Prompt Mechanisms for Online Auctions

Prompt Mechanisms for Online Auctions Speaker: Shahar Dobzinski Joint work with Richard Cole and Lisa Fleischer 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: 9 Val:10 Val: 20 Sun Mon Tue Wed Thu Fri Problem Definition identical items are for sale, the j-th item must be allocated at time j. ► Bidder i arrives at time ai (unknown in advance), and has a value of vi for getting exactly one item ai ≤ j ≤ di (before his departure time). ► Goal: maximize the sum of values of bidders that won some item. ►m The Greedy Algorithm 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. ► The ► What about truthfulness?  In this talk: the only private information of bidder i is his value vi.  In particular, a bidder cannot lie about his arrival time ai and departure time di. Truthfulness of the Greedy Algorithm 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. algorithm is monotone if for each bidder i that wins with value vi, bidder i also wins with value v’i > vi. ►The payment that a winning bidder i pays is the minimum value that he can bid and still win (the threshold value). ►An ► Theorem: The Price of Winning Val: 20 Val:9 Val:10 Val:10 0 Val:10 0 Val:10 0 Sun Mon Tue Wed Thu Fri Prompt Mechanisms 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 ► Definition: much money do I have to spend in my vacation? if a bidder refuses to pay after getting the service?  Debt Collection ► What ► In  Trusted Auctioneer 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. Our Results There exists a prompt deterministic 2competitive truthful mechanism for online auctions. ► Theorem: No prompt deterministic mechanism can achieve a (2-e)-competitive ratio. ► We ► Theorem: 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. 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. The Deterministic Mechanism Val: 20 Val:9 Val:5 20 Sun 0 0 9 Mon 0 Tue 0 Wed 0 Thu 0 Fri Promptness and Truthfulness The deterministic algorithm is prompt and truthful. ► Proof:  Monotonicity…  Promptness: ►Bidder ► Lemma: 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. The Competitive Ratio 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. ► Lemma: Proving the Competitive Ratio (cont) OPT=(o1,…,om), ALG=(a1,…,am). ► Fix item j. WLOG, let o2,o5,o8,o10 be the bidders that won some item in the optimal solution and are competing on j (by order of arrival). ► The Matching: Match o2 to a5, o5 to a8,… Match o10 to aj. ► 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. ► Let 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. Balls and Bins ►n balls are thrown to n bins, where the i’th ball is thrown uniformly at random to the interval [ai,di]. 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? 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. ► 1-1/e≈0.61 Summary 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 ► Introduced 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. The Lower Bound No prompt deterministic mechanism can achieve a (2-e)-competitive ratio. ► Proof: Claim that a player can win exactly one item, and that this item is determined the moment he arrives. ► Theorem: The Proof (cont) ► ► ► ► ► The arrival order of competitors on item j: o2,o5,o8,o10. Construction: Match o2 to a5, o5 to a8,… Match o10 to aj. 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: a5 is associated to o2, and to the last bidder in OPT that was assigned to compete on item 5.  Next we prove that the value of o2 is less than the value of a5.  When o5 arrives, the value of the candidate for j is less than the candidate for item 5 (o5 competes on item j, not on item 5).  The value of o2 is at most the value of the candidate for j when o5 arrives.  The value of the candidate of item 5 can only increase over time.