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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

- 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 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 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

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

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

- 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:
- 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 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 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

- 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.

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 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 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 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

- 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: 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: 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: 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: 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

- 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

- 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

- 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:
- 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)

- 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 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

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

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

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?

- 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

2

1

2

2

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

- 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

- 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

- 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.

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