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**Online Mechanism Design (Randomized Rounding on the Fly)**

Piotr Krysta, University of Liverpool, UK Berthold Vöcking, RWTH Aachen University, Germany

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**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 6 Opt SW = 8

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**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)

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**Combinatorial Auctions: Problem definition**

|U|= m = 8 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: b=2 1 2 3 4 vv 5 vv 6 vv 7 vv 8 vv

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**How are bidders represented ? (Demand oracles)**

Problem: The length of bidder’s valuation v(.) is exponential in m. v(.) given by demand oracles Di(Ui, p): Given item prices what is utility maximizing subset Si Ui and its valuation v(Si) ? Utility of bidder i for set Si: Demand oracle is: restricted if unrestricted if 1 2 3 4 vv 5 vv 6 vv 7 vv 8 vv

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Truthful mechanisms -- deterministic mechanism for CA: A mechanism is truthful if for each bidder i, all vi , vi’ 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.

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**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 Si := Di(Ui , pi) * 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)

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**On-line models: standard definition & 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 ≤ vi(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.

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**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(d2) 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]

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**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(d2) 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]

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**Your most profitable bundle ?**

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

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**Your most profitable bundle ?**

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

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**Your most profitable bundle ?**

Overselling Multiplicative Price Update (MPU) Algorithm [inspired by BGN 05, …] b=2 Bidders: Bidder 3 vv Your most profitable bundle ? vv vv vv vv

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**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 Set Update for each good NOTE: Bidder i gets set

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**Overselling MPU Algorithm: Analysis**

LEMMA 1. For any : * S assigns ≤ sb copies of each item, * LEMMA 2. For : THEOREM 1. S infeasible alloc. if : overselling factor infeasible

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**Overselling MPU Algorithm: Analysis**

LEMMA 1 (Part II). For any , where LEMMA 2. For , THEOREM 1. The algorithm with outputs an infeasible alloc. S: where copies of each item is assigned; 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): ☐

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**Overselling MPU Algorithm: Analysis**

LEMMA 1 (Part I). For any , alloc. S assigns ≤ sb copies of each item, where PROOF: Consider e ε 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 μL. By definition of μ, L there is ≤ 1 such bidder that receives ≤ 1 copy of e. Hence, at most copies of e assigned. ☐

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**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. ☐

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**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 Ui for demand oracles, and using RR. Idea: Provisionally assign bundles Si 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 Si become final bundles.

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**Overselling MPU Algo. with Oblivious Randomized Rounding**

Bidders: Bidder 1 vv Your most profitable bundle ? vv vv YES! (Pr=q) vv b=2 vv

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**Overselling MPU Algo. with Oblivious Randomized Rounding**

Bidders: Bidder 2 vv Your most profitable bundle ? vv vv YES! (Pr=q) vv b=2 vv

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**Overselling MPU Algo. with Oblivious Randomized Rounding**

Bidders: Bidder 3 Your most profitable bundle ? vv vv NO! (Pr=1-q) vv b=2 vv

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**MPU Algorithm with Oblivious RR**

Order of bidders 1,2,…,n is arbitrary (adversarial). 1. For each good 2. For each bidder Set Update for each good With prob. Update for each good

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**MPU Algorithm with Oblivious RR: remarks**

1. For each good 2. For each bidder Set Update for each good With prob. Update for each good The algorithm outputs allocation R; payment for Ri is Def. of Ui in line 3. ensures that R is feasible! If q=0, then the provisional alloc. S is same as MPU algo. with Ui=U. If q=0, then the output alloc. R is empty. With prob. 1-q the algo. increases prices of e in Si but does not sell Si (and thus “learns” the correct prices). If q>0, then LEMMA 1 holds, but LEMMA 2 doesn’t! We will show a stochastic version of LEMMA 2 to imply O(1/q)-apx.

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**Overselling MPU Algo. with RR: Analysis**

Recall the previous analysis: LEMMA 1. For any : * S assigns ≤ sb copies of each item, * LEMMA 2. For : Always holds Not always holds!!! opt bundle for i

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**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.

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**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.

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Thanks! Questions ?

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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.

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PROOFS

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**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.☐

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**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 (**)

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**MPU Algo. with RR: stochastic LEMMA 2’**

LEMMA 2’. If for any and any then PROOF: Sum (***) for all bidders:

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**MPU Algo. with RR: stochastic LEMMA 2’**

LEMMA 2’. If for any and any then PROOF: By LEMMA 1: Now: E[v(Ri)]=qE[v(Si)] as Pr[Ri=Si]=q, so E[v(R)]=qE[v(S)], and finally E[v(R)] ≥ q v(opt)/8. ☐

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**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 ☐

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END

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