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 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.
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 (timeSP) 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 VCGbased 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.
2.
3.
4.
5.
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
Σqivi, compared with VCG (=OPT).
 Revenue: Σpi, compared with F (2,k), defined as the
 Efficiency:
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.
B.
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
Agent 1
Agent 2
T
$5
$2
Agent 3
Agent 4
Agent 5
Agent 6
$5
$8
$4
$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
p
Agent 2
q
$5
Agent 1 wins, pays $2
$2
Agent 3
Agent 4
Agent 5
Agent 6
$5
$8
$4
$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
Agent 1
Agent 2
T
$5
$2
Agent 3
Agent 4
Agent 5
Agent 6
$5
$8
$4
$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
Agent 1
Agent 2
t
$5
T
p
q
$2
Agent 3
Agent 4
Agent 5
Agent 6
$5
$8
Agent 4 wins, pays $5
$4
$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
Agent 1
Agent 2
r
s
T
$5
$2
Agent 3
$5
Agent 4
$8
Agent 5
Agent i
$4
$10
Analysis: Strategyproofness

Stating arrival time in (ai,r] changes nothing.
0
Agent 1
Agent 2
r
s
T
$5
$2
Agent 3
Agent 4
Agent 5
Agent i
$5
$8
$4
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
Agent 1
Agent 2
r
s
T
$5
$2
Agent 3
Agent 4
Agent 5
Agent i
$5
$8
$4
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
Agent 1
Agent 2
r
s
T
$5
$2
Agent 3
Agent 4
Agent 5
Agent i
$5
$8
$4
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.
2.
3.
4.
5.
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 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


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 singleitem auction:
1.
2.


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 singleitem 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
Dave
Alice
Emily
4
3
Bob
5
Carol
Dave
Carol
X
Emily
X
2
6
X
Fred
X
1
7
Fred
Gladys
Analysis of Greedy Allocation
Alice
4 O
Bob
3G
5
Carol
Dave
O
2
6G
G
Emily
O
7
G
1
Fred
Gladys
O
2 * Greedy ≥ OPT
N.B. No need to assume random ordering in this theorem.
Greedy Mechanism: Payment Rule
Alice
3 G
4
Bob
5
Carol
7
Dave
G
G
Emily
G
2
65
3
7
Carol pays min(7,5,3) = 3.
1
Fred
Gladys
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.
1
2
2

1
2
VCG charges 1 to each agent.
The revenue of re-usable good
mechanisms

The revenue of the greedy algorithm can
be disastrous, e.g.
G 2
1
G2

1 G
2
Greedy charges 0 to all but the first agent.
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.
2.
3.
4.
5.
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.