Algorithmic Game Theory
September 8, 2008
Lecture 1: Single-Item Auctions
Lecturer: Sébastien Lahaie
Auctions and markets, in their various forms, are a central topic in algorithmic game
theory, and will be one of the main topics of this course. To begin, let’s examine the
simplest problem that auctions are used to solve: allocating a single item among various
buyers. There are several plausible auction formats even for this simple problem. The goal
of each format is to provide the item to the buyer that values it the most. The formats
differ in their payment rules (first- or second-price), and in whether bidding occurs over
just a single round or multiple rounds (sealed-bid or iterative).
first-price
second-price
sealed-bid
iterative
FPSB
Vickrey
Dutch
English
Auctions can be viewed through two lenses: game-theoretic and algorithmic. The sealedbid auctions nicely illustrate the kind of incentive concerns that arise in auction design: since
we don’t know the bidders’ values for the item, how can we be sure it eventually goes to
the one who values it the most? The iterative designs use bidding over multiple rounds to
reach a price such that supply equals demand, so that not necessarily every buyer needs to
submit a bid, possibly economizing on communication.
In the following, there is a single item held by a seller, and a set N of n buyers. The
value of the item to bidder i is vi ∈ R+ (the non-negative reals). The seller has value 0 for
the item. An allocation is denoted by the bidder j ∈ N who obtains the item. An outcome
is an allocation together with a vector of payments q = (qi )i∈N , denoting how much each
bidder is charged. The utility of outcome (j, q) to bidder i is defined as
vi − qi if j = i
ui (j, q) =
−qi
otherwise
In words, a buyer receives utility vi if it gets the item and 0 otherwise, and we subtract
from that its payment. A rational bidder is one who chooses its actions (e.g., how it will
bid) so as to maximize its utility. We will always assume that bidders are rational.
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Sealed-Bid Auctions
Let’s first consider the “sealed-bid” formats, and their incentive properties. In a sealed-bid
auction, each bidder i first submits a bid bi ≥ 0 to the seller. Importantly, they might
submit bi 6= vi is this is to their advantage. A sealed-bid auction is formally defined by an
allocation and a payment rule, mapping vectors of bids to allcoation and payments. The
first-price sealed bid (FPSB) and Vickrey auctions share the same allocation rule.
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Allocation. The item is allocated to the highest bidder.
The auctions differ on the payment rule. For clarity, reindex the bidders so that b1 ≥ b2 ≥
. . . ≥ bn .
FPSB payment. The highest bidder is charged his bid b1 . The others are charged 0.
Vickrey payment. The highest bidder is charged the second-highest bid, b2 . The others
are charged 0.
The virtue of the second-price scheme is the following, first observed by Vickrey (hence the
name of the auction) [2].
Proposition 1 In the Vickrey auction, bidder i maximizes its utility by reporting bi = vi .
Proof. Consider an arbitrary bidder i, and fix the others’ bids to b−i . (The notation b−i
is shorthand for the vector of bids for all bidders except i. It is standard and convenient
notation in game theory.) For simplicity, we assume that vi 6= b̄ (the argument is simple to
extend to the case of ties.) Let b̄ = maxj6=i bj . There are two cases:
(i) vi < b̄. Bidding vi , bidder i gets utility 0 (no item and no payment charged). Bidding
anything else, the bidder either receives utility 0 (if it still loses), or utility vi − b̄ ≤ 0
(if it outbids all the others). Thus bidding vi maximizes utility.
(ii) vi > b̄. Bidding vi , bidder i gets utility vi − b̄ > 0. Any other bid that causes bidder
i to win yields this same utility, because note that the payment remains the same at
b̄. Meanwhile, a bid that causes him to lose yields utility 0. Thus bidding vi again
maximizes utility.
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Note that in the Vickrey auction, the bidder does not need any information about the others’
values to bid optimally: bidding the true value is an always optimal strategy. If there is
a unique always optimal strategy, we say it is a dominant strategy. It is straightfoward
to check that for any bid bi 6= vi , there is a vector of the others’ bids b−i which make it
such that bidding bi yields strictly less utility than vi . (You can check this for yourself.)
Thus bidding bi = vi is a dominant strategy in the Vickrey auction. This means we can
confidently predict that each bidder will be “truthful”.
Now let’s consider the first-price sealed bid auction. Clearly, bidding one’s true value
does not maximize utility, since this ensures a utility of 0 whether the item is won or lost.
The first-price payment rule induces bidders to report less than their true values. But there
is no bidding strategy that maximizes utility no matter what the others bid. To reason
about the best bid, bidders need at least an estimate (i.e., a probability distribution) over
the others’ values for item. This means that to predict the outcome of this auction, we
need to make stronger assumptions about the bidders and their beliefs than in the Vickrey
auction. We won’t go into the equilibrium analysis here; it can be found in any standard
textbook on auction theory [1].
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Iterative Auctions
Now let’s look at the “iterative” auctions. The English or “ascending” auction is wellknown—it is used to sell artwork and antiques, and it is similar to the auction run by eBay.
The price initially starts low, say at 0, and the auction proceeds in rounds. The price is
increased by a small increment > 0 at each round if more than one bidder is still willing
to acquire the item at the quoted price. The auction ends when all but one bidder have
dropped out.
The English auction is an iterative analog of the Vickrey auction, because the final
price is precisely the second-highest bid (to within ). It therefore inherits similar incentive
properties: at each round, it is best to bid if the price is below one’s true value, and to drop
out if not—we call this price-taking behavior. But it is unclear whether this is the reason
for its popularity. After all, why go through multiple rounds when the sealed-bid approach
is much faster?
Really, the main point of the English auction is not to find the second-highest value
for incentive purposes. The mechanics of the auction are designed to find a price which
only one bidder is willing to pay. This kind of price “clears the market”. Initially, supply
does not meet demand: there is just a single item, but many buyers willing to buy it. The
auction finds a price which balances supply and demand. This is any price in the interval
(v2 , v1 ). Because the English auction starts off low, it happens to converge to the lowest of
these clearing prices, v2 . Note that, unlike in the Vickrey auction, the highest value v1 is
never revealed in the English auction.
The Dutch or “descending” auction is similar: it begins at a very high price, which is
decreased by > 0 at each round. Initially no bidder would accept the item at the quoted
price. As soon as a bidder accepts the price quoted in a round, it is awarded the item at
that price and the auction ends.
The virtue of the Dutch auction is inherently algorithmic. It is fast: the Dutch auction is
so named because the Dutch use it to sell vast quantities of flowers at wholesale. Note that
it uses less communication than the English auction: just a single bid is placed throughout
the whole procedure. On the other hand, we would not expect this to occur at the highest
possible value for the item, for the same reason as in the FPSB.
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Generalizations
The “combinatorial auctions” for multiple items that we will see later in the course can be
construed as attempts to generalize some of these single-item auctions, while trying to retain
their appealing incentive and algorithmic properties. The general goal of an auction is to
allocate the items to maximize the total value created—this is called an efficient allocation
(not to be confused with the notion of an efficient algorithm in computer science). We will
look at one-shot auctions and ask:
• Can we quickly allocate the items efficiently among the buyers?
• What should we charge buyers so that they are truthful?
One-shot auctions are centralized solutions, hence the focus is on quickly computing an
efficient, or at least approximately efficient, allocation, and on inducing the bidders to
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truthfuly report their values. We will then turn to decentralized solutions to the allocation
problem, in the form of iterative auctions, and ask:
• How can we reach clearing prices?
• How can we induce price-taking behavior?
• How much communication is needed to allocate the items?
References
[1] Vijay Krishna. Auction Theory. Academic Press, 2002.
[2] William Vickrey. Counterspeculation, auctions and competitive sealed tenders. Journal
of Finance, 16:8–37, 1961.
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