Prompt Mechanisms for Online Auctions
Richard Cole Shahar Dobzinski Lisa Fleischer
November 12, 2007
Abstract
We study the following online problem: at each time unit, one of m identical items is offered
for sale. Bidders arrive and depart dynamically, and each bidder is interested in winning one item
between his arrival and departure. Our goal is to design truthful mechanisms that maximize the
welfare, the sum of the utilities of winning bidders.
We first consider this problem under the assumption that the private information for each bidder
is his value for getting an item. In this model constant-competitive mechanisms are known, but
we observe that these mechanisms suffer from the following disadvantage: a bidder might learn his
payment only when he departs. We argue that these mechanism are essentially unusable, because
they impose several seemingly undesirable requirements on any implementation of the mechanisms.
To crystalize these issues, we define the notions of prompt and tardy mechanisms. We present
two prompt truthful mechanisms – one deterministic and the other randomized, that guarantee a
constant competitive ratio. We also prove that our deterministic mechanism is optimal for this
setting.
We then study a model in which both the value and the departure time are private information.
While in the deterministic setting only a trivial competitive ratio can be guaranteed, we use ran-
domization to obtain a prompt truthful Θ(
1
log m
)-competitive mechanism. Finally, we show that no
truthful randomized mechanism can achieve a ratio better than
1
2
in this model.
1 Introduction
1.1 Bac kground
The field of algorithmic mechanism design attempts to handle the strategic behavior of selfish agents
in a computationally feasible way. To date, most work in this field has sought to design truthful
mechanisms for static settings, such as auctions. In reality, however, the setting of many problems is
online, meaning that the mechanism has no prior information regarding the identity of the participating
players, or that the goods that are for sale are unknown in advance. Examples include sponsored search
auctions [12], single-good auctions [10], and even pricing WiFi at Starbucks [5].
This paper considers the following online auction problem: at each time unit exactly one of m
identical items is offered for sale. The item at time t is called item t.Therearen bidders, where bidder
i arrives at time a
i
and departs at time d
i
, both unknown before bidder i’s arrival. The interval [a
i
,d
i
]
will be called bidder i’s time window, and the set of items offered in i’s time window will be denoted
by W
i
. Each bidder is interested in winning at most one of the items within W
i
.Letv
i
denote the
value to the ith bidder of getting an item in W
i
. Our goal is to maximize the social welfare: the sum of
the values of the bidders that get some item within their time window. As usual in online algorithms,
are goal is to optimize the competitive ratio: the worst-case ratio between the welfare achieved by the
algorithm and the optimal welfare.
This problem is equivalent to scheduling unit-length jobs on a single machine. In the non-strategic
setting, this problem and its variants have been widely studied (e.g., [1, 8, 3]). The best determin-
istic algorithm to date guarantees a competitive ratio of ≈ 0.547 [4, 11], while it is known that no
deterministic algorithm can obtain a ratio better than
2
√
5+1
≈ 0.618 [2]. In the randomized setting, a
competitive ratio of 1 −
1
e
is achieved by [1], and no algorithm can achieve a ratio better than 0.8[2].
This scheduling problem provides an excellent example of the extra barriers we face when designing
online mechanisms. The only general technique known for designing truthful mechanisms is the VCG
payment scheme. In the offline setting we can obtain an optimal solution in polynomial time (with
bipartite matching), and then we can apply VCG. In the online setting, however, it is impossible to
find an optimal solution, and thus we cannot use VCG. Yet, truthful competitive mechanisms do exist.
The competitive ratio of these mechanisms depends on the specific private-information model each
mechanism was designed for. This paper considers two different natural models:
• The Value-Only model: Here, the private information of bidder i consists of only his value
v
i
, and the arrival time and the departure time are known to all (but both are unknown prior to
the arrival of bidder i).
• The Generalized Model: The private information of bidder i consists of two numbers: his
value v
i
and his departure time d
i
. The arrival time is public information (but unknown prior to
the arrival of bidder i).
1.2 The Value-Only Model: Is Monotonicity Enough?
The only private information of a bidder in the value-only model is his value, and thus this model falls
under the category of single-parameter environments – environments in which the private information
of each bidder consists of only one number. Fortunately, designing truthful mechanisms for single-
parameter environments is quite well understood: an algorithm is truthful if and only if it is monotone.
That is, a winning bidder that raises his bid remains a winner.
Using the above characterization, it is possible to prove that the greedy algorithm is monotone [7]
(see Section 2.4 for a description). Together with the result of [8], this gives a truthful mechanism that
is
1
2
competitive.
However, a closer look at this mechanism may make one wonder if it is indeed applicable. The
notions of prompt and tardy mechanisms we define next highlight the issue.
Definition 1.1 We say that a mechanism for the scheduling problem is prompt if a bidder that wins
an item always learns his payment immediately after winning the item. A mechanism is tardy if it is
not prompt.
The tardiness in the above mechanism [7, 8] is substantial: there are inputs for which a bidder
learns his payment only when he departs. Tardy mechanisms seem very unintuitive for the bidders,
and in addition they suffer from the following disadvantages:
• Uncertain ty: A winning bidder does not know the cost of the item that he won, and thus does
not know how much money he still has available. E.g., suppose the mechanism is used in a Las
Vegas ticket office for selling tickets to a daily show. A tourist that wins a ticket is uncertain of
the price of this privilege, and thus might not be able to determine how much money he has left
to spend during his Las Vegas vacation.
• Debt C ollection: A winning bidder might pay the mechanism long after he won the item. A
bidder that is not honest may try to avoid this payment. Thus, the auctioneer must have some
way of collecting the payment of a winning bidder.
• Trusted Auctioneer: A winning bidder essentially provides the auctioneer with a “blank check”
in exchange for the item. Consequently, all bidders must trust the honesty of the auctioneer.
2
Even if the bidders trust the auctioneer, they may still want to verify the exact calculation of the
payment, to avoid over-payments that make winning the item less profitable, or even unprofitable.
In order to verify this calculation, the bids of all bidders have to be revealed, leading to an
undesirable loss of privacy.
Notice that all of these problems are due to the online nature of the setting, and do not arise in
the offline setting. To overcome these problems, we present prompt mechanisms for the scheduling
problem. Prompt mechanisms are very intuitive to the bidders as they (implicitly) correspond to take-
it-or-leave-it offers: a winning bidder is offered a price for one item exactly once before getting the item,
and may reject the offer if it is not beneficial for him. We improve upon the greedy algorithm of [7, 8]
by showing a different mechanism that achieves the same competitive ratio, but is also prompt.
Theorem: There exists a
1
2
-competitive prompt and truthful mechanism for the scheduling problem
in the value-only model.
We show that this is the best possible by proving that no prompt deterministic mechanism can guarantee
a competitive ratio better than
1
2
.
We also present a randomized mechanism that guarantees a constant competitive ratio. The
achieved competitive ratio of the latter algorithm is worse than the competitive ratio of the determinis-
tic algorithm. Yet, the core of the proof studies a balls-and-bins problem that might be of independent
interest.
1.3 The Generalized Model
While truthful mechanisms for single-parameter settings are well characterized and thus relatively
easy to construct, truthful mechanisms for multi-parameter settings, like the generalized model, are
much harder to design. The online setting considered in this paper only makes the design of truthful
mechanisms a more challenging task.
The scheduling problem in the generalized model illustrates this challenge. Lavi and Nisan [9]
introduced the scheduling problem to the mechanism design community. They showed that no truthful
deterministic mechanism for this multi-parameter problem can provide more than a trivial competitive
ratio. As a result, Lavi and Nisan suggested looking for a weaker solution concept, set-nash, and
provided mechanisms with a constant competitive ratio under this notion. We stress that the set-nash
solution concept is much weaker than the dominant-strategy truthfulness considered in this paper.
By contrast with [9], instead of relaxing the solution concept, we use the well-known idea that
randomization can help in mechanism-design settings [14]. We provide randomized upper and lower
bounds in the generalized model for the scheduling problem.
Theorem: There exists a prompt truthful randomized Θ(
1
log m
)-competitive mechanism for the
scheduling problem in the generalized model.
The main idea of the mechanism is to extend the randomized mechanism for the value-only model to
the generalized model. Specifically, we use the random-sampling method introduced in [6] in order to
be able to “guess” the departure time of each bidder. This mechanism is also a prompt mechanism.
We notice that it is quite easy to obtain mechanisms with a competitive guarantee of the logarithm of
the ratio between the highest and lowest valuations. However, since this ratio might be exponential
in the number of items or bidders, this guarantee is quite weak. By contrast, the competitive ratio
our mechanism achieves is independent of the ratio between the highest and lowest valuations, and the
mechanism is not required to know the highest and lowest valuations in advance.
Theorem: No truthful randomized mechanism for the scheduling problem in the generalized model
can obtain a competitive ratio better than
1
2
.
3
The proof of this bound is quite complicated. We start by defining a family of recursively-defined
distributions on the input, and then show that no deterministic mechanism can obtain a competitive
ratio better than
1
2
on this family of distributions. We then use Yao’s principle to derive the theorem.
The main open question left in the generalized model is to determine whether there is a truthful
mechanism with a constant competitive ratio.
Paper Organization
In Section 2 we describe prompt mechanisms for the value-only case, and prove a that no deterministic
tardy algorithms can achieve a ratio better than
1
2
. Lower and upper bounds for the generalized case
areprovedinSection3.
2 Prompt Mechanisms and the Value-Only M odel
2.1 A Deterministic Prompt
1
2
-Competitive Mechanism
The mechanism maintains a candidate bidder c
j
for each item j. To keep the presentation simple and
without loss of generality, we assume an initialization of the mechanism in which each item j receives
a candidate bidder c
j
with a value of 0 for winning an item (i.e., v
c
j
=0).
The mechanism runs as follows: at each time t we look at all the bidders that arrived at time t.We
consider these bidders one by one in some arbitrary order (independent of the bids): for each bidder
i we look at the set of all candidates in i’s time window, and let c
j
be the candidate bidder with the
smallest bid (if there are several such candidates select one arbitrarily). Formally, c
j
∈ arg min
k∈W
i
c
k
.
We say that i competes on item j.Now,ifv
c
j
<v
i
,wemakei the candidate bidder for item j.After
all the bidders that arrived in time t have been processed, we allocate item t to the candidate bidder
c
t
.
The next theorem proves that this algorithm is monotone, i.e., a bidder that raises his bid is still
guaranteed to win. This is also a necessary and sufficient condition for truthfulness. We are still left
with the issue of finding the payments themselves. First, observe that the payment of each winning
bidder must equal his critical value: the minimum value he can declare and still win. Notice that this
value is indeed well defined if the algorithm is monotone. For each bidder i this value can be found
by using a binary search on the values in i’s domain. Clearly, this procedure takes a polynomial time.
See, e.g., [13] for a more thorough discussion. By the discussion above, it is clear that a mechanism
is prompt if and only if i’s critical value can be found by the time a i wins an item. In this case, the
payment can also be calculated in polynomial time.
Theorem 2.1 The above mechanism is prompt and truthful. Its competitive ratio is
1
2
.
Proof: To show that the mechanism is truthful we have to show that it is monotone: that is, a winning
bidder i still wins an item by raising his value v
i
to v
i
. First, observe that fixing the declarations of
the other bidders, i competes on item j regardless of his value. We now compare two runs of the
mechanism, with i declaring v
i
and with i declaring v
i
, and show that at each time the candidate for
any item j
is the same in both runs. In particular, it follows that the set of winners stays the same,
and thus the mechanism is monotone.
Look at the next bidder e that arrives after i. For a contradiction, suppose that the candidate for
some item changes after bidder e arrives. It follows that i declaring v
i
causes e to compete on an item
different than the one that e competes on when i declares v
i
. This is possible only if e is competing
on j if i declares v
i
, but if i declares v
i
, e competes on h = j. It follows that if i declares v
i
both i
and e compete on j,andthati wins j.Thus,v
i
≥ v
e
.Wheni raises his bid e competes on h.Letc
h
be the candidate for h atthetimethate arrives. We have that v
i
>v
c
h
>v
i
, and thus v
e
<v
c
h
so e
4
does not become a candidate on h, and the set of candidates stays the same. To finish the proof for
the monotonicity of the mechanism, we look at the rest of the bidders one by one, and repeat the same
arguments.
As for the promptness of the mechanism, observe that the identity of the item that i competes on
is determined only by the information provided by bidders that had already arrived by the time of i’s
arrival. The winner of any item j is of course completely determined by the information provided by
bidders that arrived by time j. Thus, we can calculate the payment of a winning bidder immediately
after he wins an item.
We now analyze the competitive ratio of the mechanism. Let OPT =(o
1
, ..., o
m
)betheoptimal
solution, and ALG =(p
1
, ..., p
m
) be the solution constructed by the mechanism. That is, o
j
is the
bidder that wins item j in OPT and p
j
is the bidder that wins item j in ALG. We will match each
bidder i that wins an item in OPT to exactly one bidder l that wins an item in ALG. Furthermore,
we will make sure that v
i
≤ v
l
, and that each bidder in ALG is associated with at most two bidders in
OPT. This is enough to prove a competitive ratio of
1
2
.
The bidders are matched as follows: for each item j,leto
j
1
, ··· ,o
j
k
j
be the bidders (ordered by
their arrival time) that won an item in the optimal solution and are competing on j.Nowmatcheach
o
j
r
to p
j
r +1
for r<k
j
.Matcho
j
k
j
to p
j
, the bidder that wins j in ALG (it is possible that p
j
= o
j
k
j
).
Observe that bidder p
j
is associated with at most two bidders that win some item in OPT: bidder
o
j
k
j
, and at most one bidder (o
j
i
) that is competing on an item j,wherej is the item that o
j
(= o
j
i+1
)is
competing on in ALG. To finish the proof, we only have to show that v
o
j
k
j
≤ v
p
j
and v
o
j
i
≤ v
p
j
.Since
o
j
k
j
and p
j
both compete for slot j (possibly they are the same bidder) and p
j
wins, v
o
j
k
j
≤ v
p
j
.Now
we show the second claim. When o
j
i+1
arrives, o
j
i
is already competing on slot j;aso
j
i+1
chooses to
compete on slot j rather than slot j
which is also in its interval, it must be that the current candidate
for slot j has value at least v
o
j
i
. But the eventual winner of slot j, p
j
, can only have a larger value; i.e.
v
o
j
i
≤ v
p
j
.
2.2 A Prompt Randomized Mechanism
In this section we present a randomized prompt O(1)-competitive mechanism for the online scheduling
problem in the value-only model. The analysis of the competitive ratio of the mechanism is related to
a variant of the following balls-and-bins question:
Balls and Bins (intervals version): n balls are thrown to n bins, where the ith ball is thrown
uniformly at random to bins in the interval W
i
=[a
i
,d
i
]. We are given that the balls can be placed in
a way such that all bins are filled, and each ball i is placed in exactly one bin in [a
i
,d
i
]. What is the
expected number of full bins?
The theorem below proves that for every valid selection of the a
i
’s and d
i
’s in expectation at least
1
10
of the bins will be full (notice that in the scheduling problem the “balls” have weights). There is a gap
between this ratio and the worst example we know: in Subsection 2.3 we present an example in which
at most
11
24
of the bins are full in expectation. Improving the analysis of the balls and bins question
will almost immediately imply an improvement in the guaranteed competitive ratio of the mechanism.
The Mechanism
1. When bidder i arrives, assign it to exactly one item in W
i
to compete on uniformly at random.
2. At time j conduct a second-price auction on item j among all the bidders that were selected to
compete on item j in the first stage.
5