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 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

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.

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

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 !

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 .

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)

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)

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

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

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)]

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.

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)

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)

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)

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).

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>

Application to spectrum auctions

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

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.

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

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

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

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)

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)

END

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

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

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