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Adaptive Limited-Supply Online Auctions

Adaptive Limited-Supply Online Auctions. Robert Kleinberg (MIT and Cornell) Collaborators: Mohammad Hajiaghayi (MIT) Mohammad Mahdian (Microsoft) David Parkes (Harvard). Background: Online Auctions.

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Adaptive Limited-Supply Online Auctions

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  1. Adaptive Limited-Supply Online Auctions Robert Kleinberg (MIT and Cornell) Collaborators: Mohammad Hajiaghayi (MIT) Mohammad Mahdian (Microsoft) David Parkes (Harvard)

  2. Background: Online Auctions • Online auction theory studies incentive-compatible mechanisms for markets in which agents arrive and depart over time, and the mechanism designer lacks foreknowledge of the future. • This talk presents a theoretical analysis of adaptive pricing in simple markets with: • one monopolistic seller; • a limited supply of identical goods; • n buyers each wanting exactly one good. • We use competitive analysis: comparing mechanism vs. benchmark on worst-case instances.

  3. Background: Online Auctions • Competitive analysis ofadaptive pricing (in the unlimited-supply case) has been heavily studied, with mechanisms achieving progressively stronger revenue guarantees, culminating in the paper presented in Hartline’s talk. • But such strategies assume the arrival order of the agents is exogenous. Agents can manipulate the mechanism if they can strategically misstate their arrival time.

  4. Background: Online Auctions • We address this by modeling agents as having three private values: arrival time, departure time, and value. • Agents can misstate any of these, but can not state an earlier arrival time than their true arrival. • A mechanism is temporally strategyproof (time-SP) if an agent’s dominant strategy is to truthfully reveal arrival, departure, value.

  5. Time-SP Auctions: Prior Work • Lavi and Nisan (2000) considered: • Limited-supply multi-unit auctions. • Valuations are constrained to an interval [p,q]. • Seller’s utility for retaining a unit of the good is p. • Their mechanism is θ(log(q/p))-competitive for efficiency and revenue. (Best possible.) • The mechanism is time-SP becauseprice increases monotonically over time.

  6. Time-SP Auctions: Prior Work • Friedman and Parkes (2003) considered VCG-based online mechanisms. • These are time-SP if the underlying online allocation algorithm is perfectly efficient. • Otherwise the VCG-based mechanism is not truthful. Agents have an incentive to report misinformation if they believe that it will improve the efficiency of the allocation.

  7. Time-SP Auctions: Prior Work • Summary: Constant-competitive online mechanisms exist in the following cases. • Agents can’t misstate their arrival time. • Valuations constrained to an interval [p,q] with q=O(p), and unsold items are worth p to the seller. • There exists a perfectly competitive online algorithm for the underlying allocation problem.

  8. Our Contributions • Instead of making these strong assumptions, we simply assume: Agents’ valuations are independent random samples from some distribution. • Distribution is unconstrained: • Need not be known to mechanism designer. • Valuations need not be bounded above or below. • The only property we use is random ordering. (All permutations of a bid set equally likely.) • No assumption about arrival-departure process!

  9. Our Contributions • Our mechanisms are constant-competitive for both efficiency and revenue. • To our knowledge, these are the first known online mechanisms to achieve time-SP without relying on non-decreasing prices. • This work is related to: • Optimal stopping (“secretary problems”) • Competitive offline auctions

  10. Our Contributions • In deriving these results, we introduce a general technique for truthful mechanism design with restricted misreports. • We prove characterization theorems addressing: • Which allocation rules can be implemented truthfully in the restricted misreporting model? • Which mechanisms are truthful in this model? • As an additionalapplication of these general techniques, we study time-SP mechanism design for scheduling a re-usable resource.

  11. Talk Outline • Formal specification of the model • Single-item auctions and secretary problems • Mechanism design with restricted misreports • Multi-item auctions and generalized secretary problems • Online auctions with re-usable goods

  12. Model and Problem Statement • One seller, n buyers, 1≤k≤∞ identical goods. • Each agent (buyer) has a type defined by: • Arrival time ai. • Departure time di ≥ ai. • Valuation vi >0. • Agent may report any type (Ai ,Di ,Vi), subject to ai ≤ Ai ≤ Di. • Must compute allocations, payments online, must allocate to agent i during [Ai ,Di], if at all.

  13. Model and Problem Statement • Will require mechanisms to satisfy time-SP: An agent’s dominant strategy is to report (ai ,di ,vi) truthfully. • Mechanisms evaluated according to • Efficiency: Σqivi, compared with VCG (=OPT). • Revenue: Σpi, compared with F (2,k), defined as the maximum revenue obtainable by setting a fixed price and selling between 2 and k items. qi = quantity allocated to agent i (either 0 or 1). pi = price charged to agent i.

  14. Special Case: Online Single-Item Auction • To design a mechanism with constant competitive ratio for efficiency, must solve: • Online selection problem: Choose when to stop and allocate the item, though future bids are not yet known. • Incentive problem: The decision rule in (A) must be implemented without giving agents an incentive to delay announcing their arrival, or to lie about their valuation or departure time.

  15. Special Case: Online Single-Item Auction • First consider the online selection problem by itself. Specialize further to the case of disjoint arrival-departure intervals. 5 2 7 1,000 3

  16. Special Case: Online Single-Item Auction • First consider the online selection problem by itself. Specialize further to the case of disjoint arrival-departure intervals. • Reduces to the secretary problem: • A totally ordered set of n elements is presented in random order. • Design a stopping rule to maximize probability of stopping on the maximal element. 5 2 7 1,000 3

  17. The Secretary Algorithm • Theorem (Dynkin, 1962): The following stopping rule picks the maximal element with probability approaching 1/e as n→∞. • Observe the first n/e elements. Set a threshold equal to the maximum seen so far. • Stop the next time this threshold is exceeded. • The asymptotic success probability of 1/e is best possible, even if the numerical values of elements are revealed.

  18. Single-Item Auction Mechanism • Secretary algorithm is clearly not time-SP. Early agents have an incentive to hide until after time t, when the (n/e)-th agent appears. • So change the mechanism: • At time t, let p≥q be the top two bids yet received. • If any agent bidding p has not yet departed, sell to that agent (breaking ties randomly) at price q. • Else, sell to the next agent whose bid is at least p.

  19. Single-Item Auction Mechanism • At time t, let p≥q be the top two bids yet received. • If any agent bidding p has not yet departed, sell to that agent (breaking ties randomly) at price q. • Else, sell to the next agent whose bid is at least p. 0 T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent 6 $10

  20. Single-Item Auction Mechanism • At time t, let p≥q be the top two bids yet received. • If any agent bidding p has not yet departed, sell to that agent (breaking ties randomly) at price q. • Else, sell to the next agent whose bid is at least p. 0 t T Agent 1 $5 p Agent 1 wins, pays $2 Agent 2 q $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent 6 $10

  21. Single-Item Auction Mechanism • At time t, let p≥q be the top two bids yet received. • If any agent bidding p has not yet departed, sell to that agent (breaking ties randomly) at price q. • Else, sell to the next agent whose bid is at least p. 0 T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent 6 $10

  22. Single-Item Auction Mechanism • At time t, let p≥q be the top two bids yet received. • If any agent bidding p has not yet departed, sell to that agent (breaking ties randomly) at price q. • Else, sell to the next agent whose bid is at least p. 0 t T Agent 1 $5 p Agent 2 q $2 Agent 3 $5 Agent 4 $8 Agent 4 wins, pays $5 Agent 5 $4 Agent 6 $10

  23. Analysis: Strategyproofness • If agent i wins, the price charged to her does not depend on her reported valuation. • Pr(agent i wins) is non-decreasing in Vi, hence no incentive to understate Vi. • Reporting Vi > vi can not increase the probability that agent i wins at a price ≤vi, hence no incentive to overstate Vi. • Price facing agent i is never influenced by Di, so no incentive to misstate Di.

  24. Analysis: Strategyproofness • Claim: Given two arrival times ai<Ai, it’s always better to report ai if possible. • Let r,s be the ([n/e]-1)-th and [n/e]-th arrival times excluding agent i. 0 r s T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent i $10

  25. Analysis: Strategyproofness • Stating arrival time in (ai,r] changes nothing. 0 r s T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent i

  26. Analysis: Strategyproofness • Stating arrival time in (ai,r] changes nothing. • Stating arrival time in (r,s) influences the transition time t but not the pricing. 0 r s T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent i

  27. Analysis: Strategyproofness • Stating arrival time in (ai,r] changes nothing. • Stating arrival time in (r,s) influences the transition time t but not the pricing. • Stating arrival time ≥ s can’t improve price. 0 r s T Agent 1 $5 Agent 2 $2 Agent 3 $5 Agent 4 $8 Agent 5 $4 Agent i

  28. Analysis: Competitive Ratio • Claim: Competitive ratio for efficiency is e+o(1), assuming all valuations are distinct. • Case 1: Item sells at time t. Winner is highest bidder among first [n/e]. With probability ~1/e, this is also the highest bidder among all n agents. • Case 2: Otherwise, the mechanism picks the same outcome as the secretary algorithm, whose success probability is ~1/e.

  29. Analysis: Competitive Ratio • Claim: Competitive ratio for revenue is e2+o(1), assuming all valuations are distinct. • Proof works by estimating probability of selling to highest bidder at second-highest price. Use same two cases as before. • Case 1: Probability ~1/e2. • Case 2: Probability ~(1/e)(1-1/e). • Can achieve competitive ratio 4+o(1) by setting transition time at (n/2)-th arrival.

  30. Talk Outline • Formal specification of the model • Single-item auctions and secretary problems • Mechanism design with restricted misreporting • Multi-item auctions and generalized secretary problems • Online auctions with re-usable goods

  31. Restricted Misreporting • Online mechanism design is a special case of mechanism design with restricted misreporting. • Given a strategic-form game: • Let V be the type space of one player. • Let ► be a reflexive, transitive binary relation on V. • Interpretation: v► v’ means, “An agent with type v can misreport its type as v’.” • A mechanism is strategyproof if an agent with type v can never improve its utility by reporting a type v’ such that v► v’.

  32. Characterizing Truthfulness • Theorem: A social choice function f: Vn → A is truthfully implementable if and only if there exist price functions pi: A × Vi × V-i → R∞, such that: • pi(x,vi,v-i) = min {pi(x,vi’,v-i) : vi► vi’ and f(vi’,v-i)=x} if that set is non-empty, and otherwise pi(x,vi,v-i)=∞. [No agent can get outcome x at a cheaper price by lying.] • f(v)  arg maxxA{vi(x) – pi(x,vi,v-i)} for all agents i and all type vectors vVn. [Each agent gets the outcome which maximizes its utility, given the price function and the type vector.]

  33. Characterizing Truthfulness II • For the online auctions we’re considering, three natural misreporting models are: (A1)vi► vi’ if and only if ai ≤ ai’ and di ≥ di’. (A2)vi► vi’ if and only if ai ≤ ai’. (A3)vi► vi’ if and only if di ≥ di’. • Let qi=1 if i receives an item, 0 otherwise. Allocation rule is monotonic if ai ≤ai’≤di’≤di implies qi ≥ qi’.

  34. Characterizing Truthfulness III • Theorem: For misreporting model (A1), the following are equivalent: • An allocation rule is truthfully implementable. • An allocation rule is monotonic. • For each agent i there is a price schedule ps(a,d,v-i) such that: • ps(a’,d’,v-i) ≥ ps(a,d,v-i) if a’ ≥ a and b’ ≤ b. • qi(v)=1 if and only if vi ≥ ps(ai,di,v-i). • Similar theorems hold for (A2), (A3). (Characterization requires an additional constraint on the timing of the allocation.)

  35. Multi-Item Auction • Recall our paradigm for designing a competitive single-item auction: • Construct allocation rule using secretary problem. • Use the characterization theorem to implement this allocation in dominant-strategy equilibrium. • With more than one item for sale, the relevant allocation problem is a multiple-choice secretary problem… • A set of n positive numbers is presented in random order. Algorithm must pick k of them (at the time they are first revealed) to maximize the expected sum.

  36. The Algorithm MultSec(k) • Assume input consists of n distinct numbers. (Ensure distinctness with random multiplier.) • MultSec(1) is the secretary algorithm. • MultSec(k) does the following: • Toss n fair coins, let m = # of heads. • Run MultSec(k/2) on first m numbers. • Set threshold x = (k/2)-th highest among first m. • Subsequently pick every number exceeding x.

  37. The Algorithm MultSec’(k) • An easy transformation makes this time-SP. • MultSec’(1) is the allocation rule for the single-item auction presented earlier. • MultSec’(k) does the following: • Toss n fair coins, let m = # of heads. • Run MultSec’(k/2) on first m bidders. • Set threshold x = (k/2)-th highest among first m. • Allocate an item to every bidder whose bid exceeds x and who is present at or after the arrival of bidder m.

  38. Multiple Secretary Algorithm:Analysis • Theorem: The expected value of the numbers chosen by MultSec(k) is at least (1-5/√k)*OPT. • Theorem: For some C>0, no algorithm can do better than (1-C/√k)*OPT. • Theorem: Competitive ratio of MultSec’(k) (for efficiency) is at least 1-10/√k.

  39. Revenue-Competitive Auction • For the objective of maximizing revenue, competitive ratio doesn’t approach 1 as k→∞. • But it also doesn’t approach infinity:for all k, we can achieve competitive ratio < 6400 using a time-SP variation on the DSOT offline auction of Goldberg et al. • More sophisticated analysis (unpublished) improves the upper bound from 6400 to 250.

  40. Revenue-Competitive Auction • Set random transition timet = Binom(n,½). Sell up to s=k/2 items at time t, to all agents present and bidding above the (s+1)-th bid. • After t, let p be the revenue-optimizing price for the bid set seen before t. Sell to any agent whose bid exceeds p until supply is exhausted. • This is 6400-competitive with F (2,k) for revenue. • To be competitive for revenue and efficiency, toss a coin at time 0 and use it to determine which of the two mechanisms to run.

  41. Scheduling Auctions:The Greedy Allocation Rule Dave Carol Emily Fred Alice 4 4 3 3 X Emily Bob 5 5 2 2 X Fred Carol 6 6 X 1 1 Gladys Dave 7 7 X

  42. Analysis of Greedy Allocation Alice 4 3 O G Emily Bob 5 2 Fred O G Carol 6 1 G O Gladys Dave 7 G O 2 * Greedy ≥ OPT N.B. No need to assume random ordering in this theorem.

  43. Greedy Mechanism: Payment Rule Alice 4 3 G Emily Bob 5 2 Fred G G Carol 6 1 7 5 3 Gladys Dave 7 G Carol pays min(7,5,3) = 3.

  44. Greedy Mechanism: Strategyproof? • The greedy mechanism is monotonic, and the pricing rule specified earlier is exactly the one specified by the characterization theorem. • Hence, assuming misreporting model (A1) [no early arrivals or late departures] it is time-SP. • If agents are allowed to report arbitrary departure times then no time-SP mechanism can be constant-competitive.[Lavi-Nisan ’05, essentially]

  45. The revenue of re-usable good mechanisms • The revenue of the greedy algorithm can be disastrous, e.g. • VCG charges 1 to each agent. 1 2 1 2 2

  46. The revenue of re-usable good mechanisms • The revenue of the greedy algorithm can be disastrous, e.g. • Greedy charges 0 to all but the first agent. 1 2 1 G G 2 2 G

  47. Revenue lower bound • Definition: An impatient bidder is an agent satisfying di=ai+1. A mechanism considers impatient bidders anonymously if it never allocates a time slot t to an impatient bidder x when another impatient bidder y has a higher value for t. • Theorem: A deterministic time-SP mechanism which considers impatient bidders anonymously can’t be constant-competitive with VCG revenue.

  48. Revenue upper bound • Theorem: There is a randomized time-SP mechanism which achieves a competitive ratio of O(log h) when all bids belong to an interval [a,b] with b/a=h. The mechanism need not know the values a, b, or h. • Proof sketch: If [a,b] is known, let p be a random power of 2 between a/2 and b, and run greedy with reserve price p.

  49. Revenue upper bound • If VCG picks agent x with value v at time t, probability is 1/(log h) that reserve price is between v/2 and v. If so, our mechanism charges at least v/2 to at least one of: • Agent x; • The winner at time t. • If interval [a,b] is unknown, randomly partition the agents and use one half to estimate a and b.

  50. Conclusions • Introduced a framework for studying pricing problems when agents can strategize about timing their entry into the market. • These problems are a special case of mechanism design with restricted misreporting. • Presented a characterization theorem identifying which social choice functions have a dominant strategy implementation. (Proof is constructive: specifies the pricing rule explicitly.) • Related these problems to secretary problems and their generalizations. • Derived a new multiple-choice secretary theorem of independent interest.

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