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Incentives in Core-Selecting Auctions Paul Milgrom1 October 18, 2006
A “core-selecting auction mechanism” is a direct mechanism for a multi-item
allocation problem that selects a core allocation with respect to the bidders’
reported values and the auctioneer’s exogenously given preferences. For every
profile of others’ reports, a bidder has a best reply that is a truncation report.
For every “bidder optimal” core imputation, there exists a profile of truncation
reports that is a full-information Nash equilibrium for every core-selecting
auction with those payoffs. Among core-selecting auctions, the incentives to
deviate from truthful reporting are minimal at every preference profile if and
only if the auction always selects a bidder optimal allocation with respect to the
reported preferences. Finally, a core-selecting auction that selects a minimum
revenue core allocation is a bidder optimal auction and make the seller’s
revenue a non-decreasing function of the bids, which eliminates distortions that
can otherwise occur in the process of bidder application and qualification. All
these results have analogues in two-sided matching theory.
Keywords: core, stable matching, marriage problem, auctions, core-selecting
auctions, menu auctions, proxy auctions, package bidding, combinatorial
bidding, incentives, truncation strategies.
JEL Codes: D44, C78
1 Financial support for this research was provided by the National Science Foundation under grant ITR-0427770. Thanks to Roger Myerson for suggesting the connection to Howard Raiffa’s observations about bargaining and to Manuj Garg for proofreading.
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I. Introduction Recent years have seen a number of new and important applications of stable matching
procedures in practical applications, including school assignments in New York and Boston and
new designs for life-saving organ exchanges. The mechanisms that have been adopted, and
sometimes even the runner-up mechanisms, are stable mechanisms that select a kind of core
allocation. More precisely, stable matches are matches with the property that no individual can do
better by staying unmatched and no pair can both do better by matching to one another; stable
mechanisms are mechanisms that select a stable match for any reported preferences.
Evidence that stable mechanisms persist in use while unstable mechanisms do not is found
in both empirical studies (Roth and Xing (1994)) and laboratory experiments (Kagel and Roth
(2000)). Stable mechanisms have the twin advantages that there is no couple that would want
either to renege after the mechanism is run in favor of some alternative pairing or to pair off
before the mechanism is run, since no such agreement can be better for both members of the
couple than the outcome of a stable matching mechanism.
Similar reasoning applies to auction mechanisms. An individually rational outcome is in
the core of an auction game if and only if there is no group of bidders who would strictly prefer
an alternative deal that is also strictly better for seller. Consequently, an auction mechanism that
delivers core allocations has the twin advantages that there is no individual or group that would
want either to renege after the auction is run in favor of some feasible alternative and or to agree
to an alternative deal before the auction, since no alternative that is feasible for any group of
participants is preferred by them all to the outcome of a core-selecting auction.
The preceding arguments rest on the idea that stable matching mechanisms or core-
selecting auctions actually result in stable or core allocations, and that in turn may depend on the
parties’ incentives to report truthfully. This paper explores incentives in core-selecting auction
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mechanisms, including both equilibrium outcomes and incentives to report truthfully, and
compares them to the corresponding results for two-sided stable matching mechanisms.
Incentives in stable matching mechanisms have been argued to be an important part of their
practical appeal,2 even though these incentives fall short of strategy-proofness. Roth (1982) has
shown that there exists no two-sided stable matching mechanism that is strategy-proof for all
participants in the marriage problem, in which men have preferences over women and women
have preferences over men. For that problem, there always exists a unique man optimal match3
and the mechanism that always selects it is strategy-proof for men,4 but the women then have an
incentive to misreport. For that mechanism, there is a full information Nash equilibrium that
results in the woman-optimal match for the actual preferences. At equilibrium, men report
truthfully while women adopt truncation strategies, ranking men in the truthful order but listing
men who are worse than their equilibrium match as unacceptable. These results hold in pair-wise
matching environments where no transfers are possible, so it is surprising to find close parallels
among package auctions operating in transferable utility auction environments.
Package auctions (also called “combinatorial auctions”) are auctions in which bidders can
bid directly for non-trivial subsets (“packages”) of the items being sold. Just as there is no
strategy-proof stable matching mechanism for all preference profiles, there is no core-selecting
package auction that is strategy-proof for all quasi-linear preferences or, indeed, for any
sufficiently large subset of the quasi-linear preferences. This is true even when the seller’s
incentive to set its reserve price truthfully is excluded. This conclusion follows by combining two
theorems. The first is the well-known result of Green and Laffont (1979), as extended by
2 For example, see Abdulkadiroglu, Pathak, Roth and Sonmez (2005). 3 As Gale and Shapley first showed, there is a stable match that is Pareto preferred by all men to any other stable match, which they called the “man optimal” match. 4 Hatfield and Milgrom (2005) identify the conditions under which this result extends to cover the college admissions problem, in which one type of participant, the colleges, can accept multiple applicants. Their analysis also covers problems in which wages and other contract terms are endogenous.
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Holmstrom (1979), that any path-connected set of general environments with quasi-linear
preferences, the only strategy-proof direct auction mechanism that selects total-value-maximizing
choices is the Vickrey mechanism.5 The second is a theorem of Ausubel and Milgrom (2002) that
the outcome of the Vickrey mechanism is a core allocation only in special cases: if all bidders’
values are drawn from a set in which the goods are substitutes, then the Vickrey outcome is a core
allocation but if the valuations are drawn from any strictly larger set,6 then there are admissible
preference profiles at which the Vickrey outcome is not a core allocation. So, for classes of
preferences larger than the substitutes class, no core-selecting auction is strategy-proof.
This impossibility does not necessarily imply that sufficient incentives for truthful
reporting in core selecting mechanisms cannot be provided in practice. Bidders in a package
auction often need to solve a combinatorial optimization problem with uncertain parameters to
find their optimal bid, so if reporting truthfully is nearly optimal in some setting, then even
rational bidders may quite reasonably choose reports close to the truth. A similar claim of
sufficient incentives has been made for the pre-1998 algorithm used by National Resident
Matching Program, which was not strategy-proof for doctors, but for which the gains to
misreporting were nevertheless difficult to realize (Roth and Peranson (1999)).7
Ausubel and Milgrom (2002) showed that their ascending proxy auction is a core-selecting
mechanism, that it selects the Vickrey outcome whenever that is in the core, that truthful
reporting is an ex post equilibrium of that mechanism when goods are substitutes, and that the full
information equilibria of the mechanism include ones that select bidder optimal outcomes with 5 The terminology used here is not completely standardized. What we here call the “Vickrey auction” is sometimes called the “generalized Vickrey auction” and is the same as the Vickrey-Clarke-Groves “pivot mechanism” specialized to the case of resource allocation. 6 The theorem of Ausubel and Milgrom (2002) asserts more: if the class of admissible goods valuations includes all the additive preferences or all the singleton valuations (in which a bidder’s value of a set of goods is just its highest value for a single good in the set), then if the class also includes any valuation for which goods are not substitutes, there exists some profile from the set such that the Vickrey outcome is not in the core. 7 There is quite a long tradition in economics of examining approximate incentives in markets, particularly when the number of participants is large. An early formal analysis is by Roberts and Postlewaite (1976).
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respect to the actual preferences, rather than just reported preferences. None of these statements,
however, assert the optimality of incentives, so we may ask whether there are core-selecting
auctions with better incentive properties. In this paper, we identify the core-selecting auctions
with the best incentives for truthful reporting.8
Unlike matching mechanisms, auction mechanisms determine both an assignment of goods,
x, and a vector of payments, p. Nevertheless, the four central results emerging from our study of
incentives in core-selecting package auctions all correspond to results about two-sided stable
matching mechanisms.9 First, for all core-selecting direct auction mechanisms and regardless of
the pure strategies adopted by other bidders, each bidder always has a best reply that is a
particular truncation strategy, that is, a strategy that exaggerates the attractiveness of the no-trade
outcome without changing the relative rankings of other outcomes.10 Moreover, the bidder’s
maximum profit is always equal to his profit from a Vickrey mechanism in which the other
bidders’ reports are the same. For the second result, let a bidder optimal allocation ( , )x p be a
core allocation with the property that no other core allocation is weakly preferred by all bidders.
Then, for any bidder optimal allocation, there is a profile of truncation strategies that is a full-
information Nash equilibrium for every core-selecting auction and for which the equilibrium
outcome is ( , )x p . Each profile of truncation strategies is also a profile of “truthful strategies” as
defined by Bernheim and Whinston (1986), who found that these are the coalition-proof
8 A paper by Parkes, Kalagnanam and Eso (2002) adopts what we will show is a very closely related approach. It derives mechanisms that minimize various measures of the distance between the vector of transfers to be paid by participants and the Vickrey mechanism transfers subject to individual rationality and ex post optimality constraints. If one takes their approach but constrains the mechanism to be core-selecting, then one is led to core-selecting mechanisms with “optimal incentives” as defined below. 9 Other close relations have been established by Kelso and Crawford (1982), who showed that the Gale-Shapley algorithm is a special case of a particular multi-item auction, and by Hatfield and Milgrom (2005), who derive general dominant strategy implementation results in a model that includes both matching and auction problems as special cases. 10 Mathematically, this is the most fundamental of the results connecting package auction theory to matching theory. It is essentially a statement about the structure of the core. In both matching and auction problems, not only is the core non-empty, but a participant j’s best core payoff is the best allocation it can achieve that is not blocked by the coalition N–j of everyone else.
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equilibria of their menu auction game.11,12 A link between their result and the new one reported
here is that the menu auction is a core-selecting auction.
A third result characterizes the core-selecting mechanisms that provide the best incentives
for truthful reporting: these are as precisely the ones that always select a bidder optimal
allocation. To make the notion of “best incentives” precise, let ( )j tε denote the maximum that
bidder j can gain by deviating from truthful reporting when the type profile is t and the other
bidders report truthfully. If a mechanism always selects bidder optimal allocations, then for any
type profile t, there exists no other core-selecting mechanism for which the incentives to deviate
at t are weakly lower for every bidder and strictly lower for some bidder. Moreover, mechanisms
that sometimes fail to choose bidder optimal allocations cannot have the aforementioned
property. For the identified auction mechanisms, truthful reporting is an ex post equilibrium if and
only if the Vickrey outcome is in the core and, in that event, the auction selects the Vickrey
outcome.
A fourth result is that any auction that minimizes the seller’s revenue among core
allocations results in seller revenue being a non-decreasing function of the bids. Revenue-
monotonicity of this sort is important because, without it, a seller might have an incentive to
exclude qualified bidders to increases its revenues and a bidder might have an incentive to
sponsor a shill, whose bids reduce prices.
11 Here, we seek to adapt and expand the language of matching theory, rather than using the existing language of auction theory, according to which these are “truthful” or “profit-target” strategies. In matching theory, Gale and Shapley (1962) defined a man optimal allocation to be a stable allocation that is weakly preferred by all men to any other stable allocation. They show that such an allocation exists. For package auctions, there can be many bidder optimal allocations which are not Pareto ranked, so we use the weaker definition, which Gale and Shapley proved is equivalent for the two-sided matching problem, is needed for the auction problem to guarantee existence and to highlight the similarities between the theories. 12 Truncation strategies have also been recommended in the business literature on bargaining. Howard Raiffa (1982) writes “In general, I would advise negotiators to act openly and honestly on efficiency concerns; tradeoffs should be disclosed (if the adversary reciprocates), but reservation prices should be kept private.” (Roger Myerson suggested this reference to me.)
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The Vickrey auction is not a core-selecting auction but, given its prominence in the
received literature, it is useful to review its defects in terms of the core and the failure of revenue-
monotonicity. These can be illustrated by a simple example with three bidders and two identical
items for sale.13 Suppose that a Vickrey auction were run in which bidder 1 bids 0 for a single
item and 10 for the pair of items. Bidders 2 and 3 bid 10 for a single item and the same 10 for the
pair. In this example in which all three bidders bid 10 for the pair of items, the outcome of the
Vickrey auction is that bidders 2 and 3 are the winners and pay prices of zero. This low revenue
outcome illustrates the first major defect of the Vickrey auction as a practical auction design.
When the Vickrey outcome is not in the core, as in this case, that always means that the seller’s
revenue is low—lower than at any core allocation (Ausubel and Milgrom (2002)).
In the same example, revenue-monotonicity fails. If the seller could find a reason to
exclude bidder 2, effectively reducing its bids to zero, then the Vickrey mechanism would award
the items to bidder 1 for a total price of 10 instead of zero. Even in single item auctions, sellers do
often retain the right to disqualify a winning bidder for practical reasons,14 and there is an extra
reason to retain for that flexibility in the Vickrey package auction. In our example, if bidder 3
were absent, bidder 2 would have an incentive to hire bidder 3 and instruct her to bid 10, despite
having a value of zero for the second item. Hiring the shill reduces the price from 10 to zero and
converts 2 from a loser to a winner! Or, perhaps bidders 2 and 3 are both real bidders but have
low values of, say, 3 apiece for a single unit. These would-be losers could engage in a profitable
joint deviation, raising their bids from 3 to 10 to become winners while reducing revenues from 6
to zero. The seller might retain a right to exclude bidders in order to defend itself against such
13 A fuller treatment of these issues can be found in Ausubel and Milgrom (2005) and Milgrom (2004). 14 It is common to verify qualifications only after an auction, perhaps to save on transaction costs, which creates opportunities to renege. Klemperer (2002) also emphasizes this possibility, arguing that “Sealed-bid auctions [may be] vulnerable to rule-changing by the auctioneer. For example, excuses for not accepting a winning bid can often be found if losing bidders are willing to bid higher.” McAdams and Schwarz (2006) defend a similar assumption and analyze a model of auctions in which the central element is the seller’s inability to commit to the auction rules.
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tactics. In contrast, in revenue-monotonic auctions, the seller has no need to defend against shills
and never has a positive incentive to exclude bidders.
The remainder of this paper is organized as follows. Section II introduces definitions and
notation and introduces the theorems about best replies and full information equilibrium. Section
III states and proves the theorem about the core-selecting auctions with the smallest incentives to
misreport. Section IV shows that the revenue-minimizing core-selecting auction is revenue-
monotonic. Various corresponding results for the marriage problem are developed in section V,
while section VI concludes.
II. Truncation Strategies, and Full Information Equilibrium We denote the bidders by 1,...,j J= , the seller by 0, and the set of all players by N. Each
bidder j has quasi-linear utility and a finite set of possible packages jX . Its value associated with
any feasible package j jx X∈ is ( ) 0j ju x ≥ . We conduct our discussion mainly in terms of
bidding applications, but note in passing that the same mathematics accommodates much more,
including some social choice problems. In the simplest case of package bidding, the relevant
outcomes consist of a set of pairs describing each bidder’s package of items and price, but for
some auction applications jx may also usefully incorporate delivery dates, warranties, and other
product attributes. Among the possible packages for each bidder is the null package, jX∅∈ and
we normalize so that ( ) 0ju ∅ = .
The auctioneer, whom we here call the seller but could alternatively be a buyer in a
procurement auction, has a feasible set of offers 0 1 ... JX X X⊆ × × with 0( ,..., ) X∅ ∅ ∈ —so the
no sale package is feasible—and a valuation function 0 0:u X → normalized so that
0 ( ,..., ) 0u ∅ ∅ = . For example, then 0u may be the auctioneer-seller’s variable cost function.
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For any coalition S, a goods assignment x is feasible for coalition S, written ˆ ( )x F S∈ , if
(1) 0x X∈ and (2) for all j, if j S∉ or 0 S∉ , then ˆ jx =∅ . That is, a bidder can have a non-null
assignment when coalition S forms only if that bidder and the seller are both in the coalition.
The coalition value function or characteristic function is defined by:
( )( ) max ( )u x F S j jj Sw S u x∈ ∈
= ∑ (1)
In a direct auction mechanism ( , )f P , each bidder j reports a valuation function ˆ ju and the
profile of reports is 1ˆ ˆ{ }Jj ju u == . The outcome of the mechanism, ( ) ( )( ) 0ˆ ˆ, ( ) ( , )J
jf u P u X +∈ ,
specifies the choice of 0ˆ( )x f u X= ∈ and the payments ˆ( )j jp P u += ∈ made to the seller by
each bidder j. The associated payoffs are given by 0 0 0( ) jj
u x pπ≠
= +∑ for the seller and
( )j j ju x pπ = − for each bidder j. The payoff profile is individually rational if 0π ≥ .
A cooperative game (with transferable utility) is a pair ( , )N w consisting of a set of players
and a characteristic function. A payoff profile π is feasible if ( )jj Nw Nπ
∈≤∑ , and in that case it
is associated with a feasible allocation. An imputation is a feasible, non-negative payoff profile.
An imputation is in the core if it is efficient and unblocked:
( ){ }( , ) 0 | ( ) and ( )j jj N j SCore N w w N S N w Sπ π π
∈ ∈= ≥ = ∀ ⊆ ≥∑ ∑ (2)
A direct auction mechanism ( , )f P is core-selecting if for every report profile u ,
( )ˆ ˆ,u uCore N wπ ∈ . Since the outcome of a core-selecting mechanism must be efficient with
respect to the reported preferences, we have the following:
Lemma 1. For every core-selecting mechanism ( , )f P and every report profile u ,
0
ˆ ˆ( ) arg max ( )x X j jj Nf u u x∈ ∈
∈ ∑ (3)
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The payoff of bidder j in a Vickrey auction is the bidder’s marginal contribution to the
coalition of the whole. In cooperative game notation, if the bidders’ value profile is u, then bidder
j’s payoff is ( ) ( )j u uw N w N jπ = − − .15
In the marriage problem, a truncation report refers to a reported ranking by person j that
preserves the person’s true ranking of possible partners, but which may falsely report that some
partners are unacceptable. For an auction setting with transferable utility, a truncation report is
similarly defined to correctly rank all pairs consisting of a non-null goods assignment and a
payment but which may falsely report that some of these are unacceptable. When valuations are
quasi-linear, a reported valuation is a truncation report exactly when all reported values of non-
null goods assignments are reduced by the same non-negative constant. We record that
observation as a lemma.
Lemma 2. A report ˆ ju is a truncation report if and only if there exists some 0α ≥
such that for all j jx X∈ , ˆ ( ) ( )j j j ju x u x α= − .
Proof. Suppose that ˆ ju is a truncation report. Let jx and ′jx be two non-null
packages and suppose that the reported value of jx is ˆ ( ) ( ) α= −j j j ju x u x . Then,
( , ( ) )α−j j jx u x is reportedly indifferent to ( ,0)∅ . Using the true preferences,
( , ( ) )α−j j jx u x is actually indifferent to ( , ( ) )α′ ′ −j j jx u x and so must be reportedly
indifferent as well: ˆ ( ) ( )j j j ju x u x α− − = ˆ ( ) ( )j j j ju x u x α′ ′− − . It follows that
ˆ ˆ( ) ( ) ( ) ( )j j j j j j j ju x u x u x u x α′ ′− = − = .
Conversely, suppose that there exists some 0α ≥ such that for all j jx X∈ ,
ˆ ( ) ( )j j j ju x u x α≡ − . Then for any two non-null packages, the reported ranking of
15 A detailed derivation can be found in Milgrom (2004).
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( , )jx p is higher than that of ( , )jx p′ ′ if and only if ˆ ˆ( ) ( )j ju x p u x p′ ′− ≥ − which
holds if and only if ( ) ( )j ju x p u x p′ ′− ≥ − .
We refer to the truncation report in which the reported value of all non-null outcomes is
ˆ ( ) ( )j j j j ju x u x α= − as the “ jα truncation of ju .”
In full information auction analyses since that of Bertrand (1883), auction mechanisms
have often been incompletely described by the payment rule and the rule that when there is a
unique highest bid, that determines the winner. Ties often occur at Nash equilibrium, however,
and the way ties are broken is traditionally chosen in a way that depends on bidders’ values and
not just on their bids. For example, in a first-price auction with two bidders, both bidders make
the same equilibrium bid, which is equal to the lower bidder’s value. The analysis assumes that
the bidder with the higher value is favored, that is, chosen to be the winner in the event of a tie. If
the high value bidder were not favored, it would have no best reply. As Simon and Zame (1990)
have explained, although breaking ties using value information prevents this from being a feasible
mechanism, the practice of using this tie-breaking rule for analytical purposes is an innocent one,
because, for any 0ε > , the selected outcome lies within ε of the equilibrium outcome of any
related auction game in which the allowed bids are restricted to lie on a sufficiently fine discrete
grid.
In view of lemma 1, for almost all reports, assignments of goods differ among core-
selecting auctions only when there is a tie; otherwise, the auction is described entirely by its
payment rule. We henceforth denote the payment rule of an auction by ˆ( , )P u x , to make explicit
the idea that the payment may depend on the goods assignment in case of ties. For example, a
first-price auction with only one good for sale is any mechanism which specifies that the winner
is a bidder who has made the highest bid and the price is equal to that bid. The mechanism can
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have any tie-breaking rule to be used so long as (3) is satisfied. In traditional parlance, the
payment rule P defines an auction, which comprises a set of mechanisms.
Definition. u is an equilibrium of the auction P if there is some core selecting mechanism
( , )f P such that u is a Nash equilibrium of the mechanism.
For any auction, consider the tie-breaking rule in which bidder j is favored. This means that
in the event that there are multiple goods assignments that maximize total reported value, if there
is one at which bidder j is a winner, then the rule selects such a one. When a bidder is favored,
that bidder always has some best reply.
Theorem 1. Suppose that ( , )f P is a core-selecting direct auction mechanism and
bidder j is favored. Let ˆ ju− be any profile of reports of bidders other than j. Denote
j’s actual value by ju and let ˆ ˆ, ,( ) ( )j j j jj u u u uw N w N jπ
− −= − − be j’s corresponding
Vickrey payoff. Then, the jπ truncation of ju is among bidder j’s best replies in the
mechanism and earns a payoff for j of jπ .
Proof. Suppose j reports the jπ truncation of ju . Since the mechanism is core-selecting, it
selects individually rational allocations with respect to reported values. Therefore, if bidder j is a
winner, its payoff is at least zero with respect to the reported values and hence at least jπ with
respect to its true values.
Suppose that some report ˆ ju results in an allocation x and a payoff for j strictly exceeding
jπ . Then, the total payoff to the other bidders is less than ˆ ˆ, ,( ) ( )j j j ju u j u uw N w N jπ
− −− ≤ − , but so
−N j is a blocking coalition, contradicting the core-selection property. Hence, there is no report
yielding a profit higher than jπ . Since reporting the jπ truncation of ju results in a zero payoff
for j if it loses and non-negative payoff otherwise, it is always a best reply when 0jπ = .
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Next, we show that the truncation report always wins for j, therefore yielding a profit of at
least jπ so that it is a best reply. Regardless of j’s reported valuation, the total reported payoff to
any coalition excluding j is at most 0ˆ ˆ, ( , ) ˆ( ) max ( )
j j ju u x x X ii N jw N j u x
− −= ∅ ∈ ∈ −− = ∑ . If j reports the
jπ truncation of ju , then the maximum value is at least 0
ˆmax ( )x X i ji Nu x π∈ ∈
− =∑
ˆ , ( )j ju u jw N π
−− , which is equal to the previous sum by the definition of jπ . Applying lemma 1
and the hypothesis that j is favored establishes that j is a winner.
Definition. An imputation π is bidder optimal if ( , )π ∈Core N u and there is no
ˆ ( , )π ∈Core N u such that for every bidder j, ˆπ π≤j j with strict inequality for at least
one bidder. (By extension, a feasible allocation is bidder optimal if the corresponding
imputation is so.)
Next is one of the main theorems, which establishes a kind of equilibrium equivalence
among the various core-selecting auctions. We emphasize, however, that the strategies require
each bidder j to know the equilibrium payoff π j , so what is being described is a full information
equilibrium.
Theorem 2. For every valuation profile u and corresponding bidder optimal
imputation π, the profile of π j truncations of ju is a full information equilibrium
profile of every core selecting auction. The equilibrium goods assignment x*
maximizes the true total value ( )∈∑ i ii N
u x , and the equilibrium payoff vector is π
(including 0π for the seller).
Proof. For any given core-selecting auction, we study the equilibrium of the
corresponding mechanism that, whenever possible, breaks ties in (3) in favor of the
goods assignment that maximizes the total value according to valuations u. If there
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are many such goods assignments, any particular one can be fixed for the argument
that follows.
First, we show that no goods assignment leads to a reported total value
exceeding 0π . Indeed, let S be the smallest coalition for which the maximum total
reported value exceeds 0π . By construction, the bidders in S must all be winners at
the maximizing assignment, so ( )00 , 0 0 0
max ( ) ( )π π−∈ =∅ ∈ −
< + − ≤∑sx X x i i ii Su x u x
0( ) π
∈ −−∑u ii S
w S . This contradicts ( , )π ∈ uCore N w , so the winning assignment has
a reported value of at most 0π : ˆ 0( ) π≤uw N . If j instead reports truthfully, it can
increase the value of any goods allocation by at most π j , so ˆ, 0( ) π π−
≤ +j ju u jw N .
Next, we show that for any bidder j, there is some coalition excluding j for
which the maximum reported value is at least 0π . Since π is bidder optimal, for any
0ε > , 0( , , ) ( , )π ε π ε π−− + ∉j j uCore N w . So, there exists some coalition εS to
block it: ( )ε
επ ε∈
− <∑ i ui Sw S . By inspection, this coalition includes the seller but
not bidder j. Since this is true for every ε and there are only finitely many coalitions,
there is some S such that ( )π∈
≤∑ i ui Sw S . The reverse inequality is also implied
because ( , )π ∈ uCore N w , so ( )π∈
=∑ i ui Sw S .
For the specified reports, 0ˆ ˆ( ) max ( )u x X i ii S
w S u x∈ ∈= ≥∑
( )0 0 0 0
max ( ) ( )x X i i ii Su x u x π∈ ∈ −
+ − ≥∑ 00( ) π π
∈ −− =∑u ii S
w S . Since the coalition
value cannot decrease as the coalition expands, ˆ 0( ) π− ≥uw N j . By definition of the
coalition value functions, ˆ ˆ,( ) ( )−
− = −j ju u uw N j w N j .
Using theorem 1, j’s maximum payoff if it responds optimally and is favored is
ˆ ˆ, , 0 0( ) ( ) ( )π π π π− −
− − ≤ + − =j j j ju u u u j jw N w N j . So, to prove that the specified report
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profile is an equilibrium, it suffices to show that each player j earns π j when these
reports are made.
The reported value of the true efficient goods assignment is at least
( )0 0 0 0
max ( ) ( ) π∈ ∈ −+ − =∑x X i i ii N
u x u x 00( ) π π
∈ −− =∑ ii N
w N . So, with the specified
tie-breaking rule, if the bidders make the specified truncation reports, the selected
goods assignment will maximize the true total value.
Since the auction is core-selecting, each bidder j must have a reported profit of
at least zero and hence a true profit of at least π j , but we have already seen that these
are also upper bounds on the payoff. Therefore, the reports form an equilibrium; each
bidder j’s equilibrium payoff is precisely π j , and that the seller’s equilibrium payoff
is ˆ 00( ) π π
∈ −− =∑u ii N
w N .
III. Minimizing Incentives to Misreport Despite the similarities among the core-selecting mechanisms emphasized in the previous
section, there are important differences among the mechanisms in terms of incentives to report
valuations truthfully. For example, when there is only a single good for sale, both the first-price
and second-price auctions are core selecting mechanisms, but only the latter is strategy-proof.
To evaluate various bidders’ incentives to deviate from truthful reporting, we introduce the
following definition.
Definition. The incentive profile for a core-selecting auction P at u is
{ }0
( )ε ε∈ −
=P Pj j N
u where ( )ˆ ˆ ˆ ˆ( ) sup ( ( , )) , , ( , )ε − − −≡ −j
Pj u j j j j j j j j ju u f u u P u u f u u is j’s
maximum gain from deviating from truthful reporting when j is favored.
Our idea is to minimize these incentives to deviate from truthful reporting, subject to
selecting a core allocation. Since the incentives are represented by a vector, we use a Pareto-like
criterion.
16
Definitions. A core-selecting auction P provides suboptimal incentives at u if there is
some core selecting auction P such that for every bidder j, ˆ ( ) ( )ε ε≤P Pj ju u with strict
inequality for some bidder. A core selecting auction provides optmal incentives if
there is no u at which it provides suboptimal incentives.
Theorem 3. A core-selecting auction provides optimal incentives if and only if for
every u it chooses a bidder optimal allocation.
Proof. Let P be a core-selecting auction, u a value profile, and π the corresponding
auction payoff vector. From theorem 1, the maximum payoff to j upon a deviation is
π j , so the maximum gain to deviation is π π−j j . So, the auction is suboptimal
exactly when there is another core-selecting auction with higher payoffs for all
bidders, contradicting the assumption that π is bidder optimal.
Recall that when the Vickrey outcome is a core allocation, it is the unique bidder optimal
allocation. So, Theorem 3 implies that any core selecting auction that provides optimal incentives
selects the Vickrey outcome with respect to the reported preferences whenever that outcome is in
the core for those reports. Moreover, because truthful reporting then provides the bidders with
their Vickrey payoffs, theorem 1 implies the following.
Corollary. When the Vickrey outcome is a core allocation, then truthful reporting is
an ex post equilibrium for any mechanism that always selects bidder optimal core
allocations.
We note in passing that any incentive profile that can be achieved by any mechanism is
replicated by the corresponding direct mechanism. There is a “revelation principle” for
approximate incentives, so one cannot do better than the results reported in theorem 3 by looking
over a larger class of mechanisms, including ones that are not direct.
17
IV. Monotonicity of Revenues The core allocations with respect to the reports that minimize the seller’s payoff are all
bidder optimal allocations, so a mechanism that selects those satisfies the conditions of theorem
3. That mechanism has another advantage as well: its revenues are non-decreasing in the bids.
Theorem 4. The seller’s minimum payoff in the core with bidder values u is non-
decreasing in u .
Proof. The seller’s minimum payoff is:
ˆ ˆ0 0min ( ) subject to ( ) for all π π π≥ ∈ − ∈
− ≥ ⊆∑ ∑u i i ui N i Sw N w S S N (4)
The objective is an expression for 0π ; it incorporates the equation ˆ ( ) π∈
=∑u ii Nw N
which therefore can be omitted from the constraint set. The objective is increasing in
ˆ ( )uw N and the constraint set shrinks as ˆ ( )uw S increases for any coalition ≠S N .
Hence, the minimum value is non-decreasing in the vector ( )ˆ ( )⊆u S N
w S . It is obvious
that the coalitional values ˆ ( )uw S are non-decreasing in the reported values u , so the
result follows.
V. Connections to the Marriage Problem Even though Theorems 1-4 in this paper are proved using transferable utility, they all have
analogs in the non-transferable utility marriage problem.
Consider Theorem 1. Roth and Peranson (1999) have shown for a particular algorithm in
the marriage problem that any fully informed player can guarantee its best stable match by a
suitable truncation report. That report states that all mates less preferred than its best achievable
mate are unacceptable. The proof in the original paper makes it clear that their result extends to
any stable matching mechanism, that is, any mechanism that always selects a stable match.
Here, in correspondence to stable matching mechanisms, we study core-selecting auctions.
For the auction problem, Ausubel and Milgrom (2002) showed that the best payoff for any bidder
18
at any core allocation is its Vickrey payoff. So, the Vickrey payoff corresponds to the best mate
assigned at any stable match. Thus, the auction and matching procedures are connected not just
by the use of truncation strategies as best replies but by the point of the truncation, which is at the
player’s best core or stable outcome.
Theorem 2 concerns Nash equilibrium. Again, the known results of matching theory are
similar. Suppose the participants in the match in some set SC play non-strategically, like the seller
in the auction model, while the participants in the complementary set S, whom we shall call
bidders, play Nash equilibrium. Then, for bidder-optimal stable match,16 the profile at which each
player in S reports that inferior matches are unacceptable is a full-information Nash equilibrium
profile of every stable matching mechanism and it leads to that S-optimal stable match. This result
is usually stated using only men or women as the set S, but extending to other sets of bidders
using the notion of bidder optimality is entirely straightforward.
For Theorem 3, suppose again that some players are non-strategic and that only the players
in S report strategically. Then, if the stable matching mechanism selects an S-optimal stable
match, then there is no other stable matching mechanism that weakly improves the incentives of
all players to report truthfully, with strict improvement for some. Again, this is usually stated only
for the case where S is the set of men or the set of women, and the extension does require
introducing the notion of a bidder optimal match.
Finally, the last result states that increasing bids or, by extension, introducing new bidders
increases the seller’s revenue if the seller pessimal allocation is selected. The matching analog is
that adding men improves the utility of each woman if the woman-pessimal, man-optimal match
is selected—a result that is reported by Roth and Sotomayor (1990).
16 This is defined analogously to the bidder optimal allocation.
19
VI. Conclusion We motivated our study of core-selecting auctions by comparing them to stable matching
mechanisms, which have been in long use in practice. In the Kagel-Roth laboratory experiments,
subjects generally stopped using unstable matching mechanisms, preferring to make the best
match they could by individual negotiations, even when congestion made that process highly
imperfect. In contrast, subjects did voluntarily continue to participate in stable matching
mechanisms, which found stable allocations that would be too hard for the subjects to identify in
the short time allowed. These experiments, however, used only some particular stable and
unstable matching mechanisms, so even the generalization to all matching mechanisms is not
experimentally proven. Is there reason to think that a variation of the experiment in which the
parties decide whether to bargain informally or to participate in an organized auction might have
a similar experimental outcome?
Surely, whether the mechanisms being tested are matching or auction mechanisms, many
details would matter. If the Vickrey auction were tested in experimental environments where the
Vickrey payoffs lie is far outside the core (which necessarily means that the seller’s payoff is low
or even zero), one might predict that the seller would not voluntarily use the auction and instead
would entertain the direct offers that it may receive from the bidders in an unorganized market. If,
in analogy to the congestion of the experimental matching markets, the time allowed for
bargaining makes it hard to find even a nearly efficient allocation, then there would be good
reason for participants to participate voluntarily in the core-selecting auction, which selects such
an allocation. Both efficiency considerations and distributional considerations are likely to be
important to the experimental outcome.
In auctions with N items for sale, the number of non-empty packages for which a bidder is
called to report is 2 1−N . That is unrealistically large for most applications if N is even a two-
20
digit number. For the general case, Segal (2003) has shown that communications cannot be much
reduced without severely limiting the efficiency of the result.
Although communication complexity is an important practical issue, it appears to differ in a
qualitative way from the incentive issues studied in this paper. In many environments, there is
information about the kinds of packages that make sense and those can be incorporated as
restrictions on bidding in an auction design. For example, an auctioneer may know that relaxing
delivery constraints cannot increase costs, or that airport landing rights between 2:00-2:15 are
valued similarly to ones between 2:15-2:30, or that complementarities in electrical generating
result from costs saved by operating continuously in time, minimizing time lost when the plant is
ramped up or down. Practical designs that take advantage of this structure can still be core-
selecting mechanisms, where feasible allocations are subjected to the predetermined constraints.
If the problem of communication complexity can be solved, then core-selecting auctions
appear to provide a practical alternative to the Vickrey design. The class includes the pay-as-bid
“menu auction” design studied by Bernheim and Whinston (1986) as well as the ascending proxy
auction studied by Ausubel and Milgrom (2002) and Parkes and Ungar (2000). Within this class,
the auctions that select bidder-optimal allocations conserve as far as possible the advantages of
the Vickrey design—matching the Vickrey auction’s ex post equilibrium property when there is a
single good, or goods are substitutes, or most generally when the Vickrey outcome happens to lie
in the core—and avoiding the low revenue and monotonicity problems of the Vickrey
mechanism.
From the perspective of theory, the most interesting part of this analysis is that all of the
main results about core-selecting auctions have analogues in the theory of stable matching
mechanisms. The deeps reasons for this similarity remain to be fully explored.
21
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