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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

Adaptive Limited-Supply Online Auctions

Robert Kleinberg (MIT and Cornell)

Collaborators: Mohammad Hajiaghayi (MIT)

Mohammad Mahdian (Microsoft)

David Parkes (Harvard)

background online auctions
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.
background online auctions3
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.
background online auctions4
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.
time sp auctions prior work
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.
time sp auctions prior work6
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.
time sp auctions prior work7
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.
our contributions
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!
our contributions9
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
our contributions10
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.
talk outline
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
model and problem statement
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.
model and problem statement13
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.

special case online single item auction
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.
special case online single item auction15
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

special case online single item auction16
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

the secretary algorithm
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.
single item auction mechanism
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.
single item auction mechanism19
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

single item auction mechanism20
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

single item auction mechanism21
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

single item auction mechanism22
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

analysis strategyproofness
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.
analysis strategyproofness24
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

analysis strategyproofness25
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

analysis strategyproofness26
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

analysis strategyproofness27
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

analysis competitive ratio
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.
analysis competitive ratio29
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.
talk outline30
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
restricted misreporting
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’.
characterizing truthfulness
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.]

characterizing truthfulness ii
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’.
characterizing truthfulness iii
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.)
multi item auction
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.
the algorithm multsec k
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.
the algorithm multsec k37
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.
multiple secretary algorithm analysis
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.
revenue competitive auction
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.
revenue competitive auction40
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.
scheduling auctions the greedy allocation rule
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

analysis of greedy allocation
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.

greedy mechanism payment rule
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.

greedy mechanism strategyproof
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]
the revenue of re usable good mechanisms
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

the revenue of re usable good mechanisms46
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

revenue lower bound
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.
revenue upper bound
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.
revenue upper bound49
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.
conclusions
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.
open problems
Open problems
  • Extend theory of restricted misreporting, e.g. by extending characterizations of truthfulness to randomized mechanisms.
  • Improve our lower bounds. Extend them to randomized mechanisms, remove the annoying “considers impatient bidders anonymously” assumption.
  • Enrich the model of agents further, e.g. by allowing value to depend non-trivially on the quantity allocated or the timing of the allocation.