Pseudocontinuous functions and existence of Nash equilibria

Journal of Mathematical Economics 43 (2007) 174–183

Pseudocontinuous functions and existenceof Nash equilibria

Jacqueline Morgan a,∗, Vincenzo Scalzo b

a Dipartimento di Matematica e Statistica, Universita di Napoli Federico II, via Cinthia, 80126 Napoli, Italyb Dipartimento di Matematica e Statistica, Universita di Napoli Federico II, via Cinthia, 80126 Napoli, Italy

Received 30 September 2004; received in revised form 26 September 2006; accepted 21 October 2006Available online 4 January 2007

Abstract

In topological spaces, we introduce a new class of functions (pseudocontinuous functions) and we presentsome characterizations and properties. In particular, we show that any preference relation endowed of utilityfunctions is continuous if and only if any utility is pseudocontinuous. A maximum theorem is proved forsuch a class of functions and connections with similar results are investigated. Finally, the existence of Nashequilibria for games with pseudocontinuous payoffs is obtained.© 2006 Elsevier B.V. All rights reserved.

Keywords: Non cooperative game; Nash equilibrium; Pseudocontinuous function; Numerical representation of a prefer-ence; Maximum theorem

1. Introduction

Following the early theorems for non cooperative games (Nash, 1950, 1951; Glicksberg, 1952),the payoffs have to be continuous in order to obtain the existence of Nash equilibria. However,several games as the oligopolies of Bertrand (1883) and Hotelling (1929) have discontinuouspayoffs and several authors have studied the existence of equilibria when the payoffs are notnecessarily continuous. Among others, we remind the reader to Dasgupta and Maskin (1986),Vives (1990), Baye et al. (1993), Tian and Zhou (1995), Cavazzuti (1996), Reny (1999) andLignola and Morgan (2002), where only the lower semicontinuity is relaxed or there are not explicitassumptions on any data. In this paper, we introduce a new sufficient topological condition onthe payoffs (called pseudocontinuity) which is strictly weaker than lower semicontinuity and than

∗ Corresponding author. Tel.: +39 081675008; fax: +39 081675009.E-mail addresses: [email protected] (J. Morgan), [email protected] (V. Scalzo).

0304-4068/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jmateco.2006.10.004

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J. Morgan, V. Scalzo / Journal of Mathematical Economics 43 (2007) 174–183 175

upper semicontinuity and which is equivalent to the continuity of the associate preference relations:if a preference relation is endowed of numerical representations (also called utility functions), thenthe preference is continuous (Eilenberg, 1941; Rader, 1963; Debreu, 1964; Bergstrom, 1975) if andonly if any numerical representation is a pseudocontinuous function. So, pseudocontinuity is thecommon topological property satisfied by a whichever numerical representation of a continuouspreference.

The paper is organized as follows. In Section 2, (upper and lower) pseudocontinuous func-tions are presented together with some characterizations and properties. In particular, it is shownthat every preference relation endowed of utility functions is lower continuous (Eilenberg, 1941;Rader, 1963; Debreu, 1964; Bergstrom, 1975) if and only if every utility is upper pseudocontinu-ous. The relationships between upper (resp. lower) pseudocontinuity and sequential upper (resp.lower) pseudocontinuity (Morgan and Scalzo, 2004) conclude the section. In Section 3, sincethe Berge’s maximum theorem (Berge, 1959) plays a crucial role in the classical proofs of theexistence of Nash equilibria, a maximum theorem is obtained for pseudocontinuous functions; anexample shows that it is not possible to improve further on the result by using explicit assumptionson any data. Then, a new existence result for Nash equilibria is presented in games with pseu-docontinuous payoff functions. Finally, Section 2 investigates the connections between gameshaving pseudocontinuous payoff functions and others classes of games already considered inliterature.

2. Pseudocontinuous functions

In this section, we introduce the class of pseudocontinuous functions together with somecharacterizations and properties.

Definition 2.1. Let Z be a topological space and f be an extended real valued function definedon Z.

• f is said to be upper pseudocontinuous at zo ∈ Z if for all z ∈ Z such that f (zo) < f (z), wehave:

lim supy→zo

f (y) < f (z);

f is said to be upper pseudocontinuous on Z if it is upper pseudocontinuous at zo, for all zo ∈ Z;• f is said to be lower pseudocontinuous at zo ∈ Z if −f is upper pseudocontinuous at zo and f

is said to be lower pseudocontinuous on Z if it is lower pseudocontinuous at zo, for all zo ∈ Z;• f is said to be pseudocontinuous if it is both upper and lower pseudocontinuous.

Any upper (resp. lower) semicontinuous function is also upper (resp. lower) pseudocontinuous.The converse is not true. In fact, in Example 4.1 are presented pseudocontinuous functions whichare neither upper nor lower semicontinuous. Moreover, we note that the class of upper (resp.lower) pseudocontinuous functions is strictly included in the class of transfer upper (resp. lower)continuous functions introduced by Tian and Zhou (1995) (see Example 3.1).

Upper pseudocontinuity is an ordinal property, as shown in the next proposition, where somecharacterizations of upper pseudocontinuity are also given. Here, we say that a propertyP satisfiedby an extended real valued function f is ordinal if φ ◦ f satisfies P, for every strictly increasingfunction φ : R −→ R, where (φ ◦ f )(z) = φ[f (z)] for all z such that f (z) ∈ R. Moreover, in thefollowing proposition we also show that an upper pseudocontinuous function defined on a com-

176 J. Morgan, V. Scalzo / Journal of Mathematical Economics 43 (2007) 174–183

pact set admits maximum points. Similar results could be obtained for lower pseudocontinuousfunctions.

Proposition 2.1. Let f be an extended real valued function defined on a topological space Z.Then, the following statements are equivalent:

(i) f is upper pseudocontinuous on Z;(ii) L = {(z, λ) ∈ Z × f (Z)/f (z) ≥ λ} is closed in Z × f (Z);

(iii) Uλ = {z ∈ Z/f (z) ≥ λ} is closed for all λ ∈ f (Z).

Moreover, the upper pseudocontinuity is an ordinal property and it guarantees the existence ofmaximum points on compact sets.

Proof. First, we prove that (i) implies (ii). Assume that (zo, λ) ∈ Z × f (Z) and f (zo) < f (z1) =λ. It follows

lim supz→zo

f (z) < f (z1).

Then, there exist a neighborhood I of zo and a neighborhood J of f (z1) such that f (z) < λ′for all z ∈ I and for all λ′ ∈ J , which implies that Z × f (Z)\L is an open set.

Now, we prove that (ii) implies (iii). Let zo �∈ Uf (z1). So (zo, f (z1)) �∈ L and there exist aneighborhood I of zo and a neighborhood J of f (z1) such that f (z) < λ for all z ∈ I and all λ ∈ J .In particular, we have f (z) < f (z1) for all z ∈ I and (iii) follows.

Assume that (iii) holds and f (zo) < f (z). If there exists z1 ∈ Z such that f (zo) < f (z1) <

f (z), being zo �∈ Uf (z1) and Uf (z1) closed, there exists a neighborhood I of zo such thatI ∩ Uf (z1) = ∅. So, supy∈I f (y) ≤ f (z1) and lim supy→zo

f (y) ≤ f (z1) < f (z). Otherwise, if]f (z0), f (z)[∩f (Z) = ∅, zo �∈ Uf (z) and Uf (z) closed imply that there exists a neighborhood I

of zo such that f (y) < f (z) for all y ∈ I. But there are not values of f in ]f (zo), f (z)[, hencesupy∈I f (y) ≤ f (zo) < f (z) and then lim supy→zo

f (y) ≤ f (zo) < f (z). Hence f is upper pseu-docontinuous at zo.

Now, let us prove that the upper pseudocontinuity is an ordinal property. Let f be upper pseudo-continuous at zo and φ be a strictly increasing real function. Assume that (φ ◦ f )(zo) < (φ ◦ f )(z).Then f (zo) < f (z) and lim supy→zo

f (y) < f (z). This implies that f (y) < α < f (z) for all y be-longs to some neighborhood I of zo and with a suitable real number α. Hence (φ ◦ f )(y) < φ(α) <

(φ ◦ f )(z) for all y ∈ I and the statement follows.Finally, if Z is compact and f is upper pseudocontinuous on Z, there exist maximum points of f

over Z in light of Theorem 2 in Tian and Zhou (1995) since an upper pseudocontinuous functionis also transfer upper continuous. �

Pseudocontinuity is strongly related to the continuity of preference relations endowed of numer-ical representations. We recall, see Eilenberg (1941), Rader (1963), Debreu (1964) and Bergstrom(1975), that a preference � defined over a topological space Z is lower (resp. upper) continuousif {y/z y} (resp. {y/y z}) is open for any z ∈ Z, where denotes the asymmetric part of�, while � is continuous if it is both lower and upper continuous. A real valued function u de-fined over Z is a numerical representation (or utility function) of � if u(z) ≥ u(y) ⇔ z � y andu(z) > u(y) ⇔ z y. Existence results of numerical representations have been first obtained forcontinuous preferences (see Eilenberg, 1941; Rader, 1963; Debreu, 1964) and then for weaklylower continuous preferences (a concept weaker than lower continuity, see Campbell and Walker,


1990). In the following proposition, we prove that a preference relation � having numerical rep-resentations is lower (resp. upper) continuous if and only if any utility function is upper (resp.lower) pseudocontinuous. So, such a preference � is continuous if and only if any utility is apseudocontinuous function.

Proposition 2.2. Let � be a preference relation on a topological space Z. Assume that � isrepresented by the utility function u. Then, � is lower (resp. upper) continuous if and only if u isupper (resp. lower) pseudocontinuous.

Proof. Assume that � is lower continuous. Let zo and z belong to Z such that u(zo) < u(z). Ifthere exists z′ ∈ Z such that u(z′) ∈]u(zo), u(z)[, then z′ zo. So, there exists a neighborhoodI of zo such that u(y) < u(z′) for all y ∈ I, which implies lim supy→zo

u(y) ≤ u(z′) < u(z). Ifu(Z)∩]u(zo), u(z)[= ∅, being z zo, there exists a neighborhood J of zo such that u(y) < u(z) forall y ∈ J . But there does not exist a value of u between u(zo) and u(z). So, one has u(y) ≤ u(zo) <

u(z) for all y ∈ J and lim supy→zou(y) ≤ u(zo) < u(z). Hence u is upper pseudocontinuous at

zo.Finally, let u be upper pseudocontinuous, z ∈ Z and y ∈ {y/z y}. So, u(y) < u(z) and, in light

of the upper pseudocontinuity of u at y, u(y) < u(z) for all y which belongs to some neighborhoodof y, that is: y is an interior point of {y/z y}. So, � is lower continuous.

Similarly, one can obtain that � is upper continuous if and only if u is lower pseudo-continuous. �

Finally, we investigate the relationships between upper (resp. lower) pseudocontinuous func-tions and sequentially upper (resp. lower) pseudocontinuous functions (introduced in Morganand Scalzo, 2004). We recall that an extended real valued function f, defined on a sequentialconvergence spaces Z, is sequentially upper pseudocontinuous at zo ∈ Z if the following holds

f (zo) < f (z) =⇒{

lim supn→∞

f (zn) < f (z),

∀ zn −→ zo,

while f is sequentially lower pseudocontinuous at zo if −f is sequentially upper pseudocontinuousat zo.

Proposition 2.3. Let f be an extended real valued function defined on a topological space Z. Iff is upper pseudocontinuous, then it is also sequentially upper pseudocontinuous with respectto the convergence structure induced by the topology. Moreover, if Z satisfies the first axiom ofcountability and f is sequentially upper pseudocontinuous, then it is also upper pseudocontinuous.

Proof. The first statement is obvious. Assume that Z is a topological space satisfying the firstaxiom of countability. Let us proof that if f is sequentially upper pseudocontinuous at zo, then itis also pseudocontinuous at zo.

First, we prove that the following property

f (zo) < f (z) =⇒ supzn→zo

lim supn→∞

f (zn) < f (z) (1)

implies the (topological) upper pseudocontinuity. Then, we show that sequential upper pseudo-continuity implies property (1) (and so, it will be equivalent to (1)).

Suppose that f is not upper pseudocontinuous at zo and that the property (1) is satisfied. So,there exists a point z ∈ Z such that f (zo) < f (z) and lim supy→zo

f (y) ≥ f (z). Let (In)n be a


countable local base of zo decreasing with respect to inclusion. Hence, we have

f (z) ≤ lim supy→zo

f (y) ≤ infn∈N

supy∈In

f (y).

Taken ε > 0, there exists a sequence (zεn)n such that zε

n ∈ In and f (z) − ε < f (zεn) for all n.

Since zεn −→ zo, we obtain

f (z) − ε ≤ lim supn→∞

f (zεn) ≤ sup

zn→zo

lim supn→∞

f (zn).

By arbitrary of ε, we obtain f (z) ≤ supzn→zolim supn→∞ f (zn), which is in conflict with (1).

So, the property (1) implies that f is upper pseudocontinuous at zo.Now, we prove that any sequentially upper pseudocontinuous function satisfies the property

(1). Let f be sequentially upper pseudocontinuous at zo and f (zo) < f (z). So, we have

α = supzn→zo

lim supn→∞

f (zn) ≤ f (z).

Assume that α = f (z). If there exists f (z′) ∈]f (zo), f (z)[, then we obtain f (z′) <

lim supn→∞ f (zn) for at least a sequence zn −→ zo, in conflict with the hypothesis of sequentialupper pseudocontinuity of f at zo. Otherwise, let f (Z)∩]f (zo), f (z)[= ∅. f being sequentially up-per pseudocontinuous, for any sequence zn −→ zo one has f (zn) < f (z) for n sufficiently large.So, we obtain f (zn) ≤ f (zo) for n sufficiently large, which implies the following contradiction:

α = supzn→zo

lim supn→∞

f (zn) ≤ f (zo) < f (z) = α.

Therefore α < f (z) and the proof is completed. �A result similar to the above proposition holds for lower pseudocontinuous functions and

sequentially lower pseudocontinuous functions.

Remark 2.1. If f is a strictly monotone function from R to R then f is also pseudocontinuous. Infact, assume that: f is strictly increasing, x, y and z are such that f (x) < f (y) < f (z), yn −→ y.Since x < y < z and taken t1 and t2 such that x < t1 < y < t2 < z, we have x < t1 < yn < t2 < z

for n sufficiently large. So, f (x) < f (t1) ≤ lim infn→∞ f (yn) and lim supn→∞ f (yn) ≤ f (t2) <

f (z), that is: f is pseudocontinuous at y.

3. A maximum theorem with pseudocontinuous functions

The Berge’s maximum theorem (Berge, 1959) is a crucial theoretical tool in several situationsconcerning individual and strategic choices. For example, given an n-player game in strategicform, that is a set of data G = {Y1, . . . , Yn, f1, . . . , fn} where Y1, . . . , Yn are non-empty subsets oftopological vector spaces and f1, . . . , fn are real valued functions defined on Y = Y1 × · · · × Yn,the set of Nash equilibria, that are profiles of strategies y∗ ∈ Y such that fi(y∗

i , y∗−i) ≥ fi(yi, y

∗−i)for any yi ∈ Yi and any i ∈ {1, . . . , n} (Nash, 1950, 1951), coincides with the set of fixed pointsof the so-called aggregate best response set-valued function B defined by

B : y ∈ Y −→ B(y) = ×n

i=1Bi(y−i) ∈ 2Y ,

where, for any i ∈ {1, . . . , n} and any y−i ∈ Y−i = ×j �=iYj

Bi(y−i) ={

yi ∈ Yi/fi(yi, y−i) = supzi∈Yi

fi(zi, y−i)

}.


Hence, existence results for Nash equilibria can be obtained by using fixed point theorems forset-valued functions as the theorem of Glicksberg (1952). This theorem, among others classicalassumptions, requires that the set-valued function has closed graph and non-empty values. So,in order to have that B is a closed graph set-valued function, a central role is played by Berge’smaximum theorem (Berge, 1959), which requires that the payoffs are continuous functions.

In order to obtain the existence of Nash equilibria by using the same arguments when the payoffsare only pseudocontinuous, one needs of a maximum theorem for pseudocontinuous functions.Such a theorem could be obtained as a corollary of Theorem 3 in Tian and Zhou (1995), where acharacterization for the closure of the graph of Bi is given, but we prefer to give in Theorem 3.1a direct and simple proof also to emphasize that pseudocontinuity is an assumption explicit onany data (differently from those in Tian and Zhou, 1995) and it is the common property satisfiedby a whichever numerical representation of a continuous preference relation. Moreover, whilepseudocontinuity guarantees also that Bi(y−i) �= ∅ for all y−i, the class of functions used by Tianand Zhou (1995) in order to have Bi with non-empty values does not guarantee the closure ofthe graph of Bi, as we show in Example 3.1. Finally, we note that the class of pseudocontinuousfunctions is not connected with the class of non continuous functions used for a maximum theoremby Zolezzi (1984) and Lignola and Morgan (1992)—see the functions in Example 4.1.

Theorem 3.1. Let E and F be two non-empty subsets of Hausdorff topological spaces and f bean extended real valued function defined on E × F . If f is pseudocontinuous on E × F and F iscompact, then the set-valued function

M : x ∈ E −→{

z ∈ F/f (x, z) = supv∈F

f (x, v)

}

has closed graph and non-empty values.

Proof. To prove that M has a closed graph is equivalent to prove that for any x ∈ E and anyz ∈ F such that z �∈ M(x), there exist a neighborhood H of x and a neighborhood I of z such thatM(u) ∩ I = ∅ for all u ∈ H .

Suppose that M does not have a closed graph. So there exist x ∈ E and z ∈ F such that z �∈ M(x)and

∀ H ∈ σE(x) ∀ I ∈ σF (z) ∃ u ∈ H such that M(u) ∩ I �= ∅, (2)

where σE(x) denotes the set of all open neighborhood of x and σF (z) denotes the set of all openneighborhood of z.

Since z �∈ M(x), there exists z ∈ F such that f (x, z) < f (x, z). We distinguish the followingtwo cases.

First case: there exists a value f (uo, vo) of f such that

f (x, z) < f (uo, vo) < f (x, z).

From pseudocontinuity of f it follows:

lim sup(u,v)→(x,z)

f (u, v) < lim inf(u,v)→(x,z)

f (u, v). (3)

So, there exist H1, H2 ∈ σE(x), I ∈ σF (z) and a neighborhood J of z such that

f (u, v) < f (u′, v′) (4)


for all u ∈ H1, v ∈ I, u′ ∈ H2, v′ ∈ J . We set H3 = H1 ∩ H2 and in light of (2), there exists

u ∈ H3 and v ∈ F such that v ∈ M(u) ∩ I. Finally, from (4) we obtain

f (u, v) < f (u, v′)

for any v′ ∈ J and we get a contradiction since v is a maximum point of f (u, ·).Second case: there are not values of f in ]f (x, z), f (x, z)[. f being upper pseudocontinuous,

similarly to the proof of Proposition 2.1, one gets

lim sup(u,v)→(x,z)

f (u, v) ≤ f (x, z) < f (x, z).

So, since f is lower pseudocontinuous, we obtain again (3) and the thesis follows as in theprevious case.

Finally, since F is compact, M(x) is non-empty for all x ∈ E in light of Proposition 2.1. �Tian and Zhou (1995) introduced the classes of transfer upper continuous and transfer lower

continuous functions which characterize the existence of maximum and minimum points, respec-tively, over compact topological spaces. In the following example, f is a transfer upper continuousfunction and a lower semicontinuous function but the set-valued function M does not have closedgraph. So, looking at explicit (on any data) sufficient conditions for which the set-valued functionM has closed graph and non-empty values, the upper pseudocontinuity of the objective functioncannot be weakened by using transfer upper continuity, even if one considers a lower semicon-tinuous function.

Example 3.1. Let f : [0, 1] × [1, 2] −→ R defined by

f (x, y) ={

y − 2x if (x, y) ∈ [0, 1[×[1, 2],

−y if (x, y) ∈ {1} × [1, 2].

The function f is transfer upper continuous (see Theorem 2 in Tian and Zhou, 1995) and lowersemicontinuous on [0, 1] × [1, 2] but f is not upper pseudocontinuous at (1, 2) and the set-valuedfunction M is not closed at 1. In fact:

M(x) ={

{2} if x ∈ [0, 1[,

{1} if x = 1.

Finally, we remark that, by using fixed point arguments over the set-valued function B, Theorem3.1 allows to give a direct proof of the existence of Nash equilibria for games having pseudocon-tinuous payoffs. In fact we have the following theorem, whose proof, now obvious, is omitted.Anyway, the following result can be also obtained as a corollary of a theorem of Reny (1999), asshown in the next section.

Theorem 3.2. Let Y1, . . . , Yn be non-empty, compact and convex subset of locally convex Haus-dorff topological vector spaces. For all i ∈ {1, . . . , n}, assume that the function fi is pseudocon-tinuous on Y and fi(·, y−i) is quasi-concave for any y−i ∈ Y−i. Then, the game G has at least aNash equilibrium.

4. Connections with previous existence results for Nash equilibria

As observed in Section 1, many authors have studied the existence of equilibria when thepayoffs are not necessarily continuous functions. However, the results given in Dasgupta and


Maskin (1986), Vives (1990), Cavazzuti (1996) and Lignola and Morgan (2002) did not relaxthe upper semicontinuity of the payoffs, differently from Theorem 3.2. Baye et al. (1993), whoobtained a characterization for the existence of Nash equilibria, Tian and Zhou (1995) and Reny(1999) considered non explicit assumptions on all the data, differently from Theorem 3.2.

Here, we present the relationship between the games having pseudocontinuous payoffsand the better reply secure games, introduced by Reny (1999). We recall that a game G ={Y1, . . . , Yn, f1, . . . , fn} is better reply secure if for every non-equilibrium y∗ and for everyvector u∗ such that the pair (y∗, u∗) belongs to the closure of the graph of the vector functionf = (f1, . . . , fn), some player i has a strategy yi such that fi(yi, y−i) > u∗

i + ε for all deviationy−i in some neighborhood of y∗−i, with a suitable ε > 0.

Proposition 4.1. Let G = {Y1, . . . , Yn, f1, . . . , fn} be a game with pseudocontinuous payoffs.Then, G is better reply secure.

Proof. Let y∗ be a non-equilibrium for G and u∗ be such that (y∗, u∗) belongs to the closure ofthe graph of the vector function f = (f1, . . . , fn). So, there exists some yi ∈ Yi, for some playeri, such that fi(y∗

i , y∗−i) < fi(yi, y

∗−i).Assume that there exists fi(zi, z−i) ∈]fi(y∗

i , y∗−i), fi(yi, y

∗−i)[. Since fi is upper pseudocon-tinuous at (y∗

i , y∗−i) one has

lim sup(vi,v−i)→(y∗

i,y∗

−i)fi(vi, v−i) < fi(zi, z−i). (5)

The pair (y∗, u∗) belonging to the closure of the graph of f, in light of (5), we have

u∗i < fi(zi, z−i). (6)

Now, fi is lower pseudocontinuous at (yi, y∗−i), so

fi(zi, z−i) < lim inf(vi,v−i)→(yi,y

∗−i

)fi(vi, v−i)

and consequently fi(zi, z−i) < fi(yi, v−i)∀ v−i ∈ Iy∗−i

, for at least a neighborhood Iy∗−i

of y∗−i.Finally, better reply security follows from (6).

Assume now that fi(Y )∩]fi(y∗i , y

∗−i), fi(yi, y∗−i)[= ∅. fi is lower pseudocontinuous at

(yi, y∗−i), then

fi(y∗i , y

∗−i) < lim inf

(vi,v−i)→(yi,y∗−i

)fi(vi, v−i).

Since there are not values of fi between fi(y∗i , y

∗−i) and fi(yi, y∗−i), we obtain

fi(y∗i , y

∗−i) < fi(yi, y

∗−i) ≤ lim inf


)fi(vi, v−i). (7)

Again, since (y∗, u∗) belongs to the closure of the graph of f and fi is upper pseudocontinuous,we obtain

u∗i < fi(yi, y

∗−i) ≤ lim inf


)fi(vi, v−i),

which induces the better reply security. �Reny (1999) considered a sufficient condition for better reply security. More precisely, two

properties for games in strategic form were considered, called payoff security and reciprocal uppersemicontinuity and it was shown that these conditions imply better reply security. Now, following


Proposition 4.1, pseudocontinuity on payoffs is a sufficient condition, explicit on any data, forbetter reply security, and it is independent of payoff security and reciprocal upper semicontinuity.In fact, in Example 4.1, a game with pseudocontinuous payoffs which is not reciprocally uppersemicontinuous is considered.

Example 4.1. Let G = {Y1, Y2, f1, f2} be the game such that Y1 = Y2 = [1, 3] and the payoffsare defined by

fi(yi, yj) =

⎧⎪⎨⎪⎩

yj(yi − 2) if (yi, yj) ∈ [1, 2[×[1, 3],

2 if (yi, yj) ∈ {2} × [1, 3],

yi + 2 if (yi, yj) ∈]2, 3] × [1, 3],

where i, j ∈ {1, 2} and i �= j. The functions f1 and f2 are pseudocontinuous on Y1 × Y2 but Gis not reciprocally upper semicontinuous. In fact, the game is reciprocally upper semicontinuous(Reny, 1999) if, whenever the pair (y, u) belongs to closure of the graph of the vector functionf = (f1, f2) and fi(y) ≤ ui for any player i, then one has fi(y) = ui for any i. Now, in thisexample, ((2,2),(4,4)) is a cluster point for the graph of f and fi(2, 2) = 2 < 4 for any i. So, G isnot reciprocally upper semicontinuous.

We conclude pointing out that, in light of Proposition 4.1, Theorem 3.2 can be obtained as acorollary from Theorem 3 in Reny (1999).

Acknowledgment

The authors thank an anonymous referee for his valuable suggestions.

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Pseudocontinuous functions and existence of Nash equilibria

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