Online Mechanism Design (Randomized Rounding on the Fly) Piotr Krysta, University of Liverpool, UK Berthold Vöcking, RWTH Aachen University, Germany

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 B Opt SW = 8

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)

Combinatorial Auctions: Problem definition Combinatorial Auction (CA): n bidders U = set of m items (goods) Each e ε 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= |U|= m = 8

How are bidders represented ? (Demand oracles) Problem: The length of bidder’s 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

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.

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)

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]

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]

Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, …] Bidders: vv vv vv vv vv Bidder 1 Your most profitable bundle ? b=2

Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, …] Bidders: vv vv vv vv vv Bidder 2 Your most profitable bundle ? b=2

Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, …] Bidders: vv vv vv vv vv Bidder 3 Your most profitable bundle ?

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

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(μbm) (LEMMA 1) Oblivious randomized rounding (RR)  used to decide (with small Pr = q) which provisional bundles S i become final bundles.

Overselling MPU Algo. with Oblivious Randomized Rounding Bidders: vv vv vv vv vv Bidder 1 Your most profitable bundle ? b=2  YES! (Pr=q)

Bidders: vv vv vv vv vv Bidder 2 Your most profitable bundle ? b=2 Overselling MPU Algo. with Oblivious Randomized Rounding  YES! (Pr=q)

b=2 Bidders: vv vv vv vv Bidder 3 Your most profitable bundle ? Overselling MPU Algo. with Oblivious Randomized Rounding  NO! (Pr=1-q)

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

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.

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 bidder’s 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.

Thanks! Questions ?

PROOFS

MPU Algo. with RR: new LEMMA 2 LEMMA 2’. If for any and any then PROOF: Fix bidder i and let be a feasible opt alloc. By for any coin flips of the algorithm by (**)

MPU Algo. with RR: new LEMMA 2 LEMMA 2’. If for any and any then PROOF: Sum (***) for all bidders:

MPU Algo. with RR: new 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. ☐

Proving (**) LEMMA. Consider CA with bundles of size at most 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 ☐

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MPU Algo. with Oblivious RR: stochastic LEMMA 4 We will give a stochastic version of Eq. (*). Observe that prices are random variables depending on algorithm’s coin flips made for bidders 1,2, …, i-1. LEMMA 4. Let q > 0 be s.t. for any and any Then REMARK: LEMMA 4 is a recipe for devising an online algo.: choose q > 0 s.t. (**) holds. The larger q is, the better the competitive ratio.

Stochastic LEMMA 4 applied to CAs with d-bundles We will apply LEMMA 4 to 3 CA’s settings: CA with bundles of size ≤ d, CA with unbounded size bundles, CA with XOS valuations. We will show here details only of the first CA setting. LEMMA 4. Let q > 0 be s.t. for any and any Then

Stochastic LEMMA 4 applied to CAs with d-bundles LEMMA 5. Consider CA with bundles of size at most d ≥ 1, and let Then for any and any bundle of at most d items: By this lemma and LEMMA 4 we obtain: THEOREM 2. The MPU algorithm with oblivious RR and q as above is for CA with bundles of size ≤ d and multiplicity b.

Final Proofs

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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.

Combinatorial Auctions A set of m indivisible items (goods) are given for sale, each available in b copies (supply) A set of n potential buyers (bidders) is also given, each with a valuation function v(.) for subsets (bundles) of goods The valuation function 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 n goods among m bidders to maximize the social welfare (SW) Example: m=2 bidders {A,B} n=2 goods {x,y} Opt SW = 8 v( {x} ) v( {y} ) v( {x,y} ) A B 3 5 6

Problem definition: Truthful mechanisms A deterministic mechanism for CA is a pair (f, q) where A mechanism (f,q) is (deterministically) truthful if for each bidder i, all v i, v i ’ and all declarations v -I of the other bidders except bidder i: A randomized mechanism is a probability distribution over deterministic mechanisms. It is universally truthful if each of these deterministic mechanisms is truthful.

Truthfulness via the direct characterization We achieve truthfulness by serving bidders one by one in a given order, and offering items at fixed (posted) prices: If is the set of items mechanism offers to bidder i, then the mechanism defines prices (independent from bidder i) and computes S i by calling the demand oracle D i (U i, p).

On-line models Assume: bidders arrive one by one, say in the order 1,2,…,n. The mechanism has to compute the allocation S i and payment q i for bidder i without knowing the valuations of bidders i+1,…,n. Arrival models: * random order of arrivals (secretary model) * arbitrary (adversarial) order of arrivals. We want to find an allocation S in A maximizing the social welfare. Competitive ratio CR (of a randomized online algo.): Σ = set of all arrival sequences of n bidders with valuations for m items For σ ε Σ: S(σ) = alloc. computed by algo., opt(σ) = opt offline alloc.

On-line models: some aspects Competitive ratio CR (of a randomized online algo.): Σ = set of all arrival sequences of n bidders with valuations for m items For σ ε Σ: S(σ) = alloc. computed by algo., opt(σ) = 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) ≤ μ 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.

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

Overselling MPU Algorithm: Analysis LEMMA 1. For any, alloc. S assigns ≤ s  b copies of each item to bidders, where LEMMA 2. For any, where and LEMMA 3. For, THEOREM 1. The algorithm with outputs an infeasible alloc. S: (1)in which copies of each item is assigned; (2)if, then

Overselling MPU Algorithm: Analysis LEMMA 1. For any, alloc. S assigns ≤ s  b copies of each item to bidders, where PROOF: Consider any item e in U. Suppose, after some step, copies of e assigned to bidders. Then the price of e ≥ After this step, the algorithm might give further copies of e to bidders whose maximum valuation exceeds μL. By definition of μ, L there is ≤ 1 such bidder that receives ≤ 1 copy of e. Hence, at most copies of e are assigned. ☐

Overselling MPU Algorithm: Analysis LEMMA 2. For any choice of, where and PROOF: Let and As bidders are individually rational:, hence: Now and imply the claim. ☐

Overselling MPU Algorithm: Analysis LEMMA 3. 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. ☐

Overselling MPU Algorithm: Analysis LEMMA 1. For any, alloc. S assigns ≤ s  b copies of each item to bidders, where LEMMA 2. For any, where and LEMMA 3. For, THEOREM 1. The algorithm with outputs an infeasible alloc. S: (1)in which copies of each item is assigned; (2)if, then

Overselling MPU Algorithm: Analysis LEMMA 2. For any, where and LEMMA 3. For, THEOREM 1. The algorithm with outputs an infeasible alloc. S: (1)in which copies of each item is assigned; (2)if, then PROOF: (1) is by LEMMA 1. By LEMMA 2: which with LEMMA 3 gives: By v(opt) ≥ L, we have the following and this implies claim (2): ☐

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

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 LEMMAs 1, 2 hold except LEMMA 3 – Equation (*) fails. (g)We will show a stochastic version of Eq. (*) to imply O(1/q)-apx.

MPU Algo. with Oblivious RR: proving LEMMA 4 LEMMA 4. Let q > 0 be s.t. for any and any Then PROOF: Fix bidder i and let be a feasible allocation. By we have for any coin flips of the algorithm by (**)

MPU Algo. with Oblivious RR: proving LEMMA 4 LEMMA 4. Let q > 0 be s.t. for any and any Then PROOF: By using, summing (***) for all bidders we get, where the last “≥” follows by feasibility of T. Taking T = opt, we obtain Now we proceed in analogy to the proof of THEOREM 1.

MPU Algo. with Oblivious RR: proving LEMMA 4 LEMMA 4. Let q > 0 be s.t. for any and any Then PROOF: By LEMMA 2: for any coin flips of the algo. Thus. By and (i) give Now: r.v. v i (S i ) depends on coin flips for bidders 1,…,i-1 and prob. of i getting S i or not is indep. from prev. bidders. Thus 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. ☐

Proving LEMMA 5 LEMMA 5. Consider CA with bundles of size at most 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 ☐

Overselling MPU Algorithm: Analysis Parameters: LEMMA 1. For any : * S assigns ≤ s  b copies of each item, * LEMMA 2. For : THEOREM 1. The algorithm outputs an infeasible alloc. S, and if, then infeasible overselling factor