Document de Travail n°2004-67 NASH

GREQAM
Groupement de Recherche en Economie
Quantitative d'Aix-Marseille - UMR-CNRS 6579
Ecole des Hautes Etudes en Sciences Sociales
Universités d'Aix-Marseille II et III
NASH-IMPLEMENTATION OF
COMPETITIVE EQUILIBRIA
VIA A BOUNDED MECHANISM
Gaël GIRAUD
Hubert STAHN
Document de Travail
n°2004-67
Nash-implementation of Competitive Equilibria
via a Bounded Mechanism
Gaël Giraud
CERMSEM1 , University of Paris I
Hubert Stahn
GREQAM2 , University of the Mediterranean Sea
1
CEntre de Recherche en Mathématiques, Statistique et Economie Mathématique, UMR 8095 of the
CNRS
2
Groupement de Recherche en Economie Quantitative d’Aix-Marseille UMR 6579 of the CNRS
Abstract
We deal, in this paper, with the problem of providing incentives for the implementation of
competitive outcomes in a pure-exchange economy with finitely many households. We construct
a feasible price-quantity mechanism, which fully implements Walras equilibria via Nash equilibria
in fairly general environments. Traders’ preferences need neither be ordered nor continuous. In
addition, the mechanism is such that no pure strategy is weakly dominated, hence is bounded
(in the sense of Jackson (1992)). In particular it makes no use of any integer game.
Address
GIRAUD Gael
STAHN Hubert
CERMSEM
GREQAM
Maison des Sciences Economiques
Université de la Méditérranée
Université de Paris 1 Panthéon-Sorbonne
Chateau La Farge
106-112, Boulevard de l’Hôpital
route des Milles
75647 Paris cedex 13, France
13 290 Les Milles, France
tel +33 144 078 296
tel +33 442 935 981
fax +33 144 078 301
fax +33 442 930 968
E-mail : [email protected]
E-mail : [email protected]
Keywords : mechanism design, strategic boundedness, weakly dominated strategy
J.E.L classification numbers : C72, D50.
1
Introduction
It is widely acknowledged that Cournotian market games, where players choose quantities to be traded, typically induce non-competitive outcomes.1 Now, if one tries to conceive
market mechanisms whose strategic equilibrium outcomes are competitive, the most common
way to proceed consists in enabling players to quote prices (sometimes in addition to choosing
quantities): this leads to Bertrandian games, whose seminal paradigm is of course Bertrand
(1883). The main difficulty with this second approach is that the strategic equilibria that sustain competitive outcomes seem to rest on very vulnerable grounds: in a standard Bertrand
game, the unique Nash equilibrium involves weakly dominated strategies. This means that this
Nash equilibrium of such games, though it induces a Pareto-efficient outcome, will not be robust
to any minimal refinement involving admissibility: as is well-known, indeed, the elimination of
weakly dominated strategies is one privileged way to get strategically stable equilibria.2 More
intuitively, such equilibria will probably not be robust to the slightest kind of uncertainty regarding one’s opponents strategy. In the mechanism literature3 , the situation is even worst: most
of the implementation results use integer or modulo games for which every strategy is weakly
dominated ! The question we therefore ask in this paper is the following: Is it possible to design
a strategic market mechanism that fully implements competitive equilibria and whose strategic
equilibria do not involve any weakly dominated strategy ?
In this paper, we provide a positive answer to this question by explicitly constructing a suitable price-quantity mechanism for a broad class of exchange economies. The solution proposed
here is even stronger than required since we show that not only is every Nash equilibrium admissible, but in fact no pure strategy is weakly dominated. In addition, the exact equivalence
between Nash outcomes and Walras equilibria holds under assumptions that are much weaker
than in literature known so far: in particular, consumers’ preferences need neither be ordered
nor continuous.
A slightly different question was already asked by Jackson (1992) as to whether it is possible
to characterize the social choice correspondences that can be implemented via bounded mechanisms. A mechanism is bounded whenever the following holds: If a given strategy, s1 , is (weakly)
dominated by, say, s2 , then there must exist at least one third strategy s3 , which dominates s2
and is itself not weakly dominated. This requirement is a way to detect mechanisms that involve
(possibly hidden) integer or modulo games. Indeed, any such “integer game” typically involves
an infinite sequence of mutually dominated strategies. Jackson (1992) was the first to provide
1
For a recent introduction to strategic market game literature, see Giraud (2003). For a survey, see Dubey
(1994).
2
See Simon et al. (1995) for an extension of the previous ideas to games with infinitely many strategies, as well
as Weyers (2000) for a recent utilization of the iterated elimination of weakly dominated strategies in stragegic
market games.
3
See Jackson (1997) for a crash introduction.
1
a criterion for the implementability of a given correspondence via bounded mechanisms, and
was followed by a stream of literature devoted to this topic.4 However, none of the articles just
mentioned provide a sufficient characterization of such “boundedly implementable” correspondences that would allow to conclude that the Walras correspondence is indeed implementable
in reasonably large class of exchange economies by means of some bounded mechanism.5 Now,
given its property stated supra, the mechanism given in this paper is evidently bounded. Thus,
this paper provides a positive answer to the open question of the implementability of the Walras
correspondence by bounded mechanisms.
The paper proceeds as follows: The next section depicts the economic environment, and
defines the mechanism. Section 3 contains some basic properties of Nash equilibria, which
essentially aim at preparing the results to follow. In section 4, we prove that every competitive
equilibrium can be implemented by some Nash equilibrium strategy profile, while section 5 is
devoted to proving that, conversely, every Nash equilibrium outcome is a Walras equilibrium.
In the fifth section, we prove that the mechanism is bounded. Finally, the last section contains
some additional comments: In particular, we sketch how to extend the main implementation
results of the paper to the case of constrained competitive equilibria whenever the boundary
condition is not satisfied. Two technical proofs are postponed to an Appendix.
2
The market mechanism
We consider a standard exchange economy E. There are L commodities, indexed by , to be
traded by N = |I| household, indexed by i. (By a slight abuse of notations, we shall sometimes
denote by L the set {1, ..., L}). Throughout the paper, we shall assume6 that N ≥ 3. Each
consumer is given a vector ωi ∈ RL
+ of initial endowments and his tastes are defined by a
L
L
binary relation %i ⊂ RL
+ × R+ . We denote by ∂R+ the boundary of the positive orthant in
L
L
L
the Euclidean space RL
+ and by R++ = R+ \∂R+ its interior. Let us also denote by Xi =
ª
©
P
i
the individual feasible set of agent i and let us assume that
X := x ∈ RL
+ :x≤
i∈I ω
P
ω i ¿ i∈I ω i .
The set of players is precisely the set I of traders. Hence, there is no Walrasian auctioneer
or any other “teneur de marché” in order to help agents in coordinating, or even in order
to fix prices. Each player i ∈ I chooses a strategy ai in a set Ai of actions. These choices
Q
are simultaneous. Given a profile a := (a1 , ..., aN ) ∈ A := i∈I Ai , the market allocates a
4
See, e.g.,Jackson et al. (1994), Ohseto (1994) and Sjöström (1994),.
Implementation of the Walras correspondence by unbounded mechanisms is of course known for a long time,
see e.g., , Hurwicz (1979), Maskin (1985), Schmeidler (1980).
6
This restriction is standard, see, e.g., Hurwicz (1979), Schmeidler (1980), Postlewaite and Wettstein (1989),
and harmless if one thinks of current markets.
5
2
¡ ¢N
consumption bundle among players according to a map ϕ : A → RL
. Let denote by ϕi (a) ∈
+
th coordinate of ϕ(a). We now describe the collection (Ai , ϕi )
RL
i∈I .
+ the i
2.1
Setting quantities and quoting prices
©
ª
L(N−1) , to be interpreted as follows:
Each individual i’s action set is Ai := (pi , z i ) ∈ RL
+×R
There is one trading-post for each individual. On her trading-post, individual i quotes a price
pi ∈ RL
+ for each commodity at which she is willing to trade. At the same time, she sends a
message to each one of her opponents j 6= i, j ∈ I, saying how much she
to trade with
³ is´willing
L
i
i
i
i
i
L(N−1)
i
i
, where zj := zj,
∈ RL represents
player j. Thus, z = (z1 , ..., zi−1 , zi+1 , ..., zN ) ∈ R
=1
the portfolio that player i suggests to trade with each player j 6= i. If zji takes only positive
[resp. negative] values, then it can be viewed as a pure demand [resp. offer]. As any player
j 6= i acts in the same way, the confrontation of these trade proposals gives rise to net trading
proposals for each pair of agents. For the couple (i, j) , i 6= j, this quantity will be zji − zij , and
agent i can be viewed as a potential net buyer [resp. a net seller] of commodity
to agent j if
zj,i
− zi,j
≥ 0 [resp.
zj,i
− zi,j
with respect
≤ 0 ].
In order for agent i to become a net buyer with respect to agent j for commodity , she
must, in addition, be willing to pay at least the price quoted by j: The trade zj,i − zi,j ≥ 0
occurs between i and j if, and only if, pi ≥ pj . Symmetrically, if i is a potential net seller with
respect to agent j for commodity , trade occurs if pi ≤ pj . To put it differently: Players i and
j will be matched with regard to commodity
provided that
´
³
(pi − pj ) zj,i − zi,j ≥ 0.
Let us denote by T i (a) ⊂ I \ {i} the subset of individuals with whom i is matched with regard to
, whenever a is played. Any action ai ∈ Ai of player i ∈ I, given the strategies a−i := (aj )j∈I\{i}
of the other players, is interpreted as meaning: “With agent j 6= i, and for every
I’m ready to buy the amount zj,i −
[resp. higher] than, or equal to,
zi,j
pi ”.
> 0 [resp. to sell if zj,i −
zi,j
= 1, . . . , L,
< 0] at any price lower
Hence, a basic action of each player can equivalently be
viewed as a limit-price order in the spirit of Mertens (2002).7
j
7
i
A market selling order obtains whenever zj,
− zi,
< 0 and pi = 0. Since the set of conceivable prices is
unbounded, there is no market buying order in our mechanism (the same restriction is encountered in Mertens
(2002), but for different reasons). This is harmless, since no rational player would ever be willing to play such a
risky strategy, while, in ‘real’ financial markets, practical constraints provide natural upper bounds for the buying
prices, so that every buying order is de facto a limit price order.
3
2.2
From actions to trades
Having defined the action space of each player, it remains to define the mapping ϕ. The construction of this outcome function can be decomposed into three steps.
First, given the action profile a ∈ A, the different quantity messages are aggregated into
¢
¡ ¢N
¡
, taking into account the matching rule just described:
single net trades z̃ i (a) i∈I ∈ RL
+
´
X ³
zj,i − zi,j
z̃ i (a) :=
∈L
j∈T i (a)
One easily sees that, by construction, ∀a ∈ A,
P
i∈I
z̃ i (a) = 0, i.e., markets clear (even out
of equilibrium). However, because each agent h 6= i has the opportunity to manipulate player’s
i net trade, it may happen that z̃ i (a) + ω i < 0 for some commodity , i.e., that z̃ i (a) does not
belong to i’s consumption set. In other words, a player may end up short in some commodities.
Whence the need for a rationing rule.
Second, a proportional rationing rule applies on aggregate net trades.
Only players who are long or short in commodity
(i.e., who verify resp. z̃ i (a) > 0 or
z̃ i (a) < −ω i ) can be rationed. Players who sell part of their endowments (i.e., for whom
−ωi ≤ z̃ i ≤ 0)) end up with the very remaining of their initial holdings ω i + z̃ i . Individuals
must effectively sell all their initial endowments ωi . And players
who are short in commodity
who are long are rationed proportionally to their contribution to the aggregate excess demand.
To be more precise, given an action profile a ∈ A and a commodity , let us partition the set of
players I into three subsets, I = I R (a) ∪ I − (a) ∪ I + (a), defined as follows:
©
ª
• I R (a) := i ∈ I : z̃ i (a) < −ω i is the subset of players who are short in commodity ,
©
ª
• I − (a) := i ∈ I : −ωi ≤ z̃ i (a) < 0 is the subset of players who sell a part of their initial
endowment of commodity ,
©
ª
• I + (a) := i ∈ I : z̃ i (a) ≥ 0 is the subset of agents who are net buyers of commodity .
P
P
The net supply of commodity , in positive terms, is given by i∈I R (a) ω i − i∈I − (a) z̃ i (a), and
P
the net demand is I + (a) z̃ i (a). Every player with a net demand is rationed according to the
proportion:
i
ρ (a) :=
with the convention
0
0
P
i∈I R (a) ω
i
P
−
P
I + (a)
i∈I − (a) z̃
i
z̃ (a)
i (a)
.
:= 0. To check that this last expression makes sense, observe that:
X
i∈I R (a)
z̃ i (a) +
X
z̃ i (a) +
i∈I − (a)
X
I + (a)
4
z̃ i (a) = 0,
so that if
P
i∈I + (a) z̃
i (a)
> 0, then ρi (a) ∈ (0, 1], and else ρi = 0 by the convention.
The rationed net trade z̄ i (a) in commodity
of player i ∈ I, when a ∈ A is played, is given
by:

i


 −ω
z̄ i (a) =
z̃ i (a)


 ρi (a) · z̃ i (a)
if i ∈ I R (a)
if i ∈ I − (a)
if i ∈ I + (a)
This rationing rule satisfies standard desirable properties (see Bénassy (1975)): Every player
obviously remains on the same side of the market after rationing has taken place, i.e. ∀ ∈ L,
P
z̄ i (a) · z̃ i (a) ≥ 0; the rationing rule is also efficient in the sense that i∈I z̄ i (a) = 0 for every
¯ ¯
¯
¯
∈ L; finally, ∀a ∈ A, ∀ ∈ L, ¯z̄ i (a)¯ ≤ ¯z̃ i (a)¯, i.e., no agent can be forced to buy or to sell
more than she was willing to.
Third, prices enter the picture and trades occur: Players who fail to fulfill their strategic
budget constraint (calculated with respect to their own, personal trading prices) are punished,
and loose their initial endowments.8 Players not kept in the red receive their rationed aggregate
net trade.
Given an action profile a ∈ A, the strategic budget constraint of individual i is defined by:
L
X
=1
pi · z̄ i (a) ≤ 0
If, for any action profile a ∈ A, one now denotes by I(a) ⊂ I the subset of players who satisfy
their strategic budget constraint, the strategic outcome function of our game is defined by:
∀a ∈ A, ∀i ∈ I, ∀ ∈ L
¡
¢
and we denote by ϕi (a) = ϕi (a)
∈L

z̄ i (a) + ωi
ϕi (a) :=
0
if i ∈ I(a)
otherwise,
the final bundle of commodities received by agent i ∈ I.
It may help the reader reminding that, given some action profile a, there are three, possibly
distinct, kinds of quantities that are to be considered: z̃ i (which is the aggregate net trade of
player i in commodity ), z̄ i (which is the same quantity after rationing occurred) and ϕi (the
final bundle of commodity
2.3
effectively received by agent i).
Some remarks on the mechanism
In order to gain intuition about how the mechanism works, let us spell out some of its basic
features.
8
Similar punishments rules are used in Dubey (1982), Peck, Shell & Spear (1992) or Weyers (1999) for instance.
5
A. Regarding the matching rule, the following various situations may happen:
(i) zj,i − zi,j > 0 and pi − pj ≥ 0. In this case, player i is the net “buyer” of the
positive amount zj,i − zi,j , and the magnitude of the bid-ask spread is pi − pj .
(ii) zj,i − zi,j > 0, but pi − pj < 0. In such a case, i still would be willing to buy the
surplus zj,i − zi,j , but is not ready to pay the price that is asked for by j. Hence, no
trade occurs.
(iii) By reverting the inequalities above, one gets the symmetric situations where j
wears the hat of the “buyer” and i is the “seller”.
¢
¡
B. Every outcome ϕi (a) i∈I of the game is weakly feasible, and this is true even out of equilib-
rium:
∀a ∈ A, ∀i ∈ I, ϕi (a) ∈ RL
+
and
X
i∈I
ϕi (a) ≤
X
ωi.
i∈I
Moreover, if nobody defaults, this outcome is balanced, in the sense of this last inequality being
verified as an equality.9
C. A household’s bankruptcy does not provoke the break-down of the whole economy. Every nonbankrupt player receives, for each commodity , a feasible bundle z̄ i (a) + ω i as final allocation,
regardless of whether her opponents did satisfy their budget constraint. This is due to the fact
P
that the rationing rule is efficient (i.e., verifies i∈I z̄ i (a) = 0 and z̄ i (a) ≥ −ω i ), and can be
interpreted as meaning that the market keeps the endowments of bankrupt players, so that it
always has the opportunity to realize the feasible bundle of non-bankrupt agents.
D. The computation of the rationed net trade z̄(a)i of commodity
for player i is independent
from the strategies played on markets different from . Thus, the interaction between markets
only intervenes when one checks each player’s strategic budget constraint. This implies that
if at least one player, say i, modifies her strategy on a given market, say , the rationed net
trades do not change on the other markets. This usefull independence property can be stated as
following :
∀i ∈ I, ∀k 6= , ∀a , a0
3
¡
¢
z̄ki (a , a− ) = z̄ki a0 , a−
(IP)
Some basic properties of Nash Equilibria
Let us recall the
9
To take but one example, Schmeidler’s game (1980) is always balanced, but need not be feasible. Moreover,
the very rules of the game prevent players from ever defaulting, even out of equilibrium.
6
Definition 1 A Nash Equilibrium (NE) associated to this mechanism is a vector (âi )i∈I ∈ A
with the property that ∀i ∈ I, there does not exist any ai ∈ Ai ,
ϕi (ai , â−i ) Âi ϕi (âi , â−i )
¡ ¢N
is individually rational if, for each agent i ∈ I, it
Definition 2 An allocation (xi )i∈I ∈ RL
+
is not true that ωi Âi xi
The next Lemma is easy.
Lemma 1 Every NE, â ∈ A, is individually rational
Proof
¡ ¢
Let â = âi i∈I be a NE such that there exists some i ∈ I, with ω i Âi ϕi (â). Then player i
¶
µ ³ ´
could defend the no-trade issue by playing ai = pi , zji
given by pi = p̂i and for every
j∈I\{i}
∈ L,
zj,i
=
ẑi,j
. After her deviation, i’s aggregate net trade will be z̃ i (ai , â−i ) = 0. In this case
she cannot be rationed nor going bankrupt, and therefore receives her initial holding as a final
outcome. This would contradict the equilibrium character of â.
Q.E.D.
Remark. Lemma 1 shows that, at a NE, no individual can be “forced” to trade if she is not
willing to do so. However, it should be clear that, to say it with the vocabulary borrowed from
zero-sum game theory, while she can always defend an autarkic outcome for herself, agent i has
no way to guarantee the no-trade issue (i.e., to find a strategy that enables her not to trade
whatever being the strategies chosen by her opponents). This is the basic difference between
the present mechanism and the Shapley-Shubik (1977) ones: in the latter, by submitting no bid
and no offer for any good, any player can refuse to trade independently of the actions of the
other traders. This last feature is, of course, responsible for the survival of no-trade as a NE in
such games.10 Conversely, the lack of such a possibility in our mechanism is responsible for the
disappearance of autarky as an automatic Nash equilibrium outcome.
In the sequel, we say that preferences %i satisfy the no-default condition if and only if ω i Âi 0.
Under this restriction, Lemma 1 immediately implies that :
Lemma 2 If %i satisfy the no-default condition then no agent goes bankrupt at a NE, â ∈ A,
and the outcome is a feasible and balanced allocation.
10
Notice that, in Weyers (2000), a double round of elimination of weakly dominated strategies is needed in
order to get rid of the autarkic Nash equilibrium.
7
Proof
Take some NE â ∈ A. The no-default condition condition and Lemma 1 imply that ϕi (â) %i
ω i Âi 0 for every i. Nobody therefore goes bankrupt otherwise ϕi (â) = 0 would hold for players
who defaulted. Moreover if nobody defaults, it follows by the very definition of the mechanism
P
P
that i∈I ϕi (â) = i∈I ωi and that ϕi (â) ∈ RL
+ for every i.
Q.E.D.
One can also strengthen this last condition by introducing a more standart boundary condition
11
given by :
∀x ∈ ∂RL
+ ∩ X,
ω i Âi x for every i
Under this restriction, one can state that :
Lemma 3 If the boundary condition holds for each agent, then at a NE â ∈ A, no agent is
¡
¢
rationed, i.e., z̄ i (â) = z̃ i (â) for every i and , and ϕi (â) i∈I À 0, i.e., the final allocation
n¡ ¢
o
P
i ≤ ω i , the set of feasible allocations
x
belongs to the interior of F := xi i∈I ∈ RL
+ :
i∈I
Proof.
If some player i was rationed, there would be some agent j (possibly distinct from i) who would
be lead to the boundary ∂RL
+ of her consumption set – a contradiction to Lemma 1 thank’s to
the boundary condition.
Q.E.D.
4
Competitive Equilibria are NE outcomes
¡
¢
Definition 3 A Competitive Equilibrium (CE) of the economy E, is a vector (x̄i )i∈I , p̄ with
the property that :
©
¡
¢
ª
i
(i) ∃i
/ ∈ I and ∃x
/ i ∈ B i (p̄) = x ∈ RL
+ : p̄ · x − ω ≤ 0 such that xi Âi x̄i
¢
P ¡
(ii) i∈I x̄i − ωi = 0
Existence of competitive equilibria with non-ordered preferences is well-known: see, e.g., Gale
& Mas-Colell (1975) or Shafer & Sonnenschein (1975). The first step of our implementation result
consists in remarking that every competitive equilibrium can be enforced by a profile a ∈ A of
strategies.
11
This condition is known to be necessary for the Maskin-monotonicity of the Walras correspondence (see
Maskin (1985)), hence for its Nash-implementability. In the conclusion of that paper, we briefly discuss how to
get rid of this condition by replacing the Walras correspondence by the correspondence of constrained Walras
equilibria
8
¡ ¢
Proposition 1 Every CE ((x̄i )i∈I , p̄, ) can be obtained by a strategy profile â = âi i∈I given by
:
µ µ
¶¶
¢
¢
1 ¡
1 ¡
i
i
â = p̄,
x̄i − ω , . . . ,
x̄i − ω
N
N
i
∀i ∈ I,
Proof.
If a is played, every agent quotes the same price. Thus ∀i ∈ I, ∀ ∈ L, T i (â) = I\{i} and:
¶
X µ1 ¡
¢
¢
1 ¡
i
j
x̄i − ω −
x̄j − ω
N
N
i
z̃ (â) =
j∈I\{i}
=
¢
1
N −1 ¡
x̄i − ω i −
N
N
X ¡
¢
x̄j − ω j = x̄i − ω i
j∈I\{i}
because markets clear at (x̄i )i∈I . Moreover, at a Walras equilibrium nobody is short, hence
z̄ i (â) = z̃ i (â). Finally, each individual’s strategic budget constraint reduces to the competitive
one, p̄ · z̄ i (â) ≤ 0, and is also evidently satisfied. Thus ∀i ∈ I, ϕi (â) = x̄i .
Q.E.D.
In order to prove that every CE can be implemented by a NE, it remains to check whether any
player i has any profitable deviation from the strategies constructed in the preceding proposition.
For this purpose, we shall need a weakening of the standard transitivity assumption:
Assumption T For every i, the following holds: For every xi ∈ Xi , if ω i ¨i xi , then
0 ¨i xi .
Evidently, Assumption T will be verified as soon as each player’s preferences are transitive,
complete and verify the no-default condition.
Going back to our proof, it is clear that agent i has no incentive to deviate solely in quantitysetting strategies: By the very definition of a CE, such a deviation would not improve her
i
situation. Hence,
µ
¶ she must also change her price-quoting strategy. If one denotes by a :=
³ ´
pi , zji
this deviation, one can make the following observations:
j∈I\{i}
¡
¢
1. After her deviation, player i will only be matched with agents j ∈ T i ai , â−i , i.e., verifying
(pi − p̄ ) · (zj,i − ẑi,j ) ≥ 0 with zj,i 6= ẑi,j . But this implies by summing over j 6= i, that,
for every :
¡
pi − p̄
¢
·
X
¡
¢
(zj,i − ẑi,j ) = pi − p̄ · z̃ i (ai , â−i ) ≥ 0.
(O1)
j∈T i (ai ,â−i )
This means that agent i, at least in terms of her new aggregate net trade, will spend [resp.
earns ] more [resp. less] money for each commodity she buys [resp. sells] than before she
deviated.
9
2. If her preferences satisfy Assumption T, a profitable deviation of agent i never brings her
into bankruptcy, i.e.
¡ ¡
¢
¢
pi · ϕi ai , â−i − ω i = pi · z̄(ai , â−i ) ≤ 0
(O2)
As her strategic budget constraint still holds (O2), and she “spends” more money than at the
Walrasian prices (O1), the allocation obtained by player i after deviation also belongs to her
competitive budget set B i (p̄) and hence cannot strictly improve her situation. We are now
ready to state:
¢
¡
Theorem 1 Under Assumption T, every CE (x̄i )i∈I , p̄ can be implemented as a NE.
Proof
Assume that agent i deviates by playing
ai
µ ³ ´
:= pi , zji
j∈I\{i}
¶
. Applying observation O1,
and recalling that players still stay on the same market side after rationing (i.e., z̄ z̃ ≥ 0),
¢
¡ i
¢
¡
P
one deduces that ∀ ∈ L, pi − p̄ · z̄ i (ai , â−i ) ≥ 0, hence, by summing over ,
∈L p − p̄ ·
P
i i −i
z̄ i (ai , â−i ) ≥ 0. It remains to use observation O2 in order to conclude that
∈L p̄ ·z̄ (a , â ) ≤ 0
or equivalently that ϕi (ai , â−i ) ∈ B i (p̄).
5
Q.E.D.
NE outcomes are competitive
Our next main result essentially says that the converse holds: every NE induces a competitive
outcome. Let us call a market
active at a given action profile a, if at least two agents agreed
upon some effective trade relative to
(i.e., ∃i ∈ I, z̃ i (a) 6= 0). On the contrary, an inactive
market is such that, when playing a, nobody effectively trades .
Consider an active market : every effective seller i has a strong incentive to trade with
the player who quoted the highest price for commodity . If she does not already trade with
such an agent, i has an opportunity to sell the same amount of commodity
to some higher-
price-quoting player, and to use the money earned by this operation to improve her situation.
But every effective seller is, by definition, matched with a least one effective buyer. By the
matching rule, this last agent must be quoting a price at least equal to the one quoted by her
trading partner. This buyer herself has, of course, no incentive to quote a strictly greater price
for commodity . Thus, this line of Bertrandian reasoning leads to the conclusion that effective
10
trades must all occur at the same price. In the sequel, preferences ºi are said monotonic if, for
L
every x ∈ Xi , one has x + RL
++ ⊂ {y ∈ R+ : y ºi x} and there is some
y = x ∀ 6=
i
and y i > x
i
i
for which:
⇒ y Âi x.
This commodity is said to be strictly improving for agent i.
Proposition 2 Suppose that each trader’s preferences %i are monotonic and satisfy the bound¡ ¢
ary condition. Let â = âi i∈I ∈ A be a NE. If is an active commodity market, then every
effective trader quotes the same price for that commodity.
Proof
Take any active commodity market
¡ ¢
and let us denote by p̂i i∈I ∈ RN
+ the NE prices for
this good. Because this market is active, there exists at least one effective buyer and one effective
seller, indexed by b and s respectively.
Now let us first assume that there exists some effective seller s with the property that
© ª
p̂s < p̂imax = maxi∈I p̂i . This cannot be a NE. Indeed, player s has an incentive to cancel
all her trades with all the players distinct from imax , and to sell z̄ s (â) < 0 to imax at price
p̂imax . This strategy gives her the same allocation, and leaves her with some money back which
can be used to improve her situation (for a detailed construction of this deviating strategy, see
© ª
Appendix A). Hence every effective seller quotes p̂imax = maxi∈I p̂i .
Let us now consider an effective buyer b. In order to realize her trade, she must quote a
price p̂b ≥ p̂j for at least one j ∈ T b (â). If this agent j is an effective seller, it is immediate that
´
³
i0
p̂b = p̂imax . Otherwise, there exists a chain of bilateral trades ẑii0 − ẑi k 6= 0, with i0k ∈ I for
k
k = 1, ..., n and i01 = s, i0n = b, which carries commodity
from an effective seller s to player b.
The first buyer in each chain is nevertheless charged at price p̂imax , and as she sells commodity
, she has no incentive to lower the price. It recursively follows that the effective buyer must
quote p̂b = p̂imax if she is to finally obtain the good. Hence all the effective traders quote the
same price.
Q.E.D.
In other words, the “law of one price” applies at a NE, if every market is active.12 The next
result now follows easily. From now on, preferences ºi will be said to be convex if, for every
x ∈ Xi , the subset {y ∈ RL
+ : y ºi x} is convex.
12
Observe that Koutsougeras (1999) showed that this is not the case in Shapley-Shubik games with multiple
trading-posts: to be more precise, consider a Shapley-Shubik game with several trading-posts for the same commodity (or, equivalently, with one trading-post per commodity but perfectly substitutable commodities). It may
happen that the same commodity is priced differently at different trading-posts at some active NE, so that the
“law of one price” does not hold. Otherwise stated, the number of trading-posts per commodity influences the
NE prices ! In our framework, we assumed that there is one trading-post per individual, but this restriction is
immaterial at every active NE, since the same commodity will get the same price at each open trading-post.
11
Theorem 2 Suppose that every trader’s i preferences ºi are monotonic, convex and satisfy the
¡ ¢
boundary condition. Let â = âi i∈I ∈ A be a NE with the property that every market is active.
In this case, if p̄ denotes the price commonly quoted by all the effective traders, the outcome
¡ i ¢
ϕ (â) i∈I associated to the price p̄ is a CE.
Proof
One notices, by Lemmata 2 and 3, that at a NE â, there can be neither rationing nor bankruptcy.
¢
¡
¡
¢
¢
¡
Thus, ϕi (â) i∈I = z̃ i (â) + ω i i∈I and pi · ϕi (â) − ωi ≤ 0 for every i. Moreover, by Proposition
2, we know that if agent i effectively trades, say commodity , then pi = p̄ . One deduces that
¢
¡
p̄ · ϕi (â) − ω i ≤ 0, i.e., ϕi (â) satisfies for each i ∈ I her competitive budget constraint at price
p̄. It remains therefore to verify that no household i admits any xi ∈ B i (p̄) with the property
that xi Âi ϕi (â). So let us assume, by way of contradiction, that such an agent i0 and a bundle
xi0 ∈ B i0 (p̄) exist. Thanks to the convexity of i0 ’s preferences, there exists some λ ∈ (0, 1],
and some x̆i0 (λ) Âi0 ϕi0 (â) with x̆i0 (λ) := λϕi0 (â) + (1 − λ)xi0 , moreover, by construction,
x̆i0 (λ) ∈ B i0 (p̄).
Now, recall that, since every market is active, one can define for each
∈ L a set I e ⊂ I of
agents who effectively trade, and |I e | ≥ 2. Let us consider the following strategy of agent i0 :
 Ã ·
!
¸
´
³


i
i
1
h
h
0
0

, ẑi0 ,
/ Ie
if i0 ∈

 p̄ , ẑi0 , + |I e | (x̆ (λ) − ω )
e ∪{i })
h∈I\(I
0
h∈I e
à ·
!
ai0 =
¸
´
³


i
i
1
h
0
0

, ẑih0 ,
otherwise.

 p̄ , ẑi0 , + |I e |−1 (x̆ (λ) − ω )
h∈I\I e
h∈I e \{i }
0
¡
¢
Observe that z̃ i ai0 , â−i0 = x̆i0 (λ). Next, remember that no rationing occurs at a NE, i.e.,
∀i ∈ I, ϕi (â) À 0. Because x̆i0 (λ) is a convex combination of ϕi0 (â) and xi0 , it is possible
¢
¡
to choose λ sufficiently small to make sure that no rationing occurs at ai0 , â−i0 . Agent i0
¡
¢
therefore receives ϕi0 ai0 , â−i0 = x̆i0 (λ) as her budget constraint is satisfied by construction.
But this contradicts the fact that â is a NE. It follows that, for all i ∈ I, ϕi (â) maximizes ºi
over B i (p̄).
Q.E.D.
It remains to consider the case where some commodity is not active.
Theorem 3 Suppose that traders’ preferences are monotonic, convex and satisfy the boundary
¡ ¢
condition. Let â = âi i∈I ∈ A be a NE with the property that at least one market is non
active. Construct the price vector p̄ as follows: For inactive commodities, choose any price
p̂i quoted by at least two players, or if such a price does not exist, select p̄ := p̂i for which
mini∈I {p̂i } < p̂i < max{p̂i } , and p̄ is played by at least one player13 . For active commodities
13
Remember that N > 3.
12
¡
¢
, take the prices p̄ as in Theorem 2. Then, ϕi (â) i∈I associated to the price p̄ is a CE.
Proof
The first part of the proof of Theorem 2, which states that ϕi (â) ∈ B i (p̄) for every i, also works
in the present case. In order to make sure that ϕi (â) maximizes ºi over B i (p̄) for all i ∈ I,
we again assume the contrary, i.e., ∃i0 ∈ I and ∃x̆i0 (λ) ∈ B i0 (p̄) with x̆i0 (λ) Âi0 ϕi0 (â). The
problem is that, for some commodities , the subset I e of agents who are effectively trading
may now be empty. So let us select three sets of commodities with regard to i0 . Firstly, let
Lei0 ⊂ L be the subset of markets on which there is an effective trade. Second, let L1i0 be the
on which there is not effective trade but on which at least one player distinct from i0 quotes
the walrasian price p̄ . Finally, L2i0 := L \ {Lei0 ∪ L1i0 } is the set of remaining goods. For
∈ Lei0 ,
one takes the same profitable deviation ai0 as the one depicted in the proof of Theorem 2. If
n
o
∈ L1i0 , one considers again the same deviation, but replaces I e by I := i ∈ I | pi = p̄ , and
if
∈ L2i0 , one constructs the following deviation:
 µ
¶
· −
¸ ³ ´
i−
i

i0
i0
h

p
if (x̆i0 (λ) − ω i0 ) < 0
,
ẑ
+
(x̆
(λ)
−
ω
)
,
ẑ

i,
i0 ,
−
h∈I\{i
,i
}
i0
0
¸ ³
¶
µ + · +
a :=
´
i
i

i0
i0
h

if (x̆i0 (λ) − ω i0 ) ≥ 0
 p , ẑi0 , + (x̆ (λ) − ω ) , ẑi, )
+
h∈I\{i0 ,i }
© ª
© ª
i−
= maxi∈I pi and p = mini∈I pi respectively.14
¡
¢
A simple computation shows that, after i0 ’s deviation, z̃ ai0 , a−i0 = x̆i0 (λ) − ω i0 . As in the
where i+ and i− are chosen so that p
i+
proof theorem 2, one can conclude that for λ small enough no rationing occurs. It remains to
check whether i0 ’s strategic budget constraint is satisfied. But (i) For ∈ Lei0 ∪L1i0 agent i0 , when
∈ L2i0 , if agent i0 acts as a buyer she trades at
deviating, trades at competitive prices ; (ii) for
a price cheaper than the walrasian price, while if she acts as a seller, she now sells commodities
at a higher price. Since we know that x̆i0 (λ) ∈ B i0 (p̄), i0 still satisfies her strategic budget
¡
¢
constraint when playing this deviation. Hence agent i0 again receives ϕi0 ai0 , â−i0 = x̆i0 (λ),
which is the desired contradiction.
Q.E.D
Wrapping up Theorems 2 and 3, we have proven that every NE induces a competitive equilibrium allocation, and that all the effective traders quote a common competitive price. What may
happen, however, is that on some markets, some traders who do not take part to trades quote
distinct prices. Nevertheless, all the prices quoted can serve to decentralize the final allocation
14
Notice that L2i0 is the set of commodities for which agent i0 is the only one who quotes the walrasian price.
/ L2i+ and ∈
/ L2i− . This means, otherwise stated, that this
When i0 is identical to i+ or i− , one observes that ∈
c
deviation with respect to commodity
does neither affect i+ nor i− .
13
c
as a CE. The next Corollary shows how to get rid of the price multiplicity that could survive
at a NE. Recall that preferences ºi are strictly convex whenever, for each x ∈ Xi , the subset
{y ∈ RL
+ : y ºi x} is strictly convex.
Corollary 1 Suppose, in addition to the assumptions of Theorem 3, that at least one trader’s
i0 preferences are strictly convex. Then, every NE is such that all the traders quote colinear
prices
This means, in particular, that when prices are normalized (say in the unit sphere), all the
players will quote the same (competitive) price at every NE.
Proof.
Take some NE â. It suffices to observe that ϕi0 (â) belongs to Xi0 , so that the subset {y ∈ RL
+ :
y ºi ϕi0 (â)} is strictly convex. Therefore, it admits a unique (up to a normalization) supporting
price vector p̄. Applying Theorem 3 enables to conclude that all the traders must quote the very
same price (up to a normalization).
Q.E.D.
6
The mechanism is bounded
In this section, we show that under somewhat more restrictive assumptions and for the mechanism presented in this paper, no pure strategy is weakly dominated. To be more precise:
Definition 4 A strategy ai of player i is weakly dominated if there exists an alternate strategy
ãi such that, for every profile of player i’s opponents a−i ∈ A−i , ϕi (ãi , a−i ) %i ϕi (ai , a−i ),
while ϕi (ãi , ā−i ) Âi ϕi (ai , a−i ) for at least one (N − 1)-tuple a−i . A strategy that is not weakly
dominated is said undominated.
For the last result of this paper, we need to strengthen our hypotheses.
Assumption (B). For every i, the restriction of ºi to Xi is:
(i) strictly convex i.e. ∀x ∈ Xi , the upper set {y ∈ RL
+ | y ºi x} is strictly convex.
(ii) upper semi-continuous, i.e., the upper set is closed in RL
+ for every x ∈ Xi ;
¡
¢
(iii) strictly monotonic: ∀x ∈ Xi , ∀y ∈ Xi ∩ × + RL
+ \{0} , y Âi x.
(iv)transitive, complete, and verifies the boundary condition.
14
Before stating the next theorem, let us make the following remark : Whenever a player quotes
a null price pi = 0 for some commodity , she enters in some kind of “integer game”: since her
individual strategic budget constraint is no more binding (at least in terms of commodity ), she
has no reason to refrain from demanding the maximal amount of commodity
that is available
in the economy (provided her preferences are monotonic with respect to , and she finds some
-seller at a zero price). But any opponent j 6= i could quote pj = 0 as well, and compete
P
with i in order to get i ω i . In order to get rid of this kind of theoretically uninteresting, and
practically unrealistic, situations, it suffices to slightly modify our mechanism, and to forbid
players to quote null prices. Notice that, when each commodity is strictly desired by at least
one individual, every competitive price will be strictly positive. Hence, the restriction to positive
prices has no impact on the results of the preceding sections. In addition, it enables to get the
next property.
Theorem 4 Under Assumption (B), and if players’ price quoting strategies are restricted to
RL
++ , no action ai ∈ Ai is weakly dominated.
As already said, this implies that our mechanism is strategically bounded, in the sense of
Jackson (1992), hence that it makes no use of integer games, modulo games or the like. The
proof proceeds in two steps, corresponding to the two next subsections. We first establish that
each strategy ãi ∈ Ai of each player i ∈ I can be viewed as a best reply to at least one, suitably
defined, strategy ã−i ∈ A−i of her opponents. In our non-ordered environment, we call a strategy
ãi a best-reply against ã−i whenever there does not exist any alternate strategy ai such that
ϕi (ai , ã−i ) Âi ϕi (ã) = ϕi (ãi , ã−i ). In a second step, we verify that for each strategy ai ∈ Ai that
is “equivalent” to ãi when opposed to ã−i (i.e., which is such that ϕi (ai , ã−i ) ∼i ϕi (ãi , ã−i ), there
exists another strategy a−i of i’s opponents with the property that ϕi (ãi , a−i ) Âi ϕi (ai , a−i ).
This will prove that ãi cannot be weakly dominated. In the rest of this section, we only work
with agent 1, but it should be clear that the results extend to any player i.
6.1
Every strategy is a best response
The intuition beyond this first step is quite simple. Any strategy ã1 of player 1 is composed by
³ ´N
of net trades. Taking p̃1 as given, one can therefore
a price p̃1 and a quantity proposal zj1
j=2
compute a vector z opt such that z opt + ω 1 is a maximal element of ºi over the constrained
competitive budget set:
¡ ¢
B1 p̃1 :=
(
x∈
RL
+
n
X
¢
¡
: p̃ · x − ω 1 ≤ 0 and x ≤
ωi
1
i=1
15
)
Since B1 (p̃1 ) is non-empty and compact, and provided º1 is upper semi-continuous, z opt exists.
¡ ¢N
It therefore suffices to construct some strategies ã−1 = ai i=2 of the other players which induce
ϕ1 (ã) = ϕ1 (ã1 , ã−1 ) = z opt + ω 1 . It will then be immediate that ã1 is a best reply against ã−1 .
not exist any a1 ∈ A1 , such that ϕ1 (a1 , ã−1 ) Â1
QN
i
i=2 A
ϕ1 (ã1 , ã−1 ).
Proposition 3 For every ã1 ∈ A1 , there exists ã−1 ∈
with the property that there does
Proof.
¶L,N
µ ³ ´
i
i
= p̃ , z̃j,
Step 1 : Construction of
j∈I,j6=1 =1,i=2
¡ i ¢L
i
Let us first set ∀i ∈ I\{1}, p̃ := p̃ =1 = p̃1 . Next, two cases have to be considered:
ã−1
Case 1. if z opt ≤ 0, choose some player i 6= 1, say for instance agent 2, who will buy the
quantity z opt while the other players simply neutralize their trades with player 1:
if z opt

opt
2
1


 z̃1, := z̃2, − z
i := z̃ 1
≤ 0 then
∀i ≥ 3, z̃1,
i,


 ∀i, j ≥ 2, and i 6= j
z̃j,i := 0
Case 2. If z opt > 0, ask again player 2 to provide player 1 with the suitable commodity
bundle. It may happen, however, that 2 does not own enough commodities to fulfill her job, so
that the other players will be asked to transfer some commodities to agent 2, proportionally to
their own endowments. More precisely:
if z opt > 0 then

2 := z̃ 1 − z opt

z̃1,

2,



 ∀i ≥ 3, z̃ i := z̃ 1
1,
i,
i
opt
i := − P ω

∀i ≥ 3, z̃2,

N
i · z

i=2 ω



∀i, j ≥ 3 and i 6= j, z̃j,i := 0
(∗) and z̃i,2 := 0
Step 2 : Some basic verifications
We need to check that the strategies ã−1 make sense. Because every player quotes the same price,
¡
¢
it is immediate that the net trade of player 1 will be z̃ 1 ã1 , ã−1 = z opt . Moreover, since z opt
P
i
opt proportionally to
does not exceed N
i=2 ω and every player contributes to the realization of z
her endowment15 , nobody is rationed. Finally, because z̃ 1 ∈ B1 (p̃1 ), player 1 does not default
whenever ã−1 is played against ã1 . Hence ϕ1 (ã1 , ã−1 ) = z opt + ω 1 .
In order to see that ã1 is a best response against ã−1 , just take any strategy a1 ∈ A1 , and
remember that, as each player i 6= 1 quotes the same price p̃1 , the same argument as in in
15
¡
¢
ω3
Because of (*) supra, a routine calculation shows that z̃ 3 ã1 , ã−1 = − Pn
i=2
16
ωi
.
¡
¢
the proof of Theorem 1 shows that p̃1 · ϕ1 (a1 , ã−1 ) − ω1 ≤ 0. Since, on the other hand, the
mechanism ϕ is feasible, ϕ1 (a1 , ã−1 ) ∈ B1 (p̃1 ). Thus it cannot be the case that ϕ1 (ã1 , ã−1 ) ≺i
ϕ1 (a1 , ã−1 ).
Q.E.D.
The following Lemma, whose proof is relegated to the Appendix, is a preparation for the
next subsection.
Lemma 4 Let a1 ∈ A1 satisfy ϕ1 (ã1 , ã−1 ) ∼1 ϕ1 (a1 , ã−1 ), and assume that º1 are strictly
convex and strictly monotonic. Then,
(i) in terms of net trades after rationing z̄ 1 (ã1 , ã−1 ) = z̄ 1 (a1 , ã−1 ) = z opt .
(ii) If a1 induces a price with the property that ∃ , p1 < p̃1 then z̃ opt ≤ 0
(iii) If the prices induced by a1 verify that ∃ , p1 > p̃1 and p1 ≥ p̃1 then z̃ opt = 0
³
´
P
(iv) if p1 = p̃1 then one observes that j6=1 zj,1 − z̃j,1 ≥ 0.
6.2
Every strategy is undominated
In order to make sure that every strategy is undominated, it remains to verify that for each
strategy a1 ∈ A1 that is equivalent to ã1 in response to ã−1 (in the sense that ϕ1 (ã1 , ã−1 ) ∼1
ϕ1 (a1 , ã−1 )), there exists an other strategy a−1 of her opponents with the property that ϕ1 (ã1 , a−1 ) Â1
ϕ1 (a1 , a−1 ). Let us denote by A1∼ ⊂ A1 , the subset of strategies which are equivalent to ã1 in
response to ã−1 .
We first consider a strategy a1 ∈ A1∼ with the property that there exists a good k for which
p1k < p̃1k . In this case, one can imagine a strategy a−1 of the other players in commodity k where,
say, player 3 lowers her price and buys the endowments of player 1. If this last agent plays ã1
instead of a1 , she prevents herself against this attack because the price of any buyer, in this
case agent 3, must be higher than the one of a supplier. By doing this, agent 1 would surely be
better off because she keeps some good of type k. One can therefore expect that:
Lemma 5 Any a1 ∈ A1∼ for which ∃k for which p1k < p̃1k cannot weakly dominate ã1 .
Proof.
17
Let us construct
a−1
µ ³ ´
= pi , zj,i
j∈I,j6=i
¶L,N
in the following way:16
=1,i=2

µ ³ ´ ¶
µ ³ ´ ¶

i
i


∀ 6= k,
p , zj,
:= p̃i , z̃j,i


j6=i i∈I\{1}
j6=i i∈I\{1}



´+
³
³ ´N
³ ´N


opt

2 := z̃ 1 − z
1 − z 1 − ω1
2
2
 

p2k := p̃1k , z1,k
+ z̃3,k
,
zj,k
:= z̃j,k

2,k
2,k
k
k

j=3
j=3

³
³
´
´
 3
1
3
1
1
3
3


pk := pk , z1,k := z3,k + ω k ,
zj,k
:= z̃j,k


j6=1,3
j6=1,3


µ
µ
¶
¶


N
N
³ ´
³ ´





i
i

 pik , zj,k
:= p̃ik , z̃j,k
 
i6=j
i=4
i6=j
i=4
¡
¢
If a1 , a−1 is played, the trading rule induces the following net trade of agent 1 in commodity
k.
¡
¢
z̃k1 a1 , a−1 =
X
i∈Tk1 (a1 ,a−1 )
N
X
¡ 1
¢
¡ 1
¢−
i
i
≤ −ω1k
zi,k − z1,k
= −ω 1k +
zi,k − z1,k
i=2,i6=3
¡
¢
This means that at least component k of φ1 (a1 , a−1 ) is 0. Let us now compute z̃k1 ã1 , a−1 :
h
¡ 1
¡
¢ ³ 1
¢+ i´
1
1
z̃k1 ã1 , a−1 = z̃2,k
− z̃2,k
− zkopt + z̃3,k
− z3,k
− ω 1k
N
X
¢
¡ 1
¢
¡ 1
1
1 +
i
z̃i,k − z̃1,k
+ z̃3,k − (z3,k + ωk ) +
i=4
´
P ³ 1
i
z̃
By Lemma 4 (ii), we know that zkopt ≤ 0 hence, by construction, N
−
z̃
i=4
i,k
1,k = 0. It follows
¡ 1 −1 ¢
¡
¢
opt
1
that z̃k ã , a
= zk and by the early definition of the rationing scheme that z̄k1 ã1 , a−1 =
zkopt . Moreover, one notices that a−1 = ã−1 for all
6= k. In this case, by Lemma 4 (i) and the
¡
¢
¡
¢
Independence Property (see section 2.3 supra), ∀ 6= k, z̄ ã1 , a−1 = z̄ ã1 , ã−1 = z opt . As a
consequence, agent 1 satisfies her budget constrain and obtains φ1 (ã1 , a−1 ) = z opt + ω 1 . If one
remembers that ω1 ∈ B(p̃1 ) and that z̃ 1 + ω 1 maximizes ºi over B(p̃1 ), the boundary condition
immediately implies that φ1 (a1 , a−1 ) ≺1 ω1 -1 φ1 (ã1 , a−1 ).
Q.E.D.
If one now takes a strategy a1 ∈ A1∼ with the property that ∃k for which p1k > p̃1k and p1 ≥ p̃1 ,
one knows by Lemma 4 (iii) that zkopt = 0. If one player, say again, agent 3, decides on the one
hand to quote the price p1 , and on the other hand to sell more commodity k to player 1, it may
happen that this last one goes bankrupt. But if agent 1 decides to play ã1 instead of a1 , she
cancels her trades with player 3 by definition of the matching rule. She therefore circumvents
bankruptcy. As a consequence, one can assert:
16
We denote by x+ := x ∨ 0 the positive part of the real number x, and by x− = x ∧ 0 its negative part.
18
Lemma 6 Any strategy a1 ∈ A1∼ with the property that ∃k for which p1k > p̃1k and p1 ≥ p̃1 does
not weakly dominate ã1 .
Proof.
In this case, let us construct a−1 in the following way :

³
´
´
³
i , zi
i , z̃ i


∀
=
6
k,
p
:=
p̃

j,
j,

i∈I\{1}
i∈I\{1}


³
´−
³ ´N
³ ´N



2
1
2
1
1
1 +ε
2
2



p
:=
p̃
,
z
:=
z̃
+
z̃
−
z
,
z
:=
z̃j,k
 
k
k
1,k
2,k
3,k
3,k
j,k

j=3
j=3

³ ´
³ ´
 3
1
3
1
3
3

pk := pk , z1,k := z3,k − ε,
zj,k
:= z̃j,k


j6=1,3
j6=1,3


µ
µ
¶
¶


N
4
´
´
³
³





i
i
i
i

 pk , zj,k
= p̃k , z̃j,k
 
i6=j
i6=j
i=4
i=1
¡
¢
Computing the net trade z̃ 1 a1 , a−1 , one observes that :
N
X
¡
¢
¡ 1
¢+
i
≥ε>0
zi,k − z̃j,k
z̃k1 a1 , a−1 = ε +
i=2,i6=3
¡
¢
¡
¢
Moreover, by lemma 4 (iii), one can even assert that z̄k1 a1 , a−1 > z̄k1 ã1 , ã−1 = zkopt = 0,
¡
¢
¡
¢
hence p1k z̄k1 a1 , a−1 > p̃1k z̄k1 ã1 , ã−1 Concerning the other commodities 6= k, one remarks
that a−1 = ã−1 (i.e every agent diferent from 1 plays the same strategy). It follows from lemma
4 (i) and the Independence Proprety (see section 2.3 supra) that
∀ 6= k, z̄ 1
¡¡ 1 −1 ¢
¢
¡¡
¢
¢
a , ã
, a− = z̄ 1 ã1 , ã−1 , a− = z opt
It remains now to observe that for
6= k
¡
¢
¡
¢
• if p1 > p̃1 , then by lemme 4(iii) z̃ 1 = 0 hence p1 z̄ 1 a1 , a−1 = p̃1 z opt = p̃1 z̄ 1 ã1 , ã−1 = 0
¡
¢
¡
¢
• if p1 = p̃1 , then obviously p1 z̄ 1 a1 , a−1 = p̃1 z̄ 1 ã1 , ã−1
It follows that :
L
X
=1
L
¡
¢ X
¡
¢
p1 z̄ a1 , a−1 >
p̃1 z̄ ã1 , ã−1 = p̃1 · z opt
=1
As %i are strictly monotonic, one also observes by construction of z opt that p̃1 · z opt = 0. But
¡
¢
this means that agent 1 goes bankrupt, hence φ1 a1 , a−1 = 0. Let us now compute the net
¡
¢
¡
¢
¡
¢
¡
¢
trade z̃ 1 ã1 , a−1 when ã1 , a−1 is played. Obviously ∀ 6= k, z̄ ã1 , a−1 = z̄ ã1 , ã−1 = z opt .
¡
¢
¢
¡
Moreover, by computation, it is immediate that z̃k1 ã1 , a−1 = zkopt = 0. If ã1 , a−1 is played,
¡
¢
¡
¢
agent 1 cannot go bankrupt and φ1 ã1 , a−1 = φ1 ã1 , ã−1 . The conclusion follows by an
argument similar to the preceding Lemma.
Q.E.D.
19
It now remains to consider the case in which player 1 does not change her price.
Lemma 7 Any strategy a1 ∈ A1∼ with the property that p1 = p̃1 does not weakly dominate ã1 .
Proof.
By Lemma 4 (iv), we know that ∀ ∈ L,
³
´
1 − z̃ 1
z
≥ 0. As a consequence, one can
i6=1
i,
i,
P
assert that ∃i0 , ∃k, with the property that zi10 ,k − z̃i10 ,k > 0 otherwise ∀i, ∀ , zi,1 = z̃i,1 . Let us
now construct a−1 in the following way :
∀ ∈ L, ∀i ≥ 2,
pi < p̃1 ,
¡ ¢
i
z1,
:= z̃i,1 and zj,i j>1 := 0
In this case, one observes by the trading rule that
z̃ 1
¡
a1 , a−1
¢
But we know that ∃i0 , ∃k, with the property that zi10 ,k − z̃i10 ,k
µ
´+ ¶L
PN ³ 1
1
.
=
i=2 zi, − z̃i,
=1
¡
¢
> 0, hence z̃ 1 a1 , a−1 > 0.
Since, after rationing, every player remains on the same market side, one can even assert that
¡
¢
z̄ 1 a1 , a−1 > 0. As prices must be strictly positive, agent 1 goes in this case bankrupt (i.e.,
µ
´+ ¶L
¡
¢
¡
¢
PN ³ 1
1
φ1 a1 , a−1 = 0) To conclude, observe that z̃ 1 ã1 , a−1 =
−
z̃
= 0, hence
z̃
i=2
i,
i,
=1
¡
¢
¡
¢
φ1 ã1 , a−1 = ω 1 which is, by the boundary condition, obviously better than φ1 a1 , a−1 = 0.
Q.E.D.
7
Extensions and concluding comments
We end this paper with a couple of remarks.
1.As things stand in the body of this paper, two buyers who ask for the same quantity will
be rationed equally eventhough one of them is offering a much higher price than the other. One
could bias the rationing in favor of the higher-priced buyer (in many ways) or lower-price sellers.
And this would be more in the spirit of the “decentralization” of trades. In particular, the
proportional rule puts buyers together in a quite central pool (see (3) e.g.). Thus, it may be of
interest to observe that the analysis of this paper goes through for much more general rationing
rules than just the proportional. The following list spells out the collection of axioms that must
be satisfied by the rationing rule for the whole analysis to hold water (the way our proofs are
written should make transparent in which place each axiom is used):17
P
Axiom 1. ∀z̃, i∈I z i = 0. (Balancedness.)
Axiom 2. For every commodity , if,
17
As in the body of the paper, z̃ is the N-tuple of net trades that serve as input of the rationing rule, and z is
the N-tuple of outputs produced by the rationing rule. In the following, designates a generic object of exchange.
20
for all i, z̃ i ∈ [−ω i ,
P
j6=i ω
j
],
(*)
then z i = z̃ i for every trader i.(The rule operates only if necessary.)
Axiom 3. Whenever (*) above is not satisfied for some good , there exists at least one
agent i such that z i = −ω i . (When operating, the rule leads some trader to the boundary of
her consumption set.)
Axiom 4. ∀i, , ∀z̃, z̃ i z i ≥ 0. (Everybody remains on the same side of the market after the
rationing rule.)
Axiom 5. ∀i, , if z̃ i ∈ [−ω i , 0], then whatever being z̃ −i , z i = z̃ i . (Only net buyers or
traders having taken a short position are rationed.)
Axiom 6. ∀i, ∀k 6= , ∀z̃ , z̃ 0 ,
z ik (z̃ , z̃− ) = z ik (z̃ 0 , z̃− ) for every z̃− . (Independence)
Axiom 7 ∀z̃, ∀i, , |z̃ i | ≥ |z i |. (Nobody can be forced to trade.)
2. In many situations of economic interest, the boundary condition (1) is not realistic. In
order to get rid of it, it suffices to replace the Walrasian correspondence by the correspondence
P
of constrained Walras equilibria 18 , defined as follows (where ω := i ω i denotes the aggregate
initial endowment vector):
´
³¡ ¢
Definition 5 A Constrained Competitive Equilibrium (CCE) of the economy E is a vector x̄i i∈I , p̄
with the property that:
¢
ª
©
¡
i
i
i
(i) ∃i
/ ∈ I, ∃x
/ i ∈ B(p̄) = x ∈ RL
+ : x ≤ ω and p̄ · x − ω ≤ 0 such that x Âi x̄
¢
P ¡ i
(ii) i∈I x̄ − ωi = 0
and to replace our set of assumption which includes the boundary condition by the following
conditions:
Assumption (CCE) For every trader i, her preferences are convex, strictly monotonic
and satisfy the no-default condition
(i) her preferences ¹i are convex, strictly monotonic and satisfy the no-default condition
(ii) For every commodity , there exists at least three traders (i, j, k) who are furnished, i.e., for whom ω h > 0, h = i, j, k.
18
For examples of Nash-implementation of this latter correspondence, see Postlewaite & Wettstein (1989) and
Wettstein (1995). Observe that every CE is a constrained competitive equilibrium, every interior CCE is a CE,
and the First Welfare Theorem applies to CCE as well.
21
One can then state the analog of Theorems 1 to 3. (We only sketch the proofs. Details are
available from the authors upon request):
Theorem 1’ If ¹i satisfy the no-default condition, every CCE ((x̄i )i∈I , p̄) can be implemented
as a NE.
Proof
Lemmata 1 and 2 still hold. Proposition 1 requires no peculiar assumptions. It remains simply
¡
¢
to notice that our mechanism is feasible. From that point of view, if ϕi ai , ã−i in the proof of
Theorem 1 belongs to B i (p̄), the standard budget set, it also belongs to B(p̄)
.
Q.E.D.
¡ i¢
Theorem 2’ Suppose that Assumption (CCE) (i) is in force. Let â = â i∈I ∈ A be a NE with
the property that every market is active. In this case, if p̄ denotes the price commonly quoted
by all the effective traders, the outcome (ϕi (â))i∈I , p̄) is a CCE.
Proof.
If the preferences are strictly monotonic, the deviation exhibited in case 1 of Appendix A proves
Proposition 2. No interiority conditions are required in this case. The extention to CCE is
therefore obvious. The extension of Theorem 2 is less obvious. Since the CCE need not be
interior, we need to slightly modify the deviating strategy of player i0 . One however notices
contrary to a CE approach that the quantity x̆i0 (λ) defined in the proof of this theorem can be
obtained for all λ ∈ (0, 1] as a part of a feasible allocation19 . Two cases have to de considered:
/ I e , one knows that there exists at least one effective buyer b which ends up
Case 1. If i0 ∈
with a strictly positive amount of commodity . Agent i0 can therefore buy good
by b or sell
this commodity for instance to s and λ can be choosen small enough in order to make sure that
this deviation is feasible on each market.
Case 2. If i0 ∈ I e , a problem rises with the preceding argument if i0 is the unique effective
buyer. But preferences are stricltly monotonic hence p̄ À 0 and ωi0 ¿ ω. It follows that there
exists for each commodity at least one agent different from io which ends up with a positive
amount of this good, say . This agent does perhaps not belong to I e . But in any case,
because proposition 2 extends, he quotes a price lower than p̄ . Agent i0 can therefore buy some
additional commodities at price p̄ .
Q.E.D.
¡ ¢
Theorem 3’ Suppose that Assumption (CCE)(i)-(ii) is in force. Let â = âi i∈I ∈ A be a NE
with the property that at least one market is non active. Construct the price vector p̄ as follows:
For inactive commodities, choose any price p̄ := pi quoted by at least two -furnished players,
19
This is basically why the restriction to interior equilibria is required in our game to implement CE. In fact if
a CE allocation is on the boundary of the feasible set, it may happen that the quantity x̆i0 (λ) is for any λ ∈ (0, 1]
a part of a feasible allocation.
22
or if such a price does not exist, select p̄ := pi for which mini∈I f {pi } < p̄ < maxi∈I f {pi },
where p̄ is played by at least one -furnished player and I f is the set of -furnished players
¡
¢
For active commodities , take the prices p̄ as in Theorem 2’. Then, (ϕi (â))i∈I , p̄ is a CCE.
Proof.
The proof extend quite readinly if one has in mind that on inactive markets every player ends
up with her initial endowments and if one considers -furnished player to trade with.
Q.E.D.
3. Of course, in order to be able to implement feasible outcomes, the designer of our
mechanism must know the initial endowments of all the traders. This is admittedly a heroic
assumption, but we know from Postlewaite & Schmeidler (1979, Appendix C) that it is unavoidable: There is no mechanism with a strategic outcome function independent from initial
endowments that yield feasible outcomes for all strategy profiles and has NE outcomes coincident
with Walrasian equilibria for a non-trivial class of economies.
4. A game-form close to the one analyzed in this paper has been used in Giraud & Rochon
(2002a,b) in order to study the impact of coalition-formation and correlation in implementation
theory. On the other hand, Giraud & Stahn (2002) also uses a similar mechanism in order
to provide a second-best efficient solution concept that could serve as an alternative to the
competitive solution
References
[1] Bénassy, J.-P. (1975), ‘Neo-Keynesian Disequilibrium Theory in a Monetary Economy’,
Rev. of Econ. Studies, 42, 503-523.
[2] Bertrand J. (1883) ‘Review of Cournot’s Recherche sur la Théorie Mathématique de la
Richesse’, Journal des Savants 499-508
[3] Dubey P. (1982) ‘Price-Quantity Strategic Market Games’, Econometrica, 50, 111-126
[4] –––. (1994) ‘Strategic Market Games: A Survey of Some Results’ In Mertens, J-F
and Sorin, S. (eds), Game-theoretic Methods in General Equilibrium Analysis, pp209-224,
Dordrecht and Boston; Kluwer Academic.
[5] Gale, D. & A. Mas-Colell (1975) ‘An Equilibrium Existence Theorem for a General Model
without Ordered Preferences’, Journ. of Math. Econ., 2, 9-15.
[6] Giraud G. (2003) ‘Strategic Market Games : an Introduction’ forthcoming in Journal of
Mathematical Economics
23
[7] Giraud G. and Rochon C. (2001) ‘Consistent Collusion-proofness and Correlation in Exchange Economies’ Université Catholique de Louvain, CORE Discussion Paper 01/18 forthcoming in Journal of Mathematical Economics
[8] –––––––––—. (2001) ‘Generic Efficiency and Collusion-Proofness in Exchange
Economies’, Université Catholique de Louvain, CORE Discussion Paper 01/19 forthcoming
in Social Choice and Welfare
[9] Giraud G. and Stahn H. (2001) ‘Efficiency and Imperfect Competition in Economies with
Incomplete Markets’ forthcoming in Journal of Mathematical Economics
[10] Hurwicz, L. (1979) ‘Outcome Functions yielding Walrasian and Lindahl Equilibria at Nash
equilibrium Points’, Rev. of Econ. Studies , 46, 217-225.
[11] Jackson M. (1992) ‘Implementation in Undominated Strategies : A Look at Bounded
Mecanism’ Rev. of Econ. Studies, 59, 757-775
[12] –––– (1997)
‘A Crash Course in Implementation Theory’ forthcoming in:
So-
cial Choice and Welfare, to be reprinted in The Axiomatic Method: Principles and
Applications to Game Theory and Resource Allocation, edited by William Thomson.
http://www.hss.caltech.edu/~jacksonm/crash.pdf
[13] Jackson M. O., Palfrey T. R. and Srivastava S. (1994) ‘Undominated Nash Implementation
in Bounded Mechanisms’, Games and Economic Behavior 6 : 474-501
[14] Koutsougeras, L. (1999)
‘Market Games with Multiple Trading Posts’, Universite
Catholique de Louvain CORE Discussion Paper: 99/18, forthcoming in Journal of Economic Theory
[15] Maskin, E. (1985), ‘The Theory of Implementation in Nash Equilibrium’ In L. Hurwicz, D.
Schmeidler, and H. Sonnenschein, Social Goals and Social Organization : Essays in Honor
of Elisha A. Pazner, Cambridge: Cambridge University Press.
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Journal of Mathematical Economics
[17] Ohseto, S. (1994), ‘Implementation of the Plurality Correspondence in Undominated Strategies by a Bounded Mechanism’ Economic Studies Quarterly, 45 (2), 97-105
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Equilibrium’, Journal of Mathematical Economics, 21, 271-299
24
[19] Postlewaite, A. and Scmeidler D. (1979), ‘Notes on Optimality and Feasibility of Informationally decentralized allocation mechanisms’ in Moeschlin O. and Pallaschke D. (eds),
Game Theory and Related Topics, North Holland.
[20] Postlewaite, A. and Wettstein, D. (1989), ‘Feasible and Continuous Implementation’, Rev.
of Econ. Studies, 56(4), 603-611
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Preferences’, Journ. of Math. Econ., 2, 345-348.
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Games’ Games and Economic Behavior 6 : 502-511
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48 (7), 1585-1593.
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Markets’ Journ. of Math. Econ. 24 201-216
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of Mathematical Economics
APPENDIX
A
Construction of the deviation in the proof of Prop. 2
To end up the proof of Prop.2, it remains to construct the profitable deviation as := (as )L=1 of
an effective seller s in the case in which p̂s < p̂imax . The seller’s preference being monotonic,
25
either she strictly improves her situation by increasing her allocation in commodity
or their
exists an other commodity, say k, satisfying this property :
Case 1: commodity
is strictly improving for agent s.
For commodity , she simply chooses:
·³ ´
´ µ
³ ¡ ¢
j
imax
s
, ẑs,
a := p , zj, j∈I\{s} = p̂
s
j∈I\{s,imax }
with
p̂imax −p̂s
·|z̄ s (â)|
p̂imax
, z̄
s
imax
(â) + ẑs,
+η
¸¶
> η > 0 and leaves all her strategies on the other markets unchanged.
Playing this deviation, her net trade is:
¡
¢
z̃ s as , â−s :=
X
j∈T s (ai ,â−s )
³
´
j
zj,s − zs,
= z̄ s (â) + η > z̄ s (â),
But, by construction |z̄ s (â)| > η, hence she remains a seller after deviation. Being not rationed
initially (see Lemma 3), this remains true after a decrease of her supply. By the independence
property (IP of section 2.3), rationed net-trades are also unchanged on the other markets. It
simply remains to check that this seller satisfies her budget constraint after deviation in order
to conclude that this agent improves her situation. In fact :
X
¡
¢
p̂sk z̄ks (â)
p · z̄ s as , â−s = p̂imax (z̄ s (â) + η) +
=
X
k∈L
k∈L,k6=
³
´
p̂sk z̄ks (â) + p̂imax − p̂s z̄ s (â) + p̂imax η
≤ 0 (by definition of η)
Case 2: commodity
is not strictly improving for agent s
In this case she sells commodity
to agent imax by playing :
·³ ´
´ µ
³ ¡ ¢
j
as := p , zj,s j∈I\{s} = p̂imax , ẑs,
j∈I\{s,imax }
and the additional money m =
imax
, z̄ s (â) + ẑs,
¸¶
´
³
p̂imax − p̂s |z̄ s (â)| > 0 earned by this deviation in order to
increase her trades in commodities k. If one remembers that, by Lemma 3, at any NE, ∀i ∈ I,
:= maxj6=s pjk , this can be done by playing :
∀ ∈ L, ϕi (â) > ε > 0 and if p̂max
k
 Ã
!
µ
·
¸¶

m

max
i

p̂k , ẑs,k +
∧ε
if Tks (â) = ∅



(N − 1) · p̂max
k
i∈I\{s}

 Ã
"
#!
ask =
³ ´

m

s
 p̂s , ẑ s + ¯
 else
¯
, ẑi,k

k
i,k

¯T s (â)¯ · p̂s ∧ ε

i∈I\(Tks (â)∪{s})
s
k
k
i∈Tk (â)
26
on market k and leaving her strategies unchanged on the other markets. It now remains to make
sure that agent s improves her situation. Two subcases must be studied.
Case 2.1 Tks (â) = ∅
By the independence property, it is immediate that ∀h 6= , k, z̄hs (a). Moreover on market
, z̃ s (â−s , as ) = z̄ s (â) and because no other player deviates, on has z̄ s (â−s , as ) = z̄ s (â). Let us
now concentrate on commodity k and let us remark that :
the highest price, hence Tks (as , â−s ) = I\{s}, and
• Player s chooses p̂max
k
´
h
i
X ³
¡
¢
i
i
m
z̃ks as , â−s =
ẑs,k
+ (N−1)p̂
max ∧ ε − ẑs,k
k
i∈I\{s}
=
h
m
p̂max
k
since, initially, Tks (â) = ∅.
i
¡
¢
∧ (N − 1) ε > z̃ks âs , â−s = 0
• Each player i 6= s, who initially did not trade with s, now obtains:
¡
¢
z̃ki as , â−s =
X
j∈Tki (âs ,â−s )
i
³
´ h
j
i
m
− ẑi,k
ẑj,k
− (N−1)p̂
max ∧ ε
k
i
¡
¢ h
m
= z̃ki âs , â−s − (N−1)p̂
max ∧ ε
k
Moreover because â is a NE, one knows, by Prop. 3, that ∀i ∈ I, ϕik (â) > ε > 0 and that
there is no rationing i.e. z̃ki (â) = z̄ki (â) = ϕik (â) − ω ik . It follows that :
h
i
¡
¢
i
m
z̃ki as , â−s = ϕik (â) − ω ik − (N−1)p̂
max ∧ ε > −ω k
k
Therefore, no player i 6= s is short in commodity k after the deviation.
As a consequence, one can assert that no agent is rationed on commodity k (since nobody goes
short). It follows that:
i
¡
¢ h m
¡
¢
z̄ks as , â−s = p̂max
∧ (N − 1) ε > z̄ks âs , â−s = 0
k
It remains to verify that agent s does not go bankrupt after her deviation in order to make sure
that she can really improve her situation. So let us observe that
¡
¢
¡
¢
p · z̄ s as , â−s = p̂imax z̄ s (â) + p̂ikmax z̄ks as , â−s +
X
h∈L,h6= ,k
p̂sh z̄hs (â)
³
´
X
≤ p̂imax − p̂sh z̄ s (â) + m +
p̂sh z̄hs (â) ≤ 0
{z
} h∈L
|
=0
27
Case 2.2 : Tks (â) 6= ∅
For the same arguments as in the preceding case, one has ∀h 6= k, z̄hs (a). Moreover on market
k, one observes that :
• Player s now chooses the same price p̂sk , so that Tks (as , â−s ) = Tks (âs , â−s ), and
z̃ks
·
X µ
¡ s −s ¢
s
a , â
=
ẑi,k +
i∈Tks (â)
m
|Tks (â)|p̂sk
¸
¶
i
∧ ε − ẑs,k
i
¡
¢ h
¡
¢
= z̃ks âs , â−s + p̂ms ∧ |Tks (â)| ε > z̃ks âs , â−s
k
• Each player i ∈
/ Tks (â) obtains the same net trade z̃ki (as , â−s ) = z̃ki (â) and ∀i ∈ Tks (â),
z̃ki
¡ s −s ¢
a , â
=
X
j∈Tki (â)\{s}
·
³
´ µ
j
i
i
s
ẑj,k − ẑi,k + ẑs,k − ẑi,k −
m
|Tks (â)|p̂sk
¡
¢
= z̃ki âs , â−s −
·
¸¶
∧ε
m
|Tks (â)|p̂sk
¸
∧ε
By a similar argument one can assert that no player i 6= s is short in commodity k after
deviation.
One therefore concludes that:
i
¡
¢
¡
¢ h
z̄ks as , â−s = z̄ks âs , â−s + p̂ms ∧ |Tks (â)| ε .
k
and with a similar argument as in case 2.1, it is easy to verify that the budget constraint is
satisfied.
Q.E.D.
B
Proof of Lemma 4
(i) Let us first remember that ∀a1 ∈ A1 , ϕ1 (a1 , ã−1 ) ∈ B1 (p̃1 ) (see step 2 of the proof of
proposition 3) If ϕ1 (a1 , ã−1 ) ∼1 ϕ1 (ã1 , ã−1 ) and ϕ1 (ã1 , ã−1 ) = z opt + ω1 , the strict convexity
of º1 induces that ϕ1 (ã1 , ã−1 ) = ϕ1 (a1 , ã−1 ) otherwise z opt + ω 1 is not a maximal element for
º1 over B1 (p̃1 ). Now, remark that the boundary condition implies ϕ1 (ã1 , ã−1 ) À 0. Hence,
agent 1 does not default when (ã1 , ã−1 ) is played. It follows that this agent does also not default
by playing (a1 , ã−1 ) because the same allocation is obtained. One deduces that z̄ 1 (a1 , ã−1 ) =
z̄ 1 (ã1 , ã−1 ) = z opt À −ω 1 .
(ii) If p1 < p̃1 and if one remembers that, by construction, ∀i 6= 1, p̃i = p̃1 , then the trading
28
rule implies that :
¡
¢
z̃ 1 a1 , ã−1 :=
X
i∈T 1 (a1 ,ã−1 )
¡
i
zi,1 − z̃1,
¢
=
N
X
¡ 1
¢−
i
≤ 0.
z̃i, − z̃1,
i=2
But, by construction of the market game, the supply side is rationed if a player is short. But,
by the boundary condition and the fact that ϕ1 (a1 , ã−1 ) = z opt + ω1 À 0 for all a1 ∈ A1∼ . (see
¡
¢
point (i) supra) this situation does not occure, hence z̃ 1 a1 , ã−1 = z̄ 1 (a1 , ã−1 ). This implies by
point (i) that z opt ≤ 0.
(iii) If one has in mind that rationed players remain on the same market side, an argument
symmetric to the one used in (ii) above applies if p1 > p̃1 . In this case, one has: z opt =
z̄ 1 (a1 , ã−1 ) ≥ 0. Now remember that z opt + ω1 is a maximal element for º1 over B1 (p̃1 ). The
preferences º1 being strictly monotonic, one has p̃1 · z opt = 0, hence, by (i), p̃1 · z̄ 1 (a1 , ã−1 ) = 0.
¡
¢
But player 1 does not default when playing (a1 , ã−1 ), one can therefore assert that p1 − p̃1 ·
ª
¡ 1
¢ 1 1 −1
©
P
1
∈ L : p1 > p̃1 , it also implies that
z̄ 1 (a1 , ã−1 ) ≤ 0. If L+ =
∈L+ p − p̃ z̄ (a , ã ) ≤ 0
because ∀ ∈
/ L+ , p1 = p̃1 . But by our first remark, we know that ∀ ∈ L+ , z̄ 1 (a1 , ã−1 ) ≥ 0,
hence z opt = z̄ 1 (a1 , ã−1 ) = 0.
(iv) One knows by (i) that z̄ 1 (a1 , ã−1 ) = z̄ 1 (ã1 , ã−1 ). But if (ã1 , ã−1 ) is played, nobody is
¯ ¡
¢¯
rationed. Moreover, if rationing occurs as (a1 , ã−1 ) is played, one knows that (1) ¯z̃ 1 a1 , ã−1 ¯ ≥
¯ 1 1 −1 ¯
¯z̄ (a , ã )¯ and (2) that the supply side is never rationed. It follows that if either z̃ 1 ≤ 0
¡
¢
¡
¢
¡
¢
¡
¢
or z̃ 1 > 0, then either z̃ 1 a1 , ã−1 = z̃ 1 ã1 , ã−1 or z̃ 1 a1 , ã−1 ≥ z̃ 1 ã1 , ã−1 respectively.
Because p1 = p̃´1 , one deduces from the matching rule and the early definition of ã−1 that
PN ³ 1
1
≥ 0.
i=2 zi, − z̃i,
Q.E.D.
29