישוב

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  1. Issues on the border of economics and computationנושאים בגבול כלכלה וחישוב Speaker: Dr. Michael Schapira Topic: Combinatorial Auctions III

  2. 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: ivi(Si)

  3. What Do We Want? • “Good” (w.r.t. efficiency) outcomes (preferably optimal) • Incentive compatibility (preferably in dominant strategies) • Low running time (in the “natural parameters”: n and m)

  4. 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?

  5. Natural Restrictions on Bidders • Defn: A valuation v is subadditive (complement-free) if for all S,TM,v(ST) ≤ v(S) + v(T). • Defn: A valuation v is submodular if for all S,TM,v(ST) + v(ST) ≤ v(S) + v(T). • Equivalent definition of submodularity: for all STM, and j not in T,v(T{j})-v(T) ≤ v(S{j})-v(S)(decreasing marginal utilites) • Fact: Submodularity implies subadditivity.

  6. 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?

  7. 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.

  8. 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) • Oiis the optimal allocation, O-i the optimal allocation of the auction without the i’th bidder.

  9. Incentive Compatibility via VCG? • 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…

  10. RM allpartitions Maximal-In-Range Mechanisms • 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?).

  11. 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.

  12. 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. Bidders Items v1(A) 1 A 2 B 3 v3(B)

  13. 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).

  14. 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|/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) > |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) > 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.

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