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Online Mechanism Design (Randomized Rounding on the Fly) Piotr Krysta, University of Liverpool, UK Berthold Vcking, RWTH Aachen University, Germany Slide 2 Combinatorial Auctions m indivisible items (goods) given for sale n potential buyers (bidders), each with a valuation function v(.) for subsets (bundles) of goods v(.) may express complex preferences, e.g.: complements: v(camera + battery) > v(camera) + v(battery) substitutes: v(Apple iPhone + Samsung Galaxy) < v(Apple iPhone) + v(Samsung Galaxy) Goal: Partition m goods among n bidders to maximize the social welfare (SW) Example: m=2 bidders {A,B} n=2 goods {x,y} v( {x} ) v( {y} ) v( {x,y} ) A 1 5 7 B 3 5 6 Opt SW = 8 Slide 3 Combinatorial Auctions: Applications Combinatorial auctions have many important applications: * Government Spectrum Auctions (UK, Germany, Sweden, USA, ) * Allocation of Airspace System Resources * Auctions for Truckload Transportation * Auctioning Bus Routes (London) Slide 4 Combinatorial Auctions: Problem definition Combinatorial Auction (CA): n bidders U = set of m items (goods) Each e U available in b 1 copies (supply) Bidder i has valuation f-n: Meaning: money i is willing to pay for S Allocations: Problem: compute allocation maximizing social welfare: vv vv vv vv vv b=2 1 8 3 2 4 5 6 7 |U|= m = 8 Slide 5 How are bidders represented ? (Demand oracles) Problem: The length of bidders valuation v(.) is exponential in m. v(.) given by demand oracles D i (U i, p): Given item prices what is utility maximizing subset S i U i and its valuation v(S i ) ? Utility of bidder i for set S i : Demand oracle is: restricted if unrestricted if vv vv vv vv vv 1 8 3 2 4 5 6 7 Slide 6 Truthful mechanisms -- deterministic mechanism for CA: A mechanism is truthful if for each bidder i, all v i, v i and all declarations v -I of the other bidders except bidder i: Randomized mechanism = prob. distribution over deterministic mechanisms. It is universally truthful if each of these mechanisms is truthful. Slide 7 Truthfulness via direct characterization & on-line algs Achieve truthfulness by serving bidders one by one in a given order, say i=1,2,,n, and offering items at fixed (posted) prices: If set of items offered to bidder i, define prices (indep. from i) and compute: * bundle S i := D i (U i, p i ) * payment (without knowing the valuations of bidders i+1,,n) Arrival models: * random order of arrivals (secretary model) * arbitrary (adversarial) order of arrivals. Goal: find alloc. S in A maximizing the social welfare. We use standard on-line competitive analysis (CR = competitive ratio) Slide 8 On-line models: standard definition & some aspects Competitive ratio CR (of a randomized online algo.): = set of all arrival sequences of n bidders with valuations for m items For : S() = alloc. computed by algo., opt() = opt offline alloc. OBSERVE: Adversarial arrival model: If valuations v() are unbounded, then R cannot be bounded. REASON: The b bidders arriving last might have huge v()s, s.t. copies cannot be given to any bidders that arrive before them. Thus: assume 1 v i (S) for every bidder i, S subset U. Random arrival model: We assume unbounded valuations. NOTE: Random arrivals used only to extract estimate of the bids range. Slide 9 Our contributions: CAs + Random arrivals model 1. General v(): for any b 1 we obtain CR (the first online result with ) 2. General v(), bundles size d: for any b 1 we obtain CR (previous O(d 2 ) only for b=1) 3. XOS v() and b = 1: we obtain a CR (the first online result for submodular/XOS v(.)) Previous results: -comp. l.b. -best known u.b. XOS v(): O(log (m) log log (m))-apx univ. truthful offline mech. [Dobzinski 09] General v(): -apx truthful in exp. offline mech. [Lavi, Swamy 05] -apx (b=1) univ. truthful [Dobzinski, Nisan, Schapira 05] -apx (deterministically) truthful [Bartal, Gonen, Nisan 05] Slide 10 Our contributions: CAs + Adversarial arrivals model 1. General v(): for any b 1 we obtain CR (the first online result with ) 2. General v(), bundles size d: for any b 1 we obtain CR (previous O(d 2 ) only for b=1) 3. XOS v() and b = 1: we obtain a CR (the first online result for submodular/XOS v(.)) Previous results: -comp. l.b. -best known u.b. XOS v(): O(log (m) log log (m))-apx univ. truthful offline mech. [Dobzinski 09] General v(): -apx truthful in exp. offline mech. [Lavi, Swamy 05] -apx (b=1) univ. truthful [Dobzinski, Nisan, Schapira 05] -apx (deterministically) truthful [Bartal, Gonen, Nisan 05] Slide 11 Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, ] Bidders: vv vv vv vv vv Bidder 1 Your most profitable bundle ? b=2 Slide 12 Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, ] Bidders: vv vv vv vv vv Bidder 2 Your most profitable bundle ? b=2 Slide 13 Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, ] Bidders: vv vv vv vv vv Bidder 3 Your most profitable bundle ? Slide 14 Warm-up: Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, ] Order of bidders 1,2,,n is arbitrary (adversarial). 1. For each good 2. For each bidder 3. Set 4. Update for each good NOTE: Bidder i gets set Slide 15 Overselling MPU Algorithm: Analysis LEMMA 1. For any : * S assigns s b copies of each item, * LEMMA 2. For : THEOREM 1. S infeasible alloc. if : infeasible overselling factor Slide 16 Overselling MPU Algorithm: Analysis LEMMA 1 (Part II). For any, where LEMMA 2. For, THEOREM 1. The algorithm with outputs an infeasible alloc. S: (1)where copies of each item is assigned; (2)if, then PROOF: (1) is by LEMMA 1 (P. I). By L. 1 (P. II): which with LEMMA 2 gives: By v(opt) L, we have the following and this implies claim (2): Slide 17 Overselling MPU Algorithm: Analysis LEMMA 1 (Part I). For any, alloc. S assigns s b copies of each item, where PROOF: Consider e U. Suppose, after some step, copies of e assigned to bidders. price of e After this step, the algorithm might give further copies of e to bidders whose maximum valuation exceeds L. By definition of , L there is 1 such bidder that receives 1 copy of e. Hence, at most copies of e assigned. Slide 18 Overselling MPU Algorithm: Analysis LEMMA 2. For, PROOF: = feasible allocation (allocates b of each item) Algo. uses demand oracle:, so By using and summing (*) for all bidders we obtain (last follows because T allocates b copies of each item) Taking T = opt implies the claim. Slide 19 Overselling MPU Algorithm with Oblivious RR Larger price update factor more feasible solution + worse approximation Smaller helps learn correct prices, but, produces in-feasible solution. Idea: Achieve feasibility and good approximation by defining appropriate sets U i for demand oracles, and using RR. Idea: Provisionally assign bundles S i of virtual copies to bidders following MPU algorithm learn correct prices Number of virtual copies b log(bm) (LEMMA 1) Oblivious randomized rounding (RR) used to decide (with small Pr = q) which provisional bundles S i become final bundles. Slide 20 Overselling MPU Algo. with Oblivious Randomized Rounding Bidders: vv vv vv vv vv Bidder 1 Your most profitable bundle ? b=2 YES! (Pr=q) Slide 21 Bidders: vv vv vv vv vv Bidder 2 Your most profitable bundle ? b=2 Overselling MPU Algo. with Oblivious Randomized Rounding YES! (Pr=q) Slide 22 b=2 Bidders: vv vv vv vv Bidder 3 Your most profitable bundle ? Overselling MPU Algo. with Oblivious Randomized Rounding NO! (Pr=1-q) Slide 23 MPU Algorithm with Oblivious RR Order of bidders 1,2,,n is arbitrary (adversarial). 1. For each good 2. For each bidder 3. Set 4. Update for each good 5. With prob. 6. Update for each good Slide 24 MPU Algorithm with Oblivious RR: remarks 1. For each good 2. For each bidder 3. Set 4. Update for each good 5. With prob. 6. Update for each good (a)The algorithm outputs allocation R; payment for R i is (b)Def. of U i in line 3. ensures that R is feasible! (c)If q=0, then the provisional alloc. S is same as MPU algo. with U i =U. (d)If q=0, then the output alloc. R is empty. (e)With prob. 1-q the algo. increases prices of e in S i but does not sell S i (and thus learns the correct prices). (f)If q>0, then LEMMA 1 holds, but LEMMA 2 doesnt! (g)We will show a stochastic version of LEMMA 2 to imply O(1/q)-apx. Slide 25 Overselling MPU Algo. with RR: Analysis Recall the previous analysis: LEMMA 1. For any : * S assigns s b copies of each item, * LEMMA 2. For : Always holds Not always holds!!! opt bundle for i Slide 26 A stochastic LEMMA 2 for CAs with d-bundles LEMMA 2. Consider CA with |bundles| d, and let Then for any and any bundle : and THEOREM 2. The MPU algorithm with oblivious RR and q as above is for CA with |bundles| d and multiplicity b. Slide 27 Summary and further questions? We design the first online (universally truthful) mechanisms achieving competitiveness for any supply b 1. New technique: we combine the online allocation of bidders with the concept of oblivious randomized rounding. Our mechanisms are simple and intuitive: each bidders demand oracle is queried only once, We achieve competitive ratios close to or even beating the best known approx. factors for the corresponding offline setting. Question: The main open problem is to design similar deterministic mechanisms. Slide 28 Thanks! Questions ? Slide 29 Problem definition: Submodular and XOS valuations We also consider special valuations Submodular (decreasing marginal utilities): XOS (fractionally subadditive): FACT: If v() if submodular then it is XOS. Slide 30 PROOFS Slide 31 Overselling MPU Algorithm: Analysis LEMMA 1 (Part II). For any, where and PROOF: Let and As bidders are individually rational:, hence: Now and imply the claim. Slide 32 MPU Algo. with RR: stochastic LEMMA 2 LEMMA 2. If for any and any then PROOF: Fix bidder i and feasible opt alloc. By for any coin flips of the algorithm by (**) Slide 33 MPU Algo. with RR: stochastic LEMMA 2 LEMMA 2. If for any and any then PROOF: Sum (***) for all bidders: Slide 34 MPU Algo. with RR: stochastic LEMMA 2 LEMMA 2. If for any and any then PROOF: By LEMMA 1: Now: E[v(R i )]=qE[v(S i )] as Pr[R i =S i ]=q, so E[v(R)]=qE[v(S)], and finally E[v(R)] q v(opt)/8. Slide 35 Proving (**) for d-bundles LEMMA. Consider CA with |bundles| d 1, and let Then for any and any bundle of at most d items: PROOF: Fix bidder i. By LEMMA 1, is in of the provisional bundles, and each of them becomes final with prob.. Consider and note that if was sold times, i.e., at most b-1 of its provisional bundles became final. Thus the prob. that is: By and union bound Slide 36 END