Econometrica, Vol. 81, No. 2 (March, 2013), 813–824
NOTES AND COMMENTS
A NOTE ON THE EQUILIBRIUM EXISTENCE PROBLEM
IN DISCONTINUOUS GAMES
BY PAULO BARELLI AND IDIONE MENEGHEL1
In this note, we prove an equilibrium existence theorem for games with discontinuous payoffs and convex and compact strategy spaces. It generalizes the classical result
of Reny (1999), as well as the recent paper of McLennan, Monteiro, and Tourky (2011).
Our conditions are simple and easy to verify. Importantly, examples of spatial location
models show that our conditions allow for economically meaningful payoff discontinuities, that are not covered by other conditions in the literature.
KEYWORDS: Nash equilibrium, discontinuous payoffs, better reply security, location
games.
1. INTRODUCTION
OUR PURPOSE IS TO STUDY the question of existence of Nash equilibrium in
games in which players’ strategy spaces are convex and compact, and payoffs
are discontinuous. A classical reference for the theory of equilibrium existence
in games with discontinuous payoffs is Reny (1999). Because Reny’s hypotheses are sufficiently general and, at the same time, easy to verify, his results
have been applied to a variety of significant economic problems. Indeed, Reny
(1999) showed that very general multi-unit pay-your-bid auctions have pure
strategy Nash equilibria. Jackson and Swinkels (2005) used better reply security to establish the existence of equilibrium in a large class of private value
auctions, including double auctions. Monteiro and Page Jr. (2008) showed that
the mixed extension of a game in which each seller competes for a buyer of unknown type by offering a catalog of products and pricing is better reply secure,
and thus has a Nash equilibrium. Duggan (2007) derived a condition for a class
of zero-sum games that include spatial models of elections with three voters in
which the main theorem in Reny (1999) can be applied to show existence of
equilibrium in mixed strategies. Barelli, Govindan, and Wilson (2012) showed
existence of equilibrium in mixed strategies in general Colonel Blotto games
and special versions of Downs’ model of elections, by verifying that a stronger
condition than better reply security holds true in these models.
1
This note is extracted from our previously circulated Barelli and Soza (2009). We thank
Stephen Morris and three anonymous referees for their comments and extensive suggestions.
We would like to thank Phil Reny for invaluable advice throughout this project. We also thank
Rabeè Tourky for help with the proofs in this revision. We thank Andrew McLennan for helpful
guidance regarding the revision. We acknowledge helpful discussions with Guilherme Carmona,
John Duggan, Hari Govindan, Pavlo Prokopovych, Maxwell Stinchcombe, Gábor Virág, and Bob
Wilson.
© 2013 The Econometric Society
DOI: 10.3982/ECTA9125
814
P. BARELLI AND I. MENEGHEL
Reny’s results have been extended in several directions, forming a research
agenda that has been quite active in the last several years. Examples include Bagh and Jofre (2006), Barelli and Soza (2009), Bich (2009), Bich
and Laraki (2012), Carmona (2009, 2011b, 2012), McLennan, Monteiro, and
Tourky (2011), Prokopovych (2011, 2012), Reny (2009), and even a survey of
recent results in Carmona (2011a).
McLennan, Monteiro, and Tourky (2011; henceforth MMT) stands out in
the list above, as it provides a novel way of ensuring existence of a pure strategy
equilibrium in a class of finite location games, where players choose nonempty
subsets of a given set. The weakening of better reply security provided by MMT,
which allows players to use multiple securing strategies, is shown to ensure
existence of equilibrium in the class described above, once the finite game is
realized as an infinite and discontinuous game. Moreover, MMT showed that
the realized game does not satisfy better reply security, and that none of the
known approaches ensuring existence of a pure strategy equilibrium in a finite
game applies. Hence MMT provided not only an abstract extension of Reny
(1999), but also an important application of their ideas.
Our results are more general than the results of Reny (1999) and MMT and
allow for potentially economically meaningful payoff discontinuities not covered by either work. We relax Reny’s requirement for a single securing strategy
and MMT’s multiple securing strategies by allowing for the securing strategies
to vary semicontinuously in response to changes in other players’ strategies.
The flexibility of the securing strategies bodes well for economic applications
where an economic agent is required to respond to changes in the strategies of
the other agents to keep his or her edge in the, say, market considered.
In fact, we provide a class of location models to showcase the practical use
of our approach in applied work. It is an important class of games, as it can be
used to model choices of product lines by competing firms, choices of coverage
of particular areas by competing cell phone companies, choices of neighborhoods/industrial districts, etc. None of the other approaches in the literature
would ensure existence in general. Yet, it is fairly easy to verify our “continuous security” condition, and the construction is informative of the nature of
the strategic interaction involved.
As for the technique of proof, our approach is based on the essential intuition behind the concept of better reply security that securing strategies are to
be robust to the other players’ small deviations. Informally, we use these securing strategies to construct selections of the strict upper contour set of the
players’ preferences, and show that, if these selections are well behaved, then
an equilibrium in pure strategies exists. The simplicity of the approach allows
us to take some steps further. First, and in line with the better reply security
logic, we can allow different players to be activated locally. That is, the securing strategy is to be contained in the strict upper contour set of only the player
being activated locally. Second, these selections are allowed to be general mappings that allow us to exploit a fixed point argument. In particular, we allow for
THE EQUILIBRIUM EXISTENCE PROBLEM
815
upper hemicontinuous selections, consistent with securing strategies that vary
in semicontinuous manners.
Section 2 is the heart of this note. We show that a condition on the payoffs,
named continuous security, is sufficient for existence of equilibrium in games
with convex and compact strategy spaces. We then provide a generalization
of better reply security, and show that it, together with quasiconcavity, implies
continuous security. In Section 3, a few simple examples in the unit square illustrate what kind of discontinuities and lack of quasiconcavity our results allow.
Section 4 presents the class of location models that illustrates the usefulness of
our approach. In particular, they show how the flexibility of continuous security
is required to reflect the underlying strategic problem of the players involved.
Section 5 provides proofs of the results.
2. EXISTENCE OF NASH EQUILIBRIA
Let N be the finite set of players. Each player i ∈ N has a pure strategy set
Xi , which is a nonempty, convex, and compact subset of a Hausdorff locally
convex topological vector space, and a bounded payoff function ui : X → R.
Product sets are endowed with the product topology and we use X to denote
i∈N Xi , and X−i to denote
j=i Xj , with typical element x−i .
Denote by G = (Xi ui )i∈N the normal form game. A pure strategy Nash equilibrium of G is a profile x∗ ∈ X such that ui (x∗ ) ≥ ui (xi x∗−i ) for all xi ∈ Xi ,
and all i ∈ N.
A correspondence ϕ : Y Z between two topological spaces Y and Z is said
to be closed (equivalently, to have closed graph) if its graph Gr(ϕ) = {(x y) ∈
X × Y : y ∈ ϕ(x)} is closed in X × Y . By the closed graph theorem (Aliprantis
and Border (2006, Theorem 17.11)), when Y is compact Hausdorff, ϕ is closed
if and only if it is upper hemicontinuous and closed-valued. For any set K ⊆ X,
the convex hull of K is denoted co K.
The following condition, continuous security, generalizes the condition multiply security presented in MMT.2
×
×
DEFINITION 2.1: A game G = (Xi ui )i∈N is continuously secure at x ∈ X if
there is αx ∈ RN , an open neighborhood Vx of x, and a closed correspondence
ϕx : Vx X with nonempty values such that
(a) ϕxi (y) ⊆ Bi (y αxi ) for every i ∈ N and every y ∈ Vx ,
/ co Bi (y αxi ),
(b) for each y ∈ Vx , there exists i with yi ∈
where Bi (x αxi ) = {yi ∈ Xi : ui (yi x−i ) ≥ αxi }.
A game is continuously secure if it is continuously secure at each x that is not
an equilibrium.
We can now state our main existence theorem. The proof is in Section 5.
2
MMT have an additional term in their definition, the restriction operator X : X Xi . The
extension of our definitions and results to include such operator is obvious.
816
P. BARELLI AND I. MENEGHEL
THEOREM 2.2: A continuously secure game G = (Xi ui )i∈N has a pure strategy
Nash equilibrium.
The difference between continuous security and multiply security in MMT is
simple. Multiply security implies that there exist a finite number of (constant)
robust profitable deviations at a neighborhood of a point that is not an equilibrium. Continuous security is more permissive, as it allows for these robust
profitable deviations to vary semicontinuously in that neighborhood, allowing
thus for an infinite number of profitable deviations as long as they satisfy the
upper hemicontinuity condition.3
Likewise, we can generalize the condition better reply security in Reny (1999).
Whenever better reply security requires constant profitable deviations in a
neighborhood of any point that is not an equilibrium, we allow for general
closed mappings.
Let Γ = Gr(u) be the graph of the game’s vector payoff function, and let Γ
be its closure.
DEFINITION 2.3: A game G = (Xi ui )i∈N is called generalized better reply secure if, whenever (x u) ∈ Γ and x is not an equilibrium, there exist a player i
and a triple {ϕxi Vx αxi } where Vx is an open neighborhood of x, ϕxi : Vx Xi
is a closed correspondence with nonempty values, and αxi > ui , such that
ui (zi y−i ) ≥ αxi for every (y zi ) ∈ Gr(ϕxi ).
Observe that, differently from the analogous conditions in Barelli and Soza
(2009), the correspondences figuring in continuous security and generalized better reply security are not required to be convex-valued. It is straightforward to
adapt the argument in Lemma 2.5 in MMT to establish that the combination
of better reply security and own-strategy quasiconcavity, used in Reny (1999),
implies continuous security.4
PROPOSITION 2.4: A generalized better reply secure and own-strategy quasiconcave game G = (Xi ui )i∈N is continuously secure. In particular, such a game has
a pure strategy Nash equilibrium.
Similarly, one can strengthen the main results of Baye, Tian, and Zhou
(1993), Tian and Zhou (1995), Bagh and Jofre (2006), Carmona (2009), and
Bich (2009). See Barelli and Soza (2009) for details.
3
Multiply security implies the existence of an upper hemicontinuous convex polytope-valued
better reply correspondence. Observe that in locally convex spaces, through the standard
Schauder projection technique, upper hemicontinuous convex-valued correspondences have
polytope-valued approximations. One could use such an approximation idea to provide an alternative proof of our main result. We thank an anonymous referee for pointing this out to us.
4
See Barelli and Soza (2009) and Carmona (2011b).
THE EQUILIBRIUM EXISTENCE PROBLEM
817
3. TWO-PLAYER EXAMPLES
We now present two simple examples in which players choose real numbers
between zero and 1 that illustrate some novel classes of discontinuities and lack
of quasiconcavity that our results allow.
EXAMPLE 3.1: There are two players with strategy sets Xi = [0 1], i = 1 2.
For some integer m ≥ 3, let ε ∈ (0 m1 ). The payoff functions are given by
u1 (x1 x2 ) =
1 if x1 = x2 ,
0 otherwise,
and
⎧
⎪
⎨ 1
u2 (x1 x2 ) = k
⎪
⎩
0
if x2 = 1,
if x1 + ε <
otherwise.
k
= x2 for k = 1 2 m − 1,
m
We can think of it as a game of product quality choices of two firms: firm 1
wants to match the choice of firm 2; firm 2 has some quality levels (e.g., 24 and
3
when m = 4) where it makes a good profit in case it beats firm 1; and also
4
has a somewhat safe quality level x2 = 1. The profile x1 = x2 = 1 is the Nash
equilibrium.
We have x2 ∈ co{y2 ∈ X2 : u2 (x1 y2 ) > u2 (x)} for some profiles x (for in1
)). Therefore, this game is not own-strategy quasiconcave
stance, x = (0 m−1
and the result of Reny (1999) cannot be applied. However, this game is continuously secure, as we show in what follows.
Let f : X → X be given by f (x) = (x2 1), and put fx = f |Vx for any x = (1 1)
and Vx an open neighborhood not including (1 1). Then we have continuous
security: for every profile outside of the diagonal, pick a neighborhood that
does not meet the diagonal; for that neighborhood, player 1 is the player for
which part (b) of continuous security is satisfied (using αx1 = 1/2 and αx2 = 0).
For any profile x in the diagonal, pick an open ball with radius ε2 centered at x
as Vx . Even though we have profiles with x2 ∈ co{y2 ∈ X2 : u2 (x1 y2 ) > u2 (x)},
it is simple to verify that part (b) of the condition is satisfied for player 2, with
αx1 = 0 and αx2 = 1/2. Observe that the securing mapping f requires infinitely
many security strategies, so MMT’s multiply security fails.
EXAMPLE 3.2: There are two players with strategy sets Xi = [0 1], i = 1 2.
Let ki : X → R, li : X → R, and mi : X → R be bounded functions satisfying:
(i) ki (s) > li (s) > mi (s) for every s ∈ [0 1], (ii) ki is non-decreasing and lower
semicontinuous, and (iii) mi upper semicontinuous, i = 1 2. Player i’s payoffs
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P. BARELLI AND I. MENEGHEL
are given by
ui (xi xj ) =
ki (xi ) if xi < xj ,
if xi = xj ,
li (xi )
mi (xj ) if xi > xj .
We have own-strategy quasiconcavity because ki is non-decreasing. For i j =
1 2, i = j, because ki is lower semicontinuous, there exists a continuous function δi : Xj → R+ with xj − δi (xj ) > 0 for all xj ∈ (0 1], δi (xj ) = 0 only if
xj = 0, and ki (xj − δi (xj )) ≥ li (xj ). Let ϕ(x) = (x2 − δ1 (x2 ) x1 − δ2 (x1 )).
For each nonequilibrium x, let Vx be an open neighborhood of x not containing an equilibrium, and set ϕx = ϕ|Vx . Then continuous security is verified:
for a profile x below the diagonal, player 1 is activated by having αx1 with
infy∈Vx k1 (y2 ) ≥ αx1 > supy∈Vx m1 (y2 ) and αx2 = 05 ; similarly, we can activate
player 2 for profiles above the diagonal, and for profiles at the diagonal we can
activate either player.6
This example illustrates the idea described in the Introduction that continuous security allows for different players to be locally activated in a given open
neighborhood. In this example, the neighborhoods Vx are broken up into two
halves: above and below the diagonal.
4. APPLICATIONS
We now move on to a class of location models, where players choose closed
subsets of a given space of locations, and payoffs depend on how compatible
the chosen sets are. Examples of such games abound: choice of coverage of
broadcasting firms, choice of markets, choice of industrial districts, choice of
product lines, choice of neighborhoods, etc. The discontinuities arise because
compatibility of the chosen sets may involve a delicate balance.7
We make use of the following implication of Theorem 2.2.
PROPOSITION 4.1: Let G = (Xi ui )i∈N be an own-strategy quasiconcave game,
with Xi a compact, convex subset of a metrizable locally convex topological vector
space, and ui : X → {0 1} upper semicontinuous. There exists a pure strategy Nash
equilibrium.
PROOF: Let Fi = {x−i ∈ X−i : maxxi ∈Xi ui (x) = 1}. If Fi = ∅, then ui ≡ 0 and
the best response correspondence is constant and equal to Xi . So assume that
5
By lower (resp. upper) semicontinuity of k1 (resp. m1 ), there must exist a neighborhood Vx
for which infy∈Vx k1 (y2 ) > supy∈Vx m1 (y2 ).
6
Observe that the functions ki and mi are allowed to be discontinuous, contrary to the case
considered by Reny (1999), and that the case considered in Bagh and Jofre (2006) is also a special
case. See Barelli and Soza (2009) for a more general analysis.
7
“Good fences make good neighbors.”
THE EQUILIBRIUM EXISTENCE PROBLEM
819
Fi = ∅. Take a sequence xn−i → x−i in Fi . For each xn−i , there corresponds an xni
with ui (xni xn−i ) = 1. Passing to a subsequence, we have (xni xn−i ) → (xi x−i ).
By upper semicontinuity, 1 = lim supn ui (xni xn−i ) ≤ ui (xi x−i ) ≤ 1, so x−i ∈ Fi ,
showing that Fi is a closed set.
Let ϕi : Fi Xi be given by
ϕi (x−i ) = xi : ui (xi x−i ) = 1 Take a sequence (xni xn−i ) → (xi x−i ) with xni ∈ ϕi (xn−i ). Then ui (xni xn−i ) = 1,
and by upper semicontinuity, ui (xi x−i ) = 1. Thus ϕi has closed graph.
Using Theorem 2.4 in Tan and Wu (2002), there is an upper hemicontinuous,
nonempty, and closed-valued extension of ϕi to X−i . Abusing notation, let ϕi
denote such extension.
Let x not be an equilibrium, and let Vx be an open neighborhood of x such
that every y ∈ Vx is not an equilibrium. (Otherwise we would be done.) Let J
be the set of players i for which there is some y ∈ Vx with y−i ∈ Fi . For each
i ∈ J, let αxi = 1/2, and let αix = 0 for i ∈
/ J. Then part (a) of continuous
security is met, while (b) follows from quasiconcavity, so the result follows from
Theorem 2.2.
Q.E.D.
Observe that it is fairly simple to exhibit games satisfying the conditions of
Proposition 4.1 and violating multiply security, better reply security, and the
other conditions in the literature on existence of Nash equilibrium in discontinuous games, for instance, the two-player game with X1 = X2 = [0 1], u1 (x) = 1
if x1 = x2 and 0 otherwise, and u2 (x) = 1 if x2 = 1 − x1 and 0 otherwise. As
such, Proposition 4.1 is of interest in itself. We will, on the other hand, apply it
to specific location models.
4.1. Location Models
Let S be a compact, metric space of “locations.” A strategy xi for player i is
a closed subset of S. We assume that the corresponding space of strategies Xi
is a convex, compact metric subset of a locally convex topological vector space,
whose topology satisfies the following condition: if xni → xi , then for any s ∈ xi ,
there exists a sequence sn with sn ∈ xni and sn → s. For instance, let Δ(S) be the
space of probability measures on S endowed with the weak* topology. Then a
choice of a closed subset of S can be identified as the choice of a probability
measure xi ∈ Δ(S) via xi ’s support. Hence Xi = Δ(S), which is (convex and)
compact metric because S is so, and belongs to the topological vector space
of signed measures, which is Hausdorff and locally convex when endowed with
the weak* topology. As the support correspondence μ → supp μ, for μ ∈ Δ(S),
is lower hemicontinuous (Aliprantis and Border (2006, Theorem 17.14)), the
condition stated above is satisfied.
Let Ci ⊂ S × X−i be the set of locations and profiles of choices of the other
players that are compatible (from the perspective of player i). Assume that Ci
820
P. BARELLI AND I. MENEGHEL
is closed, and let χCi : S × X−i → {0 1} be the (upper semicontinuous) characteristic function of Ci . The payoff of player i is ui : X → {0 1} with
ui (x) = min χCi (s x−i )
s∈xi
Observe that ui is upper semicontinuous: indeed, take xn → x with ui (xn ) = 1,
and s ∈ xi . By the condition on the topology of Xi , there is sn ∈ xni with sn → s.
So (sn xn−i ) ∈ Ci , and hence (s x−i ) ∈ Ci , because Ci is closed. But this means
that ui (x) = 1, as we wanted to show.
Finally, assume that ui is own-strategy quasiconcave. Observe that this is
true with Xi = Δ(S) as above, because if ui (xi x−i ) = ui (xi x−i ) = 1, then
ui (λxi + (1 − λ)xi x−i ) = 1 for any λ ∈ [0 1], as supp(λxi + (1 − λ)xi ) =
supp xi ∪ supp xi .
It follows that the game G = (Xi ui )i∈N satisfies the conditions of Proposition 4.1 and therefore has a pure strategy Nash equilibrium. Our next task is
to provide more details about the structure of the game that are economically
meaningful and compatible with the assumptions. We keep the identification
Xi = Δ(S), so that own-strategy quasiconcavity holds.
4.1.1. Hit-and-Miss
For each i, partition the set N into the nonempty sets Gi of “good” and Bi
of “bad” opponents of player i. Let
Ci = (s x−i ) ∈ S × X−i : ρ(δs xj ) ≤ ε for all j ∈ Gi
and ρ(δs xj ) ≥ η for all j ∈ Bi
for ε > 0 and η > 0, where ρ(· ·) metrizes the weak* topology and δs denotes
the Dirac measure on s. The interpretation is that s is compatible with x−i when
s is close to the sets chosen by the good opponents and far from the sets chosen
by the bad opponents. For instance, individuals in particular social groups may
want to live close to people in similar groups, moving to the suburbs or gated
communities. Firms may want to choose their product lines (or target market)
based on which other firms have chosen that particular product line. And so on.
As ρ(· ·) is jointly continuous (Aliprantis and Border (2006, Lemma 3.31)), the
set Ci so constructed is closed, so the general analysis applies.
Alternatively, we can also capture hit-and-miss using
Ci = (s x−i ) ∈ S × X−i : ρ(δs xj ) ≤ ε for all j ∈ Gi
and d(s supp xj ) ≥ η for all j ∈ Bi where d(· ·) is the metric in S. The interpretation is the same, the difference
/ supp xj , whereas ρ(δs xj ) ≥ η
being that d(s supp xj ) ≥ η ensures that s ∈
does not. This set of compatible pairs (s x−i ) is closed, so the general analysis
applies.
THE EQUILIBRIUM EXISTENCE PROBLEM
821
4.1.2. Exact Hits
Using the notation above, let
Ci = (s x−i ) ∈ S × X−i : xj = δs for at least one j ∈ Gi The interpretation is that s is compatible with x−i if at least one of the good
opponents chooses exactly the singleton subset {s}. Again it is simple to conceive of economic games with such a feature (a location s is compatible with
x−i if at least one individual of i’s group is sure to live there, for instance), and
this set Ci is closed, so the general analysis applies.
We can combine the cases above with exact hits for j ∈ Gi and ρ(s xj ) ≥ η
(or d(s supp xj ) ≥ η) for j ∈ Bi , to obtain closed sets of compatible pairs that
fit the general formulation. Now one wants not only at least one individual of
the group to surely live in s, but also that the individuals from the other groups
be at a safe distance from s.
4.1.3. Finite Space of Locations
In the case that S is finite, there is another way of combining hits and misses:
let
Ci = (s x−i ) ∈ S × X−i : gi (s x−i ) ≥ bi (s x−i ) where gi (s x−i ) is the number of j ∈ Gi with xj = δs and bi (s x−i ) is the number of j ∈ Bi with s ∈ supp xj . The interpretation is that s is compatible with x−i
if there are more exact hits with good opponents than (not necessarily exact)
hits with bad opponents. Because S is finite, we must have supp xj ⊂ supp xnj
whenever xnj → xj . Hence the set Ci is again closed, and the general analysis
applies.8
4.1.4. Endogenous Groups
The partition of N into Gi and Bi was taken as given above. Players themselves may choose to become part of the “good” or “bad” opponents of a particular player. This is formalized by adding an extra dimension to the strategies. Let Xi × Yi be the strategy space of player i, where Xi is as above, and
Yi is a compact, convex metric space. Let Gi : Y−i N and Bi : Y−i N, with
Bi (y−i ) = N \ Gi (y−i ). The “good” (resp. “bad”) opponents at y−i are the players in Gi (y−i ) (resp. Bi (y−i )). The resulting game is G = (Xi × Yi ui )i∈N , with
ui (x y) = min χCi (s x−i y−i )
s∈supp xi
8
Observe that this is the application presented in Section 6 of MMT.
822
P. BARELLI AND I. MENEGHEL
where Ci is the compatible subset of S × X−i × Y−i . One can think of firms
choosing product lines (the set of locations xi ) and marketing strategies simultaneously. The marketing strategy of firm i is captured by the continuous
variable yi . A profile of marketing strategies y−i determines the firms other
than firm i that fit best with firm i.
Observe that own-strategy quasiconcavity is preserved. It is straightforward
to verify that the analogues of the sets Ci in Sections 4.1.2 and 4.1.3 are closed
when Gi is upper hemicontinuous and the ones in Section 4.1.1 are closed when
Gi is a continuous correspondence.9 Again, the general analysis applies.
5. PROOF OF THEOREM 2.2
The following proof borrows some ideas from the main argument in MMT.
We do require Hausdorff and locally convex spaces exactly because we allow
for infinitely many profitable deviations. With only finitely many (or a single
one, as in Reny (1999)), one can get by without the full force of a fixed point
argument in Hausdorff locally convex topological vector spaces.
PROOF OF THEOREM 2.2: Suppose that the game G is continuously secure
and has no equilibrium. For each x ∈ X, there exists {αx Vx φx } satisfying
parts (a) and (b) of the continuous security condition. Because X is regular,
each x has a closed neighborhood V
x ⊂ Vx . Moreover, the cover {V
x }x∈X has
a finite subcover {V
k }k=1K because X is compact. Let {αk φk } be the corresponding pair.
Define the function β : X → R by the product β = i∈N βi , where
×
βi (x) = max αki x∈V
k
and notice that β is upper semicontinuous and finite-valued. Therefore, for
each x ∈ X,
there is a neighborhood Ux such that β(y) ≤ β(x) for all y ∈ Ux
and Ux ⊂ {k : x∈Vk } Vk . Define the correspondence ϕx : Ux X by the product
ϕx = i∈N ϕxi , where
×
ϕxi (y) = φki (y)
for some k such that αki = max{j : x∈Vj } αji . Observe that ϕxi is closed.
9
For instance, the analogous of the first Ci in Section 4.1.1 is
Ci = (s x−i y−i ) : ρ(s xj ) ≤ ε for all j ∈ Gi (y−i ) and ρ(s xj ) ≥ η for all j ∈ Bi (y−i ) THE EQUILIBRIUM EXISTENCE PROBLEM
823
x ⊂ Ux , and the cover
Again, for each x there is a closed neighborhood U
{Ux }x∈X has a finite subcover {U }=1L . Let ϕ be the corresponding correspondence. Let Φ : X X be the correspondence given by
Φ(x) = co
ϕ (x)
}
{∈{1L} : x∈U
Φ is nonempty, convex, and compact-valued by construction. Also, by Aliprantis and Border (2006, Theorem 17.27), it is upper hemicontinuous and hence
closed. Thus, Φ has a fixed point (by the Kakutani–Fan–Glicksberg theorem,
Aliprantis and Border (2006, Corollary 17.55)). To see that it is a contradiction,
} and J = {k ∈ {1 K} : x ∈ V
k }.
take any x and let J = { ∈ {1 L} : x ∈ U
Notice that part (a) of continuous security implies that
ϕ (x) ⊂ B(x αk )
for all ∈ J and all k ∈ J . Thus
Φ(x) ⊂ co B x max αk k∈J
However, part (b) of continuous security implies that, for each k ∈ J , we can
/ co Bi (x αki ) for some i. So Φ cannot have a fixed point.
Q.E.D.
find xi ∈
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Dept. of Economics, University of Rochester, Rochester, NY 14627, U.S.A.;
[email protected]
and
School of Economics, University of Queensland, Brisbane St. Lucia, QLD
4072, Australia; [email protected].
Manuscript received February, 2010; final revision received June, 2012.