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

Adaptive Limited-Supply Online Auctions

Robert Kleinberg (MIT and Cornell)Collaborators: Mohammad Hajiaghayi (MIT)

Mohammad Mahdian (Microsoft)

David Parkes (Harvard)

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.


Competitive analysis of adaptive 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.


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

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 because price increases monotonically over time.

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 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

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 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 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 additional application of these general techniques, we study time-SP mechanism design for scheduling a re-usable resource.

Talk Outline

1. Formal specification of the model

2. Single-item auctions and secretary problems

3. Mechanism design with restricted misreports

4. Multi-item auctions and generalized secretary problems

5. Online auctions with re-usable goods

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 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 To design a mechanism with constant

competitive ratio for efficiency, must solve:A. Online selection problem: Choose when to stop and

allocate the item, though future bids are not yet known.

B. 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.

7 1,000 325

Special Case: Online Single-Item Auction First consider the online selection problem by

itself. Specialize further to the case of disjoint arrival-departure intervals.

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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.

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

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.


At time t, let p≥q be the top two bids yet received. If any agent bidding p has not yet departed, sell to


0 T

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Agent 1 wins, pays $2




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Agent 4 wins, pays $5

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.


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 T

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$2

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Agent 2

Agent 3

Agent 4

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Agent i

sr

$10


Stating arrival time in (ai,r] changes nothing.

0 T

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Agent 1

Agent 2

Agent 3

Agent 4

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Agent i

sr


Stating arrival time in (ai,r] changes nothing. Stating arrival time in (r,s) influences the transition

time t but not the pricing.

0 T

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Agent 1

Agent 2

Agent 3

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Agent i

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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 T

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Agent 1

Agent 2

Agent 3

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sr

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 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 Outline

1. Formal specification of the model

2. Single-item auctions and secretary problems

3. Mechanism design with restricted misreporting

4. Multi-item auctions and generalized secretary problems

5. Online auctions with re-usable goods

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

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 maxxA {vi(x) – pi(x,vi,v-i)} for all agents i and all type vectors vVn. [Each agent gets the outcome which maximizes its utility, given the price function and the type vector.]

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

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

Recall our paradigm for designing a competitive single-item auction:

1. Construct allocation rule using secretary problem.2. 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)

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’(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 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

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 Auction

Set random transition time t = 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

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Alice

Bob

Carol

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Emily

Fred

Gladys

Dave Carol Emily Fred

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1

Analysis of Greedy Allocation

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5 2

6 1

Alice

Bob

Carol

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Emily

Fred

Gladys

O

O

O

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G

G

G

2 * Greedy ≥ OPT

N.B. No need to assume random ordering in this theorem.

Greedy Mechanism: Payment Rule

4 3

5 2

6 1

Alice

Bob

Carol

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Emily

Fred

Gladys

G

G

G

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Carol pays min(7,5,3) = 3.

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 the greedy algorithm can

be disastrous, e.g.

VCG charges 1 to each agent.

1

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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.

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G

G G

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

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 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

1. Introduced a framework for studying pricing problems when agents can strategize about timing their entry into the market.

2. These problems are a special case of mechanism design with restricted misreporting.

3. Presented a characterization theorem identifying which social choice functions have a dominant strategy implementation. (Proof is constructive: specifies the pricing rule explicitly.)

4. Related these problems to secretary problems and their generalizations.

5. Derived a new multiple-choice secretary theorem of independent interest.

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.

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

Documents

timesp auctions

departure time

earlier arrival time

strategyproof time sp

agents valuations

competitive online algorithm

known online mechanisms

modeling agents