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Mechanism design for electronic markets - algorithms and economics

Mechanism design for electronic markets - algorithms and economics. Rudolf Müller International Institute of Infonomics Maastricht University www.etrade.infonomics.nl. Outline. Auctions in the New Economy Auctions and discrete markets Computational and economic efficiency

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Mechanism design for electronic markets - algorithms and economics

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  1. Mechanism design for electronic markets - algorithms and economics Rudolf Müller International Institute of Infonomics Maastricht University www.etrade.infonomics.nl

  2. Outline • Auctions in the New Economy • Auctions and discrete markets • Computational and economic efficiency • Some polynomial cases • Approximation • Primal-dual algorithms • Summary

  3. Auctions in the New Economy • Due to ICT improvements auctions are in more cases a feasible pricing mechanism(e.g. C2C and B2C auction portals, spot-markets for bandwidth and electricity) • Auctions have more applications due to an extending number of indivisible (discrete), scarce goods and services (e.g. seminars) • Auctions can be used for market-based coordination (e.g. multi-agent systems)

  4. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  5. Auctions and discrete markets • Buyers 1,…, B , Sellers 1,…,S , Items 1,…,K • Buyers have private value vj(I) for subsets of items, sellers have a limited supply of items (we assume reservation price 0) • Design a mechanism that finds an efficient allocation, i.e. maximizes j J vj(Ij)among all feasible allocations

  6. Some questions • Do (item) prices exist that, if buyers maximize their utility=value - price, are market clearing and give an efficient allocation? • How can we find these prices (by an auction)? • If item prices do not work, can it be bundle prices, buyer specific bundle prices?

  7. Some questions (cont.) • Can we define a computational efficient mechanism to find this allocation (and the prices)? • Design of revelation mechanisms:How can we set incentives to buyers that they reveal their valuation (to an auctioneer), who can then guarantee efficiency?

  8. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  9. Example: 1 item, 1 seller, B buyers, integer values • Mechanism: ascending bid English auction • This is an (economic) efficient mechanism that finds a market clearing price • How much communication is needed? With increment 1 the number of necessary increments may be exponential in the data. • With a higher increment economic efficiency is not anymore guaranteed.

  10. Example (cont.) • Mechanism: Vickrey auction (second price, sealed bid) • assign the item to the bidder with highest bid • charge this bidder the second highest bid price • Market clearing price with respect to reported bids • Computational complexity: O(|B|) • Economic efficient, as bidding truth is weakly dominant

  11. General setting with one seller and several items • I set of items, B set of buyers • Every buyer has a private, non-negative valuation vjfor all subsets Ij I • Let Jb index those subsets for which b has a positive valuation, and let J = bBJb • An allocation is a vector x {0,1}|J | • An allocation is feasible if it assigns not more than the supply of every item

  12. The truth revealing Vickrey-Clarke-Groves mechanism • Every bidder b reports valuation vb • Auctioneer computes an optimal allocation x • Price: pb(x)b = vbxb – (vx- v-bs-b) = - v-bx-b + v-bs-b • subscript b : bids of bidder b, subscript -b bids from all other bidders • s-b opt allocation after excluding bids from b

  13. Proof truth revealing • Utility for bidder b under arbitrary bid:vbxb- pb(x)= vbxb+ w-b x-b - w-bs-b • Utility under a true bid:vbxtb - ptb(xt)= vbxtb+ w-bxt-b- w-bs-b • The part of the payment that depends on b’s bid is with a true bid at least as large, since vbxtb+ w-bxt-b  vbxb+ w-b x-b

  14. What about the computational efficiency ? • Assume all bidders submit bids for all subsets • We have to solve a large set packing problem • Set packing is NP-complete, but “good” news: we can solve it in polynomial time because the input is large enough!

  15. Winner determination by dynamic programming • Draw a digraph with B layers of nodes, a node for every subset on every layer, and arcs from subset S on layer b to subset S’ on layer b+1 with length v if bidder b + 1 has made a bid of v for the set S’ \ S, and compute a longest path in this digraph • What is wrong? The number of bids is not realistic!

  16. Conclusion so far • We have to use the specific structure of an application in order • to reduce the bidders type such that it can be reported (e.g., the type is polynomial in the number of items) • to structure the bidders type such that a reasonable number (e.g. polynomial in the number of items) operations leads to an efficient allocation

  17. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  18. Winner determinationproblem • Integer linear programming (IP) model max v x Ax a x  0 x integer A is the item-bid incidence matrix, a = 1 in the single-unit case. • NP-complete, since every set packing problem is a combinatorial auction

  19. Lookup set packing literature • (Hofman, Kruskal 1956) A is totally unimodular (every square submatrix has determinant equal to -1,0,1) if and only if for every v and every a the linear program max v x Ax ax  0has an integer optimal solution (or infeasible or unbounded).

  20. Example:Linearly ordered items Polynomial solvable, for super-additive bids (Rothkopfet al., 1998) NP-complete for sub-additive valuations (follows, e.g., from Spieksma, 1999)

  21. Combinatorial algorithms • Min cost flow: • cardinality based sub-additivity (Tennenholtz (2000), Müller, Schulz (2000)) • Computational Geometry • intersection graphs (intervals, trapezoids) • Dynamic Programming • Multi-unit for a limited number of distinct items (van Hoesel, Müller, 2000)

  22. The German UMTS auction First round sells 12 identical frequency ranges License consists of 2 or 3 ranges

  23. Winner determination is pseudo-polynomial The winner determination problem for identical items can be solved in (pseudo) polynomial time (van Hoesel, M. 1999) Letm(i,s)be the value of the optimal assignment if bids by bidders1,…,iare considered and at mostsitems are assigned.

  24. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  25. What if we do not optimize in VCG mechanism? • Auctioneer computes a not necessarily optimal set packing xa • pb(xa) = - w-bxa+ w-bsa-b • Theorem (Nisan, Ronen, 2000)Such a mechanism is truth revealing if and only if xa is optimal in the range of the allocation algorithm • Algorithms that are optimal in their range are not reasonable

  26. Single minded case • Every bidder b wants to have exactly one subset Ib (or a set that contains Ib ) at price vb • An allocation algorithm is called monotone if higher or equal bids on smaller or the same subset keep winning • It is called exact if a bidder wins Ib or nothing, i.e. she looses

  27. Single minded case • Theorem (Lehman, O’Callaghan, Shoham)Let a mechanism be monotone, and let J be a set of bids. Then for every other set I0 there exists a number c0  {-,} such that for all w0 < c0 the bid (I0,w0) looses in an auction with J, and for every w0 > c0 it wins in an auction with J • c0 is called the critical bid price

  28. Single minded case • Theorem (Lehman, O’Callaghan, Shoham)If an allocation algorithm is exact and monotone then charging a winning bid the critical bid price, and a loosing bid 0, gives a truth revealing mechanism.

  29. Greedy algorithm for the single minded case • Every bidder makes a bid on a single set S • Auctioneer ranks bids by average item prices and assigns greedy • Payment for winning a subset S:|S| * (average item price of the first bid that could not win because of this bid for S) • This mechanism is truth revealing (Lehmann, O’Callaghan, Shoham, 2000) • The range consists of all allocations!

  30. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  31. Using Linear Programming:The relaxation and its dual • The linear programming relaxation (P) max v x Ax 1 x  0 • The dual linear program (D) min y 1 yA  v y  0

  32. Complementary slackness • x primal, y dual optimal then • primal complementary slackness: xj (vj - yA.j) = 0 • dual complementary slackness: yi(Ai.x - 1) = 0

  33. Primal-Dual algorithms • Set prices y such that yA  v • Compute allocation x such that for all j xj (vj - yA.j) = 0 • The allocation is optimal if for all i yi(1 - Ai.x) = 0 • Otherwise the primal-dual gap isi yi(1 - Ai.x)

  34. Dutch primal-dual winner determination algorithm • Start with high prices yi = maxj vj • While item prices can be reduced and not all items are sold • Reduce item prices until they reach 0 or the next dual constraint j becomes tight • Set xj = 1 if all items in this bid are still available, otherwise xj = 0 • fix item prices for all items of this bid

  35. Primal dual and truth revealing • The Dutch primal-dual mechanism is exact, since we assign a bid as it is, or not at all. • It is monotone • Every bid has a critical bid price • Truth-revealing is a dominant strategy for single minded bidders, if we charge winning bids the critical bid price

  36. Primal-dual and ascendingprice auctions • Suppose the auctioneer sets the item prices, and bidders submit utility maximizing bids based on these prices • How good can that be? • We can achieve an efficient allocation if and only if (a variant of) the winner determination LP has an integer optimal solution (see e.g. Bikhchandani & Ostroy)

  37. Primal-dual and ascending bundle auctions • An auctioneer might set bundle prices instead of single item prices, and bidders bid again based on these. • How good can that be? • We can achieve an efficient allocation if and only if a “very large” LP has an integer optimal solution (see e.g. Bikhchandani & Ostroy (2000)) • used in iBundle by (Parkes 1999, 2000)

  38. Outline • Auctions in the new economy • Auctions and discrete markets • Computational and economic efficiency • Polynomial cases • Approximation • Primal-dual algorithms • Summary

  39. Summary • The design of mechanisms that find efficient allocations in discrete markets is a central research topic in electronic commerce • The computational complexity will in most cases be a major challenge • Combinatorial Optimization, in particular primal-dual algorithms play a central role in providing solutions

  40. Invitation • Electronic Market Design Workshop from 11-13 July 2001 • Organizers: Eric van Damme, Rakesh Vohra, Daniel Lehmann, Tuomas Sandholm, Rudolf Müller • 25 invited presentations • rudolf.muller@infonomics.nl

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