A number of heterogeneous items are to be sold to several bidders. Each bidder demands at most one item. The price of each item is not completely flexible and is restricted to some admissible interval. In such a market economy with price rigidities, a Walrasian equilibrium usually fails to exist. To facilitate the allocation of items to the bidders, we propose an ascending auction with rationing that yields a constrained Walrasian equilibrium outcome. The auctioneer starts with the lower bound price vector that specifies the lowest admissible price for each item, and each bidder responds with a set of items demanded at those prices. The auctioneer adjusts prices upwards for a minimal set of over-demanded items and chooses randomly a winning bidder for any item if the item is demanded by several bidders and its price has reached its highest admissible price. We prove that the auction finds a constrained Walrasian equilibrium outcome in a finite number of steps.Ascending auction;multi-item auction;constrained equilibrium;price rigidities;rationing

Talman, A.J.J.

Yang, Z.F.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2007–26 
 
A DYNAMIC AUCTION FOR DIFFERENTIATED ITEMS UNDER 
PRICE RIGIDITIES 
 
By Dolf Talman, Zaifu Yang 
 
 
 
April 2007 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 A Dynamic Auction for Diﬀerentiated Items under Price Rigidities
Dolf Talman1 and Zaifu Yang2
Abstract: A number of heterogeneous items are to be sold to several bidders.
Each bidder demands at most one item. The price of each item is not com-
pletely ﬂexible and is restricted to some admissible interval. In such a market
economy with price rigidities, a Walrasian equilibrium usually fails to exist. To
facilitate the allocation of items to the bidders, we propose an ascending auc-
tion with rationing that yields a constrained Walrasian equilibrium outcome.
The auctioneer starts with the lower bound price vector that speciﬁes the low-
est admissible price for each item, and each bidder responds with a set of items
demanded at those prices. The auctioneer adjusts prices upwards for a min-
imal set of over-demanded items and chooses randomly a winning bidder for
any item if the item is demanded by several bidders and its price has reached
its highest admissible price. We prove that the auction ﬁnds a constrained
Walrasian equilibrium outcome in a ﬁnite number of steps.
Keywords: Ascending auction, multi-item auction, constrained equilibrium,
price rigidities, rationing.
JEL classiﬁcation: D44.
1 Introduction
Economists have extensively studied market environments where there may be restrictions
on the prices of commodities and services. There are many economic or political reasons
for the existence of price rigidities. For instance, to prevent breakdown of stock markets,
often ceilings and ﬂoors are imposed upon the price of each stock; price controls are used to
1A.J.J. Talman, Department of Econometrics & Operations Research and CentER, Tilburg University,
P.O. Box 90153, 5000 LE Tilburg, The Netherlands, talman@uvt.nl
2Z. Yang, Faculty of Business Administration, Yokohama National University, Yokohama 240-8501,
Japan, yang@ynu.ac.jp
1reduce inﬂation or deﬂation; and minimum wages are employed to protect certain groups
of the society; see e.g., Dr` eze (1975), Dehez and Dr` eze (1984), Cox (1980), van der Laan
(1980), Kurz (1982), Azariadis and Stiglitz (1983), Weddepohl (1987), Herings et al. (1996)
among many others. When there are ﬁxed prices or price rigidities, rationing is needed
to restrict how much agents are allowed to purchase or sell. As a result, rationing will
help prices to facilitate the distribution of commodities among agents. However, eﬃciency
cannot be fully attained in general.
While the literature has focused almost entirely on economic models with divisible
goods, we will study the eﬃcient allocation of heterogeneous indivisible goods such as
houses or apartments among several agents. Each agent demands at most one item. The
price of each item is restricted to an admissible interval and thus is assumed to be not
perfectly ﬂexible. The lowest admissible price of an item can be seen as a reservation price,
while the highest admissible price of an item may be exogenously given by the government
for the purpose of either preventing speculation or protecting low-income families so that
they can aﬀord to buy necessary goods, such as an apartment. In such a market economy,
a Walrasian equilibrium usually fails to exist. To facilitate the allocation of items to the
bidders (or agents), we propose an ascending auction with rationing that produces a con-
strained Walrasian equilibrium outcome. The auctioneer starts with the lower bound price
vector that speciﬁes the lowest admissible price for each item, and each bidder responds
with a set of items demanded at those prices. The auctioneer adjusts prices upwards for a
minimal set of over-demanded items and chooses randomly a winning bidder for an item if
this item is demanded by several bidders and its price has reached its highest admissible
price. We prove that the auction ﬁnds a constrained Walrasian equilibrium outcome in a
ﬁnite number of steps. At such an equilibrium a bidder can be rationed on his demand
for an item if the price of the item is on its upper bound, if the item is assigned to some
other bidder, and if he will demand the item when the rationing is removed from him.
Moreover, if an item is not assigned in equilibrium its price must be on its lower bound.
The auction can be seen as a variant of the auction of Demange et al. (1985) for selling
multiple items without price rigidities. Another related mechanism is due to Crawford and
Knoer (1981). Maskin (2000) studied an auction for the allocation of one item when buyers
2may be signiﬁcantly budget-constrained.
This paper proceeds as follows. Section 2 sets up the model. Section 3 presents the
dynamic auction and establishes the main theorem.
2 The Model
An auctioneer wishes to sell a set of heterogeous items N = {0,1,···,n}, to a group of
bidders M = {1,2,···,m}. The item 0 is a dummy good which can be assigned to more
than one bidder. Without loss of generality, we assume that the auctioneer values every
item at zero. Every bidder i ∈ M attaches an integer monetary value to each item, namely,
each bidder i has a utility function V i : N → Z+ with V i(0) = 0. A feasible allocation π
assigns every bidder i ∈ M an item π(i) such that no item in N \ {0} is assigned to more
than one bidder. Note that a feasible allocation may assign the dummy good to several
bidders and a real item j 6= 0 may not be assigned to any bidder at all. An item j ≥ 1
is unassigned at π if there is no bidder i such that π(i) = j. A feasible allocation π∗ is
eﬃcient if
P
i∈M V i(π∗(i)) ≥
P
i∈M V i(π(i)) for every feasible allocation π.
A price vector p ∈ I R
N
+ indicates a price for every good. The price of each good
j ∈ N \ {0} is not completely ﬂexible and is restricted to an interval [p
j, ¯ pj], where p
j and
¯ pj are integers and 0 ≤ p
j < ¯ pj. The set
P = {p ∈ I R
N | p0 = 0, p
j ≤ pj ≤ ¯ pj, j = 1,···,n }
denotes the set of admissible prices. The price of the dummy good is always ﬁxed at zero.
When rationing does not take place, the demand set of bidder i ∈ M at any price vector
p ∈ I R
N
+ is given by
D
i(p) = {j | V
i(j) − pj ≥ V
i(k) − pk for every k ∈ N}.
A Walrasian equilibrium consists of a price vector p ∈ I R
N
+ and a feasible allocation π such
that π(i) ∈ Di(p) for all i ∈ M and pj = 0 for any unassigned good j at π. It is well-known
that a Walarasian equilibrium exists in the economy when there are no price rigidities.
In the case of price restrictions, a Walrasian equilibrium may not exist since the equilib-
rium price vector may not be admissible. In this case we may introduce a rationing scheme
3Ri ∈ {0,1}N for each bidder i ∈ M with Ri
0 = 1. For i ∈ M, the vector Ri dictates which
goods bidder i ∈ M can demand and which goods bidder i cannot demand, namely, Ri
j = 1
means that bidder i is allowed to demand good j, while Ri
j = 0 means that bidder i is not
allowed to demand good j. Given the rationing scheme Ri and an item j with Ri
j = 0, the
vector Ri
−j will denote that Ri
j is being ignored and bidder i is allowed to demand item j.
A list R = (R1,R2,···,Rm) of all bidders’ rationing schemes is called a rationing system.
At admissible price vector p ∈ P and rationing system R = (R1,R2,···,Rm), the
constrained demand set of bidder i ∈ M is given by
D
i(p,R
i) = {j ∈ N | R
i
j = 1 and V
i(j) − pj = max{V
i(h) − ph | R
i
h = 1}}.
Now we can adapt the classical notion of Walrasian equilibrium to the current model under
price rigidities.
Deﬁnition 2.1 A tuple (p∗,R∗,π∗) constitutes a constrained Walrasian equilibrium if
(i) π∗ is a feasible allocation, p∗ is an admissible price vector, and R∗ is a rationing
system;
(ii) π∗(i) ∈ Di(p∗,Ri∗) for i ∈ M;
(iii) p∗
j = p
j if π∗(i) 6= j for all i ∈ M;
(iv) p∗
j = ¯ pj and π∗(h) = j for some h ∈ M if Ri∗
j = 0 for some i ∈ M;
(v) j ∈ Di(p∗,Ri∗
−j) if Ri∗
j = 0.
Conditions (i) and (ii) need no explanation. Condition (iii) says that the price of an
unassigned item must be equal to its lower bound price. Condition (iv) states that a
bidder can be rationed on an item if the item is assigned to some other bidder and if its
price is on its upper bound. Condition (v) says that a bidder can be rationed on an item
only if without rationing on that item, the bidder will demand the item.
It is well-known (see e.g., Dr` eze (1975)) that an equilibrium allocation in a constrained
Walrasian equilibrium in economies with divisible goods is not eﬃcient when there are
binding rationings on purchase or sale. It is, however, easy to show by example that this
observation may not be true for economies with indivisible goods as studied here.
43 The dynamic auction
We now establish the existence of a constrained Walrasian equilibrium for the economy
under price rigidities.
Theorem 3.1 There exists at least one constrained Walrasian equilibrium in the model
under price rigidities.
We shall design a dynamic auction that can actually ﬁnd in a ﬁnite number of steps a
constrained Walrasian equilibrium in the economy. Roughly speaking, the auctioneer starts
the auction at the lower bound prices of the items being sold. Then the bidders respond
with their demand sets. The auctioneer accordingly eliminates sets of over-demanded items
by increasing their prices or by a lottery to determine a rationing system so that no price of
any item crosses its upper bound. The auction stops when there are no set over-demanded
items left, which gives a constrained equilibrium as will be shown below. As a result, this
yields a constructive proof of the above theorem.
A set of real items S ⊆ N \ {0} is over-demanded at a price vector p ∈ I R
N, if the
number of bidders who demand only items in S is strictly greater than the number of items
in S, i.e., |{i ∈ M | Di(p) ⊆ S}| > |S|. An over-demanded set S is said to be minimal if
no strict subset of S is an over-demanded set. Now we are ready to describe the dynamic
auction under price rigidities. Note that in the auction process, since the set of bidders
and the set of items are shrinking, the demand set of each bidder and the over-demanded
sets need to be adapted accordingly.
The Dynamic Auction under Price Rigidities
Step 1: The auctioneer announces the set of items N = {0,1,···,n} for sale and the
lower bound price vector p. The bidders, denoted by M = {1,···,m}, come to bid.
Let t := 0 and pt := p. Go to Step 2.
Step 2: The auctioneer asks every remaining bidder i to report his demand set Di(pt)
on the remaining items and checks whether there is any over-demanded set of items
at pt. If there is no over-demanded set of items, the auction stops. Otherwise,
5there is at least one over-demanded set. The auctioneer ﬁrst chooses a minimal over-
demanded set S of items and next checks whether the price of any item in the set
S has reached its upper bound. Let ¯ S := {j ∈ S | pt
j = ¯ pj}. If ¯ S is empty, the
auctioneer increases the price of each item in S by one unit and keeps the prices of
all other items unchanged. Let t := t+1 and return to Step 2. If ¯ S is not empty, go
to Step 3.
Step 3: The auctioneer picks an item at random from ¯ S and asks all bidders who
demand the item to draw lots for the right to buy the item. Then the (unique)
winning bidder gets the item by paying its current price and exits from the auction.
Delete this bidder from M and delete his won item from N. If M = ∅ or N = ∅, the
auction stops. Otherwise, let t := t + 1 and return to Step 2.
Before proving the convergence of the auction, we illustrate by example how the auction
actually operates.
Example 1: Suppose that there are ﬁve bidders (1, 2, 3, 4, 5) and four items (0, 1, 2,
3, 4) in a market. The lower and upper bound price vectors are p = (0,5,4,1,5), and
¯ p = (0,6,6,4,7). Bidders’ values are given in Table 1.
Table 1: Bidders’ values on each item.
Items 0 1 2 3 4
Bidder 1 0 4 3 5 7
Bidder 2 0 7 6 8 3
Bidder 3 0 5 5 7 7
Bidder 4 0 9 4 3 2
Bidder 5 0 6 2 4 10
The auction starts at the price vector p0 = (0,5,4,1,5). Then bidders report their
demand sets: D1(p0) = {3}, D2(p0) = {3}, D3(p0) = {3}, D4(p0) = {1} and D5(p0) =
{4}. The set S = {3} is a minimal over-demanded set and the auctioneer adjusts p0 to
p1 = (0,5,4,2,5). The demand sets and price vectors and other relevant data generated
by the auction are illustrated in Table 2. In Step 3, the price of item 3 has reached its
6upper bound 4. The auctioneer assigns randomly item 3, say, to bidder 2. So bidder 2
gets item 3 by paying 4 dollars and leaves the auction. Then we have M = {1,3,4,5}
and N = {0,1,2,4}. The auctioneer adjusts p3 to p4 = (0,5,4,5). In Step 6, there is no
over-demanded set of items at p6 and the auctioneer can assign item 2 to bidder 3, item 1
to bidder 4, and item 4 to bidder 5. In the end, bidder 1 gets no item and pays nothing;
bidder 2 gets item 3 and pays 4; bidder 3 gets item 2 and pays 4; Bidder 4 gets item 1
and pays 5; Bidder 5 gets item 4 and pays 7. Letting p∗ = (0,5,4,4,7), π∗ = (0,3,2,1,4),
R1∗ = (1,1,1,0,1), R2∗ = (1,1,1,1,1), R3∗ = (1,1,1,0,1), R4∗ = (1,1,1,1,1), and R5∗ =
(1,1,1,1,1), (p∗,π∗,R∗) is a constrained equilibrium.
Table 2: The data generated by the auction in Example 2.
Step Prices N M S D1(p) D2(p) D3(p) D4(p) D5(p)
0 (0,5,4,1,5) {0,1,2,3,4} {1,2,3,4,5} {3} {3} {3} {3} {1} {4}
1 (0,5,4,2,5) {0,1,2,3,4} {1,2,3,4,5} {3} {3} {3} {3} {1} {4}
2 (0,5,4,3,5) {0,1,2,3,4} {1,2,3,4,5} {3} {3,4} {3} {3} {1} {4}
3 (0,5,4,4,5) {0,1,2,3,4} {1,2,3,4,5} ∅ {4} {3} {3} {1} {4}
4 (0,5,4,5) {0,1,2,4} {1,3,4,5} {4} {4} {3} {4} {1} {4}
5 (0,5,4,6) {0,1,2,4} {1,3,4,5} {4} {4} {3} {2,4} {1} {4}
6 (0,5,4,7) {0,1,2,4} {1,3,4,5} ∅ {0,4} {3} {2} {1} {4}
We can establish the following ﬁnite convergence theorem for the auction.
Theorem 3.2 The dynamic auction ﬁnds a constrained equilibrium in a ﬁnite number
of steps.
Proof: It is clear that the auction will stop at some step t. Let p∗ and π∗ be the price
vector and the allocation at step t generated by the auction. We will show that there is a
rationing system R∗ such that (p∗,π∗,R∗) constitutes a constrained Walrasian equilibrium.
To achieve this, we construct an equivalent version of the dynamic auction that yields the
same allocation π∗ and price vector p∗ and in addition generates a rationing system R∗. At
7price vector p ∈ I R
N and rationing system R a set of real items S ⊆ N \ {0} is said to be
over-demanded if the number of bidders who given the rationing demand only items in S
is strictly greater than the number of items in S, i.e., |{i ∈ M | Di(p,Ri) ⊆ S}| > |S|. An
over-demanded set S is said to be minimal if no strict subset of S is an over-demanded set.
We can now describe the equivalent version of the dynamic auction under price rigidities.
The Equivalent Dynamic Auction under Price Rigidities
Step 1: The auctioneer announces the set of items N = {0,1,···,n} for sale and the
lower bound price vector p. The bidders, denoted by M = {1,···,m}, come to bid.
The auctioneer sets rationing system R with Ri
j = 1 for every i ∈ M and j ∈ N and
price p := p. The set W of winning bidders andO of sold items are both empty and
go to Step 2.
Step 2: The auctioneer asks every bidder i to report his demand set Di(p,R). The
auctioneer sets Ri
j = 0 for any bidder i 6∈ W and any j ∈ O∩Di(p,R), and asks such
bidder i to resubmit his demand set at p and this adjusted rationing system R. Then
the auctioneer checks whether there is any over-demanded set of goods at p and R.
If there is no over-demanded set of items, the auction stops. Otherwise, there is at
least one over-demanded set. The auctioneer chooses a minimal over-demanded set S
of items and checks whether the price of any item in the set S has reached its upper
bound. Let ¯ S := {j ∈ S | pj = ¯ pj}. If ¯ S is empty, the auctioneer increases in p the
price of each item in S by one unit and keeps the prices of all other items and the
rationing system R unchanged. Return to Step 2. If ¯ S is not empty, go to Step 3.
Step 3: The auctioneer picks an item j ∈ ¯ S at random and asks all bidders who
demand the item to draw lots for the right to buy the item. The seller sets Ri
j := 0
for every losing bidder i who demanded item j. The winning bidder and item j are
added to the sets W and O, respectively, and go to Step 2.
Clearly, both auctions produce the same price vector p∗ and the same allocation π∗. The
diﬀerence is that the second auction also produces a rationing system R∗. Following Roth
8and Sotomayer (1990, Section 8.3), it is not diﬃcult to verify that (p∗,π∗,R∗) is indeed a
constrained Walrasian equilibrium.
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9

A Dynamic Auction for Differentiated Items under Price Rigidities

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