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Prompt Mechanisms for Online Auctions Speaker: Shahar Dobzinski Joint work with Richard Cole and Lisa Fleischer

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A Running Example ► A cinema in Las Vegas is presenting a daily show. ► For simplicity, assume only one tourist can watch the show each day. ► Each tourist has a different value for a ticket. ► We do not know the tourists that will come next. ► Our goal: maximize the total value of tourists that watched the show. Val: 20 Val:10Val: 9 Sun MonTueWedThuFri

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Problem Definition ► m identical items are for sale, the j-th item must be allocated at time j. ► Bidder i arrives at time a i (unknown in advance), and has a value of v i for getting exactly one item a i ≤ j ≤ d i (before his departure time). ► Goal: maximize the sum of values of bidders that won some item.

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The Greedy Algorithm ► The greedy algorithm: at time t, assign item t to the bidder with the highest value that is available. ► The greedy achieves a competitive ratio of 2 (Kesselman, Lotker, Mansour, Patt-Shamir, Schieber, Sviridenko) At least half of the optimal offline social welfare is recovered. ► What about truthfulness? In this talk: the only private information of bidder i is his value v i. In particular, a bidder cannot lie about his arrival time a i and departure time d i.

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Truthfulness of the Greedy Algorithm ► Theorem: The greedy algorithm is truthful (Hajiaghayi, Kleinberg, Mahdian, Parkes). ► Proof: using the following characterization: An algorithm for a single-parameter setting admits payments that make it truthful if and only if the algorithm is monotone. ► An algorithm is monotone if for each bidder i that wins with value v i, bidder i also wins with value v’ i > v i. ► The payment that a winning bidder i pays is the minimum value that he can bid and still win (the threshold value).

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The Price of Winning Val:10 Val:10 0 Val:9Val: 20 Sun MonTueWedThuFri

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Prompt Mechanisms ► Definition: A mechanism is prompt if a bidder learns his payment at the moment he wins an item. Otherwise, the mechanism is tardy. ► Why tardy mechanisms are not desired? Uncertainty ► How much money do I have to spend in my vacation? Debt Collection ► What if a bidder refuses to pay after getting the service? Trusted Auctioneer ► In tardy mechanisms the bidder essentially provides the auctioneer with a blank check. ► It is natural to know the price of a good the moment you buy it.

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Our Results ► Theorem: There exists a prompt deterministic 2- competitive truthful mechanism for online auctions. ► Theorem: No prompt deterministic mechanism can achieve a (2- )-competitive ratio. ► We also present a randomized prompt O(1)- competitive mechanism. Proof involves a nice balls-and-bins question. ► We will mention later results in other models.

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The Prompt 2-Competitive Deterministic Mechanism ► Prelims: Each Bidder is going to compete on exactly one item. Let the candidate for item j be the competitor on item j with the largest value. ► The Mechanism: On the arrival of bidder i, let him compete on the item in his window where currently the candidate bidder has the lowest value. On time t, allocate item t to its candidate.

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9020 The Deterministic Mechanism Sun MonTueWedThuFri Val:9 Val: Val:5

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Promptness and Truthfulness ► Lemma: The deterministic algorithm is prompt and truthful. ► Proof: Monotonicity… Promptness: ► Bidder i can win only one item t: the one that he is competing on. ► This item is determined on the arrival of bidder i. ► Thus, whether bidder i wins is only a function of bidders that arrive by time t. ► If bidder i wins, we can calculate the payment of bidder i at time t.

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The Competitive Ratio ► Lemma: the algorithm provides a competitive ratio of 2. ► Proof (outline): We will match each bidder in OPT to exactly one bidder in ALG. Each bidder in ALG will be associated to at most 2 bidders in OPT with lower values. Enough to get a 2-competitive ratio.

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Proving the Competitive Ratio (cont) ► Let OPT=(o 1,…,o m ), ALG=(a 1,…,a m ). ► Fix item j. WLOG, let o 2,o 5,o 8,o 10 be the bidders that won some item in the optimal solution and are competing on j (by order of arrival). ► The Matching: Match o 2 to a 5, o 5 to a 8,… Match o 10 to a j. ► The two properties: A bidder in OPT is matched to exactly one bidder in ALG. A bidder in ALG is associated to at most 2 bidders in OPT with lower values.

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The Randomized Mechanism ► The Mechanism: When bidder i arrives, he competes on an item in his time window, selected uniformly at random. At time j conduct a second-price auction on item j, with the participation of all bidders that were selected to compete on item j. ► Theorem: This is a truthful prompt O(1)- competitive mechanism.

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Balls and Bins ► n balls are thrown to n bins, where the i’th ball is thrown uniformly at random to the interval [a i,d i ]. We are given that all balls can be placed in a way s.t. all bins are full. What is the expected number of full bins? ► 1-1/e≈0.61 of the bins if each ball can be thrown to all bins. ► Between 0.1 and 0.41 of the bins in the general case.

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Summary ► Introduced prompt and tardy mechanisms. ► Showed a prompt 2-competitve deterministic mechanism. ► A prompt randomized O(1)-competitive mechanism. Can the analysis of the underlying balls and bins question can be improved? ► Main open question: upper and lower bounds (not necessarily prompt) when the arrival and departure time are also private information. Our results: a logarithmic upper bound, and a lower bound of 2.

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The Lower Bound ► Theorem: No prompt deterministic mechanism can achieve a (2- )-competitive ratio. ► Proof: Claim that a player can win exactly one item, and that this item is determined the moment he arrives.

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The Proof (cont) ► The arrival order of competitors on item j: o 2,o 5,o 8,o 10. ► Construction: Match o 2 to a 5, o 5 to a 8,… Match o 10 to a j. ► Our two properties: ► A bidder in OPT is matched to exactly one bidder in ALG. ► A bidder in ALG is associated to at most 2 bidders in OPT with lower values: 2 bidders: a 5 is associated to o 2, and to the last bidder in OPT that was assigned to compete on item 5. Next we prove that the value of o 2 is less than the value of a 5. When o 5 arrives, the value of the candidate for j is less than the candidate for item 5 (o 5 competes on item j, not on item 5). The value of o 2 is at most the value of the candidate for j when o 5 arrives. The value of the candidate of item 5 can only increase over time.

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