ON THE MANIPULABILITY OF COMPETITIVE EQUILIBRIUM RULES IN
MANY-TO-MANY BUYER-SELLER MARKETS1
By
David Pérez-Castrillo2 and Marilda Sotomayor3
ABSTRACT
We analyze the manipulability of competitive equilibrium allocation rules for the
simplest many-to-many extension of the Shapley and Shubik's (1972) assignment game.
First, we extend to the one-to-many (respectively, many-to-one) models the NonManipulability Theorem of the buyers (respectively, sellers), proven by Demange
(1982), Leonard (1983), and Demange and Gale (1985) for the assignment game. Then,
we prove a “General Manipulability Theorem” that implies and generalizes two folk
theorems for the assignment game, the Manipulability Theorem and the General
Impossibility Theorem, never proven before. For the one-to-one case, this result
provides a sort of converse of the Non-Manipulability Theorem.
Keywords: matching, competitive equilibrium, optimal competitive equilibrium,
manipulability, competitive equilibrium mechanism, competitive equilibrium rule.
JEL: C78, D78
1
Marilda Sotomayor is a research fellow at CNPq-Brazil. She acknowledges partial financial support
from FIPE, São Paulo. David Pérez-Castrillo is a fellow of MOVE. He acknowledges financial support
from the Ministerio de Ciencia y Tecnología (ECO2012-31962), Generalitat de Catalunya (2009SGR169), the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2011-0075) and ICREA
Academia. Previous versions of this paper, analyzing one-to-one matching models only, circulated under
the titles “Two Folk Manipulability Theorems in One-to-one Two-sided Matching Markets with Money
as a Continuous Variable” and “Two Folk Manipulability Theorems in the General One-to-one Two-sided
Matching Markets with Money”.
2
Universitat Autònoma de Barcelona and Barcelona GSE; Dept. Economía e Hist. Económica; Edificio
B; 08193 Bellaterra - Barcelona; Spain. Email: [email protected]
3
Universidade de São Paulo; Dep de Economia; Av. Prof. Luciano Gualberto, 908; Cidade Universitária,
5508-900, São Paulo, SP, Brazil. Email: [email protected]
1 INTRODUCTION
We study two-sided matching markets where a finite number of sellers (or workers)
meet a finite number of buyers (or firms). Each seller owns and is willing to sell a set of
identical objects, and each buyer wants to buy several objects, up to her quota, from
various sellers. Both types of agents derive utility from money and their utility functions
are additively separable. Sellers and buyers are heterogeneous, in the sense that the
sellers’ object types are different from a buyer’s point of view, and a buyer’s
willingness to pay for an object may be different from that of another buyer.
The cooperative continuous many-to-many two-sided matching model that we
analyze was proposed and studied in Sotomayor (1992) and (1999) and it was called the
multiple-partners assignment game. It is the simplest generalization of the assignment
game, a model introduced in Shapley and Shubik (1972). The only difference is that in
the assignment game each seller is willing to sell one object, and each buyer wants to
buy, at most, one object.4
The many-to-many matching model can be viewed as a market (competitive
market) operating as an exchange economy. The competitive market was presented in
Sotomayor (2007). We refer to it as a many-to-many buyer-seller market (or buyerseller market, for short). In such a market, given a vector of prices, each seller is ready
to sell his objects if the price exceeds his valuation, and each buyer demands a set of
objects that maximizes her surplus. Roughly speaking, a competitive equilibrium is a
vector of prices, one for each object, and an allocation of objects to buyers such that the
demand of every buyer is satisfied, the number of a seller’s objects allocated is not
larger than the seller’s supply, and the price of every unsold object is its reservation
price. The set of equilibrium prices is non-empty and is a complete lattice whose
extreme points are the minimum and the maximum equilibrium prices, which are called
buyer-optimal and seller-optimal equilibrium prices, respectively.5, 6
4
Crawford and Knoer (1981) studied the linear model where the utility of the seller depends on the
identity of the buyer. Kelso and Crawford (1982) extended the analysis to many-to-one matching models.
5
The competitive one-to-one market was proposed in Gale (1960). Demange, Gale and Sotomayor (1986)
introduced the term “minimum competitive prices,” also called “minimum equilibrium prices” in Roth
and Sotomayor (1990). Gale (1960) proved the existence of equilibrium prices in this market and Shapley
and Shubik (1972) showed that the set of equilibrium prices form a complete lattice whose extreme points
are the minimum and the maximum equilibrium prices. Sotomayor (2007) introduced the concept of a
competitive equilibrium payoff for the multiple-partners assignment game and extended the previous
2 A competitive equilibrium mechanism for the buyer-seller markets is a function
that selects a specific competitive equilibrium allocation for every buyer-seller market.
Given a set of buyers and sellers with their respective quotas, the restriction of a
competitive equilibrium mechanism to the set formed by all buyer-seller markets that
keep the given set of agents and quotas unchanged defines a competitive equilibrium
rule associated with the mechanism. That is, for every such market, the rule selects the
specific competitive equilibrium for the corresponding market defined by the
mechanism.
When a competitive equilibrium rule is adopted for use in a particular buyer-seller
market, information about the valuation of the agents is required. Therefore, the rule
induces a strategic game in which the set of players is the given set of agents; the
preferences of these agents are derived from their true valuations; the strategies of a
player are all the possible valuations that s/he can report; and the outcome function is
defined by the competitive equilibrium rule.
This paper studies the agents’ incentives to truthfully report when a competitive
equilibrium rule is being used for a buyer-seller market. The first important result in this
respect is the Non-Manipulability Theorem for the one-to-one buyer-seller market,
proposed by Demange (1982) and Leonard (1983) and extended by Demange and Gale
(1985) to a one-to-one buyer-seller model where the utilities are continuous in the
money variable, but are not necessarily linear. These authors proved that if the buyeroptimal (or seller-optimal) competitive equilibrium rule is used, then no buyer (or
seller) can profit by misstating her (or his) true valuations.7 The Non-Manipulability
Theorem implies that the mechanism that yields the optimal competitive equilibrium for
results for this generalized environment. In that paper it was also proven, for the first time, that the set of
competitive equilibrium payoffs is a subset of the set of stable payoffs and may be smaller than this set.
6
The results stated and proved in this paper are established for the competitive market game. For the
assignment game, these results can easily be translated to the cooperative model because the core
coincides with the set of competitive equilibrium payoffs (Shapley and Shubik, 1972).
7
Demange and Gale (1985) extended the non-manipulability theorem to their model and to any
competitive equilibrium rule that maps the market defined by the true valuations to the buyer-optimal (or
seller-optimal) competitive equilibrium for this market. Under the assumption that there is no monetary
transfers between buyers (respectively, sellers), these authors proved that such a rule is collectively nonmanipulable by the buyers (respectively, sellers).
3 a given side of the one-to-one buyer-seller market is non-manipulable (i.e., it is strategyproof) by the agents on that side.
The first contribution of our paper is related to the analysis of the agents’ incentives
to manipulate the buyer-optimal or seller-optimal competitive equilibrium rules in
many-to-many buyer-seller markets. First, we show that the mechanism which produces
the buyer-optimal (respectively, seller-optimal) competitive equilibrium allocation for
the many-to-many buyer-seller market is manipulable by the buyers and by the sellers.
Second, we prove that if an agent has a quota of one, then s/he does not have an
incentive to manipulate the most preferred competitive equilibrium rule for her/his side
of the market. Third, as a consequence of the previous result, the buyer-optimal
(respectively, seller-optimal) competitive equilibrium rule for a given one-to-many
(respectively, many-to-one) buyer-seller market is strategy-proof for the buyers
(respectively, sellers).
It is worthwhile pointing out that, although the non-manipulation results for the
buyers and sellers are symmetric, their proofs are not. The result that buyers with a
quota of one never have an incentive to lie about their true valuations under the buyeroptimal competitive equilibrium rule is, based on a new result, interesting in itself: for
these buyers, the buyer-optimal competitive equilibrium rule has the characteristics of a
Vickrey-Clarke-Groves mechanism because the prices charged to them do not depend
on their statements of valuations. On the other hand, the proof of the non-manipulability
theorem for the sellers with a quota of one uses a non-intuitive and also a new result: if
a seller with a quota of one deviates from his true valuation and sells his object in the
new situation, then he cannot manipulate the seller-optimal competitive equilibrium rule
(indeed, this result is more general as it does not require the seller to have a quota of
one). The theorem then follows from the fact that a seller’s deviation is never profitable
if he does not sell his object in the new situation.
Our second contribution concerns the strategic incentives of an agent facing a
competitive equilibrium rule that is applied to a particular many-to-many buyer-seller
market. The Impossibility Theorem (theorem 7.3 of Roth and Sotomayor, 1990) asserts
that any competitive equilibrium mechanism is manipulable. The proof of the theorem
was done by providing a one-to-one assignment game, with only one seller and n > 1
buyers such that, under any competitive equilibrium mechanism, there is an agent who
has an incentive to misrepresent her/his valuation. Additionally, Demange and Gale
(1985) present several examples of one-to-one assignment markets where a competitive
4 equilibrium rule that yields the optimal competitive equilibrium for a given side of the
market provides incentives to an agent belonging to the other side to increase her/his
payoff by misrepresenting her/his valuation. The main feature of all these examples is
that they apply to assignment games which have more than one vector of equilibrium
prices (if there is only one vector of equilibrium prices, then the Non-Manipulability
Theorem implies that no agent has an incentive to misrepresent her/his valuation).
It has been believed that the phenomena observed in the particular market used in
the proof of the Impossibility Theorem and in the examples of Demange and Gale
(1985) happen in any one-to-one assignment market with more than one vector of
equilibrium prices. This belief has supported, along the years, two folk theorems for the
assignment game, which have never been proven in the literature. We state the theorems
for the more general many-to-many buyer-seller markets that we analyze in this paper:
Folk Manipulability Theorem. Consider the buyer-optimal (respectively,
seller-optimal) competitive equilibrium rule. Suppose that the market defined
by the true valuations of the agents has more than one vector of equilibrium
prices. If the buyer-optimal (respectively, seller-optimal) competitive
equilibrium rule is used then there is a seller (respectively, buyer) who can
profitably misrepresent his (respectively, her) valuations, assuming the other
agents tell the truth.
Folk General Impossibility Theorem. Suppose that the market defined by
the true valuations of the agents has more than one vector of equilibrium
prices. No competitive equilibrium rule exists for which, in the game induced,
truth-telling is a dominant strategy for every agent.
In the present work, we provide the proofs and formal statements of the two folk
theorems, extended to the many-to-many buyer-seller market, aiming to fill this gap in
the literature. Indeed, we give a simple proof of a stronger and more general
Manipulability Theorem than the result suggested by the examples of Demange and
Gale (1985), because our theorem applies to many-to-many markets and also because it
does not require the competitive equilibrium rule to produce the optimal competitive
equilibrium for one of the sides. Our result has the two folk theorems as immediate
corollaries.
5 General Manipulability Theorem. Consider any competitive equilibrium
rule. Suppose that the market M defined by the true valuations of the agents
has more than one vector of equilibrium prices. If the rule applied to M
does not yield a buyer-optimal (respectively, seller-optimal) competitive
equilibrium for M, then any buyer (respectively, seller) who does not receive
his (respectively, her) optimal competitive equilibrium payoff for M can
profitably misrepresent his (respectively, her) valuations, assuming the
others tell the truth.
For the one-to-one case, the General Manipulability Theorem provides a sort of
converse of the Demange and Gale’s (1985) Non-Manipulability Theorem. It implies
that if a competitive equilibrium rule is strategy-proof for the buyers (respectively,
sellers), then the rule maps the profile of true valuations to the buyer-optimal
(respectively,
seller-optimal)
competitive
equilibrium
price.
This
result
is
mathematically unusual because it provides a way of concluding that a competitive
equilibrium rule is, for example, strategy-proof for the buyers, based only on the direct
examination of the agents’ payoffs obtained by the profile of true valuations.
Finally, the General Manipulability Theorem implies that there is no competitive
equilibrium rule for a buyer-seller market with more than one equilibrium price vector,
for which a truthful report of valuations for buyers and sellers is a Nash equilibrium. It
also implies that no competitive equilibrium rule is strategy-proof.
Miyake (1988) considers the buyer-optimal mechanism in the one-to-one
assignment game as a multi-item generalization of Vickrey’s second price auction. In
his framework, the sellers’ reservation prices are fixed and known. He analyzes the
buyers’ incentives to misrepresent their preferences and shows that, for any vector of
sellers’ reservation prices, if a competitive equilibrium rule is different from the buyeroptimal competitive equilibrium rule, then the honest strategy profile is not a Nash
equilibrium.8 Our analysis also takes into account the sellers’ incentives and extends the
result to situations where sellers own several objects and buyers can buy objects from
several sellers. Therefore, our paper strengthens Migyake’s (1988) result and provides a
8
In his analysis of buyers’ incentives, Miyake (1988) also shows that the buyer-optimal competitive rule
in the assignment game is a unique auction that satisfies that the honest strategy is a Nash equilibrium (or
that it is a unique perfect equilibrium; or that it is a dominant strategy equilibrium).
6 simple proof. In a general one-to-many buyer-seller model with (a finite set of)
contracts, Hatfield and Milgrom (2005) show that the buyer-optimal competitive
equilibrium is strategy-proof for the buyers and Sakai (2011) shows that it is a unique
equilibrium rule that is strategy-proof for the buyers.
Restricted to the one-to-one assignment game, the Non-manipulability Theorem
suggests that buyers will play their sincere strategy if the buyer-optimal competitive
equilibrium rule is used. The Folk Manipulability Theorem (and the General
Manipulability Theorem) implies that playing sincerely is often not the optimal strategy
for at least one seller. Demange and Gale (1985) consider the strategic equilibrium of
the game induced by the rule that produces the buyer-optimal competitive equilibrium
payoff if the buyers always play their sincere strategies and if the sellers keep their
utility function fixed and only manipulate their reservation prices. The analysis of the
case in which the rule produces a competitive equilibrium payoff and there is no
restriction on the strategies selected by the agents is addressed in Sotomayor (2000).
Related results are proven for the marriage market in Roth and Sotomayor (1990) and
for the college admission market in Sotomayor (2012).
This paper is organized as follows. In Section 2 we present the framework. The
competitive equilibrium mechanisms and the competitive equilibrium rules are
introduced in Section 3. In Section 4 we present some existing results. In Section 5, we
study the extension of the Non-Manipulability Theorem to many-to-many buyer-seller
models. We state and prove the General Manipulability Theorem in Section 6 and
further discuss the one-to-one buyer-seller model in Section 7. Finally, we conclude in
Section 8.
2-THE FRAMEWORK AND PRELIMINARIES
There are two non-empty, finite, and disjoint sets of agents, P and Q, which can
be thought of as being buyers and sellers, respectively. Set P has m buyers and set Q
has n sellers. A generic element of P is denoted by pj and of Q is denoted by qk.
Each seller qk owns t(qk) identical objects of the same type, which we also denote qk.
Each buyer pj has a quota r(pj), representing the maximum number of objects she is
allowed to acquire. No buyer is interested in acquiring more than one item of a given
type. Without loss of generality, we can consider r(pj) ≤ n and t(qk) ≤ m, for all pj ∈ P
and qk ∈ Q. We denote by r and t the array of numbers r(pj)’s and t(qk)’s,
respectively.
7 Seller qk assigns a value of sk ≥ 0 to each of his objects. Buyer pj assigns a value
of αjk ≥ 0 to any of the objects of seller qk. The value αjk does not depend on the
other objects that may be acquired by buyer pj or on the other buyers that may acquire
objects of type qk. We call this condition separability across the pairs. Thus, if buyer
pj purchases an object qk at price ρk ≥ sk , then her individual payoff in this transaction
is ujk = αjk − ρk and the individual payoff of seller qk is vjk = ρk − sk. We denote by αj
the vector of the values of αjk’s; the valuation matrix of the buyers and the valuation
vector of the sellers are denoted by α and s, respectively.
For notational simplicity, a dummy buyer, denoted by p0, will be available as well
as a dummy seller q0 who owns artificial “null-objects” q0 whose valuation and price
is zero, that is, s0 = ρ0 = 0. The valuations of the dummy object are αj0 = 0 and the
dummy buyer’s valuations are α0k = sk for all pj ∈ P and qk ∈ Q. The quotas of the
dummy agents are enough to fill the quotas of the real agents in the market.
We call this model the buyer-seller model which is denoted by M(P, Q; α, s; r, t).
An instance of the buyer-seller model will be called a market. If each seller’s
reservation price is 0 and all quotas are 1, then the corresponding model is the wellknown Shapley and Shubik’s (1972) Assignment Game.9
An object qk is acceptable to buyer pj if and only if the potential gain from a
trade between buyer pj and seller qk is non-negative, that is, αjk − sk ≥ 0. Thus, an
object is not acceptable to a buyer if there is no price at which the buyer wishes to buy
and the seller wishes to sell. We denote a(α, s)jk ≡ αjk − sk if αjk − sk ≥ 0 and
a(α, s)jk ≡ 0 otherwise. Finally, for any set A ⊆ P∪Q, we use the notation ∑A for the
sum over all elements in A.
An allowable set of objects for buyer pj ∈ P is a set with r(pj) objects that does
not contain more than one object of the same seller, except that some of them may be
repetitions of the null object. A feasible matching assigns each non-null object to one
buyer (who might be the dummy buyer) so that each non-dummy buyer is assigned to
an allowable set of objects for her. Of course, a dummy buyer may be assigned to any
number of objects and the null object may be allocated to any number of buyers. If a
seller’s object is assigned to a buyer at a given matching, we say that both agents are
9
See Roth and Sotomayor (1990) for an overview of this model.
8 matched. If an object is allocated to the dummy buyer, we say that it is left unsold.
Formally,
Definition 2.1. A matching for market M(P, Q; α, s; r, t) is identified with a matrix x
= (xjk) of non-negative integer numbers, defined for all pairs (pj, qk) ∈ PxQ such that
xjk ∈ {0,1} if pj ∈ P−p0 and qk ∈ Q−q0. A matching x for M(P, Q; α, s; r, t) is
feasible if it satisfies (a) ∑P xjk = t(qk) for all qk ∈ Q−q0, (b) ∑Q xjk = r(pj) for all
pj ∈ P−p0 and (c) αjk − sk ≥ 0 if xjk = 1.
If xjk > 0 (respectively, xjk = 0), then buyer pj and seller qk are (respectively, are
not) matched at x. Condition (c) requires that the object assigned to a buyer by the
feasible matching must be acceptable to her. Given a feasible matching x, we denote by
C(pj, x) the pj’s allowable set of objects allocated to pj at x. Similarly, we denote by
C(qk, x) the set of buyers assigned to qk at x.
Definition 2.2. A matching x for market M(P, Q; α, s; r, t) is optimal if (a) it is
feasible and (b) ∑PxQ a(α, s)jk xjk ≥ ∑PxQ a(α, s)jk x´jk for all feasible matching x´.
A feasible price vector (or a vector of feasible prices) ρ for market M(P, Q; α, s;
r, t) is a function from Q to R (the set of real numbers) such that ρk ≡ ρ(qk) is
greater than or equal to sk and ρ0 = s0 = 0. A feasible allocation for M(P, Q; α, s; r, t)
is a pair (ρ, x), where ρ is a feasible price vector and x is a feasible matching.
The array of pj’s individual payoffs ujk’s at the feasible allocation (ρ, x) is a
function uj from C(pj, x) to R such that uj(qk) ≡ ujk = αjk − ρk for qk ∈ C(pj, x) if
pj ≠ p0 (in particular, uj0 = αj0 − ρ0 = 0) and u0k = 0 for qk ∈ C(p0, x). The array of
qk’s individual payoffs vjk’s at (ρ, x) is a function vk from C(qk, x) to R such that
vk(pj) ≡ vjk = ρk − sk for pj ∈ C(qk, x) if qk ≠ q0; vk(p0) ≡ v0k = sk and v0(pj) ≡ vj0 = 0.
The corresponding vector of individual payoffs (u, v) is called a feasible payoff and
(u, v; x) is called a feasible outcome. Thus, at a feasible outcome (u, v; x), ujk + vjk =
αjk − sk, for all pairs (pj, qk) with xjk > 0. We say that x is compatible with (u, v).
9 Given a feasible outcome (u, v; x), we denote uj(min) the smallest individual
payoff of buyer pj: uj(min) = Min {ujk; qk ∈ C(pj, x)} and vk(min) the smallest
individual payoff of seller qk: vk(min) = Min {vjk; pj ∈ C(qk, x)}.10
The value of an allowable set of objects S to buyer pj is the sum of the values of
the objects in S to pj. Thus, given a price vector ρ, buyer pj ∈ P has preferences over
the allowable sets of objects that are completely described by the numbers αjk’s: For
any two allowable sets of objects S and S’, buyer pj prefers S to S’ at prices ρ if
∑S (αjk − ρk) > ∑S’ (αjk − ρk). She is indifferent between these two sets if ∑S (αjk − ρk) =
∑S’ (αjk − ρk). It is interesting to note that these preferences of a buyer over allowable
sets are responsive to her preferences over individual objects.
Under the structure of preferences that we are assuming, each buyer pj is able to
determine which allowable sets of objects she would most prefer to buy at a given
feasible price vector ρ. We denote by D(pj, ρ) the set of all such allowable sets and we
call it the demand set of buyer pj at the feasible prices ρ.
Remark 2.1. Note that D(pj, ρ) is non-empty because pj always has the option of
buying r(pj) copies of the null object. Note also that if S ∈ D(pj, ρ) and qk ∈ S, then
αjk − ρk ≥ 0, so qk is acceptable to pj. Furthermore, αjk − ρk ≥ αjt − ρt for all
qt ∈ Q−S. Conversely, if S is an allowable set of objects to pj and αjk − ρk ≥ αjt − ρt
for all qk ∈ S and qt ∈ Q−S, then S ∈ D(pj, ρ). g
A competitive equilibrium is a vector of feasible prices for the objects plus an
allocation of each buyer to an allowable set of objects belonging to her demand set that
respects the quotas of the sellers. Moreover, in a competitive equilibrium, the price of
each unsold object corresponds to its value for his owner. Remark 2.1 implies that every
object allocated to a buyer is acceptable to her; thus, the resulting allocation is a feasible
matching. Formally,
Definition 2.3. A competitive equilibrium for M(P, Q; α, s; r, t) is a feasible allocation
10
uj(min) and vk(min) can depend on the maching x. However, as will become clear after Proposition
4.1, they are independent of the matching for competitive equilibrium outcomes. Therefore, for notational
simplicity, we do not include reference to the matching x in the expressions uj(min) and vk(min).
10 (ρ, x) such that (a) C(pj, x) ∈ D(pj, ρ) for all pj ∈ P; (b) if an object qk is left unsold,
then ρk = sk.
If the allocation (ρ, x) is a competitive equilibrium for M(P, Q; α, s; r, t), ρ is
called an equilibrium price vector or simply equilibrium prices and x is called a
competitive matching. In this case, we say that x is compatible with ρ and vice-versa.
The corresponding outcome (u, v; x) is called competitive equilibrium outcome and
(u, v) is called a competitive equilibrium payoff compatible with x.
The next proposition, whose proof comes immediately after Remark 2.1 and
Definition 2.3, provides an alternative characterization of competitive equilibria.
Proposition 2.1. The outcome (u, v; x) is a competitive equilibrium if and only if it is
feasible and (a) vjk = vk(min) for all pj ∈ C(qk, x), (b) uj(min) + vk(min) ≥ αjk − sk for
all pairs (pj, qk) with xjk = 0 and (c) uj(min) ≥ 0 and vk(min) ≥ 0 for all pj ∈ P and
qk ∈ Q.
Remark 2.2. At a competitive equilibrium, every seller sells all of his items for the
same price. In fact, if a seller has two identical objects, qk and qt, and ρk > ρt, for some
price vector ρ, then no buyer pj demands a set S of objects that contains object qk.
This is because by replacing qk with qt in S, pj gets a more preferable allowable set
of objects. But then, qk would remain unsold with a price higher than sk, which would
violate condition (b) of Definition 2.3.g
3. COMPETITIVE EQUILIBRIUM MECHANISMS AND COMPETITIVE
EQUILIBRIUM RULES
A competitive equilibrium mechanism for the buyer-seller market is a function that
selects a specific competitive equilibrium allocation for every instance of the buyerseller model, that is, for every possible set of buyers and sellers, for every possible
valuation matrix for the buyers and valuation vector for the sellers, and for every
possible array of quotas for the buyers and sellers.
In what follows, we consider that the sets P, Q, and the arrays of quotas r and t
are fixed. For all (α, s), we will set M(α, s) ≡ M(P, Q; α, s; r, t). Given (P, Q; r, t), a
competitive equilibrium mechanism can be used to define an associated competitive
11 equilibrium rule (competitive equilibrium rule, for short) (П, X) where П is a price
rule and X is a matching rule. The domain of (П, X) is the set of all markets M(α, s).
Since P, Q, r and t are fixed, the value of (П, X) for M(α, s) is denoted simply by
(П(α, s), X(α, s)). Then, (П(α, s), X(α, s)) is the competitive equilibrium selected by
the mechanism when it is applied to market M(α, s). The corresponding buyers’ payoff
vector is denoted by u(α, s) and the corresponding sellers’ payoff vector is denoted by
v(α, s). Then, for all (pj, qk) ∈ PxQ, u(α, s)jk = αjk − П(α, s)k and v(α, s)jk =
П(α, s)k − sk if X(α, s)jk > 0.
When a competitive equilibrium rule (П, X) is adopted for use in a particular
market M(α, s) and the agents are requested to report their valuations, then the rule
induces a strategic game Γ(П, X, α, s), where the set of players is the set of agents
P∪Q; the profile of true valuations is given by (α, s); the set of strategies for buyer
pj ∈ P is the set of vectors α’j ∈ Rm+, with α’j0 = 0; the set of strategies for seller
qk ∈ Q is the set of numbers s’k ≥ 0, with s’0 = 0; and the outcome function applied to
the profile of strategies
(α’, s’)
is given by the rule applied to the market
M(P, Q; α’, s’; r, t). Thus, if the agents report the profile of strategies (α’, s’), the
outcome function produces (П(α’, s’), X(α’, s’)), which is the value of the rule applied
to the market M(α’, s’). We can say that the game Γ(П, X, α, s) is the strategic game
induced by (П, X) with true valuations (α, s) (it is implicit that P, Q, r and t are
fixed). The preferences of the players over the allocations that can be produced by
(П, X) are given by their preferences over the corresponding array of true individual
payoffs, which are determined by their true valuations (α, s).11 Thus, the true individual
payoffs in the transaction between buyer pj and seller qk with respect to market
M(α, s) at allocation (П(α’, s’), X(α’, s’)) are as follows:
Ujk(П(α’, s’), X(α’, s’); α, s) = αjk − Пk(α’, s’) if X(α’, s’)jk > 0,
Vjk(П(α’, s’), X(α’, s’); α, s) = Пk(α’, s’) − sk if X(α’, s’)jk = 1 and pj ≠ p0 and
Vjk(П(α’, s’), X(α’, s’); α, s) = 0 if X(α’, s’)jk > 0 and pj = p0.
Therefore, if seller qk sells all of his objects at (П(α’, s’), X(α’, s’)), all his true
individual payoffs are identical. When (α, s) is the profile of true valuations for the
We will use “primes” to denote reported variables. For example, α is the true valuation matrix of the
buyers, whereas α’ is the reported valuation matrix of the buyers. The domain of the outcome function is
the set of all possible reports.
11
12 agents, we will sometimes refer to the market M(α, s) as the true market.
Definition 3.1. A competitive equilibrium rule (П, X) is manipulable via M(α, s) if
there is an agent y ∈ P∪Q and a profile of valuations (α’, s’), which differs from
(α, s) only in agent y’s valuations, such that:
(i) if y = pj for some pj ∈ P, then
∑C(pj, X(α’, s’)) Ujk(П(α’, s’), X(α’, s’); α, s) > ∑C(pj, X(α, s)) Ujk(П(α, s), X(α, s); α, s),
(ii) if y = qk for some qk ∈ Q, then
∑C(qk, X(α’, s’)) Vjk(П(α’, s’), X(α’, s’); α, s) > t(qk) (Пk(α, s) − sk).
We say that such an agent y manipulates the rule (П, X) via M(α, s). Also, a
competitive equilibrium rule (П, X) is manipulable if it is manipulable via some
market M(α, s).
From Definition 3.1 it follows that no dummy agent is able to manipulate a
competitive equilibrium rule. Furthermore, if qk sells all of his objects at (П(α’, s’),
X(α’, s’)) then qk manipulates the rule (П, X) via M(α, s) if Пk(α’, s’) > Пk(α, s).
A competitive equilibrium rule is non-manipulable if for every market M(α, s)
there is no agent who can manipulate the rule via M(α, s). Equivalently, a competitive
equilibrium rule (П, X) is non-manipulable if and only if the sincere strategy is a
dominant strategy for every agent in the game Γ(П, X, α, s), for every profile of true
valuations (α, s). It also follows that a competitive equilibrium rule (П, X) is nonmanipulable if and only if, for every market M(α, s) (or for every profile of strategies
(α, s)), (α, s) is a Nash equilibrium of the game Γ(П, X, α, s).
A competitive equilibrium mechanism is manipulable if there is some market
M(P, Q; α, s; r, t) such that the associate competitive equilibrium rule is manipulable
via M(α, s). In other words, a competitive equilibrium mechanism is non-manipulable
if and only if for every market M(P, Q; α, s; r, t) and for every pj ∈ P (respectively,
qk ∈ Q), to state αj (respectively, sk) is a dominant strategy for buyer pj (respectively,
seller qk) in the strategic game induced by the competitive equilibrium rule for M(α, s)
associated with the mechanism when the true preferences are given by (α, s).
If a competitive equilibrium mechanism (respectively, rule) is non-manipulable we
say that it is strategy-proof.
13 4. PRELIMINARY RESULTS
In this section, we denote by M the market M(P, Q; α, s; r, t) and we state several
relationships between competitive equilibrium outcomes and optimal matchings for M.
They are used in the following sections to prove our main results.
Proposition 4.1 (Sotomayor, 1999) Let (u, v; x) be a competitive equilibrium outcome
for market M. Let x’ be an optimal matching for M. Then:
a) ujk = uj(min) for all pj ∈ P and qk ∈ C(pj, x)−C(pj, x’).
b) There exists a competitive equilibrium outcome (u’, v; x’) such that u’j(min) =
uj(min) and u’jk = ujk for all pj ∈ P and qk ∈ C(pj, x)∩C(pj, x’).
We have considered that the individual payoffs of the buyers are indexed according
to the matching. It follows from Proposition 4.1 that we can represent the array of
individual payoffs of a buyer, at a competitive equilibrium outcome, as a vector in an
Euclidean space, whose dimension is the quota of the given player. In addition, this
representation is independent of the matching that is being used. In what follows, the set
u of arrays of numbers are represented by a m-tuple of vectors, still denoted by u.
Thus, under the new representation we keep the same notation
(u, v; x)
of the
competitive equilibrium outcome.
The vectorial representation of the competitive equilibrium payoffs allows us to
state the following proposition.
Proposition 4.2. (Sotomayor, 1999) If (u, v; x) is a competitive equilibrium outcome
and x’ is an optimal matching for market M, then (u, v; x’) is also a competitive
equilibrium outcome for M.
Remark 4.1. One of the implications of Proposition 4.2 is that if a seller qk has an
unsold object at an optimal matching, then the prices of all of his objects will be sk, at
any competitive equilibrium. Therefore, if ρk > sk under a competitive equilibrium
price ρ, then seller qk sells all of his items under all competitive equilibria.g
Proposition 4.3 provides the converse of Proposition 4.2.
14 Proposition 4.3. (Sotomayor, 1992) Let (u, v; x) be a competitive equilibrium outcome
for market M. Then, x is an optimal matching for M.
We can identify two particularly interesting outcomes in the set of competitive
equilibrium outcome:
Definition 4.1 A competitive equilibrium payoff is called a P-optimal competitive
equilibrium payoff if every player in P weakly prefers it to any other competitive
equilibrium payoff. That is, the P-optimal competitive equilibrium payoff gives to each
player in P the maximum total payoff among all competitive equilibrium payoffs.
Similarly, we define a Q-optimal competitive equilibrium payoff.
The existence of the P-optimal and Q-optimal competitive equilibrium payoffs is
proved in Sotomayor (2007). The competitive equilibrium price vector corresponding to
the P-optimal (respectively, Q-optimal) competitive equilibrium payoff is smaller
(respectively, larger) in each componente than any other competitive equilibrium price
vector. It is called minimum competitive equilibrium price vector (respectively,
maximum competitive equilibrium price vector). We denote by
ρ
and
ρ
the
minimum and the maximum competitive price vectors, respectively. Also, we let
( u , ρ − s)
and
( u , ρ − s)
be the P-optimal and the Q-optimal competitive
equilibrium payoffs.
If the competitive equilibrium rule (П, X) produces the P-optimal (respectively,
Q-optimal) competitive equilibrium for every M(α, s), the rule is called the P-optimal
(respectively, Q-optimal) competitive equilibrium rule.
5. THE NON-MANIPULABILITY THEOREMS
According to the Non-Manipulability Theorem for the assignment game, the
mechanism that yields the buyer-optimal (respectively, seller-optimal) competitive
equilibrium is not individually manipulable by the buyers (respectively, sellers).
Therefore, if (П, X) is the buyer-optimal (respectively, seller-optimal) competitive
equilibrium rule for a market M(P, Q; α, s; r, t), with r(pj) = t(qk) = 1 for all pj ∈ P
15 and qk ∈ Q, then stating αj (respectively, sk) is a dominant strategy for buyer pj
(respectively, seller qk) in the induced strategic game Γ(П, X, α, s).
In this section, we use examples to show that the Non-Manipulability Theorem for
the assignment game does not extend to the many-to-many buyer-seller market. Then
we give a sufficient condition on the quotas of the agents under which no seller
(respectively, buyer) can manipulate the seller-optimal (buyer-optimal) competitive
equilibrium rule. This sufficient condition is that every seller (respectively, buyer) has a
quota of one. That is, no seller can manipulate the seller-optimal competitive
equilibrium rule when this rule is used for markets where each seller only has one object
to sell, even though the buyers may want to acquire more than one object. Similarly, no
buyer has the incentive to manipulate the buyer-optimal competitive equilibrium rule
when this rule is used in markets where each buyer wants to acquire one object at most,
even though each seller may own several objects. Indeed, we prove a stronger result
which asserts that no agent can manipulate the optimal competitive equilibrium rule for
his side via a market in which this agent has a quota of one.
As we will see, although these results are symmetric, the proofs are not. The proof
of the non-manipulability theorem for the sellers with a quota of one is obtained from
the observation that a seller’s deviation is never profitable if he does not sell his object
in the new situation, followed by a non-intuitive result implied by Proposition 5.1: if a
seller with a quota of one deviates from his true valuation and sells his object in the new
situation, then he cannot manipulate the seller-optimal competitive equilibrium rule.
Indeed, Proposition 5.1 is more general as it does not require the seller to have a quota
of one.
Example 5.1 illustrates a situation in which the seller-optimal competitive
equilibrium rule is manipulated by a seller who does not sell all his objects in the new
market.
Example 5.1. (The sellers can manipulate the Q-optimal competitive equilibrium
rule) Consider the market M(P, Q; α, s; r, t), where P = {p1, p2, p0}, Q = {q1, q0},
α11 = 3, α21 = 1, r(p1) = r(p2) = 1, t(q1) = 2 and s1 = 0. At the seller-optimal
competitive equilibrium allocation, seller q1 sells one of his objects to buyer p1 and
the other object to buyer p2 at price 1, so his total payoff is 2. However, if the seller
misrepresents his valuation and reports s’1 = 2.5 then, at any competitive equilibrium
16 allocation for M(α, s’), he sells one of his objects to buyer p1 and the other object is
unsold. In this case, the true payoffs of seller q1 is v*11 = 2.5 and v*01 = 0, so the
total true payoff of the seller is 2.5. Therefore, q1 can manipulate the seller-optimal
competitive equilibrium rule.
Proposition 5.1 asserts that no seller can manipulate the seller-optimal competitive
equilibrium rule through a deviation in which he sells all his objects.
Proposition 5.1. Let (П, X) be the Q-optimal competitive equilibrium rule. Let M(α, s)
be the true market and denote ( u , ρ − s) the Q-optimal competitive equilibrium
payoff for M(α, s). Let qk ∈ Q and consider s’ such that s’l = sl for all ql ≠ qk.
Denote ( u ’, ρ ’ − s’) the Q-optimal competitive equilibrium payoff for M(α, s’). Let
(u*, v*) be the true payoff under ( u ’, ρ ’ − s’; X(α, s’)). Finally, suppose that qk sells
all of his objects at X(α, s’). Then,
ρ – sk ≥ v*k , where v*k = v*jk for all
k
pj ∈ C(qk, X(α, s’)).
Proof. Suppose ρ – sk < v*k . Then, it follows from the maximality of ρ – s that
k
(u*, v*) is not a competitive equilibrium payoff for M(α, s). We claim that there exists
a pair (pj, ql) with X(α, s’)jl = 0 such that u*jm + v*tl < αjl – sl for some qm ∈ C(pj, x)
and pt ∈ C(ql, x). First, note that all individual true payoffs of a seller ql are the same
non-negative number. In fact, this is certainly true for qk because, by hypothesis, qk
sells all his objects at X(α, s’); moreover, if ql ≠ qk, then s’l = sl and so:
v*jl = ρ ’ – s’l ≥ 0 and v*jl = v*l(min) for all pj ∈ C(ql, X(α, s’))
l
(1)
Also, u*jl = u’jl ≥ 0 if X(α, s’)jl > 0, for all pj ∈ P. Then, u*j(min) ≥ 0 for all pj
∈ P. By the feasibility of ( u ’, ρ ’ − s’; X(α, s’)) in M(α, s’) and (1), we have that
u*jl + v*jl = u ’jl + ( ρ ’l − sl) = αjl − sl if X(α, s’)jl > 0 and pj ≠ p0, and u*0l + v*0l = 0
= α0l − sl if X(α, s’)0l > 0. Therefore, (u*, v*; X(α, s’)) is feasible for M(α, s).
Hence, given that (u*, v*) is not a competitive equilibrium payoff for M(α, s),
Proposition 2.1 guarantees the existence of a pair (pj, ql) with X(α, s’)jl = 0 such that
u*jm + v*tl < αjl – sl for some qm ∈ C(pj, x) and pt ∈ C(ql, x).
17 Therefore, we can write u ’jm + ρ ’l – s’l = u*jm + (v*tl + sl) – s’l < αjl – sl + sl – s’l
= αjl – s’l, which, by Proposition 2.1, contradicts the fact that ( u ’, ρ ’ – s’) is a
competitive equilibrium payoff for M(α, s’). Hence the proof is complete.g
Proposition 5.1 implies that no seller with a quota of one can manipulate the selleroptimal competitive equilibrium rule.
Theorem 5.1. Let (П, X) be the Q-optimal competitive equilibrium rule for the market
M(α, s). Let qk ∈ Q such that t(qk) = 1. Then, the sincere strategy is a dominant
strategy for qk in the game Γ(П, X, α, s).
In particular, in many-to-one buyer-seller models, that is, in markets where all
sellers have a quota of 1 and each buyer is interested in acquiring several objects, no
seller can manipulate the seller-optimal competitive equilibrium rule. That is to say, the
seller-optimal competitive equilibrium rule for many-to-one buyer-seller models is
strategy-proof for the sellers.
Corollary 5.1. Let (П, X) be the Q-optimal competitive equilibrium rule for the market
M(α, s) where t(qk) = 1 for every seller qk. Then, telling the truth is a dominant
strategy for every seller in the game Γ(П, X, α, s).
Next, we analyze the buyers’ incentives to manipulate. Example 5.2 shows that
there are markets where the buyers may have incentives not to report their true
valuations if the buyer-optimal competitive equilibrium rule is applied.
Example 5.2. (The buyers can manipulate the P-optimal competitive equilibrium
rule) Consider the market
M(P, Q; α, s; r, t), where
P = {p1, p2, p0},
Q = {q1, q2, q3, q0}, the first row of the matrix α (which indicates the valuations of
buyer p1) is (7, 6, 4, 0), the second row of the matrix α (the valuations of buyer p2) is
(8, 6, 3, 0), s1 = s2 = s3 = 0, r(p1) = 2 and r(p2) = t(q1) = t(q2) = t(q3) = 1. Under the
P-optimal competitive equilibrium payoff, buyer p1 is matched to q2 and q3 and she
receives the individual payoffs of 5 and 4 from these transactions, respectively,
whereas buyer p2 is matched to q1 and receives the individual payoff of 5. However,
18 if buyer p1 reports that the first row is (7, 6, 7, 0) then, under the new P-optimal
competitive equilibrium payoff, buyer p1 pays 0 to sellers q2 and q3. Hence, p1
gets 6 and 4 under her true individual payoffs instead of 5 and 4 if she reports
truthfully. Therefore, she has an incentive to misrepresent her valuations.
In Example 5.2, buyer
p1
can manipulate the mechanism because she can
influence the price of one of the objects she buys by reporting a different valuation for
some other object. Under truthful reporting, the prices for the objects in the P-optimal
competitive equilibrium are ρ1 = 3, ρ2 = 1 and ρ3 = 0. The price for the first object
(that is assigned to buyer p2) is equal to 3 because, if it were lower then buyer p1
would prefer q1 to q3. Also, ρ2 = 1 so that p2 chooses q1 instead of q2. By
reporting a valuation of 7 for q3 instead of 4, buyer p1 achieves a reduction in the
equilibrium price ρ1 to ρ’1 = 3 that, in turn, pushes the equilibrium price ρ2 to ρ’2 =
0. Therefore, buyer p1 indirectly influences the price ρ2 by manipulating her valuation
for q3.
Theorem 5.3 states the absence of incentives for the buyers with a quota of 1 to
manipulate the P-optimal competitive equilibrium rule. The technique used to prove this
result is not straightforward and it is distinct from that used to prove the symmetric
result for the sellers. It requires a key theorem (Theorem 5.2), which identifies a buyer’s
payoff in the buyer-optimal competitive equilibrium rule, when this buyer has a quota
of one, as the difference between the “total value” of the market and the “total value” of
the market without that buyer. Remark 5.1 shows that this result implies that, at the
buyer-optimal competitive equilibrium allocation, the price charged to a buyer with a
quota of one does not depend on the buyer’s statements of valuations. To prove
Theorem 5.2 we need Lemma 5.1 below.
Lemma 5.1. Let ( u , v ; x) be a P-optimal competitive equilibrium outcome for
M(α, s). Let Q’ ⊆ Q with Q’ ≠ φ be such that v k > 0 for all qk ∈ Q’. Then,
(a) x(Q–Q’) ≠ {p0} and (b) there is a pair (pj, qk) ∈ x(Q–Q’)xQ’, with pj ≠ p0 and
xjk = 0, and a seller qt ∈ Q–Q’ with xjt = 1, such that u jt + v k = αjk – sk.
Proof. (a) Suppose by contradiction that x(Q–Q’) = {p0}. Given that q0 ∈ Q–Q’,
x(Q–Q’) = {p0} implies that if qk ∈ Q’ and xjk = 1, then pj ≠ p0. Moreover, every
19 pj ≠ p0 fills her quota and does that with sellers in Q’. That is, x(Q’) = P–{p0}. Then,
take a positive λ such that v k – λ > 0 for all qk ∈ Q’ and define (u, v; x) as follows:
vk = v k – λ if qk ∈ Q’ and vk = v k , otherwise;
ujk = u jk + λ if pj ∈ x(Q’) = P–{p0} and xjk = 1;
u0k = u 0k = 0 if x0k = 1.
We claim that (u, v; x) is a competitive equilibrium outcome for M(α, s). In fact,
the feasibility of (u, v; x) and condition (iii) of Proposition 2.1 are clearly satisfied. To
show that uj(min) + vk ≥ αjk – sk for all (pj, qk) ∈ PxQ, use the construction of (u, v; x)
and the fact that ( u , v ; x) is a competitive equilibrium outcome for the case when
pj ≠ p0. When pj = p0, it is enough to verify the inequality when qk ∈ Q’. In this case,
we have that uj(min) + vk = v k – λ > 0 = α0k – sk = αjk – sk. However, vk < v k for all
qk ∈ Q’, and Q’ ≠ φ, which contradicts the fact that ( u , v ; x) is a Q-optimal
competitive equilibrium outcome for M(α, s). Hence, x(Q–Q’) ≠ p0.
(b) Arguing by contradiction, suppose u jt + v k > αjk – sk for all (pj, qk) ∈ x(Q–Q’)xQ’
with pj ≠ p0 and xjk = 0, and for all qt ∈ Q–Q’ with xjt = 1. Then, there exists a
positive λ such that
v k – λ > 0 for all qk ∈ Q’ and also such that for all
(pj, qk) ∈ x(Q–Q’)xQ’, with pj ≠ p0 and xjk = 0, and for all qt ∈ Q–Q’ with xjt = 1, the
parameter λ satisfies
u jt + v k – λ > αjk – sk .
(2)
Now, define (u’, v’; x) as follows:
v’k = v k – λ if qk ∈ Q’ and v’k = v k , otherwise;
u’jk = u jk + λ if qk ∈ Q’ and xjk = 1;
u’jk = u jk if qk ∈ Q–Q’ and xjk = 1.
We claim that (u’, v’; x) is a competitive equilibrium outcome. The argument is
similar to the one used in part (a). We only need to check that u’j(min) + v’k ≥ αjk – sk
for all qk ∈ Q’ and xjk = 0. Then, let (pj, qk) ∈ PxQ’, with xjk = 0 and let
u’j(min) = u’jt, for some qt ∈ C(pj, x). If qt ∈ Q’ then the result follows from the
competitiveness of ( u , v ; x); if qt ∈ Q–Q’ the result follows from (2).
20 However, v’k < v k for all qk ∈ Q’, and Q’ ≠ φ, which contradicts the fact that
( u , v ; x) is a Q-optimal competitive equilibrium outcome for M(α, s), and the proof is
complete. g
To state Theorem 5.2 we need some notation. For a market M = M(P, Q; α, s; r, t)
and an optimal matching x for M, we set v(P, Q) ≡ ∑PxQ a(α, s)jk xjk.
Theorem 5.2. Let ( u , v ; x) be a P-optimal competitive equilibrium outcome for
M(P, Q; α, s; r, t). Then, u jk = v(P, Q) – v(P–{pj}, Q), for all pj ∈ P with r(pj) = 1
and xjk = 1.
Proof. Construct a graph whose vertices are P∪Q with two types of arcs. If xjk = 0
and u j(min) + v k = αjk – sk there is an arc from qk to pj; if xjk = 1 and u j(min) = u jk
there is an arc from pj to qk. Let p1 ∈ P such that r(p1) = 1. Let q1 ∈ Q such that
x11 = 1. Then, there is an arc from p1 to q1. We claim that there is an oriented path
starting from p1 and ending at a seller with the payoff zero (this seller might be q0). To
see this, suppose there is no such a path. Let P’∪Q’ be all the vertices that can be
reached by a directed path starting from p1. Then, v k > 0 for all qk ∈ Q’, so q0 ∉ Q’
and pj ≠ p0 for all pj ∈ P’. Also, if pj ∈ P’ then pj fills her quota, since otherwise
u j(min) = u j0 = 0 and then there is an arc from pj to q0, so q0 ∈ Q’, contradiction. By
Lemma 1, x(Q–Q’) ≠ p0 and there is a pair (pj, qk) ∈ x(Q–Q’)xQ’, with pj ≠ p0 and
xjk = 0, and qt ∈ Q–Q’ with xjt = 1, such that u jt + v k = αjk – sk. But then, αjk – sk =
u jt + v k ≥ u j(min) + v k ≥ αjk – sk, so u j(min) + v k = αjk – sk and there is an arc from
qk to pj, which implies that pj ∈P’, and u jt = u j(min), so there is an arc from pj to
qt, which implies that qt ∈ Q’, which contradicts the fact that qt ∈ Q–Q’.
Now consider a directed path c starting on p1 and ending at a seller qh with
payoff zero (note that qh might be q1). That is, c = (p1, q1, p2, q2, p3, …,qh-1, ph, qh),
where xdd = 1, u d(min) = u dd and u d(min) + v d–1 = αdd–1 – sd–1 for d = 1,…,h.
Define the matching x’ in M’ = M(P–{p1}, Q; α, s; r, t) that assigns q1 to
{p2}∪(C(q1, x)–{p1}), q2 to {p3}∪(C(q2, x)–{p2}),…, qh–1 to {ph}∪(C(qh–1, x)–{ph–1})
and qh to {p0}∪(C(qh, x)–{ph}) if qh ≠ q0, and that otherwise agrees with x on every
21 agent in P–{p1}∪Q that is not in the path. Define u* as follows: if pd ∈ c and
pd ≠ p1, then u*dd–1 = u dd and u*dk = u dk if qk ∈ C(pd, x)–{qd}; if pd ∉ c. Thus, u*dk
= u dk if qk ∈ C(pd, x). From the construction of x’ and u* it follows that (u*, v ; x’)
is a competitive equilibrium outcome for M’, which implies that x’ is an optimal
matching for M’. Therefore, ∑pj≠p1,qk∈Q a(α, s)jk x’jk = v(P–{p1}, Q). Then, we can write
∑pj≠p1,qk∈Q a(α, s)jk x’jk = ∑pj∈P–{p1},qk∈C(pj, x’) u*jk + ∑Q v k =
∑pj∈P–{p1},qk∈C(pj, x’) u jk + ∑Q v k = v(P, Q) – u 11.
Hence, u 11 = v(P, Q) – v(P–{p1}, Q), which completes the proof.g
Remark 5.1. Consider a buyer pj such that r(pj) = 1. Let x be an optimal matching
for M = M(P, Q; α, s; r, t) and let qk be such that xjk = 1. Then, we can write
v(P, Q) = ∑P,Q a(α,s)it xit = a(α, s)jk + ∑P–{pj},Q a(α, s)it xit
and Theorem 5.2 implies that
u jk = a(α, s)jk – [v(P–{pj}, Q) – ∑ P–{pj},Q a(α, s)it xit].
(3)
Therefore,
v k = [v(P–{pj}, Q) – ∑ P–{pj},Q a(α, s)it xit],
that is, buyer pj acquires object qk at the price
ρ k = [v(P–{pj}, Q) – ∑ P–{pj},Q a(α, s)it xit] + sk
(4)
The price ρ k does not depend on any valuations αjk of buyer pj. So, as in the Vickrey
second price auction for a single object, the price that a buyer with a quota of one pays
is not determined by the valuation she states. This property allows us to prove the
following theorem.
Theorem 5.3. Let (П, X) be the P-optimal competitive equilibrium rule for the market
M ≡ M(α, s). Let pj ∈ P such that r(pj) = 1. Then, telling the truth is a dominant
strategy for pj in the game Γ(П, X, α, s).
Proof. Denote X(α, s) ≡ x and let ( u , v ; x) be a P-optimal competitive equilibrium
outcome for M. If xjk = 1, then u jk = a(α, s)jk – [v(P–{pj}, Q) – ∑ P-{pj},Q a(α, s)it xit],
by (3). Suppose pj misrepresents her valuations. Let α’it = αit for all (pi, qt) ∈ PxQ
22 with pi ≠ pj. Denote X(α’, s) ≡ x’. Let ( u ’, v ’; x’) be a P-optimal competitive
equilibrium outcome for M(α’, s) and (u*, v*) be the true payoff under ( u ’, v ’; x’).
If x’jk = 1, u*jk = u jk, since pj will pay the same price given by (4). If x’jf = 1, for
qf ≠ qk, buyer pj will pay ρ ’f = [v(P–{pj}, Q) – ∑P–{pj},Q a(α, s)i tx’it] + st and her true
payoff for ( u ’, v ’; x’) will be u*jf = αjf – [v(P–{pj}, Q) – ∑P-{pj},Q a(α, s)it x’it] – st =
a(α, s)jf – [v(P–{pj}, Q) –∑P–{pj},Q a(α, s)it x’it]. But, a(α, s)jf + ∑P–{pj},Q a(α, s)it x’it ≤
a(α, s)jk – ∑P-{pj},Q a(α, s)it xit) = v(P, Q), by the optimality of x in M(α, s). Then
u jk ≥ u*jt, and pj has not profited from misstating her valuations. If pj is matched to q0
at x’ then u*j0 = 0 ≤ u jk. If pj is matched to q0 under the true valuations, then
v(P–{pj}, Q) = v(P, Q), so if pj is matched to some qf under x’, u*jf = a(α, s)jf +
∑P–{pj},Q a(α, s)it x’it) – v(P, Q) ≤ v(P, Q) – v(P, Q) = 0 = u j0. Then, in every case, the
total payoff for buyer pj when she selects her true valuations is greater than or equal to
her total true payoff when she misrepresents her valuations, whatever the valuations
selected by the other agents. Hence truth-telling is a dominant strategy for pj.g
In particular, in one-to-many buyer-seller markets, that is, in markets where all
buyers have a quota of one and each seller may own more than one object, no buyer can
manipulate the buyer-optimal competitive equilibrium rule. This is to say, the buyeroptimal competitive equilibrium rule for one-to-many buyer-seller models is strategyproof for the buyers.
Corollary 5.2. Let (П, X) be the buyer-optimal competitive equilibrium rule for the
market M(α, s) where r(pj) = 1 for every buyer pj. Then, telling the truth is a
dominant strategy for every buyer in the game Γ(П, X, α, s).
6. THE MANIPULABILITY THEOREMS
In this section, we prove the two folk theorems stated in the introduction. Both
theorems are corollaries of a stronger and more general result that we call the General
Manipulability Theorem. In these results, we consider the competitive equilibrium rules
and analyze the agents’ equilibrium behavior when the true valuations of buyers and
23 sellers determine a buyer-seller market with more than one competitive equilibrium
price vector.
Theorem 6.1. (General Manipulability Theorem) Let (П, X) be any competitive
equilibrium rule. Let M ≡ M(α, s) be a market with more than one competitive
equilibrium price vector. For Y ∈ {P, Q}, suppose that (П(α, s); X(α, s)) is not a Yoptimal competitive equilibrium for M. Then, any y ∈ Y whose vector of individual
payoffs at (П(α, s); X(α, s)) is different from her/his vector of individual payoffs under
the Y-optimal competitive equilibrium for M can manipulate (П, X) via M.
Proof. The competitive equilibrium outcome if agents select the profile (α, s) is
(u(α, s), П(α, s) – s; X(α, s)), where u(α, s) is the corresponding payoff vector for the
buyers, feasibly defined. Let ( u , ρ − s) and ( u , ρ − s) be the buyer-optimal and the
seller-optimal competitive equilibrium payoffs for M. Proposition 4.3 implies that
X(α, s) is optimal and Proposition 4.2 implies that it is compatible with ( u , ρ − s)
and ( u , ρ − s). Furthermore, ρ ≤ ρ ≤ ρ , where ρ is the equilibrium price vector of
the competitive equilibrium outcome selected by the competitive equilibrium rule.
First case: Y = P. By hypothesis, u(α, s) ≠ u . Let pj be any buyer such that
u(α, s)j ≠ u j. Then, there is some ql ∈ C(pj, X(α, s)) such that u jl > u(α, s)jl, which
implies that the set Aj ≡ {qk ∈ C(pj, X(α, s)); u jk > u(α, s)jk.} is non-empty. (Note that
u jk = u(α, s)jk. for all qk ∈ C(pj, X(α, s))−Aj ). Clearly, for all qk ∈ Aj, u jk > 0 and so
qk ≠ q0. Furthermore, for every
qk ∈ Aj, there is some positive
λk
such that
u jk > u jk − λk > u(α, s)jk ≥ 0, so u jk − λk > 0.
Now define α’ as follows: α’it = αit for all (pi, qt) ∈ PxQ with pi ≠ pj;
α’jk = αjk − ( u jk − λk) for all qk ∈ Aj; α’jk = αjk − u jk for all qk ∈ C(pj, X(α, s))−Aj;
and α’jk = 0 for all qk ∉ C(pj, X(α, s)). It is a matter of verification that α’it ≥ 0 for
all (pi, qt) ∈ PxQ; hence, α’ is well-defined. (It is enough to see that for all qk ∈ Aj,
α’jk = αjk − ( u jk − λk) > αjk − u jk = ρ k ≥ 0 and, for all qk ∈ C(pj, X(α, s))−Aj,
α’jk = αjk − u jk = ρ k ≥ 0.)
We show that pj’s total true payoff with respect to market M(α, s) at allocation
(П(α’, s), X(α’, s)) is strictly greater than her payoff at allocation (П(α, s); X(α, s)). To
24 do so, first note that ρ is a competitive equilibrium price vector for M(α’, s) because
pj still wants to demand the set C(pj, X(α, s)) at prices ρ in M(α’, s) and the demand
sets of the other buyers do not change. Then, by Proposition 4.2, X(α, s) is compatible
with every competitive equilibrium price vector for M(α’, s); and, by Proposition 4.3,
X(α’, s) is compatible with ρ . Denote Bj ≡ {qk; qk ∈ C(pj, X(α, s))−C(pj, X(α’, s))}
and B’j ≡ {qk; qk ∈ C(pj, X(α’, s))−C(pj, X(α, s))}. Clearly, |Bj| = |B’j|. Also, denote
μ j = min{ u jk; qk ∈ C(pj, X(α, s))}. By Proposition 4.1, since ( u , ρ − s) is compatible
with X(α, s) and X(α’, s), then μ j = αjm – ρ m = α’jk − ρ k, for all qm ∈ Bj and
qk ∈ B’j. By definition, α’jk = 0 for all qk ∈ B’j, so μ j = 0. Therefore, for all qk ∈ B’j
and qm ∈ Bj, Ujk(П(α’, s), X(α’, s); α, s) = αjk – Пk(α’, s) ≥ αjk – α’jk = αjk ≥ 0 = μ j =
u jm ≥ u(α, s)jm. We conclude that
Ujk(П(α’, s), X(α’, s); α, s) ≥ u(α, s)jm, ∀ qk ∈ B’j and qm ∈ Bj.
(5)
Equation (5) states that pj’s utility derived from any new transactions (those under
X(α’, s) and not under X(α, s)) is larger than the utility derived from any transactions
that do not take place in the new situation.
Now observe that, at every equilibrium price ρ’ for M(α’, s), and for all qk ∈ Aj
and qm ∉ C(pj, X(α, s)), we have that α’jk − ρ’k = ( ρ k − ρ’k) + λk > 0 ≥ −ρ’m =
α’jm − ρ’m, so α’jk − ρ’k > α’jm − ρm. This implies that every qk ∈ Aj is matched to pj
at any optimal matching for
M(α’, s), and in particular at
Aj ⊆ C(pj, X(α, s))∩C(pj, X(α’, s)). Moreover, for all
X(α’, s), that is,
qk ∈ Aj
we have that
Ujk(П(α’, s), X(α’, s); α, s) = αjk − Пk(α’, s) ≥ αjk − α’jk = u jk − λk > u(α, s)jk. Thus,
Ujk(П(α’, s), X(α’, s); α, s) > u(α, s)jk, ∀ qk ∈ Aj ⊆ C(pj, X(α, s))∩C(pj, X(α’, s))j (6)
Finally, for all
qk ∈ C(pj, X(α, s))∩C(pj, X(α’, s))−Aj
the following holds:
Ujk(П(α’, s), X(α’, s); α, s) = αjk − Пk(α’, s) ≥ αjk − α’jk = u jk = u(α, s)jk. Therefore,
Ujk(П(α’, s), X(α’, s); α, s) ≥ u(α, s)jk, ∀ qk ∈ C(pj, X(α’, s))∩C(pj, X(α’, s))−Aj. (7)
The result follows from (5), (6), (7) and the facts that |B| = |B’| (if |B| = 0 the
result also holds following (6) and (7)) and Aj ≠ φ. Hence, the proof of this case is
complete.
Second case: Y = Q. By hypothesis, П(α, s) ≠ ρ . Let qk be any seller for whom
25 ρ k > Пk(α, s) ≥ sk. Then ρ k > sk, so seller qk sells all his objects at X(α, s).
Furthermore, for some positive λ, ρ k > ρ k − λ > Пk(α, s).
Let s’ be defined as follows: s’t = st for all qt ≠ qk and s’k = ρ k − λ. That is,
under the profile (α, s’), qk replaces his true valuation by s’k, whereas the other
players keep their valuations.
First note that ρ t ≥ s’t for all qt, which implies that ( ρ , X(α, s)) is a feasible
allocation for M(α, s’). Now use the fact that ( ρ , X(α, s)) is a competitive equilibrium
in M(α, s) and that if qt does not sell some of his objects at X(α, s), then qt ≠ qk, so
ρ t = st = s’t, to obtain that ( ρ , X(α, s)) is a competitive equilibrium for M(α, s’). As
ρ k − s’k = λ > 0, it follows from Remark 4.1 that qk sells all his objects at any optimal
matching for M(α, s’); in particular, qk sells all his objects at X(α, s’). Hence, the
seller’s individual payoff from each of his objects is
Vk(П(α, s’), X(α, s’)) =
Пk(α, s’) − sk ≥ s’k − sk = ( ρ k − λ − sk) > Пk(α, s) − sk, which completes the proof. g
Theorem 6.1 implies that, for any competitive equilibrium rule
(П, X)
(not
necessarily one of the optimal competitive equilibrium rules), if (П(α, s), X(α, s)) is
not the buyer- (respectively, seller-) optimal competitive equilibrium for market
M(α, s), then in the induced game Γ(П, X, α, s), truthful behavior is not a best response
(hence, it is also not a dominant strategy) for at least one buyer (respectively, seller),
and so (α, s) is not a Nash equilibrium for Γ(П, X, α, s). Therefore, if M(α, s) has
more than one competitive equilibrium price vector, (α, s) is not a Nash equilibrium in
the induced game with a true profile of valuations (α, s).
In particular, if the buyer- (respectively, seller-) optimal competitive equilibrium
rule is used and the true market has more than one competitive equilibrium price vector,
a seller (respectively, buyer) can profitably misrepresent her (respectively, his)
valuations, assuming the others tell the truth.
We have proved the following corollary:
Corollary 6.1. (Folk Manipulability Theorem) Consider the buyer-optimal
(respectively, seller-optimal) competitive equilibrium rule. Suppose that M(α, s) has
26 more than one vector of equilibrium prices. Then, there is a seller (respectively, buyer)
who can manipulate the rule via M(α, s).
Another immediate consequence of Theorem 6.1, stated in Corollary 6.2, is that for
every market M(α, s) with more than one vector of competitive equilibrium prices,
there is no competitive equilibrium rule such that the induced game with a true profile
of valuations (α, s) gives every agent an incentive to play her/his sincere strategy. We
notice that the proof of the old impossibility result (Theorem 7.3 of Roth and
Sotomayor, 1990) consists of showing that any competitive equilibrium rule for the
particular market M*, with only one seller q1 and n > 1 buyers, such that there is
some pj ∈ P such that αj1 > αt1 for all pt ∈ P, and αj1 > s1, is manipulable either by
pj or by q1. Corollary 6.2 strengthens this result by stating that this is true for all
markets M(P, Q; α, s; r, t). That is, the old Impossibility Theorem implies that every
competitive equilibrium rule is manipulable via market M* (which has more than one
vector of equilibrium prices), whereas our Theorem 6.1 implies that every competitive
equilibrium rule is manipulable via every market that has more than one vector of
equilibrium prices.
Corollary 6.2. (Folk General Impossibility Theorem) Suppose M(α, s) has more
than one vector of equilibrium prices. No competitive equilibrium rule (П, X) for
M(α, s) exists for which, in the game induced Γ(П, X, α, s), stating (α, s) is a dominant
strategy for every agent.
7. THE CASE WHERE ALL QUOTAS ARE ONE
We can immediately derive from the previous results that every competitive
equilibrium rule for a market M(α, s) with more than one vector of equilibrium prices
is manipulable. If the quotas of all buyers and sellers are one, this result, together with
the Demange and Gale’s (1985) Non-manipulability Theorem, implies that a
competitive equilibrium rule for a given one-to-one buyer-seller market M(α, s) is nonmanipulable if and only if the market has only one equilibrium price vector. We state
the results in Proposition 7.1.
Proposition 7.1. Suppose the quotas of all agents are one. Then, there is no agent who
27 can manipulate the competitive equilibrium rule (П, X) via M(α, s) if and only if
M(α, s) has only one equilibrium price vector.
We notice that Proposition 7.1 does not extend to one-to-many or many-to-one
buyer-seller markets. Examples 7.1 (respectively, Example 7.2) illustrates a situation for
the one-to-many (respectively, many-to-one) markets.
Example 7.1. Consider the market
M(P, Q; α, s; r, t), where
P = {p1, p0},
Q = {q1, q0}, α11 = 3, r(p1) = 1, t(q1) = 2 and s1 = 0. There is only one competitive
equilibrium allocation, where seller q1 sells one object to buyer p1 at price 0, so his
total payoff is 0. However, if the seller misrepresents his valuation and reports s’1 = 2
then, at the competitive equilibrium allocation for M(α, s’), he sells one of his objects
to buyer p1 at price 2, in which case, his total true payoff will be 2. Therefore, q1 can
manipulate the seller-optimal competitive equilibrium rule even though M(α, s) has
only one equilibrium price vector.
Example 7.2. Consider the market M(P, Q; α, s; r, t), where P = {p1, p2, p0},
Q = {q1, q2, q3, q4, q0}, the two rows of the matrix α are identical and they are equal to
(7, 7, 1, 1, 0), sk = 0 for all qk ∈ Q, r(p1) = 2 and r(p2) = t(q1) = t(q2) = t(q3) = t(q4) =
1. The unique equilibrium price vector is ρ1 = ρ2 = 6 and ρ3 = ρ4 = 0, under which
buyer p1 receives the total payoff of 2. However, if buyer p1 reports that the first row
is (2, 2, 1, 1, 0) then, under any competitive equilibrium payoff, ρ1 = ρ2 =1 and
ρ3 = ρ4 = 0. Buyer p1 will be assigned either {q1, q3} or {q1, q4} or {q2, q3} or
{q2, q4}. Therefore, her true total payoff will be 7, higher than 2. Therefore, she has an
incentive to misrepresent her valuations under any competitive equilibrium rule.
Theorem 6.1 allows us to obtain the converse of the Demange and Gale’s (1985)
Non-Manipulability Theorem. We state a general result in the proposition below.
Proposition 7.2. Let (П, X) be a competitive equilibrium rule for M(α, s). No buyer
can manipulate (П, X) via M(α,s) if and only if (П, X) maps (α, s) to the buyeroptimal competitive equilibrium allocation for M(α, s)
28 The general manipulability theorem implies that the condition is necessary. The
fact that the condition is sufficient follows from the Demange and Gale’s (1985) NonManipulability Theorem.
Finally, we state the following corollary, whose proof is immediate:
Corollary 7.1. The competitive equilibrium rule (П, X) is non-manipulable (strategyproof) by the buyers (respectively, sellers) if and only if (П, X) is the buyer-optimal
(respectively, seller-optimal) competitive equilibrium rule.
8. CONCLUSION
We have studied sellers’ and buyers’ incentives if competitive equilibrium rules
are applied. First, we have shown that although the Non-Manipulability Theorem for the
one-to-one buyer-seller market does not generalize for the many-to-many buyer-seller
market, it can be extended to the model where all agents of one of the sides of the
market have a quota of one. Second, we have proven a General Manipulability Theorem
for the many-to-many buyer-seller market. The theorem states that if a competitive
equilibrium rule does not yield the buyer- (seller-) optimal competitive equilibrium for
an instance of the buyer-seller market, then any buyer (seller) who does not receive his
(her) optimal equilibrium payoff has an incentive to misrepresent his (her) preferences.
This theorem has two corollaries which are extensions of two important folk theorems
for the Shapley and Shubik’s (1972) one-to-one assignment game. Moreover, restricted
to the assignment game, it is a sort of converse of the Non-ManipulabilityTheorem.
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31