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Sloan School of Management
Working Paper 4226-01
December 2001
AN INVERSE-OPTIMIZATION-BASED AUCTION MECHANISM
TO SUPPORT A MULTI-ATTRIBUTE RFQ PROCESS
Lawrence M. Wein, Damian R. Beil
© 2001
by Lawrence M. Wein, Damian R.
Beil. All rights reserved.
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permission provided that
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http://papers.ssm. com/paper.taf?abstract_id=295066
MASSACHu.cYTS INSTITUTE
OF "^CHMOLOGY
JL'M
3 2002
LIBRARIES
AN INVERSE-OPTIMIZATION-BASED AUCTION MECHANISM
TO SUPPORT A MULTI- ATTRIBUTE RFQ PROCESS
Daniian R. Beil and Lawrence M. Wein
Operations Research Center, MIT, Cambridge,
MA
02142, beil@mit.edu
Sloan School of Management, MIT, Cambridge,
MA
02142, lwein@mit.edu
Abstract
We
consider a manufacturer
which suppUer
attributes
will
who
be awarded a contract. Each bid consists of a price and a set of non-price
(e.g., quality,
The manufacturer
lead time).
form of the suppliers' cost functions
prior information
uses a reverse, or procurement, auction to determine
on the parameter
(in
is
assumed to know the parametric
terms of the non-price attributes), but has no
values.
We
construct a multi-round open-ascending
auction mechanism, where the manufacturer announces a slightly different scoring rule
(i.e.,
a function that ranks the bids in terms of the price and non-price attributes) in each round.
Via inverse optimization, the manufacturer uses the bids from the
learn the suppliers' cost functions,
attempts to maximize his own
myopic best-response bids
rounds
(i.e.,
and then
utility.
in the last
in the final
first
several rounds to
round chooses a scoring rule that
Under the assumption that suppliers submit
their
round, and do not distort their bids in the earlier
they choose their minimum-cost bid to achieve any given score), our mechanism
indeed maximizes the manufacturer's utility within the open-ascending format.
discuss several
enhancements that improve the robustness
the model's informational and behavioral assumptions.
December
6,
2001
of our
mechanism with
We
also
respect to
INTRODUCTION
1.
Although the market
at
$746B
in
for
2004 by Kafka
eCommerce landscape
onhne business-to-business auctions
severely hinder the range of products that can be auctioned over the
procurement
ized items are being transacted by current online
high-value complex items are
process.
enormous (estimated
2000), the price-only auctions that dominate the current
et al.
Internet. In particular, within the industrial
(RFQ)
is
An RFQ
still
setting,
procurement
many
low-cost standard-
(or reverse) auctions, while
being procured via the traditional Request for Quotes
process allows the sale to be determined by a variety of attributes,
involving not only price, but quality, lead time, contract terms, supplier reputation, and
incumbent switching
costs. It also lets the
the suppliers to compete on their
own
al.
for
his preferences
and permits
specialized dimensions. Consequently, eMarketplaces
are currently being developed to partially
process (see Kafka et
manufacturer reveal
automate the
RFQ
process;
i.e,
to create
eRFQ
examples).
This paper was stimulated by about eight hours of discussions (during the
and winter
an
of 2001) with the Chief
fall
of
2000
Technology Officer (CTO) of Frictionless Commerce, who
was seeking help with designing a multi-attribute
eRFQ
mechanism. The
CTO
described
the company's multi-attribute procurement software (we are not at liberty to discuss
details)
and the perceived needs and preferences of
who own
their software)
and the supplier companies
to various aspects of both traditional
to guide our
rough
first
eRFQ
and electronic
their
customers
(i.e.,
its
the manufacturers
(i.e.,
the potential bidders) with respect
RFQ
processes. This information helped
design and our assumptions about supplier behavior. After receiving a
draft of this paper, the
CTO
impressions are briefly summarized in
their
for this setting is the multi-attribute, or
mul-
its
ideas with several customers,
§4.
The appropriate mathematical context
tidimensional, auction; consequently,
and
shared
we often
refer to the
manufacturer as the auctioneer
and the suppliers
(or bid-taker)
as bidders.
There are two primary objectives
in the auc-
and
(allocative)
tion theory literature, revenue maximization (on the part of the auctioneer)
efficiency;
in multi-attribute auctions,
is
it
appropriate to speak in terms of utility max-
imization rather than revenue maximization.
auctions, whereas utility maximization
efficient
is
Efficiency
is
often the goal in public-sector
typically strived for in private-sector auctions.
auction mechanism maximizes the total surplus, without concerning
this surplus
is
divided
among
itself
with how
the bidders and the auctioneer. In complex auctions, these two
objectives are often in conflict
(e.g.,
Bikhchandani 1999), because the auctioneer can usually
increase his utility by either withholding items for sale or by allocating items to those
do not value
it
in a multi-attribute auction,
some information pertaining
to
how he
an auctioneer needs to provide the
values the non-price attributes. While
several rather obtuse approaches are possible (e.g., the auctioneer could provide
prices
shadow
from a mathematical program without revealing the mathematical program), the
predominant approach
its
who
the most.
To engage bidders
bidders with
An
in
RFQ
straightforward nature -
is
practice -
for the
bid price and various attributes.
and the one favored by most bidders because of
auctioneer to announce a scoring rule in terms of the
This scoring rule may, or
auctioneer's true utihty function; indeed, this
is
may
not, be identical to the
the crux of the strategic problem from the
auctioneer's viewpoint.
There are two key papers on multi-attribute auctions with scoring
ing each objective.
Milgrom (2000a) has recently shown that
rules,
efficiency
is
one address-
achieved
if
the
auctioneer announces his true utility function as the scoring rule, and conducts a Vickrey
(i.e.,
second-price, sealed-bid) auction based on the resulting scores.
paper
is
The
other important
by Che (1993), who considers a two-dimensional price-quality procurement auction
in a sealed-bid setting.
He assumes
that the suppliers' quality costs are a function of a
parameter that
single
private information
is
and independent across suppliers (Branco 1997
generalizes this research to the case of correlated supplier costs).
utility,
the auctioneer -
who
He shows
that to maximize
only knows the probability distribution of the cost parameter -
announces a scoring rule that understates the value of quality, so as to limit the informational
rents collected
by the low-cost suppliers.
This paper
tion
motivated by two opinions regarding this literature:
is
the appropriate objective in
is
results, while elegant
ficiency
2000a)
may be an
is
and
many
if
utility
industrial procurement settings,
insightful, are not practically
appropriate objective
(i)
the goal (as
implementable. More
is
and
maximiza(ii)
specifically, ef-
the case of Perfect.com in Milgrom
to develop software to support an entire eMarketplace, which needs to attract both
auctioneers and bidders. Moreover, Milgrom (2000a) and Wise and Morrison (2000),
others,
develops
its
our view that for a large manufacturer that either
software in-house or buys
maximization on the part of the auctioneer
still
unlike the
if
it is
own procurement auction
the short term and
bidder
among
warn that utility-maximizing auctions may chase suppliers away from the market-
place in the long run. Nonetheless,
utility
Che's
medium
is
it
from an external vendor,
the appropriate objective, at least over
term. Also, in our utility-maximizing mechanism, the winning
earns the amount by which he exceeds his most able competitor, which
outcome
in
many RFQs,
contracts and auctions.
Moving
is
not
to our second point, even
Che's utility-maximizing results could be extended to multiple attributes, we believe his
key informational assumption - that the auctioneer knows the probability distribution of the
bidders' cost for non-price attributes is,
we do not think
it is
and to
one or several parameters.
on
not amenable to practical implementation. That
possible for a manufacturer to have a precise a priori estimate of the
suppliers' cost functions
rule based
is
reliably
summarize them
as a probability distribution over
Consequently, any attempt to strategically change the scoring
this estimated probability distribution
would have a reasonable likelihood of
backfiring
(i.e.,
of resulting in lower utility for the manufacturer), particularly because the
optimal scoring rule depends only on the top few bidders' cost functions, which requires a
precise estimate of the tail of the probability distribution.
Consequently, our goal in this research
is
to
employ Che's strategic ideas while
si-
multaneously enabling the manufacturer to learn the suppliers' cost functions. This paper
proposes a forward- and inverse-optimization-based approach {inverse optimization deduces
some
of the unobserved parameters of
an optimization problem from the observed solution)
that allows the manufacturer, via several changes in the announced scoring rule, to learn the
suppliers' cost functions
and then determine a scoring
the open-ascending auction format.
make
istics
that
1998
for details)
it
We
rule that
maximizes
choose this format because
it
his utility within
has
many
character-
superior to sealed-bid auctions from a practical standpoint (see
and because
it
is
the most
common
Cramton
type of procurement auction on the
Internet.
RFQ
processes are typically less structured than auctions.
Although changing the
scoring rule during the course of a traditional auction would be perceived by
as unfair, this practice
is
not
throughout the course of an
uncommon
RFQ
in
RFQ
processes.
The
scoring rule
process for a variety of reasons,
e.g.,
many
bidders
may be changed
the buyer
may
learn
from supplier presentations that the importance of certain attributes has been misestimated,
or the buyer
may want
a certain supplier to be awarded the contract and needs to alter the
scoring rule so as to enable this supplier to attain the highest score. Consequently, several
commercial
eRFQ
of the process.
software packages allow changes in the scoring rule throughout the course
Not only can the scoring
rule
change over time, but neither the suppliers'
bids nor the manufacturer's scoring rule needs to be binding.
Nonetheless,
RFQ
processes
maintain a certain amount of structure, because reputations are diminished by too much
non-committal behavior.
We
believe that our basic
mechanism - by learning the
suppliers'
eRFQ
manufacturer's utility in an
assumed
in
described in
is
if
the
We
2.1.
Notation.
may
represent six
P
structured than
is less
THE MECHANISM
months
buying a single item. Although
where a
=
1,
The
majority, but not
.
.
.
,^4
fifth
subscript
up
indexes the attributes,
cost (and utility) parameters per attribute,
the auction; the
we assume that
of production for a subassembly,
as a single item.
it
is
are for a single item. Because our notation requires
the
process
in §4.
assume that the auctioneer
and we model
subscripts,
RFQ
Section 3 discusses several practical con-
§2.
and concluding remarks are offered
2.
bidder,
setting, even
our model.
The auction mechanism
siderations
rule - has the potential to increase a
and strategically setting the scoring
cost information
is
and
r
it is
indexes the
=
1,
.
P-l-
,
.
.
sold to a single
RFQ processes in practice
all,
to five subscripts,
s
S
1
we use mnemonic
suppliers,
introduced in §3.2. To limit the notational complexity, we
variables sometimes appear with two subscripts,
hope that these inconsistencies are more than
For multi-attribute auctions,
(i.e.,
it is
offset
that
exogenous attributes to
and Xa
is
all
§3.5.
variables (thus the
and
quality, versus attributes that
For expositional
non-price attributes are endogenous, and defer a discussion of
Each bid
we assume
domains
We
important to distinguish between attributes that are
is
of the form {p,Xi,
the magnitude of non-price attribute a for a
sentation and analysis,
certain
by improved readability.
bidder-controllable), such as lead time
we assume
e.g.,
and other times with three subscripts.
are exogenous, such as a bidder's reputation at the time of the auction.
purposes,
p indexes
indexes the rounds of
frequently suppress subscripts that are not crucial to the immediate discussion;
endogenous
item
this
=
.
.
1,
.
.
,Xa), where
.
,
.
^.
To
p denotes
price
simplify the pre-
that the non-price attributes are continuous, nonnegative
of the cost, utility
and scoring functions are the nonnegative
real
numbers) and that larger values of Xa are more desirable from the auctioneer's point of view
and more costly from the
suppliers' point of view. Hence, attributes such as tolerance or lead
when low
time, which are desirable and costly
magnitude, need to be defined relative to a
in
worst-case upper bound. Supplier s's cost function
by lZa=i
'^as{xa,Oasi
additive across attributes, and
we assume
that Cas
assumption
is
and our example
three,
+
the form Oasi^a
I
das2Xa
+
all
in §2.6 considers a
a and
s;
we expect
§2.4. In practice,
We make
^as3-i"a^-
given
For ease of presentation,
cost parameters for
end of
easily relaxed, as described at the
would typically equal two or
Cas of
P >
a function of exactly
is
is
convex (convexity and concavity are
Cas IS increasing,
paper) and twice continuously differentiable in Xa-
strict in this
function
wherc
^asp),
is
this
that
P
three-parameter cost
the crucial assumption that the
auctioneer knows the form of the suppliers' cost functions, but does not have any information
about the parameter values
a procedure that chooses
The
{6asi,
among
•
•
,
This issue
Oasp)-
is
alternative functional forms.
assumed to be additive
auctioneer's true utility function and scoring rules are
across non-price attributes, and (except for the scoring rule in round
exactly
P
of the cost
parameters
and
for
likely
utility functions
be too arduous
given by X^^^^ Vai^a,
V'ai,
and twice continuously
different scoring rule,
and scoring
,
'^ai^a, <Pair,
prior to the final
a,
but
•
,
make
'4-'ap)
and the number
— P-
to consist of
different
implementation.
Va
(mnemonic
Each round
of rounds
The
makes the problem more
tractable, existing
is
The
true utility function
for value) is increasing,
of the auction
exactly one
is
=
1,
.
.
concave
characterized by a
more than the number
scoring rule in round r
is
.
,
P
is
of cost,
denoted by
Noticc that, for simplicity, we assume that the scoring rules
round are identical
may employ
1)
these assumptions, and nonseparable functions
~P) where
differentiable in Xa-
(f^aPr)
rules
for industrial
or utility, parameters per attribute.
Yla=\
P+
each non-price attribute. While the separability across attributes
multi-attribute software packages also
would
where we propose
revisited in §3.2,
in
form to the true
parameter values.
We
utility function for
a given attribute
relax this assumption for the scoring rule
in the final
{(pair>
^<t>aPr)
•
m
concave
"°^^°
problem
(as discussed in §2.5),
P+
round
rule in
and
round
—
Q^*"''
=
1,
.
One
.
,
.
P,
final,
is
P+
consist of
P+
1
a
»
However,
P-
—
cxd,
9casM,u...,e^,p)
^
1, i.e.,
—
r,
rule
The
as Xa -^ U
paper,
Vs,
the
g^
same
is
assumed
for the generic
^^ ^^ _^ q+ ^^^ ^^^^
dfj^
{(pair,
_^ q ^^
(paPr),
,
auctioneer initially informs the bidders that the auction will
rounds; our use of multiple rounds
repeated in the
envision this
new
increasing and
is
dx
stages.
final
is
not unlike traditional
^^=1
.
.
'"ai^a, (pair,
.
,
(paPr)
—p
r
to
processes,
—
all
1,
.
.
.
,P,
suppliers.
,xa) in an open-ascending manner; the
round when scoring rule ^a=i
mechanism taking place
RFQ
At the beginning of each round
—p
fa{-^'a)
is
announced.
electronically (e.g., over the Internet).
each round, suppliers have ample opportunity to bid, and each supplier
a
the scoring vector
delayed for expository purposes until §2.4.
Suppliers then submit bids of the form {p,xi,
We
a and
to guarantee that the solution to the optimization
the auctioneer announces the scoring rule
is
<
a^
finite positive solution;
which typically consist of multiple
same process
for fixed
must be such that the scoring
fa
—
~
more technical assumption on only the scoring vectors
Basic Outline.
2.2.
as Xa
>
in (l)-(2) possesses
Xa —> oo.
/a(-^'a)
assume that
also
scoring rule in round
r
as "^^=1
and the function
We
Xa-
1
and denote the generic (unparameterized) scoring
is
Within
forced to
make
bid during each round in order to proceed to the next round (at this point in the
we do not
rule out the possibility that a supplier simply re-submits
other activity rules and transition rules are briefly discussed in §3.4.
The
an
earlier bid);
auctioneer ranks
the bids according to the current scoring rule and displays the ranked scores, but does not
reveal the bidders' identities or detailed bids.
In contrast to a traditional open-ascending
auction, submitted bids need not exceed the current best bid. However, there
is
a
minimum
bid increment (with respect to the scoring rule) to take the lead (thereby speeding up the
auction, at least in the final round), and
we assume that the highest bidder
at the
end of
round
P+
The
1
wins the contract, at his proposed bid.
mechanism
analysis that provides the basis for our
which are described
how the
how
given their bids, and
main
how the supphers bid given the
in the next three subsections:
scoring rule and current best score,
consists of three
parts,
current
auctioneer estimates the suppliers' cost functions
the auctioneer determines an optimal scoring rule once he learns
the suppliers' cost functions.
scribe the notion of
profit,
A
supplier using
MBR
to denote the score
minimum
we need
to de-
chooses his next bid to maximize his current
assuming no other supphers change their
ending after his bid. More
the
specify supplier behavior in our model,
myopic best-response (MBR) bids and introduce the weaker condition
of undistorted bids.
S
To
Supplier Behavior.
2.3.
specifically,
in
if
and the number of
bid increment
round r
i.e.,
< P
he behaves as
s solves
if
the auction was
the current top score
suppliers, but this should cause
then supplier
is e,
bids:
is
S
(we use
no confusion) and
the following optimization problem
(note that the subscripts s and r are suppressed in the decision variables Xasr and
Pst)'-
A
p-y^ Cas{Xa,dasl,-
max
,&a3p)
(1)
^—
a=l
P,X1,...,IA
A
subject to
y^ VajXa, (pair:
(paPr)
•
-P= S+
t.
(2)
a=l
Under the same circumstances
in the final round, supplier s
would solve
(1) subject to
A
Y^Ia{Xa)-p = S +
e.
(3)
a=l
In either case,
if
the optimal objective function value in (1)
responding optimal solution
negative, then the
The
mange
MBR
et al.
MBR
is
is
the
MBR
to not submit a
assumption has been used
1986.
Wellman
et al.
bid.
new
If
is
nonnegative then the cor-
the optimal objective function value
bid.
in a variety of recent auction studies (e.g..
1999. Parkes
9
is
De-
and Ungar 2000, Galhen and Wein 2000,
although the
two studies
first
MBR
refer to
On
middle ground with regards to the bidders' rationality.
to be sophisticated
enough
to formulate
and solve
prior distributions
and the auctioneer's
and would account
would be solving
their
able and tractable
that there
may be
bidders
who do
difficult
the other hand, a
for the fact that these
modeling question,
it is
the
other players
MBR
as a reason-
important to point out
not even use an optimization-based mental model for their
bidding (this was the opinion of the
exposures to this mechanism,
On
on the other bidders' cost functions
own game-theoretic problems. While we view
compromise to a
asserts a
the one hand, they are assumed
(l)-(2) or (1), (3).
more astute bidder would formulate
utility function,
and
as "straightforward bidding"),
may
CTO
we spoke
realize that the first
purposes of learning on the part of the auctioneer, and
intentionally distort, their cost information. In §3,
and others
with),
we
that,
upon repeated
few rounds of bidding are
may
for the
place bids that withhold, or even
ways that our mechanism can
discuss
be enhanced to mitigate this danger.
Before specifying supplier behavior, we introduce a weaker condition that
torted bids.
bitrary score 5,
we say that
right side of (2) replaced
solve (1), (3) with
may withhold
(i.e.,
< P
a supplier submits a round r
If
S
it
is
by S.
undistorted
Were
bid
(p, Xi,
this bid
if
the bid in round
replacing the right side of
(3).
A
.
.
.
,.r.4)
we
call undis-
that generates the ar-
the solution to (l)-(2) with the
is
P
-t-
1,
an undistorted bid would
suppher submitting undistorted bids
absolute information about his cost parameters by not bidding aggressively
even though the
MBR bid may allow him to take the lead,
withhold this bid
for strategic reasons),
cost parameters.
We
he nonetheless
may
choose to
but does not withhold relative information about his
believe that bid distortion
(i.e.,
for
a given score, purposely choosing
a suboptimal bid to deceive the auctioneer about the relative contributions to the supplier's
cost function)
is
considerably more sophisticated and risky than the strategy of withholding
absolute information about the cost function, and
10
is
much
less likely to
be engaged
in
than
the latter strategy, particularly
know
suppliers
the
number
if
activity rules are in place (see §3.4). Moreover, because the
of rounds in the auction, they are unhkely to withhold absolute
cost information in the final round of bidding, because doing so could cause
them
to lose the
auction.
These arguments motivate our main assumption about supplier behavior: suppliers'
bids are undistorted in rounds
Our
earlier
1,
.
,
.
.
P, and are
MBR
in
round
P+
1.
assumptions about the cost function and scoring rules imply that,
for r
<
P,
the solution to (l)-(2) can be found by solving (2) for p, substituting for p into (1) to get an
unconstrained optimization problem, and solving the first-order conditions
=
for
OXa
Because the first-order condition
undistorted bids satisfy
all
a=l,...,A.
(4)
OXa
have the same magnitude
(4).
independent of the right side of
(4) is
Hence,
all
(2), it follows
that
bids by a given supplier s in a given round r
for their non-price attributes; these
their price. This observation provides a simple
way
undistorted bids differ only in
to empirically validate or invalidate the
undistorted-bid assumption.
Analogously, the solution to
IbT^
If
we
let
P
round
(1), (3) in
-I-
1
must
a
^°'
di
=
satisfy
l,...,A.
(5)
X* denote the solution to (5), then the corresponding bid price in round
P
-I-
1 is
A
P*
= E/'^«)-'5-^-
(6)
a=l
If
P*
>
X^a=i
otherwise, the
2.4.
'^as{x*a, Oasi,
MBR
is
•
•
•
.
dasp) then {p* X*,
to not submit a
Cost Estimation. Because
at least
one new bid
,
in
new
.
.
.
,
x^)
is
the
MBR
bid in round
P
-f 1;
bid.
bids are undistorted and each supplier
each round, at the end of round
11
P
is
forced to submit
the auctioneer possesses, for
each attribute a and each suppHer
in
P unknown
terms of the
scoring vectors
{(pair,
,
P
the
s,
cost parameters.
=
for f
times
1,
if
—
1;
in practice, this could
and
s,
first
and suppher
fixed attribute a
=
1,
^<^'^^asr<^a\T,-,(PaPT)
P
Pas,
the
F, these
,
P
This
a.
va(x^^f,^^ir,-.-,<t>asr)
^
^
fail
the condition. Per the above, at
we
discuss
scoring rules might be determined in practice.
if
the
number
of parameters per attribute, F, varied by attribute
then we would learn the parameter values for attribute a and supplier
round
.
rounds, the auctioneer knows the suppliers' cost functions. In §3.3,
Note that
plier,
.
.
s
be enforced by simply perturbing (perhaps several
needed) scoring vectors that would otherwise
P
1,...,P,
true cost parameters for attribute
s's
satisfied as long as, for fixed a
...,;
the end of
how the
is
If for
=
with r
(4)
induce a different x* for each round r
4>aPr)
equations can be solved to obtain suppher
requirement
equations given by
and hence the number of rounds required to learn
all
and sup-
s at the
end of
suppliers' cost functions
is
^^'^^a,s ^as-
2.5
The Optimal Scoring Function
Round P +
in
1.
With the
true cost functions in
hand, the auctioneer can determine his optimal scoring rule for round F-lfor
now
that the auctioneer chooses the generic scoring rule fa for attribute
we
introducing more terminology,
rules,
By
MBR assumption,
sufficient,
suppher
Notice that supplier
,
.
.
.
,
/^
s will
which occurs at the
s will
Ss
,
.
.
.
,
'^as(^'ai ^asi,
•
•
,
F
-f I's
we
"^^=1
is
In §2.1
to avoid
refer to /i
,
.
fai-^'a)
— P-
we imposed
possess a unique
(1), (3) to
.
.
,
/^ as
feasible.
open-ascending competition no
^asp) equals zero (where x*
is
the solution
drop-out score
A
= Y^ faiK) - Y^ Cas«, Oasl,.
a=l
(1), (3).
/^ for problem
a;
individually, as scoring
auction in the final round
submit bids that solve
/i,
and
/^, collectively
.
satisfies these conditions,
maximum
A
.
drop out of round
than when his profit p — X]a=i
to (5)),
.
for the
but not necessary, conditions on
finite positive solution; if /j
later
refer to /i,
although the actual scoring rule
the
Let us suppose
1.
a=l
12
,
Oasp)-
(7)
Our
analysis below, culminating in (13), depends only on the top two bidders,
maximum
the suppliers so that their
drop-out scores in round
upon our choice
Ss\ note that this ranking depends
>
52
on the scoring rule
minimum
Si
<
S2
for
P+
round
^-
That
we
is,
>
Depending on the detailed sequence
—
in the interval (52
e,52
+
e].
52
analysis simple,
+
,
>
.
by
at
most
e,
which
is
we ignore the
I's
>
and exclude the
effect of the
possibility that
e.
(9)
winning score
analysis cleaner,
we assume
may
anywhere
lie
that the winning
final
dwarfed by the magnitude of the bids. With this assumption,
the winning score in the open-ascending auction will be submitted by supplier
equal 52; supplier
.
To guarantee
bidder wins with a score equal to 52; this assumption miscalculates the auctioneer's
utility
.
Ja to satisfy
of bids, supplier I's
To make the
^2
(8)
require the scoring rule f\,
5i
>
constraint
bid increment on the detailed sequence of bids,
+
S*!
e
To keep our
I.
satisfy
of generic scoring rule.
we impose the
that these two suppliers can actually bid,
P +1
and we index
winning bid
is
1
and
will
the solution to
A
max
P-
P,Xl,...,XA
y]cal(Xa,6lall,...,6'alp)
'-^^
(10)
a=\
A
subject to
By
(6)
and
solution
(7), this
solves (5) for s
=
1,
is x-*^
(we
^
now
fa{xa)
-p =
S2.
(11;
include the supplier subscript in the bids), which
and
AAA
A
Pi
= J]/a(-<l)-52
= E/»«l)-E/"«2) +
a=l
a—1
E^''2«2,^a21,...,^a2p).
a=l
13
(12)
Recall that the auctioneer's true utility function
is
^a('^a'''^ai,
X3a=i
equation (12), the auctioneer's optimal scoring rule in round
P+
1
•
is
—
,^ap)
Using
P-
the feasible scoring
rule that solves
A
A
max
Y^ ^'^(^a\
'
V'al
,
,
" J^
(PaP)
A
/-^(-^'"l
)
+ J2
f^(^'a2)
a=l
A
(13)
-E^'^2«2,^a21,...,^a2p),
a=l
subject to constraints (7)-(9).
possible that the auctioneer's true valuation function
It is
violates equation (9); in spite of - or rather, because of - such "near ties," the auctioneer
extracts nearly
the surplus in the auction by revealing his true valuation function as
all
the scoring rule. In this case, solving (13) subject to (7)-(9) can do no better than simply
announcing the true valuation function as the scoring rule
and - despite
We
of the
S
violation of (9) -
its
we consider v
(see the proof of Proposition 2),
to be optimal.
re-emphasize that although equations
(13)
(7)- (9),
depend on only the top two
suppliers, the rankings of the suppliers in this optimization
of the decision variables
solve (7)-(9), (13) for
all
A
scoring rule).
(i.e.,
and
1
2(2) ordered pairs of supphers,
discussion below
2 below,
we do not
we
refer to
derive a
presentation,
a feasible scoring rule
attribute levels xi,
and
utility v{xi,
.
.
.
.
.
.
fi,
,
xa
,xa)
ease of presentation,
—
,
we
more
>
call
/a satisfying
at price p.
p,
a function
is
to
and the ordered pair that generates
approach to
efficient
P+
I.
With
the aid
this problem.
In the
a specific scoring rule, and therefore drop the assumption
that the suppliers are ordered such that Si
To structure the
is
brute force approach to the problem
the highest utility in (13) provides the optimal scoring rule in round
of Propositions
problem
We
S2
>
a bid
(8)- (9)
>
(p, Xi,
Ss-
.
.
.
,xa) enforceable
if
there exists
that causes the auction to be
say that such a rule enforces bid
won with
(p, Xi,
.
.
.
,
xa),
where v denotes the auctioneer's true valuation function. For
we omit the parameters from the
14
suppliers' cost functions
and the
auctioneer's true valuation function, and let Cs{x)
where
=
;r
(xi,
.
.
.
Proposition
,xa)-
1
and
—
'^a=i^as{xa) and v{x)
—
J2a=i'"a{xa),
subsequent nonobvious results are proved
all
in
the Online Appendix.
Proposition
i.e.,
<
Ci{x)
(Enforceability). Let supplier
1
at x,
and
Then
{ci{x)
+
7r,x)
^
Ti
+
some
J
+
c,{z)<T,{z)
Corollary
Ci{x)
Cj,
at
+
e
supplier
let
i
<
^
s
Cs{x),
1
is
n
profit
i 's
satisfy
enforceable if
>
tt
and only
's
(Minimum
where
if Cs{x)
Prices). Let supplier
^
i.
Cj(x)
+
If for
minj-_s^i{cs(£)
applying Corollary
will ignore the arbitrarily
words, Corollary
mins's^t{cs{z)
e,
t
>
cost surface Cj
some
/
j
—
1
1
—
i,
+
then enforceable prices at x approach Cj(x)
When
=
(xi
,
.
.
,
.
xa)
minimum
the
is
+
Cj(x)
(i.e.,
if
tt
for
,
Ci
bid increment.
all s
^
and for
i,
there exists a z such that
n).
Cs{x) for all s
X approach
atx
Let T, he the hyperplane tangent to supplier i's cost surface
i.
intersects supplier j
IT
be the low-cost supplier
i
e
be the low-cost supplier at
i
Tj
+
intersects supplier j
e
from above.
T^{z)}
+
5,x)
enforceable for
is
small 5 and for practical purposes consider
i's
cost surface
from above; otherwise, enforceable prices
to cases in which {p
states that supplier
's
x such that
profit,
tt,
will
be
e if
T,
+e
{p,
(5
—+
O"*",
x) enforceable. In
intersects
some
Cj,
and
Ti{z)} otherwise. Hence, this result allows us to put the price tag Ci{x)
on any /l-tuple of non-price attribute
levels,
which quantifies how
we
is
+ tt
effectively competition
can, or cannot, be exploited via a properly chosen scoring rule. This result transforms the
problem from a "what if we choose scoring rule / approach, to an "informed shopper"
approach of
utility
maximization given prices
Cj(x)
+
tt.
Notice that the prices themselves
are not market prices in the traditional sense, but rather the prices of idiosyncratic markets
distorted for hypercompetition (lowering supplier
Illustrations of enforceability with
i's
profit
tt).
two suppliers are provided
§2.6 for a two-dimensional (e.g., price, quality) problem.
15
Near
q
=
in Figures
1
and
2 in
0.86, the prices that are
enforceable in Figure 2 are lower than those of Figure
hyperplanes) to
much
permits prices
=
near q
cj
0.86 are
maximization problem
we say that
the tangent lines (one-dimensional
closer to C2 in Figure 2,
and by Corollary
present a result that incorporates the auctioneer's utility
in {7)-(9), (13). If supplier s
supplier s
is
wins the auction under an optimal scoring
optimal.
Proposition 2 (Restricting Search Over Suppliers). For suppliers
Ms =
—
maxf{t'(x)
supplier s
's
supplier
is
i
Cs(iJ')},
where v
some
we note
Corollary
Cs, s
^
i,
at
.r*
1,
To
supplier
and
=
tt
^
+
utility possible
?'),
intersects
^
s
main
1,...,5,
Ms, for
and
let
Cg
is
Then
all s.
the auctioneer reveals his true
intuition behind the proof of Propo-
i
min£^s^,{cs(5')
T^
+
Mi —
e;
s 7^
—
intersects
them T„ and
Ti{z)} \iT^
+
e
does not intersect any
in utility.
and are
A/j
—
units apart, this utility
s y^
winning.
i
Ci{x*)
+e
since any winning supplier
we
tacitly
Since Ty and T,
A/,
—
e
assume that
< Ms)
is
,
m
must receive
A/j
—
16
sandwich v
tt
bounds from above
some
s
^
i,
which case the auctioneer walks
e
> Mg
tie."
Ms, while the payoff
both cases the auctioneer's
+
In the latter case where for
at x*
By
Tj) are parallel.
— n
tt
optimal solution
In the former case, the auctioneer
c^.
which announcing v leads to a "near
wins (where
in
t
(call
its
A/j
with supplier
In the above
— {Ml — Ms);
if
and receive
respectively,
payoff
A/j
=
is tt
some
tt,-x*)
trivial case in
i
profit
for
have the
if
describe the
q, we can enforce price
utility
result follows.
i's
e if
any
away with
drop-out score
which v and q's tangent hyperplanes
Cs (for s
e
>
that the maximization problem that defines A/j has
and
+
loss of generality) that A/,
maximum
supplier s's
is
can enforce {c^{x*)
T^
=
optimal.
valuation in the scoring rule.
at
s
the auctioneer's true valuation function
is
Suppose (without
cost surface.
Note that Mg
sition 2,
this
1
closer to supplier I's true cost curve.
We now broaden our view and
rule,
much
1;
utility is at least
profit of at least
for all s 7^
To
if i
i;
if
e,
not,
the
we
see this, note that the
wins
Mi —
e,
is
no smaller than
which bounds any
The
solution to (7)-(9), (13) from above.
and
i
and
s are
considered an optimal scoring rule
is
both taken to be optimal suppliers.
In the remainder of this subsection,
method
function v
for finding the
we apply
these results to construct a three-step
P+
optimal scoring rule in round
Proposition
I.
1
and
its
Corollary
reduce the problem to one of utility maximization given prices, but the prices are determined
with respect to low-cost supplier
method's
step
first
when computing
Step
is
that Proposition 2 allows us to
r
=
argmaxs=i_..._5 M^;
Ms > M, —
e,
=
s
Va{Xa,
i/'al,
.
•
•
,
Ipap)
^
"
i
is
behind the
.
5, let
,
>!
OasP)
(14)
>
J
If
announce the true valuation function
there exists an s
7^ i
such that
and
in the scoring rule
exit the
2.
utility
given price, which by Corol-
given by
1 is
A
max
subject to
A
"Y^VaiXajIpal,-a=l
A
^
Cas{Xa, O^sl,
,'ipap)
,
TT
>
is
,9aip)
-n
(15)
A
Oasp)
>
^
Ca,(.Ta, 6ail,
,
Oaip)
+
£,
S t^
I,
(16)
(17)
e,
<
^^ Cas{za, dasi,
the hyperplane tangent to supplier
variable profit.
.
a=l
n > minmin
Tj
-'^Cai{Xa,9atl,.
a=l
a=\
i's
,
.
Cas{Xa, dasl,
the optimal suppher.
Step 2 (Find a Best Competitor): Maximize
where
.
a=l
three-step method. Otherwise, proceed to step
lary
1,
A
A
^
U=l
I
^
i
crucial idea
the identity of the low-cost supplier
fix
(Choose an Optimal Supplier): For
1
The
j.
prices.
Ms = max
Set
and competing supplier
i
In §A3,
we simpUfy
,
i's
dasp)
-
Ti{z)
(18)
,
cost surface at x
and
tt is
supplier
(15)-(18) by finding a closed-form solution
(equation (48) in §A3) to the innermost minimization in (18).
17
>
We
will call
any suppher
j
who
achieves the minimization in (18) in the optimal solution (and thereby enables
the optimal solution to be enforced) a "best competitor" to supplier
Step 3 (Choose Optimal Scoring Rule): The derivation
i.
of this scoring rule
is
given in §A4. Let x*, n* denote an optimal solution to (15)-(17), (48). There are three
cases to consider.
is
equation (17)
If
is
tight at the optimal solution, then the scoring rule
the function / constructed in Claim 5 of the "=>" direction proof of Proposition
in §A1.2. For fixed
dimension
a, this
1
optimal scoring function fa has the form
^al^C
if
OCa
<
ZaU
"^aya^a"*
if
^a
>
2q2,
fa{Xa)
where three of the ten parameters
,uJaS, •Zai, Za2
u)a\,
are actually redundant, but are
included here to preserve readability. This function enforces optimal bid
(
J2 Ca,K, e^n,between
in a two-supplier auction
i
and
j.
.
,
ea^p)
+
^T*,
X*
(20)
J
If
equation (17)
is
not tight at the optimal
solution, then - as explained in §A1.2 - the optimal scoring rule
supplier
actually losing
i
when /
is
announced to
all
S
suppliers.
must
The
scoring rule in this case depends on whether the true valuation v,
rule, satisfies or violates constraint (8). If v satisfies (8)
is
A*/
+
(1
which has
(20).
is
—
8-l-P
Otherwise,
A*/
§A1.2.
-I-
(1
—
where A*
X*)v,
is
if
(17)
A*)^,
The optimal
is
/,
P
from
v,
non-binding but v violates
where /
is
given in (19), A*
is
choice of optimal
used as a scoring
then the optimal scoring rule
given in (46) at the end of §A1.2.
parameters (seven from
if
buffer against
This scoring
rule,
plus the parameter A*) enforces
(8),
then the optimal scoring rule
given in (46), and g
is
defined in
scoring function in this pathological case - in which at most one
18
supplier can bid below the true valuation - requires 18 parameters to enforce (20); the
function g has a form similar to (19), but with the middle case repeated, for a total of
ten non-redundant parameters.
A
2.6.
S =
example that has
F=
suppher
=
and
I's
3, 9i2
=
V'3
We
2,
=
and
0,
+
=
^13
see Figure
0;
Aq^^
The
cost of quality
for the attribute
and suppher
1.
non-price attribute, which
2's cost is
3q
true value function
for a plot of the cost
1
4 rounds of bidding, the
first
4q^-^ in
=
03r
round 3
for r
=
(i.e.,
1, 2, 3.).
0„ =
2, (p2i
=
These scoring
is
and use q
is
+
we
form 9s\q
of the
^n =
q^^; i.e.,
ip^q^^
+ 9s2Q^ + ds2,q^^
More
in place of Xj.
0, 612
+ 039, where
and
call quality,
!/'!
specifically,
=
=
1,
^13
5, •02
=
=
4,
0.9,
and value functions. Our auction mechanism
three for learning and the last for optimizing.
assume that the auctioneer announces the scoring
and
and
is
P+1 =
requires
q^
cost
1
The
3 cost parameters per attribute.
mechanism with a simple numerical
illustrate our
^ =
2 suppliers,
where we suppress the subscript a
^11
We
Numerical Example.
1/2, (pu
rules
rule 2y/q in
=
3.5, 022
were chosen
round
=
1,
0.7, 0i3
arbitrarily,
3.5q°^ in round
=
4,
=
023
0.8,
but do satisfy the
requirements of §2.1 and the requirement of §2.4 related to non-redundant bids (see below);
we consider
later in §3.3
minimum
that the
for
round r
bid increment
is (jjj
=
quahty
(21) for
1,
=
is e
0.1.
Suppher
s's
We
also
undistorted bids satisfy
assume
(4),
which
is
^,1
In round
a systematic approach for determining these rules.
+
3^,2(g:.)'
+ l^e^MrY' =
02.01,.(</:J^^^-'
supplier 2's undistorted bids have quality
0.1111.
is Qj.,
=
Similarly, in
round
0.5085; in round
2,
3, 993
suppher
=
(^ji
=
0.6265,
2's quality is
0.7714 and q^^
=
g.22
1.1774
0.0215 \
1
1.6721
0.2506
1
1.7852
0.3963
/
^21
\
=
(21)
03..
and supplier
I's
quality
0.7466 and suppher
0.7681. For fixed
rounds 1-3 yield a linear system with three unknowns;
1
+
s,
I's
the equations
for supplier 2, the
system
is
/ 1.2634 \
2.6745
\^
3.3705
(22)
J
Solving (22), the estimated cost values
The equations
623-
for supplier
With knowledge
A/ +
rule
(1
-
M2 =
optimal. Because A/i
q*
,
71"*,
are omitted, but the analysis
1
identical.
is
is
the final round of bidding [v satisfies (8)).
1.6823, which
more than
Mi =
smaller than
is
we
units greater than M2,
e
= {^y^^\ and
max
=
C2(0)
Vig"^'
and
3
c'i(0.7593)
=
-
^39 - ^119 - ^129^
+
we
3,
In the
first
step, the
3.4375; hence, suppher
1
is
solve (15)-(17), (48) to find
the optimal quality level and profit at which supplier
(c2)-i(x)
^21, ^22,
of these cost parameters, the auctioneer chooses the optimal scoring
A)(' for
auctioneer finds
coincide with the true values of
621, 622, ^23
1
wins the auction.
Since
solve
^139^^
-^
g>0,7r
Onq + OuQ^ + O^q^^ >
subject to
TT
>
e,
TT
>
^219
+
-
^229'
-
^239''
g[^ii+ 3012?'
-
e2iq
+
-
+
+
^229^
+
[Onq
^129'
g
enforceable prices (per Corollary
1)
search (with a discretization grid of 0.0001 yields q*
and
W2
final step,
=
0.1304,
0.7219,
{ci(q*)
zi
=
+
Onq'"']
are
=
shown
in
0.6238 and
if
g
<
0.7593
if
g>
0.7593
Figure
vr*
=
=
47.7807, W4
0.0100,
and
=
Z2
+ n* + e, q"), though
=
13.0341, W5
=
0.6238; see Figure
1.2484,
1.
wq
=
0.3000,
1.
exhaustive
=
1,
Wi
W7
=
1.5167,
=
0.4194,
actually prices arbitrarily close to (but greater than) Ci{q*)
0.01, the auctioneer's utility
the winning suppHer's profit
is i'liq*)"^^
is tt*
+
e
=
20
=
and the
total surplus
is
+ n*
quality level
2's)
+i'3q* -Oiiq* -0i2{q*f -9a3{q*y^ -t^*
0.6327,
ws
(The rule constructed above enforces
can be enforced.) Under this scoring strategy, the losing supplier's (supplier
is
An
0.5327. In the third
the construction in §A1.2 produces parameter values A*
w-i
e,
15^139^4],
i±hi^±hx}h±l_2y -q
The minimum
+
6'23'?^^
-t =
3.0236.
2.3909,
Figure
True value, supplier
1:
and the optimal scoring
To
rule
I's cost Ci,
(•
•
•
)
which the parameters
in
unchanged, but we
function,
=
set $21
2,
for supplier
^22
=
0.1000
auctioneer
to Figure
=
is
2,
=
e.
If
we
3.2627, which
is
minimum
enforceable price p,
623
we next consider a "high-competition"
35%
+
and the true value functions are
cost
=
0;
prices are
new parameters
enforce {c\{q*)
roughly
I's
and
1,
and the resulting minimum enforceable
optimization (15)-(17), (48) under the
TX*
C2,
versus quality.
illustrate the effects of inducible competition,
example
and
supplier 2's cost
tt*
+
for
the cost curves, the true value
shown
in
Figure
2.
suppher 2 results
Running the
in q*
—
0.8482
e,q*) (see Figure 2), the payoff to the
greater than in the previous example. Referring
notice that co crosses the line tangent to
C\
at ci's "elbow,"
which allows the
auctioneer to enforce near-cost prices just past where the elbow starts.
We
ure
1)
see that, depending on the situation, the optimal scoring rule can
or overstate (Figure 2) the true valuation of quality.
can operate
in
In
downplay
summary, our scoring
(Fig-
rule
a fundamentally different manner than Che's rule. Che's optimal scoring rule
21
Figure
2:
True value, supplier
and optimal scoring
tangent to supplier
rule
(•
•
I's cost
•
)
I's cost ci,
supplier 2's cost
C2,
minimum
enforceable price p,
versus quality for the high-competition example. Ti, the line
curve at q
=
0.8640, intersects the cost curve of supplier
2.
understates the true value of quality to reduce the information rents received by the more
cost-efficient suppliers.
cost functions before
value function.
Some
In contrast, the auctioneer in our
round
P+
I,
mechanism knows the
and the optimal scoring
rule
suppliers'
might even exaggerate the
practical implications of such a scoring rule are discussed in §3.1.
To put our proposed mechanism
we
into perspective,
also consider a "straw"
mecha-
nism, where the auctioneer announces his true utility function as the scoring rule, and assume
that the suppliers submit their
but with ipiq^^
plier s's
q*
=
+
MBR
i^^q in place of /;
maximum
drop out score
bids.
we
is
This scenario adheres to equations
consider the
first,
Mg, so supplier
1
0.8008 (the solution to the right hand side of (14) for
(By Step
1
of §2.5,
we note that the winning
22
(7)-(9), (13),
"low-competition" example. Sup-
wins the auction bidding quality
s
=
1) at price C\{q*)
supplier in our proposed
+ My —
mechanism
M2.
will
be
the
be
same
as the winning suppher in the straw mechanism, but the winning bids will likely
The
different.)
I's profit is
auctioneer's utility
1.7552,
and the
is
total surplus
-
v{q*)
is
Ci(g*)
— (Mi —
A/2)
magnitude of
to assess the
assessment
beyond the scope of
is
in
rounds
we made two
1,
.
.
,
.
P
in practice; such
may
restrictive assumptions:
and they bid
their
distort their bids or not
MBR
in
(i)
round
the suppliers do not distort their bids
P+
and
I;
MBR, and
conform to
the form of the suppliers' cost functions. In this section,
to the proposed
an
this paper.
the auctioneer knows
(ii)
the form of the suppliers' cost functions, but not their parameter values. In practice,
bidders
in
PRACTICAL CONSIDERATIONS
3.
In §2,
Milgrom 2000a),
would be required
set
enhancement that might be achieved
1.6823, supplier
and a decrease
utility,
mechanism. However, an industrial data
utility
=
A/2
3.4375. Hence, as expected (see
the proposed mechanism leads to an increase in the auctioneer's
efficiency relative to the straw
=
mechanism, which either improve
assumptions, or address other practical concerns.
its
We
the auctioneer
we
may
not
some
know
discuss several enhancements
robustness with respect to these two
begin with a discussion of the scoring
rule in the last round.
3.1.
Scoring Rule in the Last Round. There are three potential problems with the
analysis of §2.5:
the determination of the optimal enforceable bid
rather difficult mathematical program; the resulting scoring rule
parameters per attribute
scoring rule
levels.
To
may
Step 2 searches
t)
satisfies (9), 18
force the losing supplier to
deal with the
(15)-(17), (48) to
if
f=
X,
first
may
may be
require solving a
too complex (8
+P
otherwise) for practical implementation; and the
submit bids with negligible non-price attribute
problem, Step 2 of our method can be replaced by restricting
where x maximizes the right side of
(14) for s
=
for the lowest possible enforceable price at x, the bid level
23
i.
This alternative
with the potential
While
to yield the largest utility for the auctioneer.
to find the optimal scoring rule,
hyperplane tangent to supplier
2
it
i's
will
do so
this alternative Step 2
an auctioneer's
are
still
i is
which supplier
s
/
is
this alternative Step
greater than or equal to the utility level from any
is
not the top bidder;
sure to do better than
intersects the
cost surface at x. Furthermore, the proof of Proposition
utility that
auction in which supplier
not guaranteed
any supplier's cost surface
if
shows that, though not necessarily optimal, the rule generated using
2 yields
is
possible via
i.e.,
full
even with this simplified approach, we
optimization over generic scoring rules in
wins.
i
The complexity
of the scoring rule (see (19)) can perhaps be finessed in practice by pro-
viding the scoring rule in graphical form (one graph per non-price attribute), together with
An
a calculation device that converts uncommitted bids into scores.
to
is
employ a parametric scoring
rule in Step
alternative approach
For this purpose, a natural parameterization
3.
that of the true value function; in this case, with the identity of suppliers
competitor to
selection
i)
in
hand from the method's
first
two
i
and j
(a best
steps, the auctioneer's scoring rule
problem becomes
A
A
max
y2va{xl^,1pal,---,^ap)
"
^Va{xl^,((>a\,-
Y2va{xl,(f)al,---,(t>ap)
-'^Casixl,OasU---,Oasp),
and the scoring vector constraints
later in §2.4
is
(23)
S
=
i,j,
(24)
a=l
a=l
mentioned
^^aj '^aj\,---, Oajp)
A
A
=
^'^i
a=l
a=l
Ss
(t>ap)
A
A
+ X^ VaiKj ,(t^al,---,4>ap) -Yl
subject to
,
a=l
a=l
(8)-(9),
is
at the
superfluous here).
end of
Since
i
§2.1 (the scoring vector constraint
and j were selected with respect
to
a generic scoring rule, they are not guaranteed to be optimal for the parameterized case.
However, this drawback - though
solve 2(2) mathematical
pair),
difficult to
quantify a priori - compensates for the need to
programs (a version of
which may not be practical
if
S
is
(8)-(9), (23)-(24) for
large. In practice,
24
every ordered supplier
some approach midway between
these two extremes could be used - for instance, examining
candidates in Steps
and
1
and §2.4
(i.e.,
derivatives at zero
pairs
from the top 10% of
2.
In addressing the third potential problem,
in §2.1
all
we
first
note that our scoring-rule restrictions
the scoring rules are strictly concave and satisfy the conditions on the
and
infinity to ensure a unique, interior
are not innocuous. In the absence of these constraints,
MBR
we have constructed nonpathological
convex or closely mimics step
cases in which the optimal scoring rule in the last round
is
supplier Vs cost surface everywhere except near x, where
it is
ignoring these constraints
may
bid response by suppliers)
precisely
e
I's
units higher. While
increase the auctioneer's utility over the short run, in the
longer run cost-mimicking can eliminate meaningful bids from the non-low-cost suppliers,
compromising competition
in -
and consequently the
more, a non-concave scoring rule
may make
credibility of
- the auction. Further-
transparent the strategic nature of our proposed
mechanism. While we have been careful to avoid cost-mimicking and convex scoring
we note that the optimal scoring
suppher
j in step 2) to
on the
for all
3.2.
and
MBR
it
2's quality level is 0.01.
may be shrewd
our
levels; e.g., in
This phenomenon
for the auctioneer to either
may
(i.e.,
first
arouse bidder
add a lower-bound constraint
attribute levels that result from his scoring rule or impose a reservation level
non-price attributes.
Cost Estimation. Each time a
of equation (4) that the auctioneer
noted
round can force the best competitor
submit bids with negligible non-price attribute
numerical example, supplier
suspicion,
rule in the last
rules,
earlier,
for all bids
if
supplier submits a bid, he generates a
new
version
can use to estimate the suppliers' cost parameters. As
these bids are undistorted, then the vector of non-price attributes
within a given round, and after
P
rounds the
the cost parameters for each attribute. However,
if
P
first-order equations
bidders intentionally
or unintentionally (e.g., lack of sophistication) distort their bids, then
25
(i.e.,
is
identical
determine
strategically)
an inconsistent
set of
first-order conditions
condition in (4)
s
is
is
generated.
=
fasri^a)
0.
If
Let us
we
define fasr{Xa)
now
= |^ — ^^,
then the first-order
consider a fixed attribute a and a fixed suppUer
and suppress these subscripts, and suppose that Br bids are submitted
that Br
is
hkely to be a small number because bids in
on the bidders' part than
in
a traditional price-only auction.
the first-order condition be given by frbi^rh)
the
unknown
RFQ
=
0.
If
cost parameters by choosing [Oasi,
,
Then
round
for bid b in
and Wrb >
0.
variance for fitting the curve frb[Xrb)
Crb
are iid normal, mean-zero
=
=
in the case
random
on bad judgment or experimentation are
later bids within
r, let
minimize the weighted-average
=
for
1
This quadratic program might also incorporate convexity constraints on
the parameter values. Note that the weights Wrb
where
note
we can estimate
least-squares quantity, '^r=i^b=i'^'rbfrb{^''^rb)i where the weights satisfy Ylb=i'^'rb
all r,
r;
processes require more work
bid distortion occurs,
(^asp) to
round
in
l/Br
for all r
and
b
would minimize the
where the true model was
variables.
However,
less likely to
if
we
=
^rb,
believe that bids based
occur as each round proceeds, then
each round should be assigned higher weights.
Finally,
methods, which employ Bayesian a priori estimates on the output data
parameter values, have been developed
frbi'^rb)
more complex
(e.g.,
e^fc)
and the
in the geophysics field (Tarantola 1987).
This same weighted-average least-squares procedure can be used to choose among several alternative cost functions. For
ditions for
a
two cost functions,
more consistent sequence
3.3.
example, we can simultaneously compute first-order con-
OasiX^""^
and 0asix
+
das2X^-:
of first-order conditions, as
and use the option that leads
measured by X^^^j '^b=i
Scoring Rules in Earlier Rounds. Under the assumptions
estimation will be achieved as long as the auctioneer announces
in the first
P
to
'^rbfrbi^rb)-
in §2, accurate
parameter
distinct scoring functions
P rounds that satisfy the conditions stated in §2.4 and at the end of §2.1.
However,
may
cause the
in practice, large fluctuations in the auctioneer's scoring rule across
rounds
bidders to suspect strategic behavior on the part of the auctioneer, which in turn
26
may
lead
the bidders to counter with their
own
strategic behavior (e.g., intentional cost distortion).
At the other extreme, a minuscule change
change
in the auctioneer's scoring rule
perhaps because the non-price attributes -
in the suppliers' bids,
may
not cause any
in contrast to our
model's assumptions - take on only a discrete set of values. In our view, the auctioneer's
goal in the early rounds
while
a
generating
still
way
is
new
to
make changes
in the scorii^g rule that are as subtle as possible,
bids from the suppliers. Ideally, these changes are
that the bidders have the impression that the auctioneer
reasons
for non-strategic
(e.g.,
is
made
in
such
tweaking his scoring rule
an improved understanding of the relative importance of the
attributes).
The
choice of the initial scoring rule
is
the most difficult, because the auctioneer
assumed to possess no bidding information. As a general
is
guideline, the auctioneer should
use his experience and historical data to choose an initial scoring function that
close to
is
the predicted final scoring rule.
We
propose the following procedure
which makes
cedure
is
for
choosing the scoring rule in rounds
effective use of the bidding information
described in four steps, the
first
2,
.
.
.
,P,
from the previous rounds. This pro-
three of which are devoted to finding (possibly
inaccurate) cost parameter estimates for the two highest-bidding suppliers in the previous
round.
determine the scoring rule
First, to
have no more than r parameters;
and four rounds
e.g.,
even
in
if
round
we plan
r,
we assume that
round 2 we assume that
functions have only two parameters. Second, in addition to the first-order
r
in
—
1.
we
also
cost functions
to use three-parameter cost functions
of bidding, to determine the scoring rule in
bid) conditions in (4),
all
assume that suppliers submitted
their
(i.e.,
MBR
all
cost
undistorted
bids in round
In particular, this assumption implies that the second-highest bidder's final score
round
r
—
1
is
his
drop-out score.
If
we denote
27
this bidder as supplier 2
and
his bid as
(P2i ^12'
•
•
•
1
^/i2)i
then this assumption generates the additional equation
A
P*2
=
^
Ca2«2>
^a21,
•
•
•
,
(25)
6'a2p)-
a=l
If all
bids are indeed undistorted, and supplier 2 submitted
equation (25) can be combined with the r
—
1
MBR
first-order equations
bids in round r
—
from the
rounds to
solve uniquely for the second-highest bidder's true cost parameters.
bids are distorted (see §3.2), then
to (25)
We
'}2,b=\'^'rb
~
li
only have censored information, namely pi
highest bidder in round r
—
1.
That
is,
^rb
>
we do not know
>
X^^^j
this bidder's
(e.g.,
or
earlier
subject
•
•
•
,
^aip), for the
drop-out score. Conse-
a drop-out score that
5%
^''''/^(^''•b),
Cqi(3:*j, ^aii,
we assume that the current
a fixed percentage
some
then
0.
quently, in the third step of our procedure
is
instead
If
we propose minimizing Xlr=i X^6=i
and the additional constraints
earlier
1,
first-place bidder has
10%) higher than the second-highest
bidder's observed drop-out score; this fixed percentage should include the estimated gap
between the top two bidders and the perceived amount by which the second-place bidder
is
"holding back"
(i.e.,
his true drop-out score
minus
his
observed drop-out score).
estimated drop-out score provides the additional equation (see
first-place bidder's cost parameters.
(7)) to
This
estimate the current
In the final step of our procedure,
we substitute these
(possibly inaccurate) parameter estimates into the three-step (optimal scoring rule determi-
nation)
3.4.
method
of §2.5,
and
find the
proposed scoring rule
round
Activity and Transition Rules. In the absence of activity
to bid aggressively in the earlier rounds of our
in auctions (see, e.g.,
FCC
for
r.
rules,
bidders are unlikely
mechanism. Activity rules are often imposed
Kelly and Steinberg 2000 and Milgrom 2000b for details pertaining to
auctions) to prevent bidders from delaying their bid submissions until the very end of
the auction. Similarly, weak-bidding suppliers are often weeded out throughout the course
of a
RFQ
process.
We
propose that after each round of the auction, the auctioneer allows
only a subset of suppliers to proceed to the next round.
28
The
criteria could
be based on
either the
number
compete
suppliers
compete
of suppHers (e.g., only five supphers
in
round
3) or
on their scores
fixed percentage of the current leader
may
(e.g.,
in
round
and only three
2
only supphers with scores within a
proceed to the next round). Our mechanism also
requires a rule for transitioning to the next round. This rule could be time-based
round
lasts
certain
a certain number of days) and/or activity based
number
Exogenous Attributes.
as quality
each
a round terminates after a
of bid- free days). Finally, to encourage competition
we propose using activity-generated overtime periods
3.5.
(e.g.,
(e.g.,
down
the homestretch,
in the final round.
and endogenous attributes such
In addition to bid price
and lead time, exogenous attributes such as a
supplier's reputation
and
his past
history with the manufacturer typically play a vital role in the allocation decision.
These
factors are easily incorporated into our model. Let Cg represent the auctioneer's total utility
derived from supplier
s's
exogenous attributes;
for
an incumbent supplier,
incorporate the fixed cost to switch to a different supplier.
Then
function (with the supplier notation suppressed) becomes X]o=i
We recommend
65,
that the auctioneer reveals to supplier
might
the auctioneer's true utility
,'4'ap)
'^'ai^a^'i^ai^
s his
+ e — p.
truthful exogenous value
but not the other suppliers' exogenous values. While we have not attempted to prove
that truthful revelation
about
e^.
is
optimal on the auctioneer's part, withholding
all
information
would be unsatisfactory to the suppliers because they would only possess a
scoring function. Moreover, a large portion of the exogenous value
standardized supplier ratings, which are readily available in
Under the assumption that the true
a straightforward manner.
units; this shift
is
The
e^ is
many
revealed to supplier
supplier's cost surface Cg
is
is
likely to
partial
be based on
industries.
s,
the analysis extends in
simply shifted vertically by
— Cs
allocated to the costs over individual attributes by taking Cas to be shifted
—Xas^s units vertically, where Xas
Cg
this utility
>
and Yla=i
value to change during the course of the
eRFQ
29
^as
=
^-
It is
even possible
for
a suppher's
process, e.g., by delivering an unexpectedly
impressive presentation or by providing perks such as tickets to sporting or cultural events.
Although by strategically assigning exogenous attribute
any
to enforce
more obvious
^— tuple
at
e
profit,
One
of the
most
difficult
the auctioneer's perspective
is
dynamic
way
on scoring rules as described
is
can contrive
likely to
be much
in §2.5.
CONCLUDING REMARKS
aspects of running a multi-attribute procurement auction from
the lack of knowledge about the suppliers' cost functions for
endogenous non-price attributes.
as other
generating competition in this
to suppliers than relying
4.
levels the auctioneer
We develop an auction mechanism that is in the same spirit
strategies for problems with imperfect information (e.g., Gittens 1989),
focuses on learning the relevant information and then switches to an optimization
which
first
mode
after sufficient learning has taken place.
In our model,
techniques to learn the suppliers' cost functions.
market) mechanisms have been in use
studies have
all
for nearly
we use inverse-optimization
Although optimization-based
a half century (Stanley et
al.
used forward-optimization, and this paper appears to be the
(or smart-
1954), existing
first
to use an
inverse-optimization-based approach. Considering the ease with which individualized data
can be collected on the Internet and the fact that an auction's outcome often depends on
the parameter values of only a few bidders, individualized learning using
optimization
may be more
fruitful
and inverse
than "collective" learning, where the bidding population's
parameters are modeled probabilistically.
be applicable
MBR
in other types of auctions
This inverse-optimization-based approach
and perhaps other settings of learning
in
may
games
(Fudenberg and Levine 1998). Moreover, consultants have argued that understanding the
cost drivers of purchased items
strategy
(e.g.,
is
the most fundamental capability of an effective sourcing
page 6 of Laseter 1998), and
cost estimation purposes.
this
approach could even be used solely
for
Although our analysis uses elementary techniques, considerable
30
theory has been developed in recent years
mathematical programming
useful for
(e.g.,
Ahuja and Orlin 2001 and references therein) that may be
more complex problems.
Aside from transaction cost savings, the prospect
procurement auction compelling
how
for the buyer.
Our
for
competition
is
what makes a
(largely geometric) analysis in §2
shows
the auctioneer, via the choice of the scoring rule in the last round, can manipulate the
maximize
rules of the competition so as to
format.
In particular,
optimal to
it is
his
first
with the largest drop-out score (see equation
in the
will
optimization in
for various aspects of inverse
announced scoring
rule,
and then to
minimize the winning suppher's
Ideally, all suppliers exit the auction
suppliers, rather
than feeling as
if
profit,
own
utility
identify the
(7))
if
within the open-ascending auction
winning suppher, which
is
the one
the auctioneer revealed his true valuation
identify his best competitor,
i.e.,
the one that
thereby leaving more utility for the auctioneer.
with a renewed sense of respect and fear for the opposing
they have been manipulated by the auctioneer.
While
our numerical examples consider only one non-price attribute, our analysis has the potential
to provide nonobvious insights about which attributes
most
and competing suppliers provide the
fruitful focus of competition.
Section 3 discusses several important practical issues that bring this mechanism closer
to practice.
The
CTO
of Frictionless
Commerce shared
the basic ideas of our mechanism
with several key customers. While they found the ideas intriguing, they did not seem ready
to use
it
(even assuming
two reasons.
First, several
it
had undergone successful human testing prior to
customers
complexity of the mechanism.
(i.e.,
manufacturers) did not
In this regard,
feel
release) for
comfortable with the
we agree that the mechanism
is
much
transparent than the efficiency-maximizing mechanism. Second, although traditional
less
RFQ
processes allow the changing of the scoring rule as the process proceeds, one customer (from
the private sector)
felt
that an equity issue would arise
31
if
activity rules (see §3.4) were in
place; in his words, "a supplier
would get upset
if
he was thrown out of the auction when
the score was based on apples, but would have done better
when
the score was later changed
This suggests that activity rules must strike a delicate balance between the
to oranges."
generally, the
CTO
thought that major changes in the scoring rule of a private-sector auction would
likely
perception of fairness and the mitigation of strategic behavior.
perception that the manufacturer was "beating up" the vendors, which suggests
fuel the
that the discussion of the early-round scoring rules in §3.3
summary, the
for this
More
CTO
first
of particular importance.
1.5 to 2 years.
He
also thought that
might make
it
attempt to implement the cost-estimation portion of our analysis as an extra
software feature that allows the auctioneer to gain valuable information
(e.g.,
estimating cost
parameters, assessing the magnitude of bid distortion) without committing to a
mechanism.
In
conjectured (in the winter of 2001) that the market would not be ready
type of mechanism for another
sense to
is
Finally, before this
mechanism could be
to incorporate discrete-valued attributes
practically implemented,
and non-smooth cost and
it
new eRFQ
would need
utility functions.
ACKNOWLEDGMENT
We
thank Rob Guttman, Chief Technology Officer of Frictionless Commerce hicorpo-
rated, for invaluable input during the
modeling phase of our project.
Baron, Ani Dasgupta and Jeremie Gallien
for helpful
We
comments on an
also
thank Opher
earlier draft of this
manuscript.
References
[1]
Ahuja, R. K.,
[2]
Bikhchandani,
J.
B. Orhn. 2001. Inverse optimization. Operations Research 49, 771-783.
S.
1999. Auctions of heterogeneous objects.
32
Games and Economic
Be-
havior 26, 193-220.
[3]
Branco, F. 1997. The design of multidimensional auctions.
RAND Journal of Economics
28, 63-81.
[4]
Che, Y.-K. 1993. Design competition through multidimensional auctions.
RAND
Jour-
nal of Economics 24, 668-680.
[5]
Cramton,
[6]
Demange,
Economy
[7]
[8]
P. 1998.
Ascending auctions. European Economic Review 42, 745-756.
G., D. Gale,
M. Sotomayor.
1986. Multi-item auctions. Journal of Political
94, 863-872.
Fudenberg, D., D. K. Levine. 1998. The Theory of Learning in Games, The
Cambridge,
MA.
Gallien,
L.
J.,
M. Wein. 2000. Design and analysis
of a smart
procurement. Sloan School of Management, MIT, Cambridge,
[9]
Gittens,
C. 1989. Multi-armed Bandit Allocation Indices.
J.
market
MIT
Press,
for industrial
MA.
John Wiley and Sons, New
York.
[10]
Kalka,
S. J., B.
Research,
[11]
Inc.,
D. Temkin, L. Wegner. 2000.
Cambridge,
[12]
Laseter, T.
[13]
Milgrom,
P.
M.
auctions go beyond price. Forrester
MA.
Kelly, F., R. Steinberg. 2000.
versal service.
B2B
A
combinatorial auction with multiple winners for uni-
Management Science
46, 586-596.
1998. Balanced Sourcing. Jossey-Bass, Inc.,
2000a.
An
San Francisco, CA.
economist's vision of the B-to-B marketplace. Executive white
paper, www.perfect.com.
33
[14]
Milgrom,
tion.
[15]
P.
2000b. Putting auction theory to work:
Journal of Political
Parkes, D.
C,
L. H.
Economy 108,
The simultaneous ascending
auc-
245-272.
Ungar. 2000. Iterative combinatorial auctions: Theory and practice.
Proceedings 17th National Conference on Artificial Intelligence (AAAI-00), 74-81.
[16]
Tarantola, A. 1987. Inverse Problem Theory. Elsevier, Amsterdam.
[17]
Stanley, E. D., D. P. Honig, L. Gainen. 1954. Linear
Naval Research Logistics Quarterly
[18]
Wellman, M.
P.,
W.
1,
E. Walsh, P. R.
protocols for decentralized scheduling.
[19]
programming
in bid evaluation.
48-54.
Wurman,
J.
K. MacKie-Mason. 1999. Auction
To appear
in
Games and Economic
Behavior.
Wise, R., D. Morrison. 2000. Beyond the exchange: The future of B2B. Harvard Business
Review, November-December, 86-96.
34
ONLINE APPENDIX
Al. Proof of Proposition
Proposition
Al.l.
convenience,
begin by proving the
"<^" Direction.
1
=
/(f)
let
We
1.
X^^^j
fa{-Ca)-
of the auction implies that Cj{x)
>
Suppose
+
We now show
tt.
Let f denote supplier j's bid induced by
auction with profit
vr
^ V/(f)/(f — i}
<
z)
TT
+
Cj(f)
=> Cj{z)
Since
Cj is
plane Tj
We
+
<
TT
convex, increasing and
vr
i
+
is
tt,
easier.
For
f).
the winner
intersects
that suppher
i
Cj.
wins the
+
<
TT
—
Cj{z)
+
C,(f)
lies
Tr
—
Cj(f )
+
=
0,(5-))
c,{x)-c,{z),
since
Cj{z)
since suppher
+
+
/
concave,
is
bids
f
T^{z)
+
Vcj(f)
iff
= V/(f),
IT.
must eventually cross the hyper-
at z, Cj
tt
i
=
Vc,(f)/(i'- f)
below Tj
7r,
TT.
next prove the other direction of Proposition
A1.2. Proof of Proposition
we prove Proposition
as (Al): C2(f)
(A2),
+
that T,
The assumption
/.
-c,(f)-(/(z)-
=^f{x)-f{z} =
—
(Ct(.f)
is
implies that
/(f)
=> Vc,(f)/(f
/^ enforces
assumption that supplier
Clearly, the
c,(f)
/i, ...,
which
direction,
"<S="
we need
>
1
Ci(f)
1
n;
where
-Htt,
.
,
=
s
and (A2): the hyperplane Ti
to find a feasible scoring rule fi,
the auction with bid (ci(f)
is
more
difficult.
"=>" Direction. The proof proceeds in two stages.
for the two-supplier case,
-I-
which
1,
-f-
1,2.
tt
We
f) (which also implies Si
>
e).
index the assumptions
intersects C2-
Ja such that S2
First,
>
t
Such an /
Given (Al) and
and supplier
is
1
wins
found via Claims
1-5.
In the second stage, the two-supplier results are extended to
(proof of the "=>" direction of Proposition
shows that
1)
35
(Ci(f)
-I-
S
7r,f)
suppliers.
Claim 7
can be enforced by
+ TT, x)
taking a convex combination of the scoring rule that enforces {ci{x)
case (with
Stage
price
i
and
j in the roles of
Claim
1.
and
1
2),
and a second scoring
in the
two-suppher
Claim
rule constructed in
contains the main construction result for the two-dimensional
1
and a non-price
two supplier
attribute),
Claim
case;
1
be our main
will
6.
(e.g.,
tool, as the
construction of / for the multi-attribute case uses an attribute-by-attribute construction
procedure. This procedure
by Claims
1-4.
Claim
For
1.
i
=
2,
1,
let g,
the line tangent to g\ at Zi
g2,
then there exists 22
as z —^ 00, for
i
=
1,
provided in Claim
is
>
>
M"*"
:
— R
6e convex, increasing
>
and
0,
let
a >
2 f'{z)
>
gi{z) as z
^
+a+
where h can he chosen arbitrarily
first find
a
22
li
+a
be
li
intersects
that f'{z) -^
and
0"*",
f{z2)=g2{z2)
h,
/>2) =
+
h,
(26)
(27)
52(^^2),
large.
+
52(^2)
[21-
g2{z2)
<
22]
>
h{z2)
smooth, convex, and increasing and
is
+ a and
gi{zi)
laid
such that
52(22)
Since §2
>
If goizi)
and
f'{z,)=g'{z,),
We
0.
smooth functions. Let
and a concave, increasing smooth function f such
f{z,)=g,{z,)
Proof.
which builds upon the groundwork
5,
51(21
+
lies
)
+
«,
and
(28)
(29)
oc.
above
li
+a
aX z\, 52 intersects
li
-\-
ot
either to the right or left of z\ (but not both).
(cl):
32 intersects
+
a.
At
+a
to the left of z^.
must
Let 2 be the closest point to
a from below;
+
a,
we can
Since (2,52(2)) and (21,51(21)
-I-
a) are points on the line
2,
92
36
=
g'i{zi)
-^
a.
li
which
at
>
l\
li
zi
hence, g^iz)
cross
intersects
li
+
l\
§2
slope of
write the
point slope formula for
li+a
and evaluate
at the point {z,g2{z))
92iz)
+
-
g[{zi)[zi
=gi{zi)
+
a.
>
+
a,
z]
at zi, yielding the equation
Replacing g[{zi) by the larger value 92(2) implies
g2{z)
>
since Zi
we can
32
find a
<
>
ryi
left
+a
h{z)
z]
g^{zi)
(30)
side of (30) can be viewed as a continuous function of z,
|2
—
there exists a
772
such that (30) holds with z in place of z as long as
continuous and approaches
is itself
92{z)
Noting that the
z.
+ g'2{z)[z, -
provided that z
<
li
z
+a
from below at
and
|5
—
Z2
=
z-r],
2:|
<
z,
An
ri2-
z|
<
>
?7i.
Since
such that
appropriate value for 22
is
then
found by taking
<
provided
(c2):
<
t]
min{7?i,
g2 intersects
intersect
l\
+
a.
li
At
+ a
z 32
(31)
7/2}-
to the right of
must cross
li
Let z be the closest point to
Zi.
+ a from
Zi
at
which
g2
above; the remaining arguments to find 22
are straightforward analogues to those of (cl).
Now
that
we have shown that a
22 satisfying (28)-(29) exists,
<
is
beginning with the case 22
21;
the complement case
we now construct
essentially the same,
and
is
/,
omitted
for brevity.
The proof
/i,
/2,
and
<
for the case 22
/s (note that the subscripts
defined over respective intervals
at 2i
and
22
synthesizes / from three concave, increasing functions
Zi
on /
in this proof
[0, 22], [22, -^i],
and
[zi,oo).
do not correspond to attributes),
For each
/j
we
enforce conditions
toward satisfying (26)-(27):
fi{z2)
^
Mz^) =
92(22)
giizi)
+
h,
and
+a+
h,
/;(22)
and
=
32(^2),
f^iz^)
37
=
for
g[{z,),
i
=
for
l,2,
i
=
and
2, 3,
(32)
(33)
as well as choose the functions' forms to ensure that the other conditions of the Claim's
statement are
We
We
satisfied.
set fi{z)
=
begin by constructing
Solving (32) with
juz'^'-'^.
711
=
i
=
/j.
+ h]zr'\ and
[g2{z2)
for 711, 712 yields
1
712
=
4^^+h
92[Z2)
Over
7i2
[0,^2]
<
1,
7n >
increasing, since
and thereby the concavity of
We
space.
is
/i
find /2
We
by
finding
first
and
/2
would be
what
produce the desired
we view
that in which
axis, the fine
to
perpendicular to
li
/2.
+a+
(^i, gi{zi)
f{{z) —^ 00 as z —^
/j
then take the resulting function,
and rotations
Choosing h > g2{z2)z2 — 92(^2) ensures that
0.
in a new, shifted
and apply a
/2,
and rotated coordinate
series of reflections, translations
The new coordinate space we
h) as the origin.
+a+h
0'^
at {zi, gi{zi)
We
+ a + h)
and to the
positive quadrant everything below the former
take
li
+a+
find convenient
h as the horizontal
as the vertical axis,
of the latter.
left
is
and
as the
The appropriate
equations for /2 are
f2{zi-
/^(2i
If
we
let f2{z)
—
721
where
li{z2)
=
gi{zi)
(to translate the
22)
li{zi)
+
=
ff2(-2)
-
=
9'2{z2)
-
a- g2{z2),
slope of {h
=
=
[^1(^2)
[^2(22)
-
+ g[{zi)[z2 —
+a-
21]
•
new coordinate space
Reflect about the origin:
92iz2)][zi
5l(~l)][^l(-2)
— /2(— 2);
2.
The
and
+a+
h),
g'lizi).
and solve the above two equations
7212^^^
722
-
=
Z2)
-
for 721, 722,
22]"^'",
we
get
and
+a- 92{Z2)]'^[ZI
-
Zo],
step-by-step transformations to yield /2 from /2
into traditional Euclidean coordinate space) are:
Shift zi units to the right
38
and
51(^1)
+a+
1.
h units
— /2( — (2 —
up:
pivoting about
+
2i))
+a+
g\{zi)
-M-{z Equation (34)
Mz) =
is
+
+Q+
giizi)
h
+
g[{zi)[z
-
72272i(-2
that in
[22, ~i]
/2
inequality
is
+
is
g\iz\) degrees,
(34)
21].
if
we
replace 2 by 2 and
we have
-i2i{-z
+
z.r^'
+
g,{z,)
+a + h + g\{z^)[z - z,];
straightforward to check that the equations (32)-(33) with
M^) =
first
zi))
^
h):
a function in the traditional Euclidean space;
is
set the result equal to /2
it
+a+
point (-1,^1(21)
tlie
Rotate counter-clockwise by sin
h; 3.
z,r^^-'
+
g[{z,), f^{z)
concave, increasing,
it
=
-
-722(722
suffices to
i
=
+
l)72i(-^
show that
true by (29) and our assumption that 22
<
721
21.
Since
2 are satisfied.
^-l)^^^-^ for verifying
>
A
and
722
>
The
1-
series of equivalences
prove that the second inequality holds:
722
=
- g\{zi)]Mz2) + a -
[92(22)
^=>
^=>
[g2{^2)
g'2iz2)[zi
<^^
which
is
If
-
-
22]
>
1,
- g'M)][z, -
Z2]
>
M^2) + a -
-
Z2]
>
giizi)
+
g[izi){z2
-22]
>
gi{zi)
+
a,
52(^2)] -'[^1
Z2]
-
g[{zi)[zi
52(^2) +52(~2)[2i
we
set 73(2)
=
7312'''^^
=
and solve
[gi{zi)
+a+
(33),
we
Zi]
+a-
g2{z2)
h]z^'^^^,
get
g'i{zi)zi
and
732
51(21)
is
-
(29),
equation (28).
131
As
by
g2{z2)]
>
for /i, 731
ensured
if
732
implies that fi increases over
<
1,
or equivalently,
To complete the
max{52(-2)~2
—
if /i
>
—
g\{zi)
—
00).
g[{zi)zi
overall construction of
g2{z2),g'\{zi)z\
[22,
a}.
—
h'
Concavity and 73(2)
gi{zi)
—
—
>
as 2
—
>
00
a.
/ we need only choose h such that h >
Finally,
note that the smoothness of the
individual /,'s and the requirements (32)-(33) ensure that /
39
+a+
is
smooth.
Taking a step back, notice that
and h -
92(^2),
Claim
a
n >
2.
TT
>
'^ai(-^'a)
is
Zi, 22, 91(21), 92(-2), g'lizi),
D
a function of seven parameters.
Consider a two-supplier auction. Under assumptions (Al) and (A2), there
and a scoring
e
Proof. Let da
Assume
/
i.e.,
/ we used only
in defining
=
Ca2{xa)
—
f that enforces supplier
rule
=
Ca\{xa), o
1,.
.
We
,A.
.
winning
1
at (ci(.f)
+
tt,x).
construct / one dimension at a time.
witliout loss of generality that the attributes are ordered such that for a
Notice that
7^ c[j2(a"a).
A >
I;
> mm{c2{v) —
n
=
+n
otherwise, Ti
-
C2{x)
exists
=
1,
.
.
.
,
A,
intersecting C2 implies that
Ti{v)}
ci(f)
if
^=
0,
which contradicts assumption (Al).
For a
tangent to
1
Za
(with gi
—
1,
.
.
.
,A, for fixed
For ^q
Cai at Xq.
=
=
Cai, 92
consider the curve Cai
a,
>
small enough,
=
and a
Ca2,
lai
=
c^i
+
—
da
Let
Sa.
lai
be the
line
and we can apply Claim
intersects Ca2,
0) to find a feasible scoring rule fa
such that,
for
a
from Claim
1
G K+,
fai^a)
=
Cal{Xa)
fai^'a)
+
=
ka
=
cli(Xa)
where h^ can be made arbitrarily
to be at least greater than
fai^a)
>
Ca2(-2a)
+
^)
=
+
da
.
.
,A,a
-
CaliXa)
if
5a
5a
-
+
ha,
and
Now,
large.
fixed,
-
c'a^iXa),
+
if
e,
fa yields a difference in
,
A-\-\,.
=
max{— d^ +
fa{Xa)
For a
Cal{Xa)
fa{Za)
/^(z^)
we take the
=
=
Ca2iZa)
{fa{Za)
-
in auction a
Ca2(^a))
arbitrarily large ha
maximum
=
da
we announce a
-
ha,
c'a2{Za),
e} (to ensure that fa{j^a)
dimension a
+
>
Cai{xa)
+
e,
and
dropout scores of
5a.
scoring rule /a(-)
=
Mai-s'^"^
such that
fa{Xa)
=
max{Cai(.Ta),Ca2(Xa)}
+
ha,
40
and
fak^a)
=
C^l(-'Ca),
(35)
{ha
some
positive real
number
greater than
then suppher
e),
2's bid for
dimension a\s
=
Za
Xa
and
faiXa)
Since fa
-
-
Cal{Xa)
"
{fa{^a)
=
Ca2{Za))
Ca2(^a)
"
Cal{Xa)
^
^a-
defined by two parameters (has two degrees of freedom) and has two equations
is
in (35) to satisfy,
such an fa can be found
(i.e.,
appropriate values of Hai and
/Ja2
can be
found).
After applying the above for a
the 6a's as small as
we
like,
=
I,
.
.
.
=
,A,let S
bookkeeping
for easier
min{^i,
=
set Sa
.
.
we can choose
,6^}. Since
.
a—
S ioi
1,
.
We
,A.
.
.
then
define
A
=
TT
/(f)
-
Ci(f)
-
{f{z)
-
= Y^
C2{z))
[fa{Xa)
-
Cal(x„)
-
-
{fa{Za)
Ca2{Za))\
,
a=l
A
= J2^^-^^-
(36)
a=l
Assumption (Al) implies that
if
6
C2(x)
—
Ci{x)
>
tt,
enforces (ci(f)
+
if,x)
when announced
Notice that the / we constructed
— A+
1,
+
.
.
.
n, x),
is
is
1
and
2,
Claim
Claim
where
ri-
>
^
^-
Hence,
assumed
1
tt.
=
a feasible scoring rule: for a
1,
.
from the assumptions on Claim
,
.
.
yl
the feasi-
I's /;
and
it
remains only to check that S\, S2
>
e,
and
5*1
tt
>
>
^2
e.
-|-
e.
The
first
condition
This concludes the proof
D
2.
3.
for
,A, the feasibility of scoring rule fa follows by simple analysis. For / enforcing
holds by our choices of ha] the second condition follows from
of
^<^
X]a=i
than n, meaning that /
strictly greater
to suppliers
bility of the scoring rule fa follows directly
(ci(.x)
is
chosen sufficiently small (possible since the earlier 6 arguments for Case
is
only that 6 was sufficiently small), equation (36)
a
which re-written
Consider a two-supplier auction. Suppose f enforces {c\{x)
41
+
n,x), n
> n >
e.
Let Cqi
=
—
Cqi
Wa, where
Wa
= Yl
[/^(^»)
-
^'l(^'0
-
(/'(^«)
-
C^2(50)]
(37)
.
1=1, ..^,A
and
ail
z is supplier 2's bid induced by f. Let
a >
such that Ca\{xa)
feasible scoring rule for
enforces (ci(f)
To
+
n
+
dimension
and supplier
which supplier
1
I's
1,
.
0) or lost
=
exists
an a
Then, using Claim
Cal{Xa)+TT
fa{ia)
=
Ca2{Za)
Announcing scoring
.
.
.
,
we can pick
we can find a
fa-i, fa, /a+ii
+a+
+
if
we ran
.
.
,a
—
<
{wa
+
(Equivalently,
fA
,
all
dimensions
1
ha,
rules fi,
ha
—
1
0) the auction.)
In essence, the shift of Cai to
1,
.
.
.
CQi(x'a)
+ +a <
7r
than m.a.x{wa
=
Cal{Xa)
- Wa +
TT
>
Cai{xa)
+
by
e
+a+
TT,
c'aiiXa) -- c'a^{Xa),
fa-i, fa, fa+i,
42
,
We now
run.
is
find a scoring rule fa
least greater
=
v4
Wa would be the amount by
such that
we can
f'a{Xa)
,
>
we ran an
if
,A,
l,a
position just before dimension a
by choosing the arbitrarily large ha at
fa(Xa)
/j,
notice that
3,
would be Wa-
this difference
2,
Suppose that there
intersects Ca2-
If
then checked the difference between the bidding positions of
a,
won {wa >
summarizes supplier
Proof.
Cai at Xa-
intersects Ca2, then
such that announcing
a, fa,
dimension auction over attributes
Cai
+n+a
Ca2{xa) andlai
better understand the idea behind Claim
1
tangent to
be the line
a,x).
except that for attribute
supplier
+ n + ct <
lai
cv
Ca2{xa) and/ai
and a
+
e,
prove Claim
Zq
6
M"*"
3.
+ +a
7r
such that,
e},
ha,
>
and
and how we chose
fai^a)
=
ha,
c'a2{Za)-
fA results in a post-auction profit
for
supplier
Wa +
of
1
fa{j^a)
"
-
Cal{Xa)
^ Wa +
"
(/a(4)
Cal{Xa)
- Wa +
Cali^a))
TT
+O+
-
ha
-
Cal{Xa)
(Ca2(ia)
+
ha
-
Ca2{Za))
^TT +
To prove the claim,
e
and Si > 52
+
e.
remains only to check that
it
The former
holds by
/i,
.
fa-i, fa, fa+i,
.
.
,
how we chose
,
a.
(38)
Ia induces 51,52
> n >
the latter holds because n
/?.„;
>
D
e.
Claim
4.
Let Ca\
=
Consider a two-supplier auction. Suppose f enforces (ci(x)
Cqi
—
Wa, whcrc Wa
is
+
7r
Proof. First, note that since / enforces {ci{x)
sions supplier
1
wins the auction by
Wa
where
z
is
+
-
faiXa)
<
7r,x),
tt
>
-
Ca2{Xa)
<
>
e.
Ca2(x'a).
+ tt, x), we know
that after running
all
dimen-
n; in equations,
CaliXa)
-
[fai^a)
"
=
Ca2(5a))
max{/a(2/)
-
fai^a)
-
Ca2{Za),
Ca2{Xa)
-
Cal{Xa).
=
Ca2{y)}
(39)
VT,
the bid by supplier 2 induced by /. Furthermore, using the definition of
fa{Xa)
tt
Then
as given in equation (37).
Ca\iXa)
+
i,
y
^ fai^a) - CaliXa) - ifa{Za) " Ca2{Za))
Then, combining equations (39) and
TT
which
verifies
Claim
hold.
5.
Then
Claim
<
TT
< Wa +
(40),
Ca2{Xa)
<
(40)
we have
"
Cal{Xa)
=
Ca2{Xa)
-
Cal{Xa),
D
4.
Consider a two-supplier auction. Suppose
tt
>
e
and assumptions (Al) and (A2)
there exists a scoring rule f that enforces {ci{x)
43
+
tt,x).
/ that enforces
(ci(x)
+
tt,x),
carry out iterations on / to construct a scoring rule / to enforce
(ci(x')
+
n,x).
Claim 2
Proof. First, use
We
n >
n.
The
general idea
supplier
at
is
to construct a feasible scoring rule
to start with /, then dimension- by-dimension decrease the profit of
(by revising fa's) until supplier
1
which point we
set
=
f
Let z be supplier
since
if
we
that by (Al) Ti
2's
bid induced by fi,
Set a
=
did, then Ti
-f
TJ-
n,
f
are ordered such that c'^^{xa)
all a,
wins the auction with profit exactly equal to
1
.
.
,
Assume
f^-
Notice that
7^ c'^2(^"^)-
would be
.
w.o.l.o.g.
we cannot have
C2,
=
c'^-^ixa)
hyperplane at
parallel to ca's tangent
cannot not intersect
that the attributes
c'^oi^a) for
x,
implying
which contradicts (A2).
I.
Procedure:
If
c^i(xa)
=
Otherwise,
a
c'^2{^a), set
let Cai
Cai (cai) at Xa-
=
a
-F 1.
and Wa be as defined
in
Claim
3,
and
let lai
{la\)
be the
line
tangent to
Also, let
Da = max{lai{y) -
Ca2{y)},
y
and
let
Da =
Now,
if
Da >
0, set
a =
0.
The
line lai
Cal{Xa)+Tr
+
Da- Wa +
n
+a =
<
Hence, we can apply Claim
3, set
fa
=
fa,
+a
TT.
intersects Cq2 (since
Da >
0),
and
Cal{Xa)+TT,
Ca2{xa)
by Claim
4.
and by announcing / enforce
{ci{x)
+
n,x), and
we are done - terminate the procedure.
Otherwise,
if
Da <
0,
we
find
ria
>
small enough that Cai {xa)
44
+ tt — Da + rja <
Ca2{xa)-
This
is
possible since
Ca2iXa)
-
(Cal{Xa)
+
Tt)
=
Ca2{Xa)
>
mill |ca2(y)
-
= —Da
The minimum
in the
second
if
is
an
interior solution to (1), (3),
the procedure runs through
da
and
set
a = —Da +
Sa-
-
since
c'^-^{xa)
/
Da <
0.
+
-n-)j,
0.
first
order conditions.
For producing a contradiction
all
the indices without terminating (explained below),
<
mm
A-'\T^D, +
< ria
la,
To check that
5a
>
0,
The
by assumption, and since /
c'^2i^a)
and so Xa >
Tt),
{Liiy)
occur either at zero or with
line will
inequality follows since Xa will satisfy neither:
enforces x, x
+
{LliXa)
we take
7rj\
note that assumption (A2) imphes
max{Ti(y) +7r - C2(y)},
<
y
A
i=l
Since 5a
>
0,
we can apply Claim
(ci(f)
+ n — Da +
Da and a imply
the definitions of
3 to construct an fa, then set fa
5a, x),
which, re-written,
for
fa, set Za
=
> Da = Da — Wa +
get that
tt
>
+
Za,
TT
and
n
+a
intersects Ca2-
Thus
fa,
such that announcing / enforces
5a,x).
(41)
the next step in the iterative procedure,
dimension a induced by
we
=
lai
is
{ci{x)+Wa-Da +
To prepare
that
set
n
= Wa — Da +
implies that
tt.
45
let Za
5a-
be supplier
Since
Wa — Da +
5a
>
n,
2's bid in
complete the groundwork
Finally, to
VL'a,
we have not changed
the fact that
/i,
.
proof of termination, by the definition of
for the
.
.
,
fa-i, fa+i,
and that / now enforces
Ja,
,
(41),
we have that
fa{Xa)
=
Setting a
a
+
-
-
CaliXa)
{fa{Za)
-
Ca2{Za))
= -Da +
(42)
(^a-
we can now repeat the same procedure with our updated
1,
/, 2,
and
TT.
Proof of termination with the desired
indices
1,
.
.
.
Suppose that the procedure
/:
without terminating with an a
,yl
=
We
0.
show that
this
iterates
through
would produce a
contradiction.
After the procedure iterates through
-
+
with
(/'.4
and:
If c'„i(xa)
D.4
5a
=
>
>
TT
c'„2(-'^a),
fa{Xa)
-
0.
all
Since no index
then Za
-
Cal{Xa)
=
the indices, / enforces {Cai{x)+u'A — DA+SA, x),
visited twice, each /^
is
c^i(xq)
if
7^ c^2(-^'a)'
updated exactly once,
Xa imphes
[fa{Za)
"
^2(^0))
since in this case the maximization that defines
and,
is
==
Ca2{Za)
Da
will
-
Cal{Xa)
occur with
= -Da,
first
(43)
order conditions;
then equation (42) holds. Hence,
A-l
WA- Da +
Sa
=
Yl [M^a) -
Ca\{Xa)
-
{fa{Za)
-
Ca2{Za))\
- Da +
Sa
by
(37),
a=l
A
<
a=l
—
Stage
for
2:
some
\
a=l
\
by
(42)- (43)
and our choice
of
(5a,
)
TT,
which contradicts wa
Q =
^
A (
^-Da + ^ [-Y^Da + ir]
index,
— Da +
(5/i
>
tt.
We
conclude that the procedure must terminate with
and an / that enforces
(ci(x)
+
7r,x).
This proves Claim
Extension to S'-supplier auction. To eliminate a
loss of generality that supplier
i
=
1
in the
variable,
statement of Proposition
46
1.
5.
D
we assume without
Claim
rule
6. Let Ci{x)
f that
bid X
and
<
it
Cs{x)
=
Vs
2,
.
.
5, where
,
.
and causes supplier
satisfies (8) -(9)
>
it
e.
win the
1 to
There exists a feasible scoring
S -supplier
auction with attribute
profit at least n.
Proof. Set das
feasible
+
=
-
Cas{x)
Cai(f),
Ds
=
J2a=\ ^a^' ^^^
with respect to the cost surface
bids from the cost functions Cg
and
Cg]
(and hence
Cg
=
Cas
-
das
+
Ds/A. SuppOSe /
Then / induces
Cg).
is
interior attribute
furthermore, since the cost functions differ from each
same
other by a constant, they both must yield the
Sas
=
Cas
-
mSiX{ faiz)
attribute bid
Zg.
Defining
Cag{z)},
z
= max{ fa{z) -
=
Sg.
Sa\
—
implies that X]a=i ^as
surfaces, with Sag
when /
is
<
announced to
profit at least
it
that / satisfies
and
construction for fixed
Cai
Cas{x)
faiZas)
We
construct a scoring rule / that
'k j
S
A V
a, s,
-
(cl):
Cag,
la\
S
+
Ca\ at Xa-
vr//l
and
=
Z\
dag
The
x.
feasible for the Cg cost
is
two conditions imply that
first
suppliers (with actual cost surfaces c^), supplier
again,
by
also ensuring that Sag
> e/A V
s,
1
wins with
we guarantee
We
we construct / dimension-by-dimension.
present the
a.
=
~
Cai,
and
Cal[X)
=
Cas\X)
—
das H
^
=
Cas{x)
-
{Cas{x)
-
Dg
n
=
We
fa{z)
Cal{xj,
Cal(f))
{cg{x]
—
+ "T ~
Ci{x))/A.
Cal(f),
Let
la\
denote the
line
consider two cases, (cl) and (c2).
s
^
7al^^"' SUch that fa{Xa)
=
does not intersect any curve
^\. Choose
+
Cag{Zag)
where the inequality follows because Dg/A
tangent to
- -^
=
(9) is satisfied;
Once
(8).
Notice that
all
+ dag- -^,
Cag{z)}
=
Cq^,
47
\.
In this case,
Ca\{Xa)
+
T^ /
A+
la\
+
vr/A
Ha, f'a{Xa)
lies
=
below
c'^ii^a),
ha S
Solving for 7ai, 7a2 yields
K"*".
=
lal
The
function fa
is
ensures that 7ai
<
+
-r
ha]x^''''\
>
increasing, since jai
1
Choosing
0.
-
Cal{Xa)
(and hence fa
is
concave, /^
>
tj A. Furthermore, for s
as well as ensuring that Sas
>
tc/A
<
Ll{z)
-
Sal
+
Sal
-
+
Set
z_a
<
In
=
lai
+
lai
-
Cas{z)}
some curve
+
71/
A
intersects
we showed that
1
h>
H,
//
6
IR+.
Notice that
Zl=Xa,
22
=
and
and
0"*",
/^
—
>•
as 2
—
>
00),
7^ 1,
+
/ai(2)
"J
Hes below
c^^,
J,
we
if
gi{zi)
+
lai{z)
<
Sai
Sai
=
=
g'2{z2)
is
- n/A
Let
tangent to fa at Xa, and fa
for s
A^ =
^
/
1,
+
tt/A
and
in [2^,a;a],
—
Cas{z)}.
set Za
>
Xq
s 7^ 1, in [xa, Za]-
we can
and
(22152(22)), 32(-^2) are a point-
find a function
/ such that (26)-(27) hold
set
g[{zi)
Cal{Xa),
TT
Cal{Xa)
is
I.
ma.Xz^s^i{lai{z)
s
Cas,
{zi,gi{zi)), g\{zi),
if
52(22)
Za,
^
oo as 2
>
no curves
Cas,
slope pair such that (28)-(29) hold, then
ior
—
Cal{Xa)}\
-
Cas, s j^ I.
T^/A intersects no curves
Claim
-
max{Cas{Xa)
,
-J,
max,{fa{z)
T^jA intersects
Xa such that
such that
"T
becausc
ial{z)
where the second inequality follows because
(c2): lai
-
< fa{z)-la\{z)--7
fa{^)-Cas{z)
concave. Hence, Sas
-fa
lc'aiiXa)Xa
> max
/!.a
+
[Cal{Xa)
+
>
-
f-
48
(A,,
=
+
c'^,j(.Xa),
C^^{Xa),
Cai{Xa)[Xa
then
-
Za])
,
g2{z2)
+
g'2{Z2)[zi
-
Z2]
and
That
is,
92(22)
+
>
g2{z2)
+ A„ +
=
Cal[Xa)
+ - - (Aa +
Ca\{:Xa)
.:Ui)
52(22)
g'2{z2)[Xa
hg^
52(22)
>
=
H_^,
Ha G
Ca\{Xa)
f'a{Za)
^
/i),
- -(A„ + c;,j(.Ta)[a-a-2j),
=
Cai(a:a)
+
C;,i(Xa)[2„-Xa]
<
lal{Za)
+
since
A
a and
^
such that
- (A^ +
+
we
J
+ ha,
show that
and hence we can
—
"
>
oo as Zq
^J)
f^{Xa)
and
=
9i(2i) the
52(^2)
+
—
>
and we can
h,,
c'^^iXa),
same but take
< ^^^ + ^ai (^^a),
22
=
a and
la\
TT
CaliXa)
+
> Ha, Ha some
--
{Aa
+
positive real
h^^ha^
ha
"T
+
4i(Xa)[Xa
-
Za])
+
ha,
/la,
-
-
and
/^(x„)
number. Setting
> max |^„, Ha, max
{Cas{Xa)
and
/
(2)
if2<Xa,
/a(^
fa{z)
49
if
2
>
Xa
-
Cal{Xa)}\
=
in
fa such that
-
TT
+
Cal{Xa)
2q,
then arguments
an increasing, concave, smooth function
=
find
0'*',
^ OO,
-
for ha
0.
(28)- (29) hold (again with tt/A in the role of
find
- 2J
Aa,
in the role of l\),
/' {za)
and
-
=
An >
C^ax{Xa)\Xa
leave Z\, 31(21),
fa{Xa)
fa{Za)
/qi
--
+
IT
If
C„i(x„)[Xa
)
+ ^,
+ TT/A-{/\a + c'^\{xa)[xa~Za\), and
as Za
- C J + Aa +
+
CaliXa)
M"*".
similar to those above
the role of
C^^{Xa)\Xa
Cai(:rJ
IS^a) = CaAXa) +
=
- ij.
=
(28)-(29) hold (with ii/A in the role of
l^iXa)
zj,
C^^{Xa)[Xa
an increasing, concave, smooth function /
for
-
=
C„i(Xa),
ensures that /„
Sas
>
—
^ oo
smooth, concave, increasing, /^
—
as z
>
0"*",
^
/^
as
2;
—
>
and
00,
i/A.
It
Sal
is
remains to show that Sai
—
A <
Let
t: I
induced by
Sas for
/.
We
some
first
s 7^
1.
>
tt/A
V
Sas
s
7^
denote supplier
Zas
Suppose
1.
not, that
suppose
is,
attribute bid in dimension a
s's
show that
71"
<
Cas{Zas)
Suppose
Cas{zas)
>
kii^as)
'Jas
—
+
T^/A.
-7-
(44)
Then
C.as\Zasj
Jax^as)
+
lal{Zas)
—
Ja\'^as)
~
'alV-^asj
<
lali^as)
+
Sal
-
Sal
"T
j
71"
which
is
is
derive a contradiction
is
if
treated similarly, and
fa{zas)
-
Cas{zas)
Sai
is
—
concave.
we have made use
We now
We
Sas-
omitted
<
fa{zas)
<
Lf{Zas)
<
laf{Za)-Ll{Za)-J + ^a,
=
Cal{Xa)
Let
—
denote the
/„/
[Lii^as)
+
^- ^a)
Ll{Zas)
~
T + ^a,
(CaliXa)
ha,
=
fa{Xa)
t^ I
A <
laf is
+- -
-
Sas-
(Aa
+
+
C^j (x
Cai{Xa)[z^
TT
(CaliXa)
The
~T,
of the fact that
Xa] the
line
tangent to fa at
z^.
by the definition of Aa,
J [x^ -
-
Xa])
-
+ j) =
/^i),
50
+ Sai
complement case
Sal
-
Z^])
-
+
ha
TT
- + Aa,
TT
J,
third inequality follows because la/
steeper than
lai
TT
TX
=
increasing function (since
-
<
treat the case Zas
for brevity.
—
use (44) along with the definition of /„ to
is
-
which contradicts Sai —
<
vr/Zl
^al(~as)
-J,
a contradiction. For the second inequahty,
tangent to fa at Xa, and fa
—
and
Zas
<
—
lai
is
an
la ^y (44) together with the
assumption that
<
Zas
only Xa, Cai{xa) and
Taking a step back, notice that
Xa-
parameters), and in case 2
c'^iixa) (three
(when defining
to z, g2{z), g'^iz)
in defining fa in case
/), z, 52(5), g'2{z)
we used
(when defining
we used
1
these three in addition
/),
and h^ ~
i.e.,
/
is
D
function of ten non-redundant parameters.
Claim
Ti
+
TT
+ tt <
^(.f)
Vs
intersects Cj, then (ci(;r)
+
Let c\[x)
7.
=
2,
n,x)
.
.
.
,
5, where
tt
>
^
// there exists a j
e.
such that
1
enforceable.
is
Claim
Proof. Let / denote the scoring rule for the two-supplier auction constructed in
such that S\{f)
is
—
=
tt
announced. Since
maximum
Sj{f), where Ss{f) denotes supplier s's
a
for all
f'a{z)
—
00 as 2
>
a
—
>
O"*"
and fai^) —^
dropout score
z —>
SiS
/
if
5
/
is
feasible
it,x)
and we
cxd,
(induces positive, finite attribute bids) in the 5-suppher auction.
(cl): S-[{f)
—
TT
>
Ss{f)
V
s 7^
1.
In this case, announcing
/ enforces
(ci(;r)
-|-
are done.
(c2):
B k
^
1
such that Si{f)
(8)-(9) constructed in
we
of our proof,
and
g.
Let g be the feasible scoring rule satisfying
Sk{f).
Claim 6 such that Si{g) — n > Ss{g)
show that there
exists a
is
(1
- X)ga)iXasW) J] C,,(x,, (A)),
[0, 1].
A* for which 5i(A*)
equahty, then X* f
= Vgix) =
> n+
—n >
+
{1
—
e
since
/ and g
Ss{X*) holds for
X*)g
is
s
when A/
-i-
(1
Vci(x), suppher
This implies that 5i(A)
because Si{f),Si{g)
(45)
a=l
the attribute bid of supplier
Notice that since V/(x)
A G
true for convex combinations
is
A
= Y^iXfa +
a=l
for all
In the remainder
convex combination of / and g that enforces
A
f
1.
Define
Ss{X)
where Xs{X)
for all s 7^
Notice that since / and g are feasible, the same
(ci(;?)-|-7r, x).
of /
will
— n <
=
XSi{f)
^
I,
I's
A)^
is
announced.
attribute bid fi(A) will equal
+ {l-X)Si{g) >
satisfy (8).
all s
—
By
vr-f e for
these observations,
and holds
for
aU A €
if
some suppher
(0, 1],
there exists
s
/
1
with
a feasible scoring rule that satisfies (8)-(9) and enforces
51
(ci(x)
+
To
7r,f) in the 5-supplier auction.
continuous function of
We
find such a A*,
begin by showing that Xas{^)
show that Xa{0) < Xa{\) < Xa{l)
continuous in
is
for all
A G
We
By
[0, 1];
<
for all
<
the definition of Xa{0), Za
XaiO), g'^iza)
A €
By
[0, 1].
>
<
>
Xa(l), /^(za)
>
for all
<
reversed).
We
Xa(l),
<
and
Xa{0)
<
prove Xa{0)
is
Xa{X)
the above analysis and /^,
+
(1
-
>
X)g'^{za)
c'^i^a),
and similarly the
g'^,
<
defini-
we have that
Xq(1),
andhence Xf^{za) + {l-X)g'a{za) >
— c^
and
— c'^{za) must
c'^{za)
strictly decreasing, for fixed
he to the right of
a;Q(0),
which
A
what
is
to show.
A
if
\5\
a
convenience we drop
suppose that Xa{0)
c^(-a)- Since Xa{0)
c'^iza),
Next, we consider perturbing A to A + (5, for
negative
for
(the proof in the case Xa{l)
Xa{0) implies g'^iza)
c'^{za)and f'^{za)
the root Xa{\) of \fa{za)
we wanted
is
the other inequality follows similarly.
tion of a'a(l) implies that for 2^
for 2a
[0, 1]
where
A,
first
and Xa{l)
essentially the same, with the roles of Xa{0)
A G
show that Ss{X)
first
A.
the subscript s in the general proof that follows.
for all
we
<
First,
=
1),
and show that
for
any fixed
\5\
small (where 6 must be positive
> 0,3
rj
S
>
if
A
such that \xa(\)—Xa{X + S)\
=
<
0,
r]
6.
note that for Za €
[2:^(0), 0:0(1)]
|((A
+
=
+
6)f'^
(1
and
6 fixed,
-
-
A
5)g',
-
-
c'J (z,)
(A/^
+
(1
-
A)^;
-
c'J {Za)
ma-9'a){Za)\,
^a€[Xa(0),Xo(l)]
<
m,
00 exists since
[;ra(0),
where some
B<
words, over
[^^^(O), ;ra(l)],
A/^
+
(1
-
X)g'^ -c'^
the curve (A
+ 6B and
decreasing function A/^
+
(1
is
+ 5)f'^ +
+
(1
-
X)g'^
-
c^
A/^
-
Xa(l)]
compact and
(1
/^
— A - 5)g'^ - c'^
— g'^
is
continuous. In other
hes between the two curves
A)^; -c'^- 6B. Let Wa{S) be the root of the strictly
+ SB. By
52
the
bound given above, and the
fact that
Xa{X
same
+
-
Wa{-S)\
<
desired result
we can
if
6B < -(A/^ +
fact that Xa{X)
is
(1
-
find 6 such that
- 0(:Ca(A) +
X)g'a
{Xf'^
<
''^a{—\^\)
5.
By
sum
+
By
is
(1
—
A)^^
—
\5\
<
Ca){xa{X))
S,
is
\5\
<
5
impUes
+
min{A|5,(A)
(1
—
—
for
X)g'a
is
—
A)^^
positive
— c'^, and
rj
q
>
0.
/2)
<
0.
u;,(|(5|)e(xa(A),.Ta(A)
c'^){xa{X)
and the continuity of Xa{X)
—
ti/2),
+ r?/2)
we have
follows.
a composition of continuous functions, and the
=
—
tt
5i(A) -TT
>
for
^^(O) for
some
the continuity of the 5s(A)'s and the fact that 5^(1)
noted below equation (45),
(1
right side
continuous in A -
is
all s.
our choice of g we have that 51(0)
=
(A/^
+
continuous, the righthand side of (45)
is
continuous in A for
A*
By
ri/2,Xa{X)) for
of continuous functions
Ss{X)
i.e.,
5B <
also ensuring that
£ {Xa{X) —
Since (A/a
Since the
S).
+ (1 - X)g'^- c'^ + S B){xa{X) +
Smce(A/^ + (l-A)5;-c; + (5B)(x„{A)) = (^B>0, we have that
\S\
The
r?/2).
a root of the decreasing function A/^
Furthermore, choosing 6 in this way imphes that
for
+
7/.
Pick S such that
by the the
Wa{—S) and Wa{S) sandwich Xa{X
liave that
we have our
true for Xa{X),
is
\Wa{S)
G [xa{0),Xa{l)], wc
S)
such a
A*,
A*/
+
(1
—
s
all s 7^ 1.
^
>
Set
< A <
1,
5i(l)
—
tt,
such a A*
A*)^ enfoixes (ci(x)
^-supplier auction. Notice that the scoring rule in this case
seven come from the function /, ten come from the function
is
g,
(46)
1}.
+
exists.
As
n,x) in the
a function of 18 parameters:
and the
last is
the parameter
A*.
D
A2.
that
Proof of Proposition
Ml >
plus the
e;
2.
In the statement of the Proposition,
otherwise, the auctioneer's valuation
minimum
not interesting).
bid increment
We
e,
is
we
tacitly
assume
everywhere below the suppliers' costs
meaning that no auction would be pursued (which
is
break the proof of Proposition 2 into three cases, and begin with an
53
observation that will be useful in
all
three.
made
Recall that the definition of optimality was
Let u* be the
(9), (13).
with
V,
and
Ci
C2
of (7)-(9), (13), here
Sk
>
i,
we
+
first
show that u* < Mi —
(i.e.,
max
(7)-(9), (13),
Towards proving the op-
Ss
=
f{xl)
—
/
are ordered
are the top bidders and
is
vixl)-f{xl) +
subject to
of (7)-
In contrast to the formalization
e.
For scoring rules / such that k and
the appropriate mathematical program
e,
optimum
we make no assumption on how the supphers' subscripts
given scoring rule.
Si
of the general version of (7)-(9), (13),
instead of their parameterized counterparts).
timality of supplier
for a
optimum
in relation to the
St
(47)
s
Cs{xl),
=
k,l
Si>e.
Examining the objective function
H^k)-
fixl)
+
Si
(47),
we
see that
= v{xl)-Ck{xl)-{f{xl)-Ck{xl)) +
=
v{xl)
-
Ckixl)
-
Sk
<
v{xl)
-
Cki'xl)
-
e
<
max{i'(f)
—
Ck{x)}
+
Si,
since Sk
—
Si,
>
Si
+
e,
e,
X
= Mk-e.
Hence, we've shown that u*
< M, —
e.
This observation motivates the
provides conditions in which the auctioneer
is
first case,
which
guaranteed to do no worse than u* by simply
announcing the true valuation function v as the scoring
rule; in this case,
we
consider v to
be optimal (as discussed below equation (13)).
(cl):
3
/c
/
?
such that
the scoring rule.
We
Mk > Mj —
e.
Suppose we announce the true valuation function v as
need to show that, no matter who
54
{i
or k) wins, the post-auction utility
to the auctioneer
is
—e
at least Afj
(and hence both suppUers are optimal). Suppose supplier
i
wins the auction. Since we assume the winning suppher wins at the losing supplier's highest
possible score, the profit at which supplier
i
wins the auction
at
is
most
— Sk <
S^
Because
e.
the auctioneer announced his true valuation function as the scoring rule, the utihty to the
maximum
auctioneer equals the winner's
least 5i
—
= Mj —
e
dropout score minus the winner's
Next, suppose that supplier k wins the auction.
e.
out at the highest possible losing score; since 5,
profit,
and the post-auction
utility of
that supplier k actually bids,
auctioneer
(e is
the
is still
NU —
at least
minimum
i.e.,
c.
<
>
If
supplier
i's
bid increment), and
is
> Mi —
Sk
= Mk <
instead Sk
e.
i.e., is
Supplier
e
drops
t
Here we have assumed
e.
then the utihty
e,
for the
winning score must have magnitude at
Mk <
at
Sk, supplier k wins the auction with zero
the auctioneer
Sk
profit,
implies that
Mj <
least
e
2e.
In the remaining two cases of the proof, announcing the true valuation function does
not generate at least
M^ —
e
under an optimal solution to
hyperplane tangent to
Tj
+
(c2):
that
is
A'/j
A'/j
tangent to
— e>Ms
(c,(:r*)
+
i;
q
and we show that supplier
in utility for the auctioneer,
(7)-(9),
Because v
at x*.
Let
(13).
M, =
v{x*)
concave and
is
c, is
—
Cj(x*),
and
let T,
i
wins
be the
we have
convex, notice that
at f*.
for all s 7^
i,
and 3
j 7^
i
such that T^
+e
e,x*) can be enforced, yielding a payoff of
intersects
M^ —
e for
Cj.
Corollary
1
implies
the auctioneer. Hence,
i
optimal.
(c3):
Mi —
e
> Mg
for all s
reUes on hyperplanes. Let
or tangent to Cg for
all s 7^
tt
z.
^
=
i,
and $ j
y^ i
minrs^i{cs(£)
We
use Ti
+
A/j
such that T^
+
e
intersects
— Tj(5)}. The hyperplane
and
Tj
+ tt
to
Our argument
-|-7r
must he below
bound v from above, and
below, respectively. Consider a scoring rule for which supplier s
55
T,
Cj.
7^ i
wins. Let
w
Cg
from
be supplier
s's
winning attribute
The
bid.
auctioneer's utility
<
v{w)-Cs{u')
most v{w) — Cs{w), where
at
is
{T,{w)
+ Ah)-{T,{w) +
Cs{w)]
= M^ < Mi
Tr),
which imphes that
ms.yi{v{w)
Applying Corollary
Mj —
payoff of
supplier
the
—
TT
1,
+
(cj{.t*)
+
n
-
d,x*)
enforceable as 5 —>
is
optimum
u*
is
yielding for fixed S a
0"*",
< Mi — n — maxs^, Mg shows
S for the auctioneer. Choosing 5
winning we can improve on the best case
i
-tt.
for
which supplier
achieved for some scoring rule / for which supplier
s j^
i
that with
Hence,
wins.
wins (supplier
i
i
is
optimal).
A3.
We
Simplification of (15)-(18).
simphfy the mathematical program (15)-(18) by
finding a closed-form solution to the innermost minimization in (18).
convex and increasing
r^^
lOr ^
a
_
-
dCas{Za,easl,...,easP)
A
T
].,... ,/i,
\
dZa
or at z*
=
0,
^
_
-
2*
innermost minimization occurs at
dCq.jZqfi^^l
,.
..fig.p)
-r
\
In words,
if
such that,
A.P
A.l
dZa"
_
Za=Xa
g^^
otherwise.
flCq, (^g
I
I
.
^
^
IZa=0
a point exists at which the
\Za—^a
tangent to
Cas is parallel to Cq^'s
exist,
dimension
By
_
g^^
dCq^{Za,ea,l,...fia.p)\
does not
in Xa, for fixed s the
Since the Cas are
a,
tangent at Xa, we take this point as
then the minimization occurs at the
and thus we take
incorporating
z*,
we
z*
=
left
>
mm
get
A
y^ Cgkjzl, 9aku-
,
Oakp)
t
_
\
dCai{Za,9ati,
^^a
a=\
of -Ti[z*).
.
.
.
,
dgkp)
a=l
,
last
- y^ Cat{Xa, 6'afci,
a=l
E,
The
endpoint of the feasible interval in
f
<
^
in place of (18).
such a point
if
0.
A
TT
z*;
two factors within the
.
.
.
,
Ogtp)
\
Zn=Xa
(48)
'
right side of (48) are the
expanded version
Notice that in solving (15)-(17), (48) we need only search over
56
fs
for
which
supplier fs cost
is
below the auctioneer's true valuation,
value reaches
its
upper bound of Mi
Optimal Scoring Rule
A4.
(17), (48),
and
any
argmius^i
and note that
TTs,
Step
By
=
tt*
be the scoring rule constructed
c^,
iXj
(otherwise
and
induces supplier
i
to bid attribute level x*,
also that Tj
that,
if
/
is
-I-
tt*
announced
to all
auction with profit at most
tt*
{6
<
is,
the optimal scoring rule
5) in the S-supplier
(c2); Tj intersects Cj for
can be applied
satisfies
from below
S
suppliers, Sg
(by Sj
some
/
j 7^
in a special way.
the conditions of Claim
5 to enforce
(c.i(;f*)
the conditions of Claim
that causes supplier
To
bid z
^
i
to
6,
i,
<
Si
not optimal). For 8
+ tt* + S, x*)
— tt* — 6 when /
—
^
i.
Tj
-|-
for all s y^
tt*
Si
>
0,
let j
=
small, let /
in the two-supplier
announced. Since /
is
is
tangent to / at x*,
Together these observations imply
— TT* — 5). That
arbitrarily small,
and v
satisfies (8).
In particular,
6; this
+ tt*,x*)
is,
/
and supplier
i,
/ enforces
is
{ci{x*)
i
wins the
+ tt* + 6,x*)
considered optimal). That
we need
-I-
to
In this case,
that Claim 7
v
implies that Claim 7 can be applied to construct an
(1
-
A*)i',
show that
where /
t;
is
is
the scoring rule constructed in
auction between
to bid x*,
i
and
j.
To rephrase
a feasible scoring rule satisfying (8)-(9)
win the auction with attribute bid x* and
i
we show
we show that the true valuation function
in the two-supplier
see that v induces supplier
x*)
for all s
Si
is
Si
Ti{z)} and
in (19).
optimal scoring rule of the form X* f
Claim
=
auction (since 5
is
is
we must have that
Cg
+S
= mm;{cs{z) —
tt^
tt*
=
our choice of /, Sj
bounds
at x*
c,
5 to enforce (ci(x*)
i
j.
and when the objective
be the optimal solution to (15)-
x'*,7r*
Let
i.
auction between
and
Let
3.
^
s
Claim
in
if
<
t.
be the hyperplane tangent to
let T^
Ti does not intersect
(cl):
in
—
,Oaip)
'^^^iCai{xa,9aii,
Also, the Search can be terminated
,4'ap)-
'^a=i'"<^i^a,4'ai,
i.e.,
profit at least
tt*.
we suppose not (suppose that v induces
and derive a contradiction by showing that the payoff can be improved (and
57
moving
enforceability maintained) by
/
X*
and
Za,
let
d
—
Za
—
.t*.
Va{x*^
—
since Va
Cai
is
TT*
,
Cj
{xl, ...,
x*,
TT* is
new
+
TT*
3,
xl+5d,
Ca^{xl
+
and
.
.
.
is
x^
,
..., x*^)) is enforceable,
optimal. Hence, x*
intersects Cj,
tt*.
we have Mj —
for supplier
5
i,
and v
=
tt
+
6d)
Sd,
e is
x* toward
Let a be such that
z.
1,
>
Va{x*J
.
.
.
-
Car{xl)
(49)
,x^) that
This implies that
we move
(cj(xj,
.
.
to,
.
,xl
the hyperplane
+ 5d,
n* and
still
.
.
,x*^)
+
fact that
to bid x*.
tight for our optimal solution in this case;
we could decrease
.
which together with (49) contradicts the
i
we can
Clearly, for 5 small,
at Za-
so announcing v induces
z,
>
<
maximized
will still intersect the surface Cj.
the optimality of
step
<
for
-
6d)
point (xj,
Notice that constraint
Tj
+
strictly concave,
ensure that at the
tangent to
Then,
away from
slightly
if
not, since
maintain enforceability, contradicting
Furthermore, because we did not exit the three-step method of §2.5 at
e
> Mg
satisfies
for all s
^
equation
i.
(9).
That
is,
announcing v yields a
profit of at least
Since equation (8) holds by assumption,
we
e
are
done.
(c3):
Tj intersects Cj for
some
j
/
i,
and v does not
satisfy (8).
eral construction of Proposition I's "=>" direction proof (§A1.2)
(Cj(.'?*)
-I-
TT*,
X*).
58
In this case, the gen-
can be applied to enforce
Date Due
Lib-26-67
3 9080 02246 0544