%Paper: ewp-game/9805001
%From: Vijay Krishna <vkrishna@psu.edu>
%Date: Tue, 05 May 1998 13:48:33 -0400
%Date (revised): Mon, 26 Jul 1999 13:11:05 -0500 (CDT)


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\begin{document}

\author{Jean-Pierre Beno\^{i}t \\
%EndAName
New York University \and Vijay Krishna \\
%EndAName
Penn State University}
\title{Multiple-Object Auctions with Budget Constrained Bidders\thanks{%
We acknowledge research support from the C. V. Starr Center at NYU and the
National Science Foundation.}}
\date{May 22, 1999}
\maketitle

\begin{abstract}
A seller with two objects faces a group of bidders who are subject to budget
constraints. The objects have common values to all bidders, but need not be
identical and may be either complements or substitutes. In a simple complete
information setting we show: (1) if the objects are sold by means of a
sequence of open ascending auctions, then it is always optimal to sell the
more valuable object first; (2) the sequential auction yields more revenue
than the simultaneous ascending auction used recently by the FCC if the
discrepancy in the values is large, or if there are significant
complementarities; (3) a hybrid simultaneous-sequential form is revenue
superior to the sequential auction; and (4) budget constraints arise
endogenously.
\end{abstract}

\section{Introduction}

In the last few years governments in many parts of the world have
aggressively sought to privatize socially held assets. This wave of
privatization has included the sale of industrial enterprises in the former
Soviet Union and Eastern Europe, the sale of public transportation systems
in Britain and Scandinavia and the sale of radio spectra in the US. Both the
enormity and the obvious importance of these sales have naturally led to an
examination of how best to accomplish this task. Auctions have been
proposed, and have frequently been used, as time-honored, convenient and
attractive mechanisms for such sales. This, in turn, has led to a revival of
interest in the theory of auctions, particularly those involving multiple
objects.

The magnitude of these privatization sales has meant that in many instances
it is realistic to consider that buyers may run up against liquidity or
borrowing constraints. The presence of these financial constraints
introduces important differences into traditional auction theory. For
instance, Pitchik and Schotter (1986) and Che and Gale (1996, 1998) have
shown that when bidders are subject to such constraints, the conclusion of
the celebrated revenue equivalence theorem no longer holds.

When multiple objects are auctioned in the presence of budget constraints,
it may be advantageous for a bidder to bid aggressively on one object with a
view to raising the price paid by his rival and depleting his budget so that
the second object may then be obtained at a lower price. In effect, a bidder
may wish to ``raise a rival's costs'' in one market in order to gain
advantage in another. Thus, unlike in traditional auction theory, a
particular bidder's payoff is affected by the price paid by a rival bidder.%
\footnote{%
This strategic linkage across goods is emphasized by Pitchik and Schotter
(1986, 1988).} It is important to note that this effect comes into play only
when several objects are sold.

Budget constraints seem to have played a significant role in the auctions
for radio spectrum licenses conducted recently by the Federal Communications
Commission (FCC) (see McMillan (1994)). In particular, Salant (1997) writes
that the assessment of rival bidders' budget constraints was a primary
component of the pre-bidding preparation of GTE's bidding team. Moreover,
confirming that the kinds of considerations mentioned above played a
significant role, he describes the strategic advantages arising from bidding
for licenses of secondary importance so as to ``make rivals spend more on
some markets, leaving them with less to spend in other markets,'' (Salant
(1997), p. 561).

In this paper we study a simple model in which two objects are sold to a
group of financially constrained bidders.\footnote{%
In the first part of the paper we suppose that the constraints faced by the
bidders are exogenously given, possibly the result of liquidity or credit
constraints. Later, we show that budget constraints may arise endogenously .}
Our model is one of \emph{complete }information: the values of the objects
and the budgets facing the bidders are all commonly known to all
participants. Without budget constraints, this setting would be too trivial
to be of any interest. With budget constraints, however, this ``bare bones''
structure turns out to be sufficiently rich to capture and highlight a
number of interesting issues. The model is very close to that studied by
Pitchik and Schotter (1988). Their experiments confirm that the strategic
considerations introduced by budget constraints play an important role in
practice, as well as in theory.

\subparagraph{Sequential Auctions and the Order of Sale}

Suppose that two precious paintings are being auctioned sequentially, say in
two successive English auctions. In a complete information setting without
budget constraints, whether painting $A,$ the more valuable one, is sold
before or after painting $B$ does not affect the price that either fetches.
>From the perspective of the seller, the order of sale is irrelevant.
However, when buyers are financially constrained the order of sale can
affect the total revenue accruing to the seller. We show below that if
object $A$ is more valuable than object $B,$ the revenue derived from
selling them sequentially in the order $AB$ is at least as large as the
revenue from selling them in the order $BA$ (Proposition \ref{order} below).

\subparagraph{Simultaneous Auctions}

An alternative auction design is to sell the objects by means of two
simultaneous auctions. This was the design favored by the FCC in the
spectrum auctions mentioned above, but to the best of our knowledge its
theoretical properties have not been studied. In our simple setting we are
able to compare the revenue obtained from the simultaneous auction with the
optimal sequential auction when the objects are complements. We find that
the optimal sequential auction outperforms the simultaneous auction in terms
of revenue when (a) the values of the two objects are substantially
different; or when (b) there are significant complementarities (Proposition 
\ref{syn}).

We then introduce a hybrid auction form that combines essential elements
from the simultaneous and sequential forms. We show that this hybrid form is
revenue superior to the sequential form (Proposition \ref{hybrid}).

\subparagraph{Endogenous Budgets}

The FCC auction also serves to introduce the second main theme of this
paper: that budget constraints may arise \emph{endogenously }as a result of
rational calculation on the part of bidders.

Some of the larger bidders in the FCC auction consisted of consortia of
large telecommunication companies: WirelessCo (a consortium including Sprint
and the three largest cable companies) and PCS PrimeCo (a consortium
including NYNEX, Bell Atlantic and US West). Other participants, bidding as
individual companies (for example, GTE, Bell South and PacTel), were
themselves rather large local exchange carriers. In what sense were these
companies financially constrained? While some of the smaller companies could
be said to face constraints on borrowing or liquidity, this is less clear
for these larger bidders. Indeed, Salant (1997, p. 553) writes that the
bidding team first assessed the values of licenses that GTE was eligible to
bid on and the ``budget parameters \emph{determined} by GTE's management
were in part based on these valuations.'' [Emphasis added.] Thus it seems
that for the larger bidders the budget constraints were endogenously
determined. The companies gave their bidding teams instructions not to spend
more than a specified amount.

We study below a simple model in which budget constraints are endogenously
determined prior to the sale of the objects. In other words, the choice of a
budget is itself a strategic decision. Our main result is that budget
constraints\emph{\ always }arise : the resulting game has an (essentially)
unique equilibrium that involves constraints that bind (Proposition \ref
{endo}). Thus our model provides an explanation of how budget constrained
bidding may be the result of a conscious choice rather than the result of
exogenous factors like liquidity constraints or capital market imperfections.

\subparagraph{Miscellaneous Results}

When multiple objects are sold, budget constraints can have some
unanticipated consequences. For example, a reserve price can raise the
seller's revenue even though it is set at such a low level that it is never
binding in equilibrium. Similarly, there may be a trade-off between
efficiency and revenue even though the setting is one of complete
information. In a separate section, we illustrate some of these phenomena by
means of examples.

\subparagraph{Related Literature}

An early discussion of some issues concerning budget constraints and
auctions is contained in Rothkopf (1977). His analysis is not
game-theoretic, rather it focuses on how the computation of best responses
is affected by budget constraints.

Palfrey (1980) has studied the effects of budget constraints in a multiple
object setting with complete information. Specifically, he analyzes
sealed-bid discriminatory (``pay as you bid'') auctions and characterizes
equilibrium in the two object, two bidder case. He also points out
difficulties when the number of objects or bidders is larger. In particular,
equilibrium may not exist and when it does, may not be unique.

Engelbrecht-Wiggans (1987) studies the different forms that financial
constraints may take. Bidders may be constrained to spend no more than a
certain amount absolutely. Alternatively, they may be constrained to spend
no more than a certain amount on average. Engelbrecht-Wiggans (1987) studies
how these different forms affect the range of admissible strategies and
derives some equivalence relations.

Our work is closely related to two papers by Pitchik and Schotter (1986,
1988), which also analyze sequential auctions with budget-constrained
bidders. The basic setting and modes of reasoning are the same. However, we
extend their model and analysis in many directions. We allow for more than
two bidders and the possibility of synergies between the objects.\footnote{%
These extensions are not without consequences. In particular, as Pitchik and
Schotter (1986) themselves anticipate, some of their results are affected by
the presence of additional bidders. See Section 3 below for details.} We
also consider simultaneous open auctions. Another important difference is
that in our model (in Section 5 below) budgets may be determined
endogenously. As will become clear, our goals and the questions we ask are
also quite different. Pitchik and Schotter (1988), in particular, is focused
towards deriving testable predictions of equilibrium bidding behavior which
can then be measured against data from experiments.

Pitchik (1995) studies two-bidder sequential auctions of two objects when
there is incomplete information. Bidder types determine both valuations and
budgets. She also studies how the sequence of sale affects both the total
revenue and the prices of the individual objects. We discuss the
relationship of her results to ours in Sections 3 and 7 below.

Che and Gale (1996 and 1998) study single object auctions with budget
constraints under incomplete information. They show that even in the
traditional setting with independent private values and symmetric bidders,
revenue equivalence between various auction formats no longer holds. The
second paper, Che and Gale (1998), provides conditions under which the first
price auction outperforms a second price auction in terms of revenue. They
also show how the use of lotteries and all-pay auctions can benefit the
seller when bidders are budget constraint.

Lewis and Sappington (1998) consider agency problems when agents have budget
constraints, again showing that contracts that incorporate all-pay features
may be advantageous.

We emphasize that in the existing literature budget constraints are always
taken to be given exogenously.

The results described in the introduction are presented below. We have
relegated all formal proofs to an appendix.

\section{The Model and an Example}

There are $n$ bidders with budgets $y_{1}\geq y_{2}\geq y_{3}\geq ...\geq
y_{n}$ and two objects, $A$ and $B.$ The value derived from obtaining $A$
alone is $V^{A},$ while the value of $B$ alone is $V^{B}$, where $V^{A}\geq
V^{B}>0.$ The value of the two objects as a bundle is denoted by $V^{AB},$
which may be smaller or larger than $V^{A}+V^{B}.$ These known values are
common to the bidders. It is convenient to write: 
\[
V^{AB}=V^{A}+V^{B}+\alpha 
\]
where $\alpha $ is the \emph{synergy, }and this may be positive or negative
or zero. When the synergy is positive we say that the objects are \emph{%
complements}; when the synergy is negative the objects are \emph{substitutes}%
. We assume that $\alpha >-V^{B},$ so that the marginal value of either
object is always positive.

\subparagraph{An Example}

We begin with a simple example which illustrates the sorts of considerations
that arise in multiple object auctions with financially constrained bidders.
The example also shows that budget constraints may have an important impact
even when these constraints are relatively liberal.

Suppose that the objects are sold sequentially by means of two successive
English auctions and that there are three bidders with budgets $y_{1},$ $%
y_{2}$ and $y_{3},$ respectively. There are no synergies ($\alpha =0$), so
the value of obtaining both objects is simply the sum of the two values. In
the absence of any budget constraints each object would, of course, fetch
its full value in an English auction and the seller would receive a total
revenue of $V^{A}+V^{B}.$

Now consider the following parameter values:

\begin{example}
Values: $V^{A}=50,$ $V^{B}=40,V^{AB}=90.$ Budgets: $y_{1}=100,$ $y_{2}=80,$ $%
y_{3}=20$.
\end{example}

Observe that the budget constraints are fairly weak. In particular, bidder $%
1 $ is effectively unconstrained since $y_{1}>V^{A}+V^{B},$ so that $1$ can
afford both objects at prices that equal their respective values. Bidder $2$
is constrained but $y_{2}>V^{A}>V^{B}$, and thus $2$ can afford to buy
either object at a price equal to its value. Suppose that the objects are
sold in the order $A$ followed by $B.$

In the second auction it is a weakly dominant strategy for each bidder to
bid the minimum of his remaining budget and $V^{B}.$ If bidder $1$ wins
object $A$ in the first auction, then in the second auction both bidders $1$
and $2$ will have residual budgets that exceed $V^{B}.$ Thus, $B$ will then
sell for $V^{B}=40$ and the second auction will yield no surplus$.$
Therefore, it is dominated for bidder $2$ to drop out of the bidding for $A$
before the price reaches $50.$ If bidder $1$ wins $A$ for $50,$ his net
total gain will be $0.$ Bidder $1$ is better off letting $2$ win the first
object for (just below) $50$ and then winning the second object for (just
above) $(80-50)=30.$ The total revenue to the seller is $80$ and the full
value of the objects is not realized.

\subparagraph{Equilibrium concept.}

A word on the equilibrium concept that we use is in order. In the second
auction, we suppose that no bidder plays a weakly dominated strategy. Thus,
in every subgame each player bids up to the minimum of the value of the
object to him and his current budget. These outcomes are taken as given in
the first auction, and we look for an equilibrium in undominated strategies
in the resulting reduced game.

For formal reasons, it is necessary to assume that there is a smallest unit
of currency in which bids are made, say, one cent, although we do not keep
precise track and in our reasoning equate, say, \$5.99 with \$6. See Pitchik
and Schotter (1986, 1988) for a similar treatment.

\section{Sequential Open Auctions and the Order of Sale}

In this section we ask which order of sale is advantageous from the
perspective of the seller. It is instructive to begin with an example.

\begin{example}
\label{ord}Values: $V^{A}=50,$ $V^{B}=39,$ $V^{AB}=89.$ Budgets: $y_{1}=55,$ 
$y_{2}=30,$ and $y_{3}=20.$
\end{example}

Since $y_{3}=20,$ both objects sell for at least $20$, so that, in
particular, object $B$ cannot yield a surplus greater than $19.$ Thus, when
the objects are sold in the order $AB,$ bidder $1$ would be willing to bid
up to $p_{1}^{A}=31$ in the first auction whereas bidder $2$'s budget is
only $30.$ Bidder $1$ wins $A$ for $30,$ and then bidder $2$ wins $B\;$for $%
y_{1}-30=25.$ Total revenue in the order $AB$ is $R^{AB}=$ $55.$

In the order $BA,$ bidder $2$ must bid up to $p_{2}^{B}=25$ in the first
auction because if he drops out before that bidder $1$ will win both
objects. At a price of $25$, bidder $1$ will drop out of the bidding for
good $B$ and then go on to win good $A$ in the second auction for a price of 
$20$. Total revenue in the order $BA$ is $R^{BA}=45,$ and $R^{BA}<R^{AB}.$

Notice that in both orders bidder $1$ wins $A$ and bidder $2$ wins $B,$ and
so the allocation is the same. What matters, however, is that bidder $1$
wins the \emph{first} good in the order $AB$, whereas he wins the \emph{%
second} good in the order $BA$. The seller is generally better off when $1$
wins the first good. To see this, consider the simple two bidder case: $i)$ $%
V^{A}>V^{B}>y_{1}>y_{2}>0$ and $ii)$ $y_{1}<2y_{2}.$ Assumption $i)$ implies
that in the second auction a bidder will be willing to bid up to his budget
constraint, while $ii)$ implies that bidder $1$ does not have enough money
to win both goods and so each bidder wins one good. Suppose bidder $i$ wins
the first good for a price $p.$ The second good will then sell for $y_{i}-p$%
, so that total revenue will be $p+y_{i}-p=y_{i}.$ Since $y_{1}>y_{2}$ the
seller is better off if bidder $1$ wins the first good. Since, as one would
expect, bidder $1$ is more prone to win the first good in the order $AB$
than in the order $BA$, the order $AB$ is better.

In the previous example each bidder wins exactly one object in each order
since $y_{1}<2y_{2}.$ In the next example bidder $1$ has enough money to
outbid $2$ on both objects.

\begin{example}
\label{ord2}Values: $V^{A}=60,$ $V^{B}=40,$ $V^{AB}=112.$ Budgets: $%
y_{1}=70, $ $y_{2}=30,$ $y_{3}=5.$
\end{example}

In the order $AB$ bidder $1$ wins both objects, each for $30$, and gains a
net surplus of $52.$ The total revenue is $R^{AB}=60.$ In the order $BA$
bidder $1$ is willing to win $B$ for a price up to $27$. At that price, if $%
1 $ wins $B$ he will then go on to win $A$ for $30,$ yielding him a total
surplus of $55.$ On the other hand, if he drops out at $27$ he will win $A$
for a price of $5$, once again resulting in a surplus of $55.$ Since bidder $%
2$ will lose both auctions if he loses the first, $2$ is willing to bid up
to $30$ on $B$. However, $1$ has no incentive to push $2$ beyond a price of $%
27$ since, in any case, he will not win the second object $A$ for less than $%
y_{3}=5$.Thus bidder $1$ will drop out at $27$. Bidder wins $A$ for $27$ and
Bidder $1$ goes on to win $A$ for $5$. The total revenue is now $R^{BA}=32.$
Once again, the revenue from the order $AB$ is higher.

In this example bidder $1$ wins both objects in the order $AB$ and only $A$
in the order $BA.$ When he wins both objects each is sold for a price of $%
y_{2}$. When he wins only one, the price of at least one of the objects is
lower than $y_{2}.$ Thus the seller is better off when bidder $1$ wins both
objects. Again, it can be argued that bidder $1$ is more prone to win both
when the order is $AB$ rather than $BA.$

Our first result shows that Examples \ref{ord} and \ref{ord2} are instances
of a general phenomenon.\footnote{%
For degenerate parameter values there may be more than one equilibrium
revenue in a given order (for instance, $y_{1}=30,y_{2}=10,V^{A}=20,V^{B}=10$
with the order $AB$). We ignore these cases.}

\begin{proposition}
\label{order}The revenue to the seller from selling the objects sequentially
in the order $AB$ is at least as great as the revenue from selling them in
the order $BA.$ For some parameter values, the inequality is strict.
\end{proposition}

In a related incomplete information setting, Pitchik (1995) also examines
how the seller's revenue is affected by the order of sale. In her model,
each bidder is identified by a one-dimensional ``type'' and both the
valuations and the budgets are affine functions of a bidder's type. In
addition, budgets are assumed to be at ``intermediate'' levels so that in
equilibrium each bidder wins exactly one object for all realizations of the
types. Pitchik (1995) then shows that the optimal order of sale depends on
whether the ``de facto valuation'' for the first object is an increasing or
decreasing function of the type. Her results are not directly comparable to
Proposition 1 because hers is a private value setting and it is not the case
that a particular object, say $A$, is considered more valuable by all types.

More closely related is a result of Pitchik and Schotter (1986) and it is
worthwhile to compare this to Proposition 1. Let $p^{A}$\ and $q^{B}$\ be
the prices of $A$\ and $B$, respectively, when the order of sale is $AB.$\
Similarly, let $p^{B}$\ and $q^{A}$\ be the prices in the order $BA.$\
According to Proposition 1, $p^{A}+q^{B}\geq p^{B}+q^{A}.$\ Pitchik and
Schotter (1986) show that when there are only two bidders, the price of an
object is higher the earlier it is sold in the sequence, that is, $p^{A}\geq
q^{A}$\ and $p^{B}\geq q^{B}.$\ This result, however, does not hold
generally when there are three or more bidders.\footnote{%
Suppose $V^{A}=100,$ $V^{B}=18;$ and budgets are $y_{1}=50,$ $y_{2}=20,$ $%
y_{3}=4.$ In the order $AB,$ prices are $p^{A}=20$ and $q^{B}=18.$ In the
order $BA,$ prices are $p^{B}=16$ and $q^{A}=4.$ Thus $p^{B}<q^{B}.$}

What factors determine the order of sale in real-world auctions? There is
some ``psychological'' intuition that suggests that it may be advantageous
to ``warm up the room'' with some lower valued objects before bringing the
auction to a climax with the more valuable masterpiece. But psychology also
suggests that it may be preferable to sell more valuable items first in
order to establish ``lively'' bidding for the rest of the items (see Cassady
(1967, pp. 84-85) for a discussion of these tactics.) Budget constraints, on
the other hand, introduce pure economic considerations of the kind
identified in this paper. Auction houses tend to balance these various
effects.\footnote{%
In private communication (Sotheby's (1998)) a representative of Sotheby's
related to us various factors that they consider in determining the order of
sale. In particular, budget issues were explicitly mentioned. Both of the
behavioral concerns mentioned above also played a role. In some instances,
there was a natural historical order associated with the items (paintings
from an early period versus a later period) which dominated all other
considerations.} As a concrete example, in an auction of eight paintings by
Paul C\'{e}zanne conducted by the auction house of Sotheby's in November
1997, the prices of the paintings, in order of sale, were as follows:

\[
\begin{tabular}{ccccccccc}
\multicolumn{9}{c}{$\text{\textsc{C\'{e}zanne Auction}}$} \\ 
Painting & $A$ & $B$ & $C$ & $D$ & $E$ & $F$ & $G$ & $H$ \\ 
Price (in \$ millions) & \$0.3 & \$1.0 & \$5.5 & \$1.1 & \$3.6 & \$5.9 & 
\$1.0 & \$0.7
\end{tabular}
\]

Assuming that more valuable paintings go for higher prices, this does not
reveal an overall monotonic pattern, although the values do decline at the
end.\footnote{%
Proposition \ref{order} concerns two object auctions. Section 6 below
contains an example (Example \ref{3good}) of an auction with more than two
objects.} This is not atypical of the results from sales at Sotheby's (see
Cassady (1967, pp. 295-298), for other examples).

\section{Simultaneous Ascending Auction}

In the recent auction of radio spectrum licenses the FCC adopted a novel
auction design based on the recommendation of a group of economists. The
licenses were sold using a \emph{simultaneous ascending auction}. As in an
English auction, there were multiple rounds of bidding that continued until
no bidder wished to raise the bid. All auctions were conducted
simultaneously and bidding on all objects remained open as long as there was
bidding activity on any one of them.

A primary reason for adopting this design was the presence of positive
synergies between licenses in adjoining areas. While combination bids, that
is, bids on bundles of objects were not allowed, it was felt that the
simultaneous nature of the auction would allow bidders to assemble bundles
that exploited the synergies in an efficient manner.\footnote{%
See McMillan et al. (1997) and Cramton (1997) for both a description and the
rationale underlying the choice of the auction rules. Ausubel et al. (1997)
use bidding data from the auction to confirm the presence of positive
geographic synergies.}

In this section we study a simple version of the simultaneous ascending
auction when budget constraints are present and there are positive synergies
between the objects ($\alpha \geq 0$). Specifically, we consider a
simultaneous auction with the following rules:

\begin{enumerate}
\item  There are multiple rounds of bidding conducted in the open.

\item  In each round for \emph{every }object, a bidder either

\begin{enumerate}
\item  raises the previous high bid on the object; or

\item  does not announce a bid on the object.
\end{enumerate}

\item  At no time can the total of a bidder's outstanding high bids exceed
his budget.

\item  The auction continues until, in two successive rounds \emph{no}
object has had its bid raised.
\end{enumerate}

Note that with these rules, a bidder who has been outbid on, say, object $A$
can first pursue the bidding on $B$ before deciding whether to continue
further bidding on $A$. Allowing for this possibility was an important
feature of the FCC auction.\footnote{%
The FCC\ auction had an ``activity rule'' which precluded bidders from
remaining dormant for too long. Incorporating such a rule would not change
the results of this section. Indeed, as can be seen from the proofs, these
results are robust to many rule modifications.}

In the simultaneous auction, we look for Nash equilibria once (weakly)
dominated strategies have been \emph{iteratively} removed.

In a simple two-object setting with complementarities and budget
constraints, we compare the performance, in terms of both revenue and
efficiency, of the simultaneous format outlined above with that of the
sequential format of the previous section. While the equilibrium of the
sequential auction is almost always unique, it is not surprising that the
simultaneous auction often has multiple equilibria. Nevertheless, in some
circumstances a ranking of the auction methods is possible.

We begin with an example.

\begin{example}
\label{simula}Values: $V^{A}=20,$ $V^{B}=10,$ $V^{AB}=30.$ Budgets: $%
y_{1}=25,$ $y_{2}=6,$ $y_{3}=1.$
\end{example}

In the sequential $AB$ auction bidder $1$ wins both $A$ and $B$, each at a
price of $6.$ The total revenue from the sequential auction $R^{AB}=12$.

One equilibrium of the sequential auction has the following bidding behavior
(where $\left( b_{i}^{A},b_{i}^{B}\right) $ denotes the bids of bidder $i$
on $A$ and $B$):

\emph{Round I}: $1$ bids $\left( 1.01,5.99\right) $; 2 bids $\left(
0,6\right) $; 3 bids $\left( 1,0\right) $

No further bids are placed.

Notice that bidder $3$ will always bid the price of both goods up to $1$ so
that bidder $1$ can never win $A$ for less than $1$. In the equilibrium,
bidder $1$ wins $A$ for (approximately) $1,$ bidder $2$ wins $B$ for
(approximately) $6$. Total revenue $R^{sim}$ is (approximately) $7$.

Now, observe that any strategy of bidder $1$ that ever lets bidder $2$ win
object $A\ $is dominated, since $1$ can always win $A$ for a surplus of at
least $14.$ If the price of $A$ rises above $2$, it is dominated for $1$ to
use a strategy that results in him winning only $A$, since he can always
obtain both objects for a surplus of at least $18$. Given this, it is
(iteratively) dominated for bidder $2$ to bid above $2$ on $A$ in the first
round. Neither $1$ nor $2$ will ever bid more than $2$ on object $A,$ and
the revenue in the $AB$ auction is greater than the revenue in \emph{every}
equilibrium of the simultaneous auction.

At this point, a few words on rule \#3 of the simultaneous auction,
preventing a bidder from exceeding his budget, may be in order. At first
blush, it might appear that this rigs things against the simultaneous
auction. After all, in the equilibrium of the sequential $AB$ auction in
Example \ref{simula},\ bidder $2$ ends up bidding his entire budget on both
goods and so, in a sense, exceeds his budget. Note, however, that this type
of behavior is also possible in the simultaneous auction, and in exactly the
same manner. For instance, (out of equilibrium) bidder $2$ could open with a
bid of $6$ on good $A,$ be outbid on this good by $1,$ and then bid $6$ on
good $B$. Note also that in the simultaneous auction bidder $2$ never bids
above $2$ on good $A,$ an amount far below his budget. Finally, the next
example will assuage any lingering doubts as to whether our formulation rigs
things against the simultaneous auction -- for some parameter values some
equilibria of the simultaneous auction are revenue superior.

\begin{example}
\label{simalpha}Values: $V^{A}=9,$ $V^{B}=7,$ $V^{AB}=16+\alpha $ where $%
\alpha >0.$ Budgets: $y_{1}=25,$ $y_{2}=6$.
\end{example}

When $0<\alpha <3$, in the sequential $AB$ auction bidder $2$ wins $A$ for $%
6,$ $1$ wins $B$ for $0$ and $R^{AB}=6.$ In the simultaneous auction one
equilibrium has bidding behavior:

\emph{Round I}: 2 bids $\left( 4,1.99-\frac{2}{3}\alpha \right) $

\emph{Round II}: 1 bids $\left( 4.99+\frac{1}{3}\alpha ,2-\frac{2}{3}\alpha
\right) $

\emph{Round III}: 2 bids $\left( 5+\frac{1}{3}\alpha ,\cdot \right) $

No further bids are placed. Bidder $1$ wins $B$ for $2-\frac{2}{3}\alpha ,$ $%
2$ wins $A$ for $5+\frac{1}{3}\alpha $ and the total revenue $R^{sim}=7-%
\frac{1}{3}\alpha >R^{AB}$.

These bids are supported in the following manner. Following $2$'s opening
bid, in \emph{Round II} : $i)$ $1$ bids $\left( 4.99+\frac{1}{3}\alpha ,2-%
\frac{2}{3}\alpha \right) $ if $2$ opened with $\left(
b_{2}^{A},b_{2}^{B}\right) =\left( 4,1.99-\frac{2}{3}\alpha \right) $ in 
\emph{Round I}, otherwise $ii)$ $1$ bids $\left( 5.99,b_{2}^{B}+0.01\right) $
if $V^{A}-b_{2}^{A}<V^{B}-b_{2}^{B}$, and bids $\left(
b_{1}^{A}+0.01,5.99\right) $ if $V^{A}-b_{2}^{A}\geq V^{B}-b_{2}^{B}$.
Bidder $2$'s strategy in \emph{Round III} is to just outbid bidder $1$ on
whichever good is higher priced. Notice that attempting to win the lower
priced good would cause $2$ to lose both goods. Notice also that for any
opening bid of player $2,$ these continuation strategies yield $1$ the
greatest surplus he could possibly obtain, given that $2$ would not let him
win both goods for less than $12.$

When $3<\alpha <5$, in the sequential $AB$ auction bidder $1$ wins both
goods and revenue $R^{AB}=12.$ In the simultaneous auction, since $1$ can
guarantee himself a surplus of at least $4+\alpha $ by winning both goods,
it is (iteratively) dominated for $2$ to open with a bid above $5-\alpha $
on good $A$, as this will induce $1$ to take both goods. Bidder $1$
recognizes this, and neither bidder will ever bid above $5-\alpha $ on good $%
A$. In any equilibrium of the simultaneous auction, bidder $1$ wins only $A$
for a price $p\leq 5-\alpha $. Thus, all equilibria of the simultaneous
auction have a total revenue of at most $11-\alpha <$ $R^{AB}$.

When $\alpha \geq 5$ bidder $1$ wins both objects in either auction and the
two yield the same revenue.

Thus, whereas some equilibria of the simultaneous auction are better than
the equilibrium of the sequential auction when the synergies are small, this
is no longer true when the synergy increases (in fact, all equilibria of the
simultaneous auction are strictly inferior when $3<\alpha <5$).

Since $V^{AB}>V^{A}+V^{B}$ an allocation is efficient only if both objects
go to the same bidder. Thus, when $3<\alpha <5$ the sequential $AB$ is
efficient whereas the simultaneous auction is not.

As noted earlier, one motivation for adopting the simultaneous auction was
the feeling that it would be efficient in the presence of positive
synergies. Example \ref{simalpha} casts some doubt on this motivation.%
\footnote{%
It should be emphasized, however, that there is complete information in our
setting, whereas in the FCC auction the designers had to take into account
incomplete information.}

The reasoning in Examples \ref{simula} and \ref{simalpha} is general and it
can be shown that:

\begin{itemize}
\item  If bidder $1$ wins only object $A$ or wins both $A$ and $B$ in the
sequential $AB$ auction, then the revenue from the sequential $AB$ auction
is at least as great as the revenue from any equilibrium of the simultaneous
auction. (See Lemma \ref{simA} in the Appendix.)
\end{itemize}

The sufficient condition stated above concerns the equilibrium of the
sequential $AB$ auction and thus is not in terms of the primitives of the
model. The next proposition restates this in terms of the primitives. This
proposition follows from the above bullet point, since bidder $1$ will want
to win good $A$ when $V^{A}$ is large compared to $V^{B}$. Similarly, when
the synergy $\alpha $ is large bidder $1$ will want to win both objects.

\begin{proposition}
\label{syn}Suppose that $A$ and $B$ are complements. The revenue from the
sequential $AB$ auction is no less than, and sometimes strictly greater
than, that in any equilibrium of the simultaneous auction if either (a) $%
V^{A}$ is large relative to $V^{B};$ or (b) the synergy parameter $\alpha $
is large enough.
\end{proposition}

\subsection{A Hybrid Form}

Returning to Example \ref{simula}, notice that the simultaneous auction
results in each bidder winning a single object: bidder $1$ wins $A$ and
bidder $2$ wins $B.$ This is the same as the equilibrium outcome of the
sequential $BA$ auction. In effect, the simultaneous format allows the
bidders to allocate the objects among themselves in the more favorable (from
their perspective) $BA$ order.

Now consider a modification of the simultaneous auction with the following
rules.

Rules 1 to 4 of the simultaneous format apply and in addition:

\begin{enumerate}
\item[5.]  If there are two successive rounds with no increments in the
bids, object $A$ is sold to the current highest bidder and then the auction
on $B$ continues.
\end{enumerate}

This modification accomplishes two things. First, in a sense it ultimately
imposes the order $AB$ upon the bidders. Second, it prevents collusive
equilibria where, for instance, in the first round $1$ bids one cent on good 
$A,$ $2$ bids one cent on good $B,$ and they ``agree'' not to place any
further bids, where the agreement is supported by threats to resume
``normal'' bidding if either player bids in the second round. We will refer
to the auction using rules 1 to 5 as the \emph{hybrid }auction\emph{\ }since
it has elements of both the sequential and the simultaneous formats.

Let us return to Example \ref{simula} under the hybrid rules. First, for the
same reasons as in the simultaneous auction, bidder $1$ will win $A.$ Once $%
A $ is sold, however, now the auction on $B$ re-opens and its price will be
bid up to $6.$ Anticipating this, the players will bid the price of $A$ up
to $6$ also. Thus, in this example, the unique equilibrium under the hybrid
rules mimics the equilibrium of the $AB$ auction.

At this stage it may appear that the hybrid auction is revenue equivalent to
the $AB$ auction. The following example (which is just Example \ref{simalpha}
when $\alpha =1$) shows that, in fact, it may out-perform the sequential
auction.

\begin{example}
\label{simulb}Values: $V^{A}=9,$ $V^{B}=7,$ $V^{AB}=17.$ Budgets: $y_{1}=25,$
$y_{2}=6$.
\end{example}

In the sequential $AB$ auction bidder $2$ wins $A$ for $6$ and $1$ wins $B$
for $0.$ The total revenue from the sequential auction $R^{AB}=6.$

One equilibrium of the hybrid auction has bidding behavior:

\emph{Round I}: $1$ bids $\left( 5.99,2\right) ;$ $2$ bids $\left(
4,1.99\right) .$

\emph{Round II}: $2$ bids $\left( 6,\cdot \right) $.

No further bids are placed. $A$ is sold to bidder $2$ for $6,$ $B$ is sold
to bidder $1$ for $2.$ The total revenue from this equilibrium of the hybrid
auction $R^{hyb}=8>R^{AB}.$

This equilibrium is, in fact, revenue maximal. In every equilibrium $1$ wins
good $B,$ since he has more money than $2$ once the $B$ auction re-opens.
Since $2$ can never win $B$, he can be driven up to a price of $6$ on $A,$
so that $1$ will never let $2$ win $A$ for less than $6.$ (Thus, $2$ has no
strategy that earns him a surplus greater than $2$ and his strategy in the
above equilibrium is undominated)$.$ It is (iteratively) dominated for
bidder $2$ to bid more than $2$ on $B,$ since bidder $1$ would then win both
objects for a surplus of $5$, rather than win only $B.$ Bidder $1,$
recognizing this, will also not bid more than $2$ on good $B.$ In every
equilibrium of the hybrid game, $2$ wins $A$ for $6$ and $1$ wins $B$ for at
most $2$.

The above equilibrium yields more than the equilibrium of the sequential
auction. In fact, \emph{every} equilibrium of the hybrid auction yields at
least as much as the equilibrium of the sequential auction.\footnote{%
Note that this result depends upon the fact that it is the auction for $B$
that reopens, and not the auction for $A$. Suppose $B$ sells first and the
auction for $A$ then reopens. In Example 4, every equilibria of this
modified auction is strictly worse than the sequential auction.}

The improvement the hybrid displays in Examples \ref{simula} and \ref{simulb}
is completely general.

\begin{proposition}
\label{hybrid}Suppose that $A\;$and $B$ are complements. The revenue from
the hybrid auction is at least as great as that from the sequential $AB$
auction.
\end{proposition}

\subparagraph{Efficiency}

The hybrid auction dominates the sequential auction in terms of revenue.%
\footnote{%
Proposition $2$ implies that when $V^{A}$ is large relative to $V^{B}$ the
hybrid is superior to the simultaneous auction. When $V^{A}$ is small
relative to $V^{B}$ some equilibria of the simultaneous auction are worse
than all the equilibria of the hybrid auction, but the sets may also overlap.%
} We now turn to the question of efficiency.

Recall that since we are working in a model with common values the only
possible inefficiency is that each bidder is allocated one object when there
are strictly positive synergies.

First, suppose that in the sequential auction the allocation is efficient,
that is, both objects are allocated to bidder $1.$ Then the allocation in
the hybrid auction is also efficient (this follows from Proposition \ref
{hybrid}). Second, suppose that in the sequential auction bidder $1$ wins
only $A.$ Then it must be that bidder $1$'s budget is insufficient to also
win $B$. Under these circumstances, in the hybrid auction bidder $2$ will
also not let bidder $1$ win both objects. Finally, suppose that in the
sequential auction bidder $1$ wins only $B.$ Then in the hybrid auction, as
in Example \ref{simulb}, it is dominated for bidder $2$ to raise the price
of $B$ so much that bidder $1$ is better off obtaining both objects and in
every equilibrium of the hybrid auction each bidder obtains one object, as
in the sequential auction.

Thus the hybrid auction is equivalent to the sequential auction in terms of
allocative efficiency.

\section{Endogenous Budget Constraints}

In this section, we consider a model where the budgets are endogenously
determined rather than exogenously given. Suppose there are only two bidders
and the objects are sold sequentially in some fixed order, either $AB$ or $%
BA $, that is announced in advance. Prior to the auction, the bidders
simultaneously choose budgets $y_{1}$ and $y_{2}\ $which are commonly known
prior to the auction and remain fixed.

As a first step, consider a situation in which only a \emph{single} object
with common value $V$ is to be sold. Observe that there cannot be an
equilibrium pair of budgets $(\overline{y}_{i},\overline{y}_{j})$ such that $%
\overline{y}_{i}<V$ and $\overline{y}_{j}<V.$ Furthermore, it is dominated
for any bidder to choose a budget $y_{i}<V.$ With a single object,
commitments to ``small'' budgets cannot arise endogenously. Now consider the
following example:

\begin{example}
\label{endoexam}Values: $V^{A}=10,V^{B}=6,V^{AB}=16.$
\end{example}

In the first stage, bidders simultaneously choose their budgets. Next, the
objects are sold sequentially in the order $AB.$ First, observe that in
equilibrium both bidders cannot come with budgets exceeding $V^{A}+V^{B}=16$%
. With budgets $\overline{y}_{1}\geq 16$ and $\overline{y}_{2}\geq 16,$
neither bidder gets any surplus. If bidder $2$ were instead to choose $%
y_{2}=9.98,$ however, bidder $1$ would let $2$ win object $A$ for $9.98$ and
then win $B$ for free. Thus, if bidder $1$ has a large budget, bidder $2$
prefers to come with a ``small'' budget. It may be verified that it is an
equilibrium for the bidders to choose (small) budgets of $\overline{y}%
_{1}=11 $ and $\overline{y}_{2}=5$ and for bidder $2$ to then win the first
object for $5$ and bidder $1$ the second for free. The choice of a budget of 
$5$ by $2$ keeps bidder $1$ indifferent between winning $B$ and winning both
objects, since either outcome results in a surplus of $6.$ Given $1$'s
budget of $11,$ bidder $2$ can do no better than the surplus of $5$ he gets
from a budget of $5.$

Our main finding is that with \emph{multiple} objects, in \emph{every }%
equilibrium at least one bidder chooses to be budget constrained. Moreover,
equilibrium payoffs are essentially unique and each good goes for less than
its value. With multiple objects, commitments to ``small'' budgets arise
endogenously. Note that bidders choose small budgets even though there are
no additional transaction costs associated with choosing a larger budget.
The latter may be the case, for instance, if a bidder had to obtain a line
of credit from a bank and more liberal letters of credit had higher costs
associated with them.

In a budget constrained equilibrium, at least one bidder cannot afford to
buy both goods at their full value. We call an equilibrium \emph{strongly
budget constrained }if at least one bidder has a budget strictly less than $%
V^{A}.$

It is clear that if there are strong positive synergies ($\alpha $ is very
large), the two object auction reduces to a ``single'' object auction in
which the bundle $AB$ is for sale. Thus, in what follows, we assume that $%
\alpha <V^{B}.$ Our main result is:

\begin{proposition}
\label{endo} The two-bidder game with endogenous budgets has a pure strategy
equilibrium. Every pure strategy equilibrium is strongly budget constrained
and all the equilibria are payoff equivalent (except for a relabelling of
the bidders). Each good sells for less than its value.
\end{proposition}

In order to gain some intuition as to why equilibria are strongly budget
constrained, suppose that the order of sale is $AB$ and consider the special
case where $\alpha =0.$ In any equilibrium of the game with endogenous
budgets, say $\left( \overline{y}_{i},\overline{y}_{j}\right) ,$ each bidder
must be getting a positive surplus. This is because if bidder $j$, say, were
to choose a budget of $y_{j}=V^{A}-\varepsilon $ one of two things may
happen. If $y_{j}>\overline{y}_{i},$ then certainly $j$ will get a positive
surplus. If $y_{j}\leq \overline{y}_{i},$ (at worst) bidder $i$ will let
bidder $j$ win good $A$ for a price of $V^{A}-\varepsilon $ in order to
obtain good $B$ for free (as in Example \ref{endoexam}), since this is
better for $i$ than either winning $A$ for $V^{A}-\varepsilon $, or winning
both objects. Hence, in equilibrium, each bidder has a positive surplus and
thus must be winning exactly one object. This means that at least one bidder
has a budget smaller than $V^{A}+V^{B}.$ Lemmas \ref{endosub} and \ref
{endocomp} in the Appendix provide an exact characterization of the
equilibrium and show that, in fact, at least one bidder is strongly\emph{\ }%
budget constrained. Proposition \ref{endo} then follows immediately.

Proposition \ref{order} shows that in the game with endogenous budgets if
the seller cannot commit to an order of sale prior to the choice of budgets,
the optimal order is $AB.$ By using Lemmas \ref{endosub} and \ref{endocomp}
in the Appendix it is easy to verify that the same is true even if the
seller commits to an order of sale prior to the choice of budgets.

\subparagraph{Flexible budgets}

We emphasize that Proposition \ref{endo} does not presuppose that the
budgets are binding commitments on the part of the bidders. To see this,
suppose that the participants in the auction are firms which send
representatives with fixed budgets to bid on their behalf. Let each firm
supply its agent with a cellular phone so that, for the nominal price of a
phone call, the agent can call for extra funds at any time. Then the
representatives' initial budgets are flexible and can be relaxed at any
time. Nevertheless, Proposition \ref{endo} continues to hold; the phones
have no effect on the equilibrium outcome. Consider Example \ref{endoexam}.
As discussed above, one equilibrium involves the firms dispatching their
representatives to the auction with budgets of 11 and 5. The presence of
cellular phones does not affect this (or any other) equilibrium. Given the
equilibrium budgets and the (small) cost of obtaining additional funds,
neither representative will find it profitable to call for more money after
the first auction is over, or at any time during the two auctions. This
reasoning is general and is a consequence of the precise equilibrium budgets
derived in Lemmas \ref{endosub} and \ref{endocomp}.

Suppose there are three bidders instead of two. Then there are still
strongly budget constrained equilibria, although the uniqueness result does
not hold.

As an example consider the case where $\frac{1}{2}V^{A}<V^{B}$ and $%
V^{AB}=V^{A}+V^{B}.$ Suppose the order of sale is $AB.$ It may be verified
that budget choices satisfying $\overline{y}_{1}\geq V^{A}+V^{B},$ $%
\overline{y}_{2}=V^{A}-\left( V^{B}-\frac{1}{2}V^{A}\right) ,$ and $%
\overline{y}_{3}=\frac{1}{2}V^{A}$ constitute an equilibrium in undominated
strategies. In the subgame following the budget selections, bidder $2$ wins
object $A$ for $\overline{y}_{2}$ and bidder $1$ wins object \ $B$ for $%
\overline{y}_{3}$. However, there are also other undominated equilibria. For
example, budgets satisfying $\overline{y}_{1}=\overline{y}_{2}=\overline{y}%
_{3}\geq V^{A}+V^{B}$ constitute an equilibrium as well.

\section{Miscellany}

In this section we highlight, by means of a series of examples, some
interesting phenomena that arise in multiple-object budget constrained
auctions.

\subparagraph{Revenue-enhancing constraints}

Suppose there are two bidders and $A$ and $B$ are sold in the order $AB.$
Bidder $1$ values $A$ at $14$ and $B$ at $3.$ Bidder $2$ values $A$ at $8$
and $B$ at $3.$ Thus, we have:

\begin{example}
\label{beneficial}Values: $V_{1}^{A}=14,$ $V_{2}^{A}=8;V_{1}^{B}=V_{2}^{B}=3.
$ Budgets: $y_{1}=13,$ $y_{2}=11.$
\end{example}

Observe that for the purposes of this example we are departing from our
assumption that bidders have common values. In the absence of any budget
constraints $A$ sells for a price of $8$ and $B$ sells for $3.$ The total
revenue to the seller is $11.$

Now suppose that $y_{1}=13$ and $y_{2}=11.$ Bidder $1$ wants to win $A$ at
prices below $11.$ Bidder $2$ wants to win $A$ at prices below $8,$ but will
force the price of $A$ up to $11$ in order to deplete $1$'s budget. This
leaves bidder $1$ with a remaining budget of $2$ so that $B$ sells for $2.$
Total revenue is now $13.$

Thus, in this example the seller actually \emph{benefits} from the fact that
the bidders are financially constrained, a phenomenon that is impossible
when there is only a single object.

Even with multiple objects, budget constraints cannot increase revenue in a
common values setting, since then revenue in the unconstrained case is $%
V^{A}+V^{B}$, which is as large as possible given that no bidder will buy an
object for more than he values it (with no synergies). When the buyers have
differing valuations, however, the possibility of revenue-enhancing
constraints arises. We now give some simple sufficient conditions. These are
essentially that the bidders' valuations of $B$ are similar but that their
valuations of $A$ differ, that the buyer's incomes are close to each other,
and that one player is constrained with respect to his valuations.

To capture these conditions simply, suppose that $V_{1}^{B}=V_{2}^{B}\equiv
V^{B}$, $V_{1}^{A}>V_{2}^{A},$ $y_{1}=y_{2}\equiv y$, and $%
V_{1}^{A}+V^{B}>y>V_{2}^{A}+V^{B}$. Note that were budgets unconstrained ($%
y>V_{1}^{A}+V^{B}$), the total revenue would be $V_{2}^{A}+V^{B}$. With the
given budgets, however, bidder $1$ wins $A$ for $\min \left\{ \frac{1}{2}%
\left( V_{1}^{A}-V^{B}+y\right) ,y\right\} $, bidder $2$ wins $B$ for $\max
\left\{ \frac{1}{2}\left( y-\left( V_{1}^{A}-V^{B}\right) \right) ,0\right\}
,$ and the total revenue is $y>V_{2}^{A}+V^{B}$.

\subparagraph{Efficiency versus revenue}

Our next example illustrates that in determining the order of sale there may
be a trade-off between revenue and efficiency. Again, we depart from our
basic model and suppose that bidders have private values.\footnote{%
In a single object auction, if the bidder who values the object the most is
budget constrained, the equilibrium allocation may be inefficient. See
Maskin (1992) for a discussion of this issue and some remedies.}

\begin{example}
Values: $V_{1}^{A}=22$ , $V_{1}^{B}=7;$ $V_{2}^{A}=20$, $V_{2}^{B}=8.$
Budgets: $y_{1}=100,$ $y_{2}=18$ $y_{3}=2$.
\end{example}

Efficiency dictates that object $A$ go to bidder $2$ and object $B$ to
bidder $1.$

First, suppose the objects are auctioned in the order $AB.$ Then bidder $2$
wins $A$ for $18$ and bidder $1$ wins $B$ for $2.$ The total revenue to the
seller is $20.$ The allocation is inefficient even though the environment is
one of complete information.

Next, suppose the order of sale is $BA.$ Then bidder $2$ wins $B$ for $8.$
Bidder $1$ wins $A$ for $10.$ While the total revenue is only $18,$ the
allocation is efficient.

Thus the order $AB$ results in a higher revenue but the allocation is
inefficient. In the $BA\;$order, the situation is reversed.

\subparagraph{Reserve prices}

Reserve prices are frequently employed in auctions. Two straightforward
explanations for their usage are $i)$ it may be that the seller derives a
positive value from the object himself and the reserve price guarantees that
it will not be sold below this value and $ii)$ a reserve price can enhance
revenues in circumstances where the reserve price is above second-highest
valuation but the below highest.

In our setting a small reserve price may help the seller even if the seller
has no value for object and even though the reserve price is below the
valuations of both bidders.

\begin{example}
\label{res}Values: $V^{A}=10,$ $V^{B}=4,$ $V^{AB}=14.$ Budgets: $y_{1}=9,$ $%
y_{2}=6.5$
\end{example}

The objects are sold in the order $AB.$ Suppose that the seller imposes a
minimum selling price of $r=1$ on $B$ (or on both $A$ and $B$). Since $B$
then yields a surplus of at most $3,$ bidder $1$ will win $A$ at a price of $%
6.5.$ Bidder $2$ then wins $B$ for $y_{1}-6.5=2.5.$ Thus, the equilibrium
prices are $p^{A}=6.5$ and $p^{B}=2.5,$ respectively. The total revenue with
a reserve price of $r=1$ is $9.$

Note that in equilibrium the reserve price is not binding. An observer
presented with the results of this auction would be tempted to conclude that
the imposition of a reserve price had proved irrelevant. Consider, however,
the auction without the reserve price. In equilibrium, bidder $2$ wins $A$
for $6.5$ and bidder $1$ wins $B$ for free. Total revenue is $6.5.$ Far from
being irrelevant, the reserve price increases revenue by an amount greater
than the reserve price itself!

Generally speaking, when there is uncertainty about the nature of the
bidders, a reserve price increases revenues in some instances at the cost of
sometimes preventing a beneficial trade from occurring. Example \ref{res}
shows that when there are budget constraints this problem may be mitigated
by the fact that a small reserve price can have a large effect on revenue.

A\ non-binding reserve price can increase revenue in a variety of
circumstances.

If bidder $1$ does not have enough wealth to win both objects ($y_{1}<2y_{2}$%
), then when $y_{1},y_{2}<V^{A}$ and $2y_{2}-y_{1}<V^{A}-V^{B}<y_{2}$ (as in
the above example), a reserve price $r$ such that $y_{2}-\left(
V^{A}-V^{B}\right) <r<y_{1}-y_{2}$ will be non-binding and enhance revenue.

If bidder $1$ has enough wealth to win both objects ($y_{1}>2y_{2}$), then
when $V^{B}>y_{2}>\frac{V^{A}}{2}$, a reserve price $r$ such that $%
2y_{2}-V^{A}<r<y_{2}$ has the same features.

\subparagraph{More than two objects}

This paper has considered two-object auctions. The analysis of auctions with
more than two objects poses substantial difficulties, largely because of the
large variety of phenomena that can occur in the various subgames.

We conclude with an example with three objects, $A,$ $B$ and $C$ without
synergies, and two conjectures regarding such auctions.

\begin{example}
\label{3good}Values: $V^{A}=45,$ $V^{B}=42,$ $V^{C}=18.$ Budgets: $y_{1}=21,$
$y_{2}=15.$
\end{example}

First, suppose the objects are sold in the order $ABC$ by means of three
English auctions. In equilibrium, bidder $2$ wins $A$ for $p^{A}=12$ and
then bidder $1$ wins both $B$ and $C,$ each at a price of $3.$ Bidder $2$
gets a surplus of $V^{A}-p^{A}=33.$ If bidder $2$ were to drop out of the
first auction at a price of $12$ then in the resulting $BC$ auction that
constitutes the subgame, the residual budgets would be $y_{1}-p^{A}=9$ and $%
y_{2}=15,$ respectively. In this subgame, $2$ would win $B$ for $9$ and then 
$1$ would win $C$ for $6.$ Thus by dropping out of the first auction at $12,$
bidder $2$'s surplus would be $V^{B}-9=33,$ which is the same as his surplus
in equilibrium. Bidder $1$, on the other hand, is better off dropping out of
the first auction at $p^{A}=12$ and winning $B$ and $C.$ The total revenue
from the sequential $ABC$ auction, $R^{ABC}=$ $18.$

Next, suppose that the objects are sold in the order $BCA.$ In equilibrium,
bidder $2$ wins $B$ for $p^{B}=8$ and then bidder $1$ wins both $C$ and $A,$
each at a price of $7.$ Bidder $2$ gets a surplus of $V^{B}-p^{B}=34.$ If he
were to drop out at a price of $8$ then in the resulting $CA$ auction, the
residual budgets would be $y_{1}-p^{B}=13$ and $y_{2}=15,$ respectively. In
this subgame, $1$ would win $C$ for $2$ and then $2$ would win $A$ for $11.$
Thus by dropping out of the first auction at $8,$ bidder $2$'s surplus would
be $V^{A}-11=34,$ which is the same as his surplus in equilibrium. Bidder $1$
is again better off dropping out of the first auction at $p^{B}=8$ and
winning $C$ and $A.$

Thus, the total revenue from the sequential $BCA$ auction, $R^{BCA}=22.$

Example \ref{3good} shows that Proposition \ref{order} does not generalize
in a straightforward manner: with three objects it is no longer the case
that the optimal order of sale is always one in which the values are
declining. In the example, the order $BCA$ yields more revenue than the
order $ABC$ (and in fact, $BCA$ is the revenue maximizing order).

We make two conjectures about multiple-object auctions.

First, we conjecture that it is not optimal to sell the least valuable
object $C$ first.

Second, with three objects fix the (declining) values of $B$ and $C$ and
consider an increase in the value of $A.$ We conjecture that when $A$ is
valuable enough, the optimal order of sale is again one in which the values
are declining. Note that for large enough $V^{A},$ all bidders want to win $%
A.$ Therefore, if $A$ is sold first it goes to bidder $1$ for $y_{2}$
regardless of the order of sale of the subsequent objects. Since the price
of $A$ is unaffected by the subsequent ordering of $B$ and $C,$ we can
deduce from Proposition \ref{order} that the order $ABC$ is superior to the
order $ACB$. Thus, establishing this conjecture is equivalent to showing
that if $A$ is valuable enough it should be sold first.

\subparagraph{Bundling and Combination Bids}

It is well-known that a monopoly seller may gain by selling multiple
products in bundles (see, for instance, Adams and Yellen (1976 )). It is
easy to see that in our setting, however, bundling can only be detrimental.
Selling $A$ and $B$ as a bundle in a single auction will fetch a price of $%
y_{2}$ (as long as $y_{2}<V^{AB}$). Selling them separately may result in a
higher revenue. In Example \ref{ord} the revenue from either order $AB$ or $%
BA$ is higher than from a single bundled auction.

A related issue concerns the possibility that bidders may bid on
combinations of objects. This possibility was considered by the designers of
the FCC auction but ultimately was not adopted. At least part of the reason
was that, with a large number of objects, the number of possible combination
bids would become computationally unmanageable (McMillan (1994)). In our
model, allowing combination bids in the simultaneous auction would be
detrimental in terms of revenue, largely for the same reasons that bundling
would be detrimental.

\section{Incomplete Information}

In this paper we have considered a \emph{complete information }model of
multiple object auctions with budget constraints. In this section, we
introduce an incomplete information version of the model. Our purpose is not
to study the incomplete information model in detail; rather it is to
indicate the issues that arise and to link the model to existing auction
theory.

As before, the values of the two objects, $V^{A}$ and $V^{B}$, are common
and commonly known. Suppose, for the sake of simplicity, that there are only
two bidders, $1$ and $2$, with \emph{privately} known budgets $y_{1}$ and $%
y_{2}$, respectively. Specifically, suppose that the budgets are identically
and independently distributed according to the distribution function $F.$
Thus, in this model, the incomplete information pertains to the budgets and
not the values.\footnote{%
This is a special case of the model studied by Pitchik (1995) in which both
values and budgets are privately known.}

Does the order of sale $AB$ still produce more revenue than the order $BA$?
The basic intuition behind this result in the complete information model of
Section 2 is that the wealthier bidder $1$ is more prone to win the first
good in the order $AB$ than in the order $BA.$ The seller is generally
better off when $1$ wins the first good and so the $AB$ order is superior to
the $BA$ order.

This basic intuition would appear to still hold when there is incomplete
information. Confirming this, however, presents substantial technical
difficulties. Suppose that in the symmetric incomplete information model
outlined above there is a symmetric equilibrium bidding function $\beta
\left( y\right) .$ Then it is very unlikely that $\beta $ will be an
increasing function. The reason for this is already apparent in the complete
information model: in many cases it is the bidder with less wealth (bidder $%
2 $) who wins the first object (in Example \ref{ord2}, for instance). Thus
one should not expect that the symmetric bidding strategy $\beta $ in the
incomplete information setting will be monotonically increasing and standard
differential equation techniques used to determine equilibrium strategies do
not work.\footnote{%
For instance, suppose $V^{A}=30,$ $V^{B}=20$ and the budgets are drawn from
a uniform distribution on $[10,11].$ It can be shown that in the order $AB$
there is no symmetric increasing equilibrium in the first auction.}

Pitchik (1995) provides sufficient conditions under which there is an
increasing equilibrium in which, regardless of the auction order, the
wealthier bidder always wins the first object and the poorer bidder then
wins the second for the residual budget of the (initially) wealthier bidder.
She shows that in these circumstances, the order of sale does not affect
total revenue. Under these circumstances, the order of sale is irrelevant in
our complete information setting as well, confirming that at least in one
simple case the results generalize. A general extension to include the cases
where the order of sale matters is hampered by the need to consider
non-monotonic equilibria. The accompanying technical difficulties are
well-known in the literature on auctions.

\section{Conclusion}

The presence of budget constraints introduces several considerations into
multi-object auctions. The bidders' strategic calculations are altered as
they have a motivation to deplete the budgets of their rivals. On the other
side of the market, the seller now desires that the wealthiest bidder win
the first auction. The interaction of these two factors leads to phenomena
such as the seller's preference for selling the more valuable good first and
the possibility of a non-binding reserve price increasing revenue.

We have analyzed auctions in which bidders do not have enough resources to
purchase \emph{all} the items for sale at their full value. Traditionally,
auction theory not only assumes that bidders have unlimited budgets but also
that the price a bidder is willing to pay for one object is independent of
the amount of money he has spent on other objects. In more familiar terms,
the traditional analysis of auctions precludes the presence of income
effects. A moment's reflection will convince the reader of the
implausibility of this assumption in many, if not most, settings. Consider,
for instance, a sequential auction of numerous valuable paintings. The
absence of income effects implies that a bidder's subsequent behavior will
be identical whether he wins the first painting auctioned off for one
million dollars or somehow wins it for free. More concretely, in the recent
auction of National Football League television broadcasting rights, CBS won
the right to broadcast AFC games for eight years for four billion dollars
and effectively did not enter the subsequent competition for Monday Night
Football. Undoubtedly, the fact that CBS won the AFC games lowered its
valuation of Monday Night Football. Nonetheless, would CBS' bidding behavior
on Monday Night Football really have been the same if it had managed to pay
only one hundred million dollars for the AFC contract?

Positing budget constraints in multi-object auctions can be viewed as
allowing for income effects, albeit in a very particular way. From this
perspective, our analysis is a step towards a more general consideration of
auctions with income effects.

\appendix

\section{Appendix}

This appendix collects various formal proofs.

\subsection{Equilibrium Bidding Behavior}

We first study some features of equilibrium bidding behavior when the two
objects are sold sequentially by means of open ascending auctions.\ The
basic analysis is similar to that in Pitchik and Schotter (1988) but is
complicated by the presence of more than two bidders and the possibility of
synergies.

For $i=1,2$ and $j=3-i,$ let $\pi _{i}\left( W_{i},z_{i};W_{j},z_{j}\right) $
be bidder $i$'s equilibrium payoff in an English auction for a single object
when bidder $i$ values the object at $W_{i}$ and has a budget of $z_{i}$,
whereas bidder $j$ values the object at $W_{j}$ and has a budget of $z_{j}.$
(Of course, the payoff $\pi _{i}\left( W_{i},z_{i};W_{j},z_{j}\right) $
depends on bidder $3$'s income $z_{3}$ as well, but we fix $z_{3}=y_{3}$ and
economize on notation by suppressing the dependence of $\pi _{i}$ on $z_{3}$%
.) The functions $\pi _{i}$ determine equilibrium payoffs in the second
auction. $W_{i}$ differs from $W_{j}$ when the synergy is non-zero.

Note that $i)$ $\pi _{i}\left( W_{i},z_{i};W_{j},z_{j}\right) $ is
non-decreasing in $W_{i}$ and $z_{i}$; $ii)$ $\pi _{i}\left(
W_{i},z_{i};W_{j},z_{j}\right) $ is non-increasing in $W_{j}$ and $z_{j}$;
and $iii)$ if $\min \left\{ W_{i},z_{i}\right\} <\min \left\{
W_{j},z_{j}\right\} $ then bidder $i$ loses the auction and so $\pi
_{i}\left( W_{i},z_{i};W_{j},z_{j}\right) =0.$

In what follows, we denote by $I$ the object that is sold in the first
auction and by $II$ the object that is sold in the second auction. Thus, for
example, if the objects are sold in the order $AB$ then $I=A$ and $II=B.$

Define: 
\[
\widehat{p}_{i}^{I}=\sup \left\{ p:\left( V^{I}-p\right) +\pi _{i}\left(
V^{II}+\alpha ,y_{i}-p;V^{II},y_{j}\right) >\pi _{i}\left(
V^{II},y_{i},V^{II}+\alpha ,y_{j}-p\right) \right\} . 
\]
For any $p<\widehat{p}_{i}^{I},$ the left hand side of the inequality is
bidder $i$'s payoff if he wins the first auction at $p$ and the right hand
side is bidder $i$'s payoff if he drops out of the first auction at $p$ and
as a result, bidder $j$ wins the first auction at $p.$\footnote{%
The supremum is well defined since $p=0$ satisfies the inequality and so the
set is not empty. Furthermore, $p=V^{I}+\max \left\{ \alpha ,0\right\} $ is
an upper bound of the set.} Of course, if $p<y_{3}$ then even if bidder $i$
were to drop out of the first auction at $p,$ bidder $3$ would bid the price
that bidder $j$ pays up to $y_{3}.$ Define:

\[
\overline{p}_{i}^{I}=\max \left\{ \widehat{p}_{i}^{I},y_{3}\right\} . 
\]

The following properties of $\overline{p}_{i}^{I}$ will be useful below.

\begin{description}
\item[Property 1]  $\overline{p}_{i}^{A}\geq \overline{p}_{i}^{B}.$
\end{description}

This is quite intuitive and just says that each bidder is willing to bid
more for the first object when it is $A$ than when it is $B.$

\begin{description}
\item[Property 2]  Either $\overline{p}_{1}^{I}\leq \overline{p}_{2}^{I}\leq
V^{I}$ or $\overline{p}_{1}^{I}\geq \overline{p}_{2}^{I}\geq V^{I}.$
\end{description}

The second property delineates two cases. If each bidder wins one object
then bidder $2$ is willing to bid higher for the first object than bidder $%
1. $ This is because bidder $1$'s residual budget in the second auction
after winning the first is greater than bidder $2$'s in similar
circumstances. Thus, the surplus from winning just the second object is less
for bidder $2$ than for bidder $1,$ and so $2$ is willing to bid higher on
the first. In the second case, since the players want to bid above $V^{I}$
they must be planning to win both goods (hence $\alpha >0$). Since, after
winning the first good, $1$ would have more money left over to bid on the
second good than would $2$ in the same position, $1$ is willing to bid
higher. Note that in Example \ref{simula}, bidder $1$ wins good \ $A$ (and \ 
$B$). When $1$ wins the first good for $p^{I}$ in a sequential auction, it
is because $2$ runs up against his budget constraint. Thus, $1$ can never be
forced to pay more than $p^{I}$ for good $I$ and it is a short step to see
that the sequential auction is revenue maximizing.

(Formal proofs of these properties are omitted but available from the
authors.)

Define 
\begin{equation}
p_{i}^{I}=\min \left\{ \overline{p}_{1}^{I},y_{i}\right\} .  \label{pIi}
\end{equation}
Property 1 then implies that: 
\begin{equation}
p_{i}^{A}\geq p_{i}^{B}.  \label{ineq2}
\end{equation}

If $p_{i}^{I}>y_{3},$ bidder $i$ will never drop out of the bidding for
object $I$ at prices below $p_{i}^{I}.$ Bidder $3$ will always bid the price
of $I$ up to $y_{3}.$ If bidder $1$ stops bidding below $y_{3},$ bidder $2$
will stay in to win $I$ at $y_{3},$ since otherwise he will not win either
object. We have,

\begin{enumerate}
\item  If $p_{1}^{I}<p_{2}^{I},$ bidder $2$ wins object $I$. Bidder $1$ (or
bidder $3$) forces the price in the first auction up to $\min \left\{
p_{2}^{I},\max \left\{ p_{1}^{I},y_{2}-y_{3}\right\} \right\} $ and bidder $%
2 $\ pays that price. Note that at prices beyond $p_{1}^{I},$ bidder $1$
does not want to win object $I$, and is bidding only to deplete $2$'s
income. At prices above $y_{2}-y_{3}$ there is no incentive to further
deplete $2$'s income since the price of object $II$ will not fall below $%
y_{3}$ in any case.

\item  If $p_{1}^{I}>p_{2}^{I}$ bidder $1$ wins object $I.$ From Property 2,
if $V^{I}$ $\geq p_{1}^{I}$ then $p_{2}^{I}=y_{2}.$ In that case, bidder $2$
runs up against his budget constraint even though he would like to continue
bidding higher, and thus \emph{bidder }$1$\emph{\ wins object }$I$ for $%
p_{2}^{I}=y_{2}.$ If $V^{I}$ $<p_{1}^{I}$ then even if $1$ wins object $I$
at a price below $p_{1}^{I},$ he still has enough money left over to win
object $II.$ Therefore, $2$ has no incentive to try and push the price
beyond $p_{2}^{I}$ and $1$ wins object $I$ for $p_{2}^{I}.$

\item  If $p_{1}^{I}=p_{2}^{I}$ then \emph{either bidder }$1$\emph{\ or
bidder }$2$\emph{\ wins object }$I$ for $p_{1}^{I}=p_{2}^{I}.$
\end{enumerate}

The equilibrium price of object $I$ can thus be written as: 
\begin{equation}
p^{I}=\min \left\{ p_{2}^{I},\max \left\{ p_{1}^{I},y_{2}-y_{3}\right\}
\right\} .  \label{eqmp}
\end{equation}

Since $p_{i}^{A}\geq p_{i}^{B},$ we have $p^{A}\geq p^{B},$ that is, the
equilibrium price of the first object in the order $AB$ is at least as high
as in the order $BA.$

\subsection{Proof of Proposition 1}

We break up the proof into three separate claims. To shorten the proof, we
restrict our attention to the case $y_{1}>y_{2}>y_{3},$ $V^{B}>y_{3}.$ We
also assume that $y_{2}<V^{A}+V^{B},$ since the reverse inequality is the
trivial unconstrained case.

Define $\alpha _{+}=\max \left\{ \alpha ,0\right\} $ and $\alpha _{-}=\min
\left\{ \alpha ,0\right\} .$ Object $X$ is worth at most $V^{X}+\alpha _{+}$
and at least $V^{X}+\alpha _{-}.$

\textbf{Proof.} We break up the proof into three separate claims.\bigskip

\noindent \textsc{claim 1}$:$\textsc{\ }\emph{If bidder }$1$ \emph{wins only
object }$B$ \emph{in the order }$BA,$ \emph{then the revenue from order }$AB$
\emph{is at least as large as that from }$BA.$

Notice that since each bidder wins one object each, $p^{B}\leq V^{B}.$

Since bidder $1$ wins $B$ in the order $BA$ it must be that $p_{1}^{B}\geq
p_{2}^{B}.$

If $p_{1}^{B}>p_{2}^{B}$, bidder $1$ wins object $B$ for $y_{2}.$ However,
bidder $1$ would rather drop out at $y_{2}$ and win object $A$ for $y_{3}$
in the second auction than win only object $B$ for $y_{2}.$ Therefore, it
must be that $p_{1}^{B}=p_{2}^{B}=p^{B}<y_{2}.$

Suppose $p^{B}=V^{B}.$ Since in this case bidder $1$ would just as soon win $%
B$ for a surplus of $0$ as drop out at $p=V^{B},$ it must be that $%
y_{2}-V^{B}\geq V^{A}$ and $\alpha \geq 0$, so that $y_{2}\geq V^{A}+V^{B}$
and the total revenue is $V^{A}+V^{B}$ in the order $BA$ and at least this
much in the order $AB.$

Suppose $p_{1}^{B}=p_{2}^{B}=p^{B}<V^{B}$. If $\widehat{p}_{1}^{B}<y_{3},$
then bidder $2$ wins $B$ for $y_{3},$ so we must have $\widehat{p}%
_{1}^{B}\geq y_{3}$ and so $p_{i}^{B}=\widehat{p}_{1}^{B}.$ From the
definition of $p_{i}^{B},$ for $i=1,2,$ $V^{B}-p^{B}=V^{A}-\max \left\{ \min
\left\{ V^{A}+\alpha _{-},y_{i}-p^{B}\right\} ,y_{3}\right\} $. Since $%
y_{1}>y_{2},$ $\max \left\{ \min \left\{ V^{A}+\alpha
_{-},y_{i}-p^{B}\right\} ,y_{3}\right\} =\min \left\{ V^{A}+\alpha
_{-},y_{3}\right\} .$ But since $p^{B}\geq y_{3}$ and $V^{A}>V^{B}$ we
cannot have $V^{B}-p^{B}=V^{A}-y_{3},$ so this case is ruled out. Thus, $%
p^{B}=V^{B}+\alpha _{-}$ and the price of the second object is $V^{A}+\alpha
_{-}.$ Since $V^{A}+\alpha _{-}\leq $ $y_{i}-p^{B},$we have $y_{i}\geq
\left( V^{A}+\alpha _{-}\right) +\left( V^{B}+\alpha _{-}\right) ,$ and in
the order $AB$ revenue is at least $\left( V^{A}+\alpha _{-}\right) +\left(
V^{B}+\alpha _{-}\right) .$ $\square $\bigskip

\noindent \textsc{claim 2}$:$\textsc{\ }\emph{If bidder }$1$ \emph{wins only
object }$A$ \emph{in the order }$BA,$ \emph{then the revenue from the order }%
$AB$ \emph{is at least as large as that from the order} $BA.$

If one bidder wins both objects in the order $AB,$ then total revenue is 
\[
\min \left\{ \max \left\{ V^{A},\min \left\{ V^{A}+\alpha
_{+},y_{2}-V^{B}\right\} \right\} ,y_{2}\right\} +\min \left\{
V^{B},y_{2}\right\} , 
\]
whereas if bidder $1$ wins $A$ alone, the total revenue is at most $\min
\left\{ V^{A},y_{2}\right\} +\min \left\{ V^{B},y_{2}\right\} .$ Therefore,
suppose that in both orders each bidder wins one object.

In the order $BA,$ bidder $1$ pays $\max \left\{ \min \left\{ V^{B}+\alpha
,y_{2}-p^{B}\right\} ,y_{3}\right\} $ for $A.$

If bidder $1$ pays $V^{A}+\alpha _{-}$ for $A$ in the order $BA,$ then $%
y_{2}-\left( V^{B}+\alpha _{-}\right) \geq \left( V^{A}+\alpha _{-}\right) .$
Since $y_{1}\geq y_{2},$ in both orders total revenue is $\left(
V^{A}+\alpha _{-}\right) +\left( V^{B}+\alpha _{-}\right) .$ In the order $%
AB,$ if $B$ sells for $V^{B}+\alpha _{-}$ then bidder $2$ must have bid the
price on $A$ up to $\min \left\{ V^{A}+\alpha _{-},y_{2}\right\} $ and
revenue is $\min \left\{ V^{A}+\alpha _{-},y_{2}\right\} +\left(
V^{B}+\alpha _{-}\right) .$ In the order $BA$, object $A$ can be bid up to
at most $\min \left\{ V^{A}+\alpha _{-},y_{2}\right\} ,$ so that revenue is
at most $\left( V^{B}+\alpha _{-}\right) +\min \left\{ V^{A}+\alpha
_{-},y_{2}\right\} .$

If bidder $1$ pays $\max \left\{ y_{2}-p^{B},y_{3}\right\} $ for $A$ in the
order $BA,$ then the total revenue is $p^{B}+\max \left\{
y_{2}-p^{B},y_{3}\right\} .$ If bidder $i$ wins only $A$ in the order $AB$
total revenue is $p^{A}+\max \left\{ y_{i}-p^{A},y_{3}\right\} .$ Since $%
p^{B}\leq p^{A}$ and $y_{2}\leq y_{i},$ revenue is at least as large in the
order $AB$ as in the order $BA.$ $\square \bigskip $

\noindent \textsc{claim 3}$:$\textsc{\ }\emph{If bidder }$1$ \emph{wins both
objects when the order is} $BA,$ \emph{then the revenue from the order }$AB$ 
\emph{is the same as that from the order} $BA.$

The total revenue when bidder $1$ wins both objects in the order $BA$ is: 
\begin{equation}
R^{BA}=\min \left\{ \max \left\{ V^{B},\min \left\{ V^{B}+\alpha
_{+},y_{2}-V^{A}\right\} \right\} ,y_{2}\right\} +\min \left\{
V^{A},y_{2}\right\}  \label{revBA}
\end{equation}
and the total revenue when bidder $1$ wins both objects in the order $AB$
is: 
\begin{equation}
R^{AB}=\min \left\{ \max \left\{ V^{A},\min \left\{ V^{A}+\alpha
_{+},y_{2}-V^{B}\right\} \right\} ,y_{2}\right\} +\min \left\{
V^{B},y_{2}\right\} .  \label{revAB}
\end{equation}

It is easy to verify that if $V^{A}+V^{B}+\alpha _{+}\geq y_{2}$ then either 
$R^{BA}=R^{AB}=$ $y_{2}$ or $R^{BA}=R^{AB}=\min \left\{ V^{A},y_{2}\right\}
+\min \left\{ V^{B},y_{2}\right\} $. If $V^{A}+V^{B}+\alpha _{+}<y_{2}$ then 
$R^{BA}=R^{AB}=V^{A}+V^{B}+\alpha _{+}.$

The gain to bidder $1$ from dropping out of the first auction in the order $%
BA$ at the price $p^{B}$ is, by definition, $\pi _{1}\left(
V^{A},y_{1};V^{A}+\alpha ,y_{2}-p^{B}\right) .$ Similarly, the gain to
bidder $1$ from dropping out of the first auction in the order $AB$ at the
price $p^{A}$ is $\pi _{1}\left( V^{A},y_{1};V^{A}+\alpha
,y_{2}-p^{B}\right) .$

It can be argued that\footnote{%
A formal proof of this step is available from the authors.} 
\begin{equation}
\pi _{1}\left( V^{A},y_{1};V^{A}+\alpha ,y_{2}-p^{B}\right) \geq \pi
_{1}\left( V^{B},y_{1};V^{B}+\alpha ,y_{2}-p^{A}\right) .  \label{piVA}
\end{equation}

The fact that bidder $1$ wins both objects in the order $BA$ implies that 
\[
V^{A}+V^{B}+\alpha -R^{BA}\geq \pi _{1}\left( V^{A},y_{1};V^{A}+\alpha
,y_{2}-p^{B}\right) 
\]
and so 
\[
V^{A}+V^{B}+\alpha -R^{AB}\geq \pi _{1}\left( V^{B},y_{1};V^{B}+\alpha
,y_{2}-p^{A}\right) 
\]
so that bidder $1$ would rather win both objects in the order $AB$ also.

Hence the revenue in both orders is the same. $\square $ \bigskip

Claims 1 to 3 complete the proof. $\blacksquare $

\subsection{Proofs of Propositions 2 and 3}

We begin by establishing two results that show revenue ranking of the
sequential auction versus the simultaneous auction depends on the
equilibrium outcome of the sequential auction.

\begin{lemma}
\label{simA}Suppose $\alpha \geq 0.$ If bidder $1$ wins only object $A$ or
wins both $A$ and $B$ in the sequential $AB$ auction, then the revenue from
the sequential $AB$ auction is at least as great as the revenue from any
equilibrium of the simultaneous auction.
\end{lemma}

\noindent \noindent \textbf{Proof. }First, suppose bidder $1$ wins both
objects in the sequential $AB$ auction. Then the total revenue is: 
\[
R^{AB}=\min \left\{ \max \left\{ V^{A},\min \left\{ V^{A}+\alpha
,y_{2}-V^{B}\right\} \right\} ,y_{2}\right\} +\min \left\{
V^{B},y_{2}\right\} 
\]
so that: 
\[
R^{AB}=\left\{ 
\begin{array}{ll}
\min \left\{ V^{A},y_{2}\right\} +\min \left\{ V^{B},y_{2}\right\} & \text{%
if }y_{2}\leq V^{A}+V^{B} \\ 
y_{2} & \text{if }V^{A}+V^{B}<y_{2}\leq V^{A}+V^{B}+\alpha \\ 
V^{A}+V^{B}+\alpha . & \text{if }V^{A}+V^{B}+\alpha <y_{2}.
\end{array}
\right. 
\]

If $y_{2}\leq V^{A}+V^{B}$, then the revenue in the simultaneous auction
cannot be greater than $R^{AB}$ . This is because it is dominated for bidder 
$1$ to let bidder $2$ wins both objects and thus, in neither auction will
bidder $2$ bid above the value. Similarly, if $V^{A}+V^{B}+\alpha <y_{2},$
then $R^{AB}$ is maximal.

We show that $R^{AB}$ is also maximal when $V^{A}+V^{B}<y_{2}\leq
V^{A}+V^{B}+\alpha .$ Suppose not. Then there is an equilibrium of the
simultaneous auction in which the total revenue exceeds $y_{2}>V^{A}+V^{B}.$
Thus one of the bidders, say $1,$ must be winning both objects. The most
that bidder $2$ could bid on $A$ is $y_{2}-V^{B}$ and the most he could bid
on $B$ is $y_{2}-V^{A}.$ First, suppose that in the first round, bidder $2$
is bidding less than $V^{X}$ on some object $X.$ Then bidder $1$ could bid $%
y_{2}-V^{X}$ on the other object, win it at that price and then win $X$ for $%
V^{X}.$ Next, suppose that in the first round bidder $2$ is bidding $%
q^{A}>V^{A}$ on object $A$ and $q^{B}>$ $V^{B}$ on object $B.$ But now
bidder $1$ can bid (slightly above) $y_{2}-q^{B}$ on object $A$ and
(slightly above) $q^{B}$ on object $B.$ This will cause bidder $2$ to drop
out of both auctions and thus bidder $1$ will win both auctions for a total
of $y_{2}.$ In either case the total revenue in the simultaneous auction
does not exceed $y_{2}.$

Thus, we have shown that if bidder $1$ wins both objects in the sequential $%
AB$ auction the revenue from the simultaneous auction cannot be greater than 
$R^{AB}.$

Next, suppose that bidder $1$ wins only $A$ in the sequential $AB$ auction.
We now show that each bidder's surplus in the simultaneous auction must be
at least as large as that in the sequential auction.

Suppose bidder $1$ opens the simultaneous auction with a bid of $p^{A}$ for $%
A$ and $0$ for $B$. If $p^{A}=y_{2}$ then $1$ wins good $A$ thereby
obtaining a surplus at least as large as in the sequential auction. If $%
p^{A}<y_{2}$ then $V^{A}-p^{A}=V^{B}-\min \left\{ V^{B},y_{3}\right\} $ and $%
y_{3}>y_{1}-p^{A},$ or else $2$ would have bid beyond $p^{A}$ in the
sequential auction. Therefore, if $2$ pushes the bid beyond $p^{A}$ in the
simultaneous auction, $1$ can drop out of the bidding for $A$ and win good $%
B $ for $\min \left\{ V^{B},y_{3}\right\} $. In both cases, bidder $1$
guarantees himself a surplus at least as large as his surplus in the
sequential auction.

Suppose bidder $2$ opens the simultaneous auction with a bid of $p^{A}$ for $%
A$ and $0$ for $B$. If bidder $1$ lets $2$ win $A$ for this price, then $2$%
's surplus, $V^{A}-p^{A}$ is at least as large as in the sequential auction.
If bidder $2$ does not win $A$ then he can win $B$ at a price not exceeding
that in the sequential auction since by outbidding bidder $2$ on $A$ bidder $%
1$'s residual budget is smaller.

Since each bidder's surplus in the simultaneous auction is at least as large
as in the sequential auction, the revenue cannot be larger.

Thus we have shown that when bidder $1$ wins only object $A$ in the
sequential $AB$ auction the revenue from the simultaneous auction is no
greater than $R^{AB}.$

This completes the proof. $\blacksquare $

\textbf{Proof of Proposition 2}

We first show that the sequential auction is revenue superior if (a) holds.

Fix $V^{B}.$ For large enough $V^{A},$ $V^{A}-y_{2}>V^{B}-y_{3}$ and bidder $%
1$ prefers winning $A$ in the order $AB$ to winning only $B.$ From Lemma \ref
{simA}, $R^{AB}\geq R^{sim}.$

We now show that the sequential auction is revenue superior if (b) holds and
that as $\alpha $ increases from $0$ the inequality may strict for
equilibria of the simultaneous auction.

Suppose $y_{1}>2y_{2},$ and that when $\alpha =0$, $1$ wins only one good in
the $AB$ auction. Let $\overline{\alpha }$ satisfy 
\[
V^{A}+V^{B}+\overline{\alpha }-2y_{2}=V^{B}-y_{3}. 
\]
For all $\alpha $ $>\overline{\alpha }$ bidder $1$ wins both goods in the $%
AB $ auction and $R^{AB}=y_{2}+\min \left\{ V^{B},y_{2}\right\} $. Let $%
\widehat{\alpha }$ solve: 
\[
V^{A}+V^{B}+\widehat{\alpha }-2y_{2}=V^{A}-y_{3}. 
\]

Suppose that $\widehat{\alpha }>\alpha >\overline{\alpha }.$ In one
equilibrium of the simultaneous auction, in the first round $2$ bids $y_{2}$
on good $B$ and $1$ bids (just below) $y_{2}$ on good $B$ and (just above) $%
y_{3}$ on good $A$ . $2$ cannot profitably deviate and win good $A,$ since $%
\alpha $ $>\overline{\alpha }$ so that $1$ prefers winning both goods to
winning $B$ for $y_{2}.$ Bidder $1$ cannot profitably deviate by winning
both goods since $\widehat{\alpha }>\alpha $. Let $\overline{p}$ solve 
\[
V^{A}+V^{B}+\overline{\alpha }-2y_{2}=V^{A}-\overline{p}. 
\]
Note that $\overline{p}<y_{2}.$ If the price of good $A$ ever rises above $%
\overline{p}$ in the simultaneous auction, then $1$ will win both goods
since this is better than winning either $A$ for $\overline{p}$ or $B$ for $%
y_{3}.$ Therefore, it is dominated for $2$ to bid above $\overline{p}$ on $A$
and every equilibrium of the simultaneous auction earns no more than $%
\overline{p}+\min \left\{ V^{B},y_{2}\right\} <y_{2}+\min \left\{
V^{B},y_{2}\right\} =R^{AB}.$

If bidder $1$ wins both goods in the $AB$ auction when $\alpha =0$, he will
still win both goods when $\alpha >0,$ and from Lemma \ref{simA} the $AB\;$%
auction is always at least as good as the simultaneous auction.%
%TCIMACRO{
%\TeXButton{End Proof}{\endproof%
%}}%
%BeginExpansion
\endproof%
%
%EndExpansion

\textbf{Proof of Proposition 3}

The proposition follows from Lemmas \ref{hybA} and \ref{hybB} below.

\begin{lemma}
\label{hybA}Suppose $\alpha \geq 0.$ If bidder $1$ wins only object $A$ or
wins both $A$ and $B$ in the sequential $AB$ auction, then the revenue from
the sequential $AB$ auction is the same as that in the hybrid auction.
\end{lemma}

\noindent \textbf{Proof. }Suppose player $1$ wins only A for $p^{A}$. Then
bidder 1's surplus in the $AB$ game is $V^{A}-p^{A}\geq V^{B}-\min \left\{
V^{B},\max \left\{ y_{2}-p^{A},y_{3}\right\} \right\} $ and bidder 2's
surplus is $V^{B}-\min \left\{ V^{B},\max \left\{ y_{1}-p^{A},y_{3}\right\}
\right\} ,$ where $V^{A}-p^{A}\geq V^{B}-\min \left\{ V^{B},\max \left\{
y_{1}-p^{A},y_{3}\right\} \right\} .$ Given that the hybrid auction is
followed by an auction for $B$ once the bidding stops, neither player will
follow a strategy which involves dropping out of the bidding for $A$ at a
price $p<p^{A},$ since this yields at most $V^{B}-\min \left\{ V^{B},\max
\left\{ y_{2}-p,y_{3}\right\} \right\} \leq V^{A}-p^{A}<V^{A}-p.$ On the
other hand, neither player will bid more that $p^{A}$ for $A$ either, or
else he would have done so in the sequential auction. Thus, $A$ will also
sell for $p^{A}$ in the hybrid auction and the revenue will be the same as
in the hybrid auction.

Similarly, if $1$ wins\ $A\;$for $P^{A}$ in the $AB$ game, and then goes on
to win $B\;$as well, $A$ will again sell for $P^{A}$ in the hybrid game and
revenues will be the same. $\blacksquare $

\begin{lemma}
\label{hybB}Suppose $\alpha \geq 0.$ If bidder $1$ wins only object $B$ in
the sequential $AB$ auction, then the revenue from the sequential $AB$
auction is no greater than the revenue from any equilibrium of the hybrid
auction.
\end{lemma}

\noindent \textbf{Proof. }First, consider the sequential $AB$ auction.

Since bidder $1$ wins only $B$ in the sequential $AB$ auction, $%
p_{1}^{A}\leq p_{2}^{A}$ and bidder $2$ wins $A$ for $\min \left\{
p_{2}^{A},\max \left\{ y_{2}-y_{3},p_{1}^{A}\right\} \right\} .$

If $y_{2}>V^{A}+V^{B}$ and $\alpha >0$ then both objects would have been won
by the same bidder and thus either $y_{2}\leq V^{A}+V^{B}$ or $\alpha =0.$

\begin{enumerate}
\item  If $\alpha =0$ and $y_{2}>$ $V^{A}+V^{B}$ then $p^{A}=V^{A},$ $%
p^{B}=V^{B}$ and $R^{AB}=V^{A}+V^{B}$, which is maximal in this case. If $%
y_{2}=$ $V^{A}+V^{B}$ then, for $\alpha \geq 0$, again $R^{AB}=V^{A}+V^{B}.$

\item  If $y_{2}<$ $V^{A}+V^{B},$ then $p^{B}<V^{B}$ and the total revenue 
\begin{eqnarray*}
R^{AB}=\min \left\{ p_{2}^{A},\max \left\{ y_{2}-y_{3},p_{1}^{A}\right\}
\right\} \\
+\max \left\{ y_{2}-\min \left\{ p_{2}^{A},\max \left\{
y_{2}-y_{3},p_{1}^{A}\right\} \right\} ,y_{3}\right\} .
\end{eqnarray*}

\begin{enumerate}
\item  If $p_{1}^{A}\geq y_{2}-y_{3},$ then $\min \left\{ p_{2}^{A},\max
\left\{ y_{2}-y_{3},p_{1}^{A}\right\} \right\} =p_{1}^{A}$ and since $%
V^{B}>y_{3}\geq y_{2}-p_{1}^{A}$ we have $R^{AB}=p_{1}^{A}+y_{3}.$

\item  If $p_{1}^{A}<y_{2}-y_{3}$ then $\min \left\{ p_{2}^{A},\max \left\{
y_{2}-y_{3},p_{1}^{A}\right\} \right\} =\min \left\{
p_{2}^{A},y_{2}-y_{3}\right\} .$ So the total revenue is $R^{AB}\leq y_{2}.$
\end{enumerate}
\end{enumerate}

Now consider the hybrid auction:

\begin{enumerate}
\item  If $\alpha =0$ and $y_{2}\geq V^{A}+V^{B}$ then the object that is
sold second goes for its value and therefore so does the first and $%
R^{sim}=V^{A}+V^{B}.$ If $\alpha >0$ and $y_{2}=V^{A}+V^{B}$ then again
total revenue in the simultaneous auction is $R^{sim}=V^{A}+V^{B}.$ In both
of these cases, $R^{sim}=R^{AB}$.

\item  If $y_{2}<$ $V^{A}+V^{B}$ and bidder $1$ wins both objects, then
revenues are $\min \left\{ V^{A},y_{2}\right\} +\min \left\{
V^{B},y_{2}\right\} $ which are maximal. So suppose each bidder wins one
object.

\begin{enumerate}
\item  If $p_{1}^{A}\geq y_{2}-y_{3},$ then by definition of $p_{i}^{A},$
both bidders prefer winning object $A$ for $p_{1}^{A},$ to winning object $B$
for $y_{3}\geq y_{2}-p_{1}^{A}$ $.$ Since bidder $3$ will not let $B$ sell
for less than $y_{3},$ $R^{sim}\geq p_{1}^{A}+y_{3}.$

\item  If $p_{1}^{A}<y_{2}-y_{3},$ let $I$ be the first object to sell in
the simultaneous auction, $q^{I}$ its selling price, and let $i$ denote the
bidder that wins $I$. The second object, $II$, sells for $q^{II}=\min
\left\{ V^{II},\max \left\{ y_{j}-q^{I},y_{3}\right\} \right\} .$ If $%
q^{II}=V^{II},$ then $q^{I}=y_{2}$ and $R^{sim}\geq y_{2}.$ If $q^{II}=\max
\left\{ y_{j}-q^{I},y_{3}\right\} ,$ then $R^{sim}=q^{I}+\max \left\{
y_{j}-q^{I},y_{3}\right\} \geq y_{2}.$
\end{enumerate}
\end{enumerate}

This completes the proof. $\blacksquare $

\subsection{Proof of Proposition 4}

We first establish some properties of any equilibrium of the game with
endogenous budgets.

\begin{lemma}
\label{endoprop}\noindent In any equilibrium with budgets $(\overline{y}_{i},%
\overline{y}_{j})$, where $\overline{y}_{j}\leq \overline{y}_{i}$: \newline
(a) each bidder wins exactly one object $X$, for a price less than $%
V^{X}+\alpha _{-};$\newline
(b) bidder $j$\ wins the second object; and \newline
(c) the first object sells for $\overline{y}_{j}.$
\end{lemma}

\noindent \textbf{Proof. (a) }We show that for small enough $\varepsilon $,
a choice of $y_{j}=V^{I}+\alpha _{-}-\varepsilon ,$ earns $j$ a positive
surplus in the auction subgame, for any value of $\overline{y}_{i}.$ If $%
V^{I}+\alpha _{-}-\varepsilon >\overline{y}_{i},$ $j$ can win object $I.$ If 
$V^{I}+\alpha _{-}-\varepsilon \leq \overline{y}_{i},$ at worst $i$ will let 
$j$ win $I$ for $V^{I}+\alpha _{-}-\varepsilon ,$ since this will yield $i$
a surplus of $V^{II},$ while winning only $I$ would net $\varepsilon -\alpha
_{-}<V^{II}$ for small enough $\varepsilon ,$ and winning both objects would
net $i$ 
\begin{eqnarray*}
&&V^{I}+V^{II}+\alpha -\left( V^{I}+\alpha _{-}-\varepsilon \right) -\min
\left\{ V^{II},V^{I}+\alpha _{-}-\varepsilon \right\} \\
&=&\left( V^{II}+\varepsilon -\alpha _{-}\right) +\left( \alpha -\min
\left\{ V^{II},V^{I}+\alpha _{-}-\varepsilon \right\} \right) \\
&<&V^{II}
\end{eqnarray*}
for small enough $\varepsilon .$

\textbf{(b) }From (a) we know that $i$ is winning exactly one object for
less than $V^{X}+\alpha _{-}$. Suppose $\overline{y}_{j}<\overline{y}_{i}$
and $i$ is winning the first object. Since $\overline{y}_{j}<\overline{y}%
_{i} $ and $y_{3}=0,$ if $i$ likes winning object $I$ at $p^{I}$ at least as
much as dropping out and winning the second object, $j$ must strictly prefer
winning $I$. Since $j$ concedes $I$ to $i,$ it must be that he is at his
budget constraint so that $p^{I}=\overline{y}_{j}.$ Now, $j$ would be better
off choosing a slightly bigger budget to make $i$ either spend more on the
object, or concede $I$ to him, contradicting the fact that $(\overline{y}%
_{i},\overline{y}_{j})$ constitute equilibrium budget choices. Therefore, $i$
must be winning the second object (and without loss of generality, can be
assumed to be winning the second object when $\overline{y}_{j}=\overline{y}%
_{i}$). This establishes the claim.

\textbf{(c) }From (b) we know that $j$ wins $I$. Let $p^{I}$ denote the
price paid by bidder $j$ for $I$ and suppose that $p^{I}<\overline{y}_{j}.$
On the one hand, $j$ is willing to bid up to $p^{I}$. On the other hand,
since from (a) the price of the second object is less than $V^{II}+\alpha
_{-},$ if bidder $i$ is not pushing $j$ any higher than $p^{I}$ it must be
that he cannot. Thus, $j$ must be just indifferent between winning $I$ at a
price $p^{I}$ and winning $II$ at a price $(\overline{y}_{i}-p^{I}).$
Therefore, 
\begin{equation}
V^{I}-p^{I}=V^{II}-\left( \overline{y}_{i}-p^{I}\right) .  \label{claim3}
\end{equation}
But if instead, $j$ chose a budget $y_{j}>V^{I}+V^{II}$, $i$ would bid up to 
$\min \{V^{I},\overline{y}_{i}\}$ on the first object and $j$ would earn $%
V^{II}-(\overline{y}_{i}-\min \{V^{I},\overline{y}_{i}\})$ on the second$,$
which is more than (\ref{claim3}). Therefore, $p^{I}=\overline{y}_{j}.$ $%
\blacksquare \bigskip $

In proving Proposition \ref{endo}, we consider the case of substitutes
(Lemma \ref{endosub} below) and complements (Lemma \ref{endocomp})
separately.

\subsubsection{Substitutes}

\begin{lemma}
\label{endosub}Suppose $\alpha \leq 0.$ The budgets $(\overline{y}_{i},%
\overline{y}_{j})$ and the prices $(\overline{p}^{I},\overline{p}^{II})$
constitute an equilibrium outcome of the game with endogenous budgets if: 
\[
\begin{array}{lll}
\begin{array}{l}
\overline{y}_{i}\geq V^{I}-V^{II} \\ 
\overline{y}_{j}=V^{I}-V^{II}
\end{array}
& 
\begin{array}{l}
\overline{p}^{I}=V^{I}-V^{II} \\ 
\overline{p}^{II}=0
\end{array}
& \text{if }\frac{1}{2}V^{I}\geq V^{II}+\frac{1}{2}\alpha \\ 
&  &  \\ 
\begin{array}{l}
\overline{y}_{i}\geq \frac{1}{2}V^{I}+V^{II}+\frac{3}{2}\alpha \\ 
\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha \right)
\end{array}
& 
\begin{array}{l}
\overline{p}^{I}=\frac{1}{2}\left( V^{I}+\alpha \right) \\ 
\overline{p}^{II}=0
\end{array}
& \text{if }\frac{1}{2}V^{I}<V^{II}+\frac{1}{2}\alpha
\end{array}
\]
All equilibria result in the same payoffs (except for a relabelling of the
bidders).
\end{lemma}

\noindent \textbf{Proof. }Considering the two cases separately, we first
verify that $(\overline{y}_{i},\overline{y}_{j})$ constitute equilibrium
budget choices. \bigskip

\noindent \textbf{Case 1:} $\frac{1}{2}V^{I}\geq V^{II}+\frac{1}{2}\alpha .$
Given budgets $\overline{y}_{i}\geq V^{I}-V^{II}$ and $\overline{y}%
_{j}=V^{I}-V^{II},$ it is an equilibrium outcome of the auction subgame for
bidder $j$ to win $I$ for a price of $V^{I}-V^{II}$ and for bidder $i$ to
win $II$ for free. The equilibrium payoff of bidder $j$ is $V^{II}$ and that
of $i$ is also $V^{II}.$

Now, suppose $\overline{y}_{i}\geq V^{I}-V^{II}.$ If bidder $j\neq i$ wins
both objects then his payoff is less than 
\begin{eqnarray*}
&&(V^{I}+V^{II}+\alpha )-(V^{I}-V^{II})-\min \left\{
V^{II},V^{I}-V^{II}\right\} \\
&=&2V^{II}+\alpha -\min \left\{ V^{II},V^{I}-V^{II}\right\}
\end{eqnarray*}
and for either realization of the minimum, this is no greater than $V^{II}$.
If $j$ wins only the second object his payoff is at most $V^{II}.$ Finally, $%
i$ will not let \ $j$ win the first object alone for less than $%
V^{I}-V^{II}. $ Thus, bidder $j$ cannot profitably deviate if $\overline{y}%
_{i}\geq V^{I}-V^{II}.$ Similarly, if $\overline{y}_{j}=V^{I}-V^{II},$
bidder $i$ cannot profitably deviate. \bigskip

\noindent \textbf{Case 2:} $\frac{1}{2}V^{I}<V^{II}+\frac{1}{2}\alpha .$
With budgets $\overline{y}_{i}\geq \frac{1}{2}V^{I}+V^{II}+\frac{3}{2}\alpha 
$ and $\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha \right) $, it is an
equilibrium outcome of the auction subgame for bidder $j$ to win $I$ for a
price of $\frac{1}{2}\left( V^{I}+\alpha \right) $ and for bidder $i$ to win 
$II$ for free. The equilibrium payoff of bidder $i$ is $V^{II}$ and that of
bidder $j$ is $\frac{1}{2}V^{I}-\frac{1}{2}\alpha .$

Suppose $\overline{y}_{i}\geq \frac{1}{2}V^{I}+V^{II}+\frac{3}{2}\alpha .$
If bidder $j$ chooses a $y_{j}<\frac{1}{2}\left( V^{I}+\alpha \right) ,$
then $i$ will win both objects. Therefore, suppose that $y_{j}>\frac{1}{2}%
\left( V^{I}+\alpha \right) .$ If $j$ wins $I$ and only $I$ then he pays at
least $\frac{1}{2}\left( V^{I}+\alpha \right) .$ (This is because bidder $i$
can force the price up to $\frac{1}{2}\left( V^{I}+\alpha \right) $ when $j$%
's budget is $\frac{1}{2}\left( V^{I}+\alpha \right) ,$ and $i$ can still
force the price to at least this level when $j$'s budget is larger.) If
bidder $j$ wins only $II$ then he pays at least 
\begin{eqnarray*}
\min \{V^{II}+\alpha ,\overline{y}_{i}-\left( V^{I}+\alpha \right) \} &\geq
&\min \{V^{II}+\alpha ,V^{II}+\frac{1}{2}\alpha -\frac{1}{2}V^{I}\} \\
&=&V^{II}+\frac{1}{2}\alpha -\frac{1}{2}V^{I}
\end{eqnarray*}
since $\alpha >-V^{I}$ and so his net gain is no greater than $\frac{1}{2}%
V^{I}-\frac{1}{2}\alpha $. Bidder $j$ can win both objects only if $\alpha
=0 $ and this then nets him $0$ since $\overline{y}_{i}\geq \max
\{V^{I},V^{II}\}.$ Thus $\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha
\right) $ is a best response to any $\overline{y}_{i}\geq \frac{1}{2}%
V^{I}+V^{II}+\frac{3}{2}\alpha $.

Suppose $\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha \right) $ and
bidder $i$ chooses a $y_{i}\neq \overline{y}_{i}.$ If $i$ wins both objects
it must be that $\alpha =0$ and then he pays a total of $2\overline{y}%
_{j}=V^{I}$ for a net gain of $V^{II},$ which is his equilibrium payoff. If
he wins only $I,$ then this is a profitable deviation only if $I$ is
obtained at a price below $V^{I}-V^{II}<\frac{1}{2}V^{I}+\frac{1}{2}\alpha $%
. But $i$ cannot obtain $I$ at such a price, since bidder $j$ would continue
to raise the bid. Finally, $i$ is already winning $II$ for a price of $0.$
Thus $\overline{y}_{i}\geq \frac{1}{2}V^{I}+V^{II}+\frac{3}{2}\alpha $ is a
best response to $\overline{y}_{j}.$

Thus, we have shown that in both cases $(\overline{y}_{i},\overline{y}_{j})$
as specified are equilibrium budget choices. The fact that in each case the
equilibrium payoffs are the same from all such equilibria is immediate.
\bigskip

Suppose $(\overline{y}_{i},\overline{y}_{j})$ are equilibrium budget choices
with $\overline{y}_{j}\leq \overline{y}_{i}.$ Now, since $j$ is not choosing
a budget $y_{j}<\overline{y}_{j},$ it must be the case that doing so causes
him to not win object $I$, since if he still won $I$ it would perforce be at
a price below $\overline{y}_{j},$ and from Lemma \ref{endoprop} (c), this is
better for him. Thus, in the subgame with budgets $(\overline{y}_{i},y_{j})$
either :

$(i)$ there is an equilibrium such that bidder $i$ wins the first object
instead of the second; or

$(ii)$ there is an equilibrium such that bidder $i$ wins both
objects.\bigskip

\noindent \textsc{claim 1:} $(i)$ \emph{holds only if }$\frac{1}{2}V^{I}\geq
V^{II}+\frac{1}{2}\alpha $ \emph{and } $\overline{y}_{j}=V^{I}-V^{II}.$

If any choice of $y_{j}<\overline{y}_{j}$ causes $i$ to win the first object
instead of the second, it must be that when $j$ chooses a budget of $%
\overline{y}_{j},$ bidder $i$ is just indifferent between winning $II$ and
winning $I.$ Hence, $V^{II}=V^{I}-\overline{y}_{j}$ or equivalently, $%
\overline{y}_{j}=V^{I}-V^{II}$ and by construction $\overline{y}_{i}\geq 
\overline{y}_{j}=V^{I}-V^{II}.$

Since bidder $i$ does not prefer to win both objects rather than just $II$,
we have: 
\begin{eqnarray*}
V^{I}+V^{II}+\alpha -\overline{y}_{j}-\min \left\{ V^{II},\overline{y}%
_{j}\right\} &\leq &V^{II} \\
\alpha +V^{II}-\min \left\{ V^{II},V^{I}-V^{II}\right\} &\leq &0
\end{eqnarray*}

If $V^{II}\leq V^{I}-V^{II}$ then $\alpha \leq 0$ implies that $\frac{1}{2}%
V^{I}\geq V^{II}+\frac{1}{2}\alpha .$ If $V^{II}>V^{I}-V^{II}$ then again $%
\frac{1}{2}V^{I}\geq V^{II}+\frac{1}{2}\alpha .$ Thus in either case we have
the necessary condition that $\frac{1}{2}V^{I}\geq V^{II}+\frac{1}{2}\alpha
. $ This establishes the claim. $\square \bigskip $

\noindent \textsc{claim 2:} $(ii)$ \emph{holds only if }$\frac{1}{2}%
V^{I}\leq V^{II}+\frac{1}{2}\alpha $ \emph{and } $\overline{y}_{j}=\frac{1}{2%
}\left( V^{I}+\alpha \right) .$

If any choice of $y_{j}<\overline{y}_{j}$ causes $i$ to win both objects
instead of the second, it must be that when $j$ chooses a budget of $%
\overline{y}_{j},$ bidder $i$ is just indifferent between winning $II$ and
winning both objects. Hence,$\ V^{II}=V^{I}+V^{II}+\alpha -2\overline{y}%
_{j}, $ or equivalently, 
\[
V^{II}=V^{I}+V^{II}+\alpha -\overline{y}_{j}-\min \left\{ V^{II},\overline{y}%
_{j}\right\} 
\]

Now there are two cases to consider.

If $\overline{y}_{j}>V^{II}$, then $\overline{y}_{j}=V^{I}-V^{II}+\alpha $
and thus 
\[
\frac{1}{2}V^{I}>V^{II}-\frac{1}{2}\alpha . 
\]

Since by choosing a $y_{i}$ large enough, bidder $i$ could have won object $%
I $ at a price of $\overline{y}_{j}=V^{I}-V^{II}+\alpha ,$ it must be that
this is no better than winning object $II$ for free. Thus $V^{II}\geq
V^{I}-\left( V^{I}-V^{II}+\alpha \right) $ or equivalently, that $\alpha
\geq 0$. Since we have assumed $\alpha \leq 0$ this can only happen if $%
\alpha =0.$ But then this is the same as the case considered in Claim 1.

If $\overline{y}_{j}\leq V^{II},$ then $\overline{y}_{j}=\frac{1}{2}\left(
V^{I}+\alpha \right) $.

Since by choosing a $y_{i}$ large enough, bidder $i$ could have won object $%
I $ at a price of $\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha \right) ,$
it must be that this is no better than winning object $II$ for free. Thus $%
V^{II}\geq \frac{1}{2}V^{I}-\frac{1}{2}\alpha $ or equivalently, 
\[
\frac{1}{2}V^{I}\leq V^{II}+\frac{1}{2}\alpha . 
\]

This establishes the claim. $\square $

Now suppose $(\overline{y}_i,\overline{y}_j)$ are equilibrium budget choices.

If $\frac{1}{2}V^{I}>V^{II}+\frac{1}{2}\alpha ,\ $then from Claim 2, $\left(
ii\right) $ cannot hold. Since at least one of $\left( i\right) $ or $\left(
ii\right) $ must hold, $\left( i\right) $ must hold and then from Claim 1, $%
\overline{y}_{j}=V^{I}-V^{II}.$

If $\frac{1}{2}V^{I}<V^{II}+\frac{1}{2}\alpha \ $then from Claim 1, $\left(
i\right) $ cannot hold. Again, since at least one of $\left( i\right) $ or $%
\left( ii\right) $ must hold, $\left( ii\right) $ must hold and then from
Claim 2, $y_{j}=\frac{1}{2}\left( V^{I}+\alpha \right) .$

If $\frac{1}{2}V^{I}=V^{II}+\frac{1}{2}\alpha $ then either $\left( i\right) 
$ or $\left( ii\right) $ may hold. In either case, Claims 1 and 2 imply that 
$\overline{y}_{j}=V^{I}-V^{II}=\frac{1}{2}\left( V^{I}+\alpha \right) .$

The prices are then determined by Lemma \ref{endoprop} and this implies that
the payoffs are unique. $\blacksquare $

\subsubsection{Complements}

\begin{lemma}
\label{endocomp}Suppose $\alpha >0.$ The budgets $(\overline{y}_{i},%
\overline{y}_{j})$ and the prices $(\overline{p}^{I},\overline{p}^{II})$
constitute an equilibrium outcome of the game with endogenous budgets if and
only if: 
\[
\begin{array}{lll}
\begin{array}{l}
\overline{y}_{i}\geq V^{I}+2\alpha  \\ 
\overline{y}_{j}=V^{I}-V^{II}+\alpha 
\end{array}
& 
\begin{array}{l}
\overline{p}^{I}=V^{I}-V^{II}+\alpha  \\ 
\overline{p}^{II}=0
\end{array}
& \text{if }\frac{1}{2}V^{I}\geq V^{II}-\frac{1}{2}\alpha  \\ 
&  &  \\ 
\begin{array}{l}
\overline{y}_{i}\geq \frac{1}{2}V^{I}+V^{II}+\frac{3}{2}\alpha  \\ 
\overline{y}_{j}=\frac{1}{2}\left( V^{I}+\alpha \right) 
\end{array}
& 
\begin{array}{l}
\overline{p}^{I}=\frac{1}{2}\left( V^{I}+\alpha \right)  \\ 
\overline{p}^{II}=0
\end{array}
& \text{if }\frac{1}{2}V^{I}<V^{II}-\frac{1}{2}\alpha 
\end{array}
\]
All equilibria result in the same payoffs (except for a relabelling of the
bidders).
\end{lemma}

The formal proof of Lemma \ref{endocomp} is similar to that of Lemma \ref
{endosub} and is omitted for the sake of brevity. It is available from the
authors.

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\end{document}
