1 / 27

University of Tokyo 東京大学 松井知己 Tomomi Matsui Iwate Prefectural University 岩手県立大学

Sealed Bid Multi-object Auctions with Necessary Bundles and its Application to Spectrum Auctions ver. 1.0. University of Tokyo 東京大学 松井知己 Tomomi Matsui Iwate Prefectural University 岩手県立大学 渡辺隆裕 Takahiro Watanabe. Multi object Auction.

gage-bauer
Download Presentation

University of Tokyo 東京大学 松井知己 Tomomi Matsui Iwate Prefectural University 岩手県立大学

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Sealed Bid Multi-object Auctionswith Necessary Bundlesand its Application to Spectrum Auctionsver. 1.0 University of Tokyo 東京大学 松井知己 Tomomi Matsui Iwate Prefectural University 岩手県立大学 渡辺隆裕 Takahiro Watanabe

  2. Multi object Auction • Multi-object Auction: trading oil leases, furniture, pollution rights, airport time slots, spectrum licenses, and delivery routes, etc. • Bidders’ preferences are defined on sets of objects. • (combinatorial auction, simultaneous auction) • Results: • (1) Analysis from the point of view of game theory. • (2) Apply the result to spectrum auction.

  3. Main result • We introduce following assumptions; • each bidder has a positive reservation value only for one special subset of objects (necessary bundle) • admissible bid is a pair of one subset of objects and its price, • Game theoretic approach: • show the existence of a Nash equilibrium when bidding unit is sufficiently small • Application to spectrum auction: • polynomial algorithm for the problem to maximize auctioneer’s revenue • explicit description of a Nash equilibrium

  4. Combinatorial optimization Game theory Multi-agent system Purpose of this talk • purpose of this talk = Find friends ! Multi-object Auction Communication searched on internet ⇒ found PRIMA2001 ⇒ submitted paper ⇒ give a talk ⇒ find friends ⇒ further work !

  5. Definitions (bidders) • Game theoretic descriptions • N ={1,2,…, n}: players (bidders) • M = {1,2,…, m}: objects • bundle: subset of objects • sealed bid auction: submit bids simultaneously • open bid auction (English, Japanese, Dutch,…) • strategies (admissible bids) of player i: • (Bi, bi) ∈2M×R+: (bundle, bidding price) • The bidding price bi is the amount of money that player i is willing to pay for bundle Bi .

  6. Assumption (strategies) • Assumption 1 • each player i submits only onepair ofbundle and its price (Bi, bi) ∈2M×R+: • restricted but practical • combinatorial auction: • each player i submitsbidding prices of all the bundlesfi: 2M→R+ • general but impractical • (2M is a huge family)

  7. bidding unit • ε : bidding unit (bidding grid) • Each bidding price is • a non-negative multiple of ε. • Zε={εj |jis a non-negative integer} • strategies (admissible bids) of player i: • (Bi, bi) ∈2M×Zε • profile of bids : vector of bids of all the players • ((B1,b1),(B2,b2),…, (Bn, bn))=(B, b) • B =(B1, B2,…, Bn ) • b = ( b1, b2 ,…, bn)

  8. Definitions (auctioneer) • Given a profile of bids • ((B1,b1),(B2,b2),…, (Bn, bn))=(B, b), • auctioneer determines the set of winners which maximizes auctioneer’s revenue. • Bundle Assignment Problem BAP(B, b) • (winner determination problem) • max. bTx=b1x1+ b2x2+‥+bnxn • s. t. Ax ≦1, x ∈{0,1}N. • A=(aji) 0-1matrix {0,1}M×N • aji=1 ⇔ object j is in bundle Bi

  9. Bundle assignment problem • BAP(B, b) • max. bTx=b1x1+ b2x2+‥+bnxn • s. t. Ax ≦1, x ∈{0,1}N. • ●A=(aji) 0-1matrix {0,1}M×N • aji=1 ⇔ object j is in bundle Bi • ●xi=1 ⇔ auctioneer assigns • bundle Bi to player i. • ●Ax ≦1: each object must belong to at most one player

  10. Bundle assignment problem • Bundle assignment problem has many names as • winner determination problem, • max. weight set packing problem, • max. weight independent set problem, and • max. weight clique problem. • theoretically hard: NP-hard • practically tractable: many commercial codes solve BAP efficiently (e.g. CPLEX) • [Andersson, Tenhunen and Ygge (2000)]

  11. multiple-optimal solutions • If BAP has multiple-optimal solutions, then auctioneer chooses an optimal solution uniformly at random. • Further work: Construct an algorithm for selecting an optimal solution of BAP uniformly at random. • The problem is much harder than BAP.

  12. Definitions (bidders) • Vi(S): Each player i has a non-negative reservation valueVi(S) for each bundle S. • Vi : 2M →Zδ (non-negative multiple of δ) • Assumption 2: Each bidder has a positive reservation value only for one special bundle. • ⇒ necessary bundle • necessary (bundle, price) of player i : (Ti ,vi) • Vi(S) = vi⇔ (S⊇Ti) • Vi(S) = 0 ⇔ (otherwise)

  13. Nash equilibrium • profile of bids : • ((B1,b1),(B2,b2),…, (Bn, bn))=(B, b) • Utility of player i : Ui (B, b) • Ui (B, b)=(Vi(Bi) ー bi ) Pr[player i is selected] • profile (B*, b*) is a Nash equilibrium • ⇔ ∀i∈N,∀(Bi, bi) ∈2M×Zε, • Ui (B*, b*) ≧ Ui ((B*-i, b*-i), (Bi, bi)) • ((B*-i, b*-i), (Bi, bi)) : profile obtained from (B*, b*) by replacing strategy of player i with (Bi, bi)

  14. Main results • Theorem 2: If the bidding unit ε is sufficiently small, then Nash equilibrium exists. • size of bidding unit ε≦δ(n2n+1) • δ:unit of reservation value, • n: number of players • <proof: omitted> • profile (B*, b*) is a Nash equilibrium • ⇔ ∀i∈N,∀(Bi, bi) ∈2M×Zε, • Ui (B*, b*) ≧ Ui ((B*-i, b*-i), (Bi, bi)) • ((B*-i, b*-i), (Bi, bi)) : profile obtained from (B*, b*) by replacing strategy of player i with (Bi, bi)

  15. description of a Nash equilibrium • Classify the bidders • Passed bidders: bidders contained in every optimal solution of BAP(B, b). • Questionable bidders: bidders contained in not all but at least one optimal solution of BAP(B, b). • Rejected bidders: bidders never appearing in any optimal solutions of BAP(B, b). • Optimal solution set: Ω (B, b): • set of all the optimal solutions of BAP(B, b).

  16. Nash equilibrium • Theorem 1: Following profile (B*, b*)is a Nash equilibrium; • questionable bidder i: (B*i, b*i)= (Ti, vi) • rejected bidder i: (B*i, b*i)= (Ti, vi) • passed bidder i: B*i =Ti, • b*i: minimal vector in Zε • satisfying Ω (B*, b*)=Ω (T, v) (solution sets are equivalent) • (Ti, vi ): (necessary bundle, reservation value) • <proof: omitted>

  17. Application to spectrum auctions

  18. Spectrum auction • Spectrum auction: • objects: spectrums (frequency channel for cellular phone) are arranged in linear order • necessary bundles (Ti): • sequences of consecutive channels • channels : T1 T2 T3 T4

  19. Longest path problem • BAP corresponding to spectrum auction satisfies the conditions; • coefficient matrix A is totally unimodular, • liner relaxation of BAP has an integer valued optimal solution, • equivalent to longest path problem.

  20. v1 v2 v3 v4 nodes: barrier of channels arcs: ( j, j+1), necessary bundles arc weight = reservation value longest path problem • directed graph T1 T2 T3 T4

  21. longest path problem • longest path problem T1 T2 T3 T4 v1 v2 v3 v4

  22. longest path problem • longest path problem Finding a longest path = winner determination T1 T2 T3 T4 v1 v2 v3 v4

  23. random selection • Generally, solving BAP and random selection from multiple-optimal solutions are hard. • Spectrum auction: • bundle assignment ⇒ longest path problem • random selection ⇒ random path generation • Key idea: • BAP(B, b) ⇒linear relaxation ⇒ dual problem • bundle assignment: dynamic programming • random selection: path counting algorithm • explicit description of a Nash equilibrium: complementality slackness theorem for linear programming problems • (detail is omitted)

  24. conclusion • Assumption 1 (multi-object auction) • each player i submits only onepair ofbundle and its price (Bi, bi) ∈2M×R+ • Assumption 2: Each bidder has a positive reservation value only for one special bundle, • called necessary bundle. • Theorem 2:If the bidding unit ε is sufficiently small, then Nash equilibrium exists. • Theorem 1: • (Characterization of a Nash equilibrium)

  25. END

  26. Main results • Theorem 2: If the bidding unit ε is sufficiently small, then (pure strategy) Nash equilibrium exists. • mixed strategy: Nash showed that every strategic form n-persons game with finite number of strategies has a mixed strategy Nash equilibrium. • size of bidding unit ε≦δ(n2n+1) • δ:unit of reservation value, • n: number of players

  27. random selection • BAP(B, b) max{bTx| Ax ≦1, x ∈{0,1}N} • linear relaxationmax{bTx | Ax ≦1, x≧0} • dual problem min{yTx | yTA ≧b, y≧0 } • y*:optimal dual solution • M*={j∈M | y*Tai=bi } ai :i th column vector • Lemma:M* is the set of passed and • questionable bidders. • Ordinary dynamic programming procedure • ⇒ random selection of longest paths

More Related