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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

References Abdulkadiroglu, Atila, Parag Pathak, Alvin Roth and Tayfun Sonmez (2005). "The Boston Public

School Match." AEA Papers and Proceedings: 368-371.

Ausubel, Lawrence and Paul Milgrom (2002). "Ascending Auctions with Package Bidding."

Frontiers of Theoretical Economics 1(1): Article 1.

Ausubel, Lawrence and Paul Milgrom (2005). "The Lovely but Lonely Vickrey Auction".

Combinatorial Auctions. P. Cramton, Y. Shoham and R. Steinberg. Cambridge, MA,

MIT Press.

Bernheim, B. Douglas and Michael Whinston (1986). "Menu Auctions, Resource Allocation and

Economic Influence." Quarterly Journal of Economics 101: 1-31.

Bertrand, J. (1883). "Théorie Mathématique de la Richesse Sociale." Journal des Savants 69:

499-508.

Gale, David and Lloyd Shapley (1962). "College Admissions and the Stability of Marriage."

American Mathematical Monthly 69: 9-15.

Green, Jerry and Jean-Jacques Laffont (1979). Incentives in Public Decision Making. North

Holland: Amsterdam.

Hatfield, John and Paul Milgrom (2005). "Matching with Contracts." American Economic Review

95(4): 913-935.

Holmstrom, Bengt (1979). "Groves Schemes on Restricted Domains." Econometrica 47: 1137-

1144.

Kagel, John and Alvin Roth (2000). "The dynamics of reorganization in matching markets: A

laboratory experiment motivated by a natural experiment." Quarterly Journal of

Economics: 201-235.

Kelso, Alexander and Vincent Crawford (1982). "Job Matching, Coalition Formation, and Gross

Substitutes." Econometrica 50: 1483.

22

Klemperer, Paul (2002). "What Really Matters in Auction Design." Journal of Economics

Perspectives 16(1): 169-190.

McAdams, David and Michael Schwarz (2006). "Credible Sales Mechanisms and

Intermediaries." American Economic Review forthcoming.

Milgrom, Paul (2004). Putting Auction Theory to Work. Cambridge, Cambridge University Press.

Parkes, David, Jayant Kalagnanam and Marta Eso (2002). "Achieving Budget-Balance with

Vickrey Based Payment Schemes in Exchanges." Proceedings of the Seventeenth

International Joint Conference on Artificial Intelligence (IJCAI-01): 1161-1168.

Parkes, David and Lyle Ungar (2000). "Iterative Combinatorial Auctions: Theory and Practice."

Proceedings of the 17th National Conference on Artificial Intelligence: 74-81.

Raiffa, Howard (1982). The Art and Science of Negotiation. Cambridge, Harvard University

Press.

Roberts, John and Andrew Postlewaite (1976). "The Incentives for Price-Taking Behavior in

Large Exchange Economies." Econometrica 44(1): 115-129.

Roth, Alvin E. (1982). "The Economics of Matching: Stability and Incentives." Mathematics of

Operations Research 7: 617-628.

Roth, Alvin E. and Elliott Peranson (1999). "The Redesign of the Matching Market for American

Physicians: Some Engineering Aspects of Economic Design." American Economic

Review 89: 748-780.

Roth, Alvin E. and Marilda Sotomayor (1990). Two-Sided Matching: A Study in Game-Theoretic

Modeling and Analysis. Cambridge, Cambridge University Press.

Roth, Alvin E. and X Xing (1994). "Jumping the Gun: Imperfections and Institutions Related to

the Timing of Market Transactions." American Economic Review 84: 992-1044.

Segal, Ilya (2003). "The Communication Requirements of Combinatorial Auctions".

Combinatorial Auctions. P. Cramton, Y. Shoham and R. Steinberg, Princeton University

Press.

23

Simon, Leo K. and William R. Zame (1990). "Discontinuous Games and Endogenous Sharing

Rules." Econometrica 58: 861-872.