# On Games and Equilibria with Coherent Lower Expectations

Authors: Giuseppe De Marco; Maria Romaniello
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Different solution concepts for strategic form games have been
introduced in order to weaken the consistency assumption that
players’ beliefs, about their opponents strategic choices, are correct in equilibrium. The literature has shown that ambiguous
beliefs are an appropriate device to deal with this task. In this
note, we introduce an equilibrium concept in which players do not
know the opponents’ strategies  in their entirety but only thecoherent lower expectations of some random variables that
depend on the actual strategies taken by the others. This
equilibrium concept generalizes the already existing concept of
equilibrium with partially specified probabilities by extending
the set of feasible beliefs and allowing for comparative
probability judgements. We study the issue of the existence of the
equilibrium points in our framework and find  sufficient
conditions  which involve the continuity of coherent lower
expectations and a Slater-like condition for the systems of
inequalities defining beliefs.


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## Body

## 1. Introduction

The concept of Nash equilibrium relies on two fundamental ideas. Firstly, the agents choose their optimal strategies according to the beliefs they have about the strategic choice made by the other players. Secondly, beliefs are consistent; that is, agents have correct expectations about the actual strategies chosen by their opponents. One of the main criticisms to the Nash equilibrium concept has always been the strength of such consistency condition as, in many settings, it is not clear why players must have exactly correct beliefs about each other. Therefore, different solution concepts have been introduced in order to weaken such consistency condition by taking into accountperturbed beliefs. In a random belief equilibrium, [1], players’ beliefs about others’ strategy choices are randomly drawn from a belief distribution that is dispersed around a central strategy profile called the focus. From a different perspective, other solutions concepts are based on the assumption that players have ambiguous beliefs about opponents behavior. A strand of this literature follows theChoquet expected utility approach [2–5]; that is, beliefs are represented by capacities (nonadditive measures) and expectations are expressed in terms of Choquet integrals. In themaxmin expected utility approach [6–9] beliefs are described by sets of probability distributions, so that ambiguity averse agents evaluate these sets by the worst feasible expectations. The approach proposed by Lehrer [9] is based on the partial specification of players’ actions; that is, players’ beliefs about the strategic choice of their opponents are given by partially specified probabilities. More precisely, players do not know the mixed strategy profile chosen by their opponents in their entirety but only the expectations of some (specified) random variables that depend on the actual choice taken by the others. This is the case, for instance, of players who are able to assess the probability of some subsets of pure strategies but do not know in which way these probabilities are distributed within those subsets.We point out in this paper that partially specified probabilities are a particular case ofcoherent lower expectations which, in turn, emerge as a key concept in the literature onimprecise probabilities; as a consequence, equilibria with partially specified probabilities can be immediately generalized to a concept of equilibrium in which each agent knows the coherent lower expectations, instead of the expectations, of the random variables which are specified. This is done in the present paper by introducing the so-called concept ofequilibrium with coherent lower expectations. At first sight, this concept is similar to Lehrer’s one as ambiguity stems from the actual strategies played but, at the same time, it differs from Lehrer’s concept as the true strategy profile is not necessarily contained in the belief of every player. Most importantly, it is more flexible than Lehrer’s one since lower expectations can model comparative probability judgements, such as “a given event is at least as probable as …,” which cannot always be reconducted to partially specified probabilities. This is the case, for instance, of players who are able to assess that the probability of some subsets of pure strategies is at least as probable as a given value which might depend on the strategy profile.Although the features of our concept of equilibrium are rather general, this is not reflected in very restrictive assumptions for the existence of the corresponding equilibrium points. This is our main result: equilibria with coherent lower expectations exist, provided that coherent lower expectations are continuous and a Slater-like condition for the beliefs is satisfied: the system of linear inequalities which defines the beliefs of each player has interior points belonging to the simplex. More precisely, we firstly point out that an existence result is obtained rather easily once it is shown that equilibria with coherent lower expectations have an equivalent formulation as equilibria of games under ambiguous belief correspondences [10]; in particular, we apply the existence result for this concept as presented in De Marco and Romaniello [11] (Theorem 6). However, this approach does not allow having explicit sufficient conditions on the founding concepts of the particular model studied in this paper, which are the coherent lower expectation functions and the set of specified random variables. Our main result, which is presented in Proposition 7, is precisely to find conditions on such founding concepts in order that the corresponding game under ambiguous belief correspondences satisfies the existence theorem in De Marco and Romaniello [11]. A final example shows that Proposition 7 is useful also for applications since it turns out that it is much more simple to check whether the sufficient conditions in Proposition 7 are satisfied than looking at the assumptions of the general existence theorem in De Marco and Romaniello [11].As a final remark, the paper by Groes et al. [8] proposes an approach similar to ours and that of Lehrer [9] since expectation about opponents’ strategic choices is described by lower probabilities. However, Walley [12] points out at page 134 that “coherent lower expectations are more general and more informative than coherent lower probabilities,”1 so that our concept generalizes also the one in Groes et al. [8].The paper is organized as follows. In Section2, we define the model and the equilibrium concept; then, we relate it to the equilibrium concepts in De Marco and Romaniello [10] and Lehrer [9]. Section 3 is devoted to the equilibrium existence theorem.


## 2. Ambiguous Beliefs and Equilibria

### 2.1. The Equilibrium Concept

We consider a finite set of playersI={1,…,n}; for every player i, Ψi={ψi1,…,ψiki} is the (finite) pure strategy set of player i, Ψ=∏i∈I‍Ψi and Ψ-i=∏j≠i‍Ψj. Denote with Xi the set of mixed strategies of player i, where each strategy xi∈Xi is a vector xi=(xi(ψi))ψi∈Ψi∈R+ki such that ∑ψi∈Ψi‍xi(ψi)=1. Denote with X=∏j=1n‍Xj and with X-i=∏j≠i‍Xj, then each mixed strategy profile x∈X can also be denoted by x=(xi,x-i), where xi∈Xi and x-i∈X-i. Finally, we assume that each player i is endowed with a payoff function fi:Ψ→R.Agents have ambiguous expectations about the strategy choice of their opponents. In particular, the information available to playeri about player j’s strategic choice is given by the coherent lower expectations (see Walley [12] and references therein) of specified random variables over Ψj. We denote with Yij the set of the random variables over Ψj which are specified to player i. More precisely, thecoherent lower expectations of player i about player j’s strategic choice are assigned by a function Pij:Yij×Xj→R provided that (1)PijYij,xj=minyj∈Kijxj⁡EyjYij,where Eyj[Yij] is the expectation of Yij under yj, that is, Eyj[Yij]=∑d=1kj‍yj(ψjd)Yij(ψjd), and the set-valued map Kij:Xj⇝Xj is defined by (2)Kijxj=yj∈Xj∣EyjYij⩾PijYij,xj∀Yij∈Yij∀xj∈Xj.In this case, given a mixed strategy xj of player j, player i does not know xj in its entirety but knows the coherent lower expectations Pij(Yij,xj) of each Yij∈Yij and the ambiguous belief of player i about player j’s strategy choice is the set Kij(xj) of all player j’s mixed strategies that are consistent with the coherent lower probability Pij(Yij,xj).Then, the utility function of playeri is defined by (3)Uix=minyjj≠i∈∏j≠iKijxj⁡xiψi∏j≠iyjψjfψ∀x∈X.So, we can consider the following strategic form game: (4)Γ=I;Xii∈I,Uii∈I.Hence, we introduce the following.Definition 1.
LetYij be the set of random variables defined over Ψj whose coherent lower expectations are specified to player i, for every pair (i,j). Then, any Nash equilibrium x∈X of the corresponding game Γ (defined by (4)) is said to be anequilibrium with coherent lower expectations with respect to the sets (Yij)i,j; that is, x∈X is an equilibrium with coherent lower expectations if, for every player i, (5)Uixi,x-i⩾Uixi′,x-i∀xi′∈Xi.


### 2.2. The Equivalent Formulation under Ambiguous Belief Correspondences

Now we show that equilibria with coherent lower expectations have an equivalent formulation as equilibria in games under ambiguous beliefs correspondences as firstly introduced in De Marco and Romaniello [10]2. This formulation is key because it allows applying directly the existence result in De Marco and Romaniello [11] to obtain an existence result for equilibria with coherent lower expectations3.Denote withΩ={(f1ψ,…,fnψ)∣ψ∈Ψ} the set of all the outcomes of the game and denote with P the set of all the probability distributions over Ω. Define the ambiguous belief correspondence over outcomes Bi:X⇝P as follows: (6)ϱ∈Bix⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ.Assume that each playeri has payoff Fi:X→R defined by (7)Fix=minϱ∈Bix⁡Eiϱ∀x∈X,where Ei[ϱ]=∑ψ∈Ψ‍ϱ(ψ)fi(ψ); then we consider the following game under ambiguous beliefs correspondences B=(B1,…,Bn) and pessimistic players: (8)ΓB=I;Xii∈I;Fii∈I.This game is a classical strategic form game and we callequilibria under ambiguous belief correspondencesBi the classical Nash equilibria of ΓB. That is, (xi,x-i) is an equilibrium under ambiguous belief correspondences Bi if, for every player i, (9)Fixi,x-i⩾Fixi′,x-i∀xi′∈Xi.It immediately follows from the definition that a strategy profile x is an equilibrium with coherent lower expectations if and only if it is an equilibrium under ambiguous belief correspondences defined by (6).


### 2.3. Equilibria with Partially Specified Probabilities

This subsection highlights the relation of our equilibrium notion with respect to Lehrer’s concept ofequilibrium with partially specified probabilities. Let YijL be the set of random variables defined over Ψj whose expectations are specified to player i; that is, given a mixed strategy xj, player i does not know xj in its entirety but knows the expectation of each Yij∈YijL. Then, the ambiguous belief KijL(xj) of player i about player j’s strategy choice is the set of all player j’s mixed strategies that are consistent with xj and YijL in the following way: (10)KijLxj=yj∈Xj∣EyjYij=ExjYij∀Yij∈YijL,where Eyj[Yij] and Exj[Yij] represent the expectation of Yij with respect to yj and xj, respectively. Then, the utility function of player i is constructed as (11)UiLx=minyjj≠i∈∏j≠iKijLxj⁡xiψi∏j≠iyjψjfψ∀x∈Xand the game is (12)ΓL=I;Xii∈I,UiLi∈I.Definition 2 (see Lehrer [9]).
LetYij be the set of random variables defined over Ψj whose expectations are specified to player i, for every pair (i,j). Then, any Nash equilibrium x∈X of the corresponding game ΓL (defined by (12)) is said to be anequilibrium with partially specified probabilities with respect to the sets (Yij)i,j.It is clear that equilibria with partially specified probabilities are a special case of equilibria with coherent lower expectations. More precisely, letx be an equilibrium with partially specified probabilities with respect to the sets YijL for every pair (i,j) and let KijL(xj) be the corresponding set of beliefs as defined in (10). Now, let Yij be the set of random variables defined as follows: (13)Yij∈Yij⟺Yij∈YijLor-Yij∈YijL;then, for every xj∈Xj, let the coherent lower probabilities Pij(·,xj):Yij→R be defined by Pij(Yij,xj)=Exj[Yij] for all Yij∈Yij. It follows that the corresponding set of beliefs Kij(xj) defined by (2) coincides with KijL(xj), so that equilibria with partially specified equilibria are equilibria with coherent lower expectations.Remark 3.
We have shown that partially specified probabilities can be reconducted to coherent lower expectations. Conversely, lower expectations can model comparative probability judgements which cannot always be described by partially specified probabilities; as already noticed in the Introduction, this is the case, for instance, of judgements like “… a given event is at least as probable as ….” The advantages and drawbacks of coherent lower expectations are studied extensively in the literature (see, e.g., Walley [12] and references therein); here we just point out that, in our case, coherent lower expectations can model situations in which players are able to assess that the probability of some subsets of pure strategies is at least as probable as a given value which might depend on the strategy profile. Consider the following simple example: suppose that an agent, say i, can only perceive the minimum probability under which his opponent, say j, will play each pure strategy. This means that player i has the following beliefs about player j’s strategic choice: (14)Kijxj=yj∈Xj∣yjψjd⩾mind∈1,…,kj⁡xjψjd∀d∈1,…,kj∀xj∈Xj.Now, the previous belief Kij(xj) comes out from coherent lower expectations by setting, in formula (2), (15)Yij=χψjd∣d=1,…,kj,where each χ(ψjd) is the characteristic function of the event {ψjd} and (16)Pijχψjd,xj=mind∈1,…,kj⁡xjψjd.


## 2.1. The Equilibrium Concept

We consider a finite set of playersI={1,…,n}; for every player i, Ψi={ψi1,…,ψiki} is the (finite) pure strategy set of player i, Ψ=∏i∈I‍Ψi and Ψ-i=∏j≠i‍Ψj. Denote with Xi the set of mixed strategies of player i, where each strategy xi∈Xi is a vector xi=(xi(ψi))ψi∈Ψi∈R+ki such that ∑ψi∈Ψi‍xi(ψi)=1. Denote with X=∏j=1n‍Xj and with X-i=∏j≠i‍Xj, then each mixed strategy profile x∈X can also be denoted by x=(xi,x-i), where xi∈Xi and x-i∈X-i. Finally, we assume that each player i is endowed with a payoff function fi:Ψ→R.Agents have ambiguous expectations about the strategy choice of their opponents. In particular, the information available to playeri about player j’s strategic choice is given by the coherent lower expectations (see Walley [12] and references therein) of specified random variables over Ψj. We denote with Yij the set of the random variables over Ψj which are specified to player i. More precisely, thecoherent lower expectations of player i about player j’s strategic choice are assigned by a function Pij:Yij×Xj→R provided that (1)PijYij,xj=minyj∈Kijxj⁡EyjYij,where Eyj[Yij] is the expectation of Yij under yj, that is, Eyj[Yij]=∑d=1kj‍yj(ψjd)Yij(ψjd), and the set-valued map Kij:Xj⇝Xj is defined by (2)Kijxj=yj∈Xj∣EyjYij⩾PijYij,xj∀Yij∈Yij∀xj∈Xj.In this case, given a mixed strategy xj of player j, player i does not know xj in its entirety but knows the coherent lower expectations Pij(Yij,xj) of each Yij∈Yij and the ambiguous belief of player i about player j’s strategy choice is the set Kij(xj) of all player j’s mixed strategies that are consistent with the coherent lower probability Pij(Yij,xj).Then, the utility function of playeri is defined by (3)Uix=minyjj≠i∈∏j≠iKijxj⁡xiψi∏j≠iyjψjfψ∀x∈X.So, we can consider the following strategic form game: (4)Γ=I;Xii∈I,Uii∈I.Hence, we introduce the following.Definition 1.
LetYij be the set of random variables defined over Ψj whose coherent lower expectations are specified to player i, for every pair (i,j). Then, any Nash equilibrium x∈X of the corresponding game Γ (defined by (4)) is said to be anequilibrium with coherent lower expectations with respect to the sets (Yij)i,j; that is, x∈X is an equilibrium with coherent lower expectations if, for every player i, (5)Uixi,x-i⩾Uixi′,x-i∀xi′∈Xi.


## 2.2. The Equivalent Formulation under Ambiguous Belief Correspondences

Now we show that equilibria with coherent lower expectations have an equivalent formulation as equilibria in games under ambiguous beliefs correspondences as firstly introduced in De Marco and Romaniello [10]2. This formulation is key because it allows applying directly the existence result in De Marco and Romaniello [11] to obtain an existence result for equilibria with coherent lower expectations3.Denote withΩ={(f1ψ,…,fnψ)∣ψ∈Ψ} the set of all the outcomes of the game and denote with P the set of all the probability distributions over Ω. Define the ambiguous belief correspondence over outcomes Bi:X⇝P as follows: (6)ϱ∈Bix⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ.Assume that each playeri has payoff Fi:X→R defined by (7)Fix=minϱ∈Bix⁡Eiϱ∀x∈X,where Ei[ϱ]=∑ψ∈Ψ‍ϱ(ψ)fi(ψ); then we consider the following game under ambiguous beliefs correspondences B=(B1,…,Bn) and pessimistic players: (8)ΓB=I;Xii∈I;Fii∈I.This game is a classical strategic form game and we callequilibria under ambiguous belief correspondencesBi the classical Nash equilibria of ΓB. That is, (xi,x-i) is an equilibrium under ambiguous belief correspondences Bi if, for every player i, (9)Fixi,x-i⩾Fixi′,x-i∀xi′∈Xi.It immediately follows from the definition that a strategy profile x is an equilibrium with coherent lower expectations if and only if it is an equilibrium under ambiguous belief correspondences defined by (6).


## 2.3. Equilibria with Partially Specified Probabilities

This subsection highlights the relation of our equilibrium notion with respect to Lehrer’s concept ofequilibrium with partially specified probabilities. Let YijL be the set of random variables defined over Ψj whose expectations are specified to player i; that is, given a mixed strategy xj, player i does not know xj in its entirety but knows the expectation of each Yij∈YijL. Then, the ambiguous belief KijL(xj) of player i about player j’s strategy choice is the set of all player j’s mixed strategies that are consistent with xj and YijL in the following way: (10)KijLxj=yj∈Xj∣EyjYij=ExjYij∀Yij∈YijL,where Eyj[Yij] and Exj[Yij] represent the expectation of Yij with respect to yj and xj, respectively. Then, the utility function of player i is constructed as (11)UiLx=minyjj≠i∈∏j≠iKijLxj⁡xiψi∏j≠iyjψjfψ∀x∈Xand the game is (12)ΓL=I;Xii∈I,UiLi∈I.Definition 2 (see Lehrer [9]).
LetYij be the set of random variables defined over Ψj whose expectations are specified to player i, for every pair (i,j). Then, any Nash equilibrium x∈X of the corresponding game ΓL (defined by (12)) is said to be anequilibrium with partially specified probabilities with respect to the sets (Yij)i,j.It is clear that equilibria with partially specified probabilities are a special case of equilibria with coherent lower expectations. More precisely, letx be an equilibrium with partially specified probabilities with respect to the sets YijL for every pair (i,j) and let KijL(xj) be the corresponding set of beliefs as defined in (10). Now, let Yij be the set of random variables defined as follows: (13)Yij∈Yij⟺Yij∈YijLor-Yij∈YijL;then, for every xj∈Xj, let the coherent lower probabilities Pij(·,xj):Yij→R be defined by Pij(Yij,xj)=Exj[Yij] for all Yij∈Yij. It follows that the corresponding set of beliefs Kij(xj) defined by (2) coincides with KijL(xj), so that equilibria with partially specified equilibria are equilibria with coherent lower expectations.Remark 3.
We have shown that partially specified probabilities can be reconducted to coherent lower expectations. Conversely, lower expectations can model comparative probability judgements which cannot always be described by partially specified probabilities; as already noticed in the Introduction, this is the case, for instance, of judgements like “… a given event is at least as probable as ….” The advantages and drawbacks of coherent lower expectations are studied extensively in the literature (see, e.g., Walley [12] and references therein); here we just point out that, in our case, coherent lower expectations can model situations in which players are able to assess that the probability of some subsets of pure strategies is at least as probable as a given value which might depend on the strategy profile. Consider the following simple example: suppose that an agent, say i, can only perceive the minimum probability under which his opponent, say j, will play each pure strategy. This means that player i has the following beliefs about player j’s strategic choice: (14)Kijxj=yj∈Xj∣yjψjd⩾mind∈1,…,kj⁡xjψjd∀d∈1,…,kj∀xj∈Xj.Now, the previous belief Kij(xj) comes out from coherent lower expectations by setting, in formula (2), (15)Yij=χψjd∣d=1,…,kj,where each χ(ψjd) is the characteristic function of the event {ψjd} and (16)Pijχψjd,xj=mind∈1,…,kj⁡xjψjd.


## 3. Equilibrium Existence

### 3.1. Preliminaries on Set-Valued Maps

We start by recalling well-known definitions and results on set-valued maps which we use below. Following Aubin and Frankowska [13]4, recall that if Z and Y are two metric spaces and C:Z⇝Y is a set-valued map, then (17)(i)Liminf⁡z→z′⁡Cz=y∈Y∣limz→z′⁡dy,Cz=0,(ii)Limsup⁡z→z′⁡Cz=y∈Y∣Liminf⁡z→z′⁡dy,Cz=0and Liminfz→z′C(z)⊆C(z′)⊆Limsupz→z′C(z). Moreover, we introduce the following.Definition 4.
Given the set-valued mapC:Z⇝Y, then(i)
Cislower semicontinuous in z′ if C(z′)⊆Liminfz→z′C(z); that is, C is lower semicontinuous in z′ if for every y∈C(z′) and every sequence (zν)ν∈N converging to z′ there exists a sequence (yν)ν∈N converging to y such that yν∈C(zν) for every ν∈N; moreover, C is lower semicontinuous in Z if it is lower semicontinuous for all z′ in Z;(ii)
C isclosed in z′ if Limsupz→z′C(z)⊆C(z′); that is, C is closed in z′ if, for every sequence (zν)ν∈N converging to z′ and every sequence (yν)ν∈N converging to y such that yν∈C(zν) for every ν∈N, it follows that y∈C(z′); moreover, C is closed in Z if it is closed for all z′ in Z;(iii)
C isupper semicontinuous in z′ if for every open set U such that C(z′)⊆U there exists η>0 such that C(z)⊆U for all z∈BZ(z′,η)={ζ∈Z∣ζ-z′<η};(iv)
C iscontinuous (in the sense of Painlevé-Kuratowski) in z′ if it is lower semicontinuous and upper semicontinuous in z′.Finally, recall the following result: if Z is closed, Y is compact and the set-valued map C:Z⇝Y has closed values; then, C is upper semicontinuous in z∈Z if and only if C is closed in z 5.The next definition will also be used.Definition 5.
LetZ be a convex set; then the set-valued map C:Z⇝Y is said to be concave if (18)tCz¯+1-tCz^⊆Ctz¯+1-tz^∀z¯,z^∈Z,∀t∈0,1,while it isconvex6 if (19)Ctz¯+1-tz^⊆tCz¯+1-tCz^∀z¯,z^∈Z,∀t∈0,1.


### 3.2. The Existence Theorem

As a direct application of Theorem 3.6 in De Marco and Romaniello [11] we have the following.Theorem 6.
Assume that, for every playeri,(i)
Bi is a continuous set-valued map with not empty and closed images for every x∈X;(ii)
Bi(·,x-i) is a convex set-valued map in Xi for every x-i∈X-i.Then, the game ΓB as defined in (4) has at least an equilibrium.Proof.
The proof follows directly by applying Theorem 3.6 in De Marco and Romaniello [11], since the utility function in (7) is a particular case of variational preferences in which the index of ambiguity aversion is identically equal to 0.The previous theorem does not allow understanding which are the explicit conditions that must be imposed on the grounding concepts of the model (coherent lower probabilities and specified random variables) in order to have the existence of equilibria. The next proposition is the main contribution of this paper as it gives explicit sufficient conditions on the coherent lower probabilitiesPij and on the specified random variables in Yij which guarantee that the assumptions of the previous theorem hold, so that equilibria with coherent lower expectations exist. Moreover, Example 8 will show that the next proposition is useful also in the applications as it makes it more simple to check whether equilibria exist.Proposition 7.
LetKij be defined by (2), for every j≠i, and let the corresponding Bi be defined by (6); then the following results hold. (1)
Assume that, for every playerj≠i,
(i)
the coherent lower probability functionPij(Yij,·) is continuous in the point xj∈Xj for every Yij∈Yij,(ii)
there existsyj∈Kij(xj) such that (20)yjψjd>0∀d∈1,…,kj,EyjYij>PYij,xj∀Yij∈Yij.Then, the belief correpspondenceBi is a continuous set-valued map with a not empty and closed image in the point x=(x1,…,xn)∈X. Hence, if (i) and (ii) hold for every j≠i and for every xj∈Xj, then Bi is a continuous set-valued map with not empty and closed images for every x∈X.(2)
Bi(·,x-i) is a convex set-valued map in Xi, for every x-i∈X-i.Proof.
First of all recall that(21)ϱ∈Bix⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ.The assumptions imply that Kij(xj)≠∅ so Bi(x)≠∅ for every x∈X.
Step
1 (
B
i
is lower semicontinuous). Let |Yij|=γj be the cardinality of Yij; it follows that Kij(xj) is completely defined by the following system of kj+γj linear inequalities in the kj unknowns yj=(yj(ψj1),…,yj(ψjkj)): (22)yjψjd⩾0∀d∈1,…,kjEyjYij⩾PYij,xj∀Yij∈Yijand by the linear equality (23)yjψj1+⋯+yjψjkj=1.It follows immediately that Kij(xj) is a closed set and therefore compact set.
Now, denote withC the (kj+γj)×kj matrix of the coefficients in system (22) and with 0kj the row null vector of dimension kj. Let b(xj) be the (kj+γj)-dimensional row vector defined as follows: (24)bxj=0kj,PijYij,xjYij∈Yij.If yjT and bT(xj) are the transpose of yj and b(xj), respectively, then system (22) can be denoted as follows: CyjT≧bT(xj). Denote also with 1kjyjT=1 the linear equation (23). Theorem 2 in Robinson [14] tells that if a system of linear equalities and inequalities is such that the coefficient matrix of the equalities has full rank and there exists a feasible point satisfying all the strict inequalities, then the system isstable as defined at page 755 in Robinson [14]7. This is our case, because assumption (ii) guarantees that there exists a vector y¯j such that (25)Cy¯jT>bTxj,1kjy¯jT=1,so the system is stable which, in our case, means that for every vector yj satisfying CyjT≧bT(xj) and 1kjyjT=1 and every ε>0 there exists δ(ε)>08 such that for every (kj+γj)-dimensional row vector b~ with b~-b(xj)<δ(ε) there exists y~j satisfying (a) Cy~jT≧b~T and 1kjy~jT=1 and (b) y~j-yj<ε.
SoKij is a lower semicontinuous set-valued map in xj. In fact, let (xj,ν)ν∈N be a sequence converging to xj and yj∈Kij(xj). For every ν∈N, argminz∈Kij(xj,ν)z-yj is not empty since Kij(xj,ν) is not empty and compact. Let (yj,ν)ν∈N be a sequence such that yj,ν∈argminz∈Kij(xj,ν)z-yj for every ν∈N; we show that yj,ν→yj. Since Pij(Yi,j,·) is continuous in xj for every Yij∈Yij, then the vector valued function b(·) defined by (24) is continuous. So, given δ(ε)>0, there exists ν¯ such that for every ν⩾ν¯ it follows that b(xj,ν)-b(xj)<δ(ε). Therefore, since Robinson’s [14] stability holds, then for every ε>0 there exists ν¯ such that for every ν⩾ν¯ there exists y~j,ν∈Kij(xj,ν) satisfying y~j,ν-yj<ε. Having yj,ν-yj⩽y~j,ν-yj<ε, we get that for every ε>0 there exists ν¯ such that for every ν⩾ν¯ it follows that yj,ν-yj<ε. So yj,ν→yj and Kij is lower semicontinuous in xj. Moreover, if (i) and (ii) are satisfied for every xj∈Xj, then Kij is lower semicontinuous in Xj.
Now, letϱ∈Bi(x) be the probability distribution defined as in (21) by (26)ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ,withyj∈Kijxjj≠i;let (xν)ν∈N be a sequence converging to x. Since for every j≠i, Kij is lower semicontinuous in xj, then there exists a sequence (yj,ν)ν∈N converging to yj with yj,ν∈Kij(xj,ν) for every ν∈N. Define ϱν as follows: (27)ϱνψ=xi,νψi∏j≠iyj,νψj∀ψ∈Ψ.It follows that ϱν→ϱ and ϱν∈Bi(xν) for every ν∈N. So Bi is lower semicontinuous in x. Moreover, if (i) and (ii) are satisfied for every x∈X, then Bi is lower semicontinuous in X.
Step
2 (
B
i
is upper semicontinuous with closed images). Let (xj,ν)ν∈N be a sequence converging to xj and let (yj,ν)ν∈N be a sequence converging to yj, with yj,ν∈Kij(xj,ν) for every ν∈N. We show that yj∈Kij(xj). In fact, from Cyj,νT≧bT(xj,ν) and 1kjyj,νT=1 and the continuity of b(·) we get CyjT≧bT(xj) and 1kjyjT=1 by taking the limit as ν→∞. So, yj∈Kij(xj) and Kij is closed in xj. Moreover, if (i) and (ii) are satisfied for every xj∈Xj, then Kij is closed in Xj.
Now, let(xν)ν∈N be a sequence converging to x and let (ϱν)ν∈N be a sequence converging to ϱ, with ϱν∈Bi(xν) for every ν∈N. By definition (21), ϱν(ψ)=xi,νψi∏j≠i‍yj,ν(ψj) for all ψ∈Ψ, where yj,ν∈Kij(xj,ν) for all j≠i. Since yj,ν→yj and Kij is closed, then yj∈Kij(xj) for every j≠i. So ϱν→ϱ, where ϱ(ψ)=xiψi∏j≠i‍yj(ψj) for all ψ∈Ψ. Therefore, (21) implies that ϱ∈Bi(x) and Bi is closed (hence upper semicontinuous) in x. Moreover, if (i) and (ii) are satisfied for every x∈X, then Bi is closed (hence upper semicontinuous) in X. Finally, if (xν)ν∈N is the constant sequence, with xν=x for every ν∈N, then the closedness of Bi in x implies that the set Bi(x) is closed.
Step
3 (
B
i
(
·
,
x
-
i
)
is a convex set-valued map, for every
x
-
i
∈
X
-
i
). By definition, (28)ϱ∈Bitxi′+1-txi′′,x-i⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=txi′ψi+1-txi′′ψi∏j≠iyjψj∀ψ∈Ψ.It clearly follows that (29)ϱψ=txi′ψi∏j≠iyjψj+1-txi′′ψi∏j≠iyjψj∀ψ∈Ψ.Let ϱ′ and ϱ′′ be defined by (30)ϱ′ψ=xi′ψi∏j≠iyjψj,ϱ′′ψ=xi′′ψi∏j≠iyjψj∀ψ∈Ψ.Then ϱ=tϱ′+(1-t)ϱ′′, ϱ′∈Bi(xi′,x-i), and ϱ′′∈Bi(xi′′,x-i). Since ϱ is arbitrary, then (31)Bitxi′+1-txi′′,x-i⊆tBixi′,x-i+1-tBixi′′,x-i.
It follows easily that the previous inclusion holds for everyt∈[0,1] and every xi′ and xi′′ in Xi. Hence, Bi(·,x-i) is a convex set-valued map in Xi, for every x-i∈X-i.The previous proposition shows that the convexity property of the correspondenceBi follows from the definition while the continuity of Bi follows from the continuity of the functions Pij and from a Slater-like condition9: the system of linear inequalities which defines the beliefs of each player has interior points belonging to the simplex (assumption (ii) in the statement of Proposition 7). In order to better understand how much restrictive this latter condition is in our framework, we give an illustrative example which is built upon the example given in Section 2. It is shown in the example below that (ii) is always satisfied except for one point. However, we prove that in this point the correspondence Bi is continuous as well; then, it follows that (ii) is not a necessary condition for the continuity of Bi.Finally, the example below shows the relevance of Proposition7 in possible applications as it turns out that checking whether (ii) holds is much simpler than checking directly the continuity of Bi.Example 8.
Consider the case of a 2-player game constructed as in the example given in Section2, in which agent i can only perceive the minimum probability under which his opponent j will play each pure strategy; that is (32)Kijxj=yj∈Xj∣yjψjd⩾mind∈1,…,kj⁡xjψjd∀d∈1,…,kj∀xj∈Xj.We recall that Kij(xj) comes out from coherent lower expectations by setting, in formula (2), (33)Yij=χψjd∣d=1,…,kj,where each χ(ψjd) is the characteristic function of the event {ψjd} and (34)Pijχψjd,xj=mind∈1,…,kj⁡xjψjd.Firstly recall that the minimum of a finite number of continuous functions is a continuous function; then it follows thatPij(χ(ψjd),·) is continuous in the entire set Xj.Now, given a pointxj∈Xj, denote with μ=mind∈{1,…,kj}xj(ψjd). There are two possible cases, either μ<1/kj or μ=1/kj. The case μ>1/kj is not possible because it would imply that ∑d=1kj‍xj(ψjd)>1 which is a contradiction since xj is a probability distribution over the pure strategies of player j.Note thatμ=1/kj implies that xj(ψjd)⩾1/kj for every d=1,…,kj. If there exists l∈{1,…,kj} such that xj(ψjl)>1/kj, then ∑d=1kj‍xj(ψjd)>∑d=1kj‍(1/kj)=1, but this is a contradiction since xj is a probability distribution. So, it must be that xj(ψj1)=⋯=xj(ψjkj)=μ. Hence, μ=1/kj if and only if xj(ψj1)=⋯=xj(ψjkj)=μ. Hereafter, we denote with x¯j the strategy in Xj defined by x¯j(ψj1)=⋯=x¯j(ψjkj)=μ.Suppose first thatμ<1/kj; let ε=(1-kjμ)/kj, and define yj(ψjd)=μ+ε for every d∈{1,…,kj}. Since ε>0, then yj(ψjd)>0 for each d; moreover, it immediately follows that ∑d=1kj‍yj(ψjd)=1. Therefore yj=(yj(ψj1),…,yj(ψjkj))∈Xj. By construction, (35)Eyjχψjd=yjψjd>μ=Pijχψjd,xj∀d∈1,…,kj.It follows that condition (ii) in the assumption of Proposition 7 is satisfied for every point xj∈Xj such that μ<1/kj. That is, the corresponding set-valued map Bi is continuous in every point (xi,xj)∈Xi×Xj such that xj≠x¯j.Suppose now thatμ=1/kj; that is xj=x¯j. Let yj∈Kij(x¯j); then yj(ψjd)⩾1/kj for every d∈{1,…,kj}. If there exists l∈{1,…,kj} such that yj(ψjl)>1/kj, then ∑d=1kj‍yj(ψjd)>1, but this is a contradiction since yj∈Xj. Hence Kij(x¯j)={x¯j}, which immediately implies that assumption (ii) in Proposition 7 is not satisfied as x¯j(ψjd)=Pij(χ(ψjd),x¯j) for all d∈{1,…,kj}. Nevertheless, we prove that Kij is continuous in x¯j. Let (xj,ν)ν∈N be a sequence converging to x¯j. By definition, it follows that xj,ν∈Kij(xj,ν) for every ν and hence Kij is lower semicontinuous in x¯j as xj,ν→x¯j. On the other hand, let (xj,ν)ν∈N be a sequence converging to x¯j and (yj,ν)ν∈N a sequence converging to yj such that yj,ν∈Kij(xj,ν) for every ν. Since Xj is compact, then yj∈Xj. Moreover, yj,ν∈Kij(xj,ν) implies that (36)Eyj,νχψjd=yj,νψjd⩾Pijχψjd,xj,ν∀d∈1,…,kj.Taking the limit for ν→∞, it follows from the continuity of the functions Pij that (37)Eyjχψjd=limν→∞⁡Eyj,νχψjd⩾limν→∞⁡Pijχψjd,xj,ν=Pijχψjd,x¯j∀d∈1,…,kj,which implies that yj∈Kij(x¯j) (in particular yj=x¯j). Therefore, Kij is a closed and hence continuous set-valued map in x¯j. Following the same steps of the proof of Proposition 7, we can prove that if Kij is continuous in x¯j then Bi is continuous in (xi,x¯j) for every xi∈Xi. As a final remark, the example emphasizes that it is more simple to check our Slater-like condition rather than applying the definition of upper and lower semicontinuity for correspondences, even in this particular case in which there is a unique point x¯j in which the Slater-like condition is not satisfied and the set-valued map Kij in this point is single valued. It is clear that, in more complex cases, calculations would have been even more demanding.Remark 9.
In order to keep the presentation of Proposition7 more simple, we did not consider the case in which there exist implicit linear equalities in the system of linear inequalities. However, the generalization to this case is rather straightforward. It consists in adding the assumption that the coefficient matrix of the system of the implicit equalities together with the equation in (23) has full rank. The proof is substantially identical to the one given above.


## 3.1. Preliminaries on Set-Valued Maps

We start by recalling well-known definitions and results on set-valued maps which we use below. Following Aubin and Frankowska [13]4, recall that if Z and Y are two metric spaces and C:Z⇝Y is a set-valued map, then (17)(i)Liminf⁡z→z′⁡Cz=y∈Y∣limz→z′⁡dy,Cz=0,(ii)Limsup⁡z→z′⁡Cz=y∈Y∣Liminf⁡z→z′⁡dy,Cz=0and Liminfz→z′C(z)⊆C(z′)⊆Limsupz→z′C(z). Moreover, we introduce the following.Definition 4.
Given the set-valued mapC:Z⇝Y, then(i)
Cislower semicontinuous in z′ if C(z′)⊆Liminfz→z′C(z); that is, C is lower semicontinuous in z′ if for every y∈C(z′) and every sequence (zν)ν∈N converging to z′ there exists a sequence (yν)ν∈N converging to y such that yν∈C(zν) for every ν∈N; moreover, C is lower semicontinuous in Z if it is lower semicontinuous for all z′ in Z;(ii)
C isclosed in z′ if Limsupz→z′C(z)⊆C(z′); that is, C is closed in z′ if, for every sequence (zν)ν∈N converging to z′ and every sequence (yν)ν∈N converging to y such that yν∈C(zν) for every ν∈N, it follows that y∈C(z′); moreover, C is closed in Z if it is closed for all z′ in Z;(iii)
C isupper semicontinuous in z′ if for every open set U such that C(z′)⊆U there exists η>0 such that C(z)⊆U for all z∈BZ(z′,η)={ζ∈Z∣ζ-z′<η};(iv)
C iscontinuous (in the sense of Painlevé-Kuratowski) in z′ if it is lower semicontinuous and upper semicontinuous in z′.Finally, recall the following result: if Z is closed, Y is compact and the set-valued map C:Z⇝Y has closed values; then, C is upper semicontinuous in z∈Z if and only if C is closed in z 5.The next definition will also be used.Definition 5.
LetZ be a convex set; then the set-valued map C:Z⇝Y is said to be concave if (18)tCz¯+1-tCz^⊆Ctz¯+1-tz^∀z¯,z^∈Z,∀t∈0,1,while it isconvex6 if (19)Ctz¯+1-tz^⊆tCz¯+1-tCz^∀z¯,z^∈Z,∀t∈0,1.


## 3.2. The Existence Theorem

As a direct application of Theorem 3.6 in De Marco and Romaniello [11] we have the following.Theorem 6.
Assume that, for every playeri,(i)
Bi is a continuous set-valued map with not empty and closed images for every x∈X;(ii)
Bi(·,x-i) is a convex set-valued map in Xi for every x-i∈X-i.Then, the game ΓB as defined in (4) has at least an equilibrium.Proof.
The proof follows directly by applying Theorem 3.6 in De Marco and Romaniello [11], since the utility function in (7) is a particular case of variational preferences in which the index of ambiguity aversion is identically equal to 0.The previous theorem does not allow understanding which are the explicit conditions that must be imposed on the grounding concepts of the model (coherent lower probabilities and specified random variables) in order to have the existence of equilibria. The next proposition is the main contribution of this paper as it gives explicit sufficient conditions on the coherent lower probabilitiesPij and on the specified random variables in Yij which guarantee that the assumptions of the previous theorem hold, so that equilibria with coherent lower expectations exist. Moreover, Example 8 will show that the next proposition is useful also in the applications as it makes it more simple to check whether equilibria exist.Proposition 7.
LetKij be defined by (2), for every j≠i, and let the corresponding Bi be defined by (6); then the following results hold. (1)
Assume that, for every playerj≠i,
(i)
the coherent lower probability functionPij(Yij,·) is continuous in the point xj∈Xj for every Yij∈Yij,(ii)
there existsyj∈Kij(xj) such that (20)yjψjd>0∀d∈1,…,kj,EyjYij>PYij,xj∀Yij∈Yij.Then, the belief correpspondenceBi is a continuous set-valued map with a not empty and closed image in the point x=(x1,…,xn)∈X. Hence, if (i) and (ii) hold for every j≠i and for every xj∈Xj, then Bi is a continuous set-valued map with not empty and closed images for every x∈X.(2)
Bi(·,x-i) is a convex set-valued map in Xi, for every x-i∈X-i.Proof.
First of all recall that(21)ϱ∈Bix⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ.The assumptions imply that Kij(xj)≠∅ so Bi(x)≠∅ for every x∈X.
Step
1 (
B
i
is lower semicontinuous). Let |Yij|=γj be the cardinality of Yij; it follows that Kij(xj) is completely defined by the following system of kj+γj linear inequalities in the kj unknowns yj=(yj(ψj1),…,yj(ψjkj)): (22)yjψjd⩾0∀d∈1,…,kjEyjYij⩾PYij,xj∀Yij∈Yijand by the linear equality (23)yjψj1+⋯+yjψjkj=1.It follows immediately that Kij(xj) is a closed set and therefore compact set.
Now, denote withC the (kj+γj)×kj matrix of the coefficients in system (22) and with 0kj the row null vector of dimension kj. Let b(xj) be the (kj+γj)-dimensional row vector defined as follows: (24)bxj=0kj,PijYij,xjYij∈Yij.If yjT and bT(xj) are the transpose of yj and b(xj), respectively, then system (22) can be denoted as follows: CyjT≧bT(xj). Denote also with 1kjyjT=1 the linear equation (23). Theorem 2 in Robinson [14] tells that if a system of linear equalities and inequalities is such that the coefficient matrix of the equalities has full rank and there exists a feasible point satisfying all the strict inequalities, then the system isstable as defined at page 755 in Robinson [14]7. This is our case, because assumption (ii) guarantees that there exists a vector y¯j such that (25)Cy¯jT>bTxj,1kjy¯jT=1,so the system is stable which, in our case, means that for every vector yj satisfying CyjT≧bT(xj) and 1kjyjT=1 and every ε>0 there exists δ(ε)>08 such that for every (kj+γj)-dimensional row vector b~ with b~-b(xj)<δ(ε) there exists y~j satisfying (a) Cy~jT≧b~T and 1kjy~jT=1 and (b) y~j-yj<ε.
SoKij is a lower semicontinuous set-valued map in xj. In fact, let (xj,ν)ν∈N be a sequence converging to xj and yj∈Kij(xj). For every ν∈N, argminz∈Kij(xj,ν)z-yj is not empty since Kij(xj,ν) is not empty and compact. Let (yj,ν)ν∈N be a sequence such that yj,ν∈argminz∈Kij(xj,ν)z-yj for every ν∈N; we show that yj,ν→yj. Since Pij(Yi,j,·) is continuous in xj for every Yij∈Yij, then the vector valued function b(·) defined by (24) is continuous. So, given δ(ε)>0, there exists ν¯ such that for every ν⩾ν¯ it follows that b(xj,ν)-b(xj)<δ(ε). Therefore, since Robinson’s [14] stability holds, then for every ε>0 there exists ν¯ such that for every ν⩾ν¯ there exists y~j,ν∈Kij(xj,ν) satisfying y~j,ν-yj<ε. Having yj,ν-yj⩽y~j,ν-yj<ε, we get that for every ε>0 there exists ν¯ such that for every ν⩾ν¯ it follows that yj,ν-yj<ε. So yj,ν→yj and Kij is lower semicontinuous in xj. Moreover, if (i) and (ii) are satisfied for every xj∈Xj, then Kij is lower semicontinuous in Xj.
Now, letϱ∈Bi(x) be the probability distribution defined as in (21) by (26)ϱψ=xiψi∏j≠iyjψj∀ψ∈Ψ,withyj∈Kijxjj≠i;let (xν)ν∈N be a sequence converging to x. Since for every j≠i, Kij is lower semicontinuous in xj, then there exists a sequence (yj,ν)ν∈N converging to yj with yj,ν∈Kij(xj,ν) for every ν∈N. Define ϱν as follows: (27)ϱνψ=xi,νψi∏j≠iyj,νψj∀ψ∈Ψ.It follows that ϱν→ϱ and ϱν∈Bi(xν) for every ν∈N. So Bi is lower semicontinuous in x. Moreover, if (i) and (ii) are satisfied for every x∈X, then Bi is lower semicontinuous in X.
Step
2 (
B
i
is upper semicontinuous with closed images). Let (xj,ν)ν∈N be a sequence converging to xj and let (yj,ν)ν∈N be a sequence converging to yj, with yj,ν∈Kij(xj,ν) for every ν∈N. We show that yj∈Kij(xj). In fact, from Cyj,νT≧bT(xj,ν) and 1kjyj,νT=1 and the continuity of b(·) we get CyjT≧bT(xj) and 1kjyjT=1 by taking the limit as ν→∞. So, yj∈Kij(xj) and Kij is closed in xj. Moreover, if (i) and (ii) are satisfied for every xj∈Xj, then Kij is closed in Xj.
Now, let(xν)ν∈N be a sequence converging to x and let (ϱν)ν∈N be a sequence converging to ϱ, with ϱν∈Bi(xν) for every ν∈N. By definition (21), ϱν(ψ)=xi,νψi∏j≠i‍yj,ν(ψj) for all ψ∈Ψ, where yj,ν∈Kij(xj,ν) for all j≠i. Since yj,ν→yj and Kij is closed, then yj∈Kij(xj) for every j≠i. So ϱν→ϱ, where ϱ(ψ)=xiψi∏j≠i‍yj(ψj) for all ψ∈Ψ. Therefore, (21) implies that ϱ∈Bi(x) and Bi is closed (hence upper semicontinuous) in x. Moreover, if (i) and (ii) are satisfied for every x∈X, then Bi is closed (hence upper semicontinuous) in X. Finally, if (xν)ν∈N is the constant sequence, with xν=x for every ν∈N, then the closedness of Bi in x implies that the set Bi(x) is closed.
Step
3 (
B
i
(
·
,
x
-
i
)
is a convex set-valued map, for every
x
-
i
∈
X
-
i
). By definition, (28)ϱ∈Bitxi′+1-txi′′,x-i⟺∃yjj≠i∈∏j≠iKijxjs.t.ϱψ=txi′ψi+1-txi′′ψi∏j≠iyjψj∀ψ∈Ψ.It clearly follows that (29)ϱψ=txi′ψi∏j≠iyjψj+1-txi′′ψi∏j≠iyjψj∀ψ∈Ψ.Let ϱ′ and ϱ′′ be defined by (30)ϱ′ψ=xi′ψi∏j≠iyjψj,ϱ′′ψ=xi′′ψi∏j≠iyjψj∀ψ∈Ψ.Then ϱ=tϱ′+(1-t)ϱ′′, ϱ′∈Bi(xi′,x-i), and ϱ′′∈Bi(xi′′,x-i). Since ϱ is arbitrary, then (31)Bitxi′+1-txi′′,x-i⊆tBixi′,x-i+1-tBixi′′,x-i.
It follows easily that the previous inclusion holds for everyt∈[0,1] and every xi′ and xi′′ in Xi. Hence, Bi(·,x-i) is a convex set-valued map in Xi, for every x-i∈X-i.The previous proposition shows that the convexity property of the correspondenceBi follows from the definition while the continuity of Bi follows from the continuity of the functions Pij and from a Slater-like condition9: the system of linear inequalities which defines the beliefs of each player has interior points belonging to the simplex (assumption (ii) in the statement of Proposition 7). In order to better understand how much restrictive this latter condition is in our framework, we give an illustrative example which is built upon the example given in Section 2. It is shown in the example below that (ii) is always satisfied except for one point. However, we prove that in this point the correspondence Bi is continuous as well; then, it follows that (ii) is not a necessary condition for the continuity of Bi.Finally, the example below shows the relevance of Proposition7 in possible applications as it turns out that checking whether (ii) holds is much simpler than checking directly the continuity of Bi.Example 8.
Consider the case of a 2-player game constructed as in the example given in Section2, in which agent i can only perceive the minimum probability under which his opponent j will play each pure strategy; that is (32)Kijxj=yj∈Xj∣yjψjd⩾mind∈1,…,kj⁡xjψjd∀d∈1,…,kj∀xj∈Xj.We recall that Kij(xj) comes out from coherent lower expectations by setting, in formula (2), (33)Yij=χψjd∣d=1,…,kj,where each χ(ψjd) is the characteristic function of the event {ψjd} and (34)Pijχψjd,xj=mind∈1,…,kj⁡xjψjd.Firstly recall that the minimum of a finite number of continuous functions is a continuous function; then it follows thatPij(χ(ψjd),·) is continuous in the entire set Xj.Now, given a pointxj∈Xj, denote with μ=mind∈{1,…,kj}xj(ψjd). There are two possible cases, either μ<1/kj or μ=1/kj. The case μ>1/kj is not possible because it would imply that ∑d=1kj‍xj(ψjd)>1 which is a contradiction since xj is a probability distribution over the pure strategies of player j.Note thatμ=1/kj implies that xj(ψjd)⩾1/kj for every d=1,…,kj. If there exists l∈{1,…,kj} such that xj(ψjl)>1/kj, then ∑d=1kj‍xj(ψjd)>∑d=1kj‍(1/kj)=1, but this is a contradiction since xj is a probability distribution. So, it must be that xj(ψj1)=⋯=xj(ψjkj)=μ. Hence, μ=1/kj if and only if xj(ψj1)=⋯=xj(ψjkj)=μ. Hereafter, we denote with x¯j the strategy in Xj defined by x¯j(ψj1)=⋯=x¯j(ψjkj)=μ.Suppose first thatμ<1/kj; let ε=(1-kjμ)/kj, and define yj(ψjd)=μ+ε for every d∈{1,…,kj}. Since ε>0, then yj(ψjd)>0 for each d; moreover, it immediately follows that ∑d=1kj‍yj(ψjd)=1. Therefore yj=(yj(ψj1),…,yj(ψjkj))∈Xj. By construction, (35)Eyjχψjd=yjψjd>μ=Pijχψjd,xj∀d∈1,…,kj.It follows that condition (ii) in the assumption of Proposition 7 is satisfied for every point xj∈Xj such that μ<1/kj. That is, the corresponding set-valued map Bi is continuous in every point (xi,xj)∈Xi×Xj such that xj≠x¯j.Suppose now thatμ=1/kj; that is xj=x¯j. Let yj∈Kij(x¯j); then yj(ψjd)⩾1/kj for every d∈{1,…,kj}. If there exists l∈{1,…,kj} such that yj(ψjl)>1/kj, then ∑d=1kj‍yj(ψjd)>1, but this is a contradiction since yj∈Xj. Hence Kij(x¯j)={x¯j}, which immediately implies that assumption (ii) in Proposition 7 is not satisfied as x¯j(ψjd)=Pij(χ(ψjd),x¯j) for all d∈{1,…,kj}. Nevertheless, we prove that Kij is continuous in x¯j. Let (xj,ν)ν∈N be a sequence converging to x¯j. By definition, it follows that xj,ν∈Kij(xj,ν) for every ν and hence Kij is lower semicontinuous in x¯j as xj,ν→x¯j. On the other hand, let (xj,ν)ν∈N be a sequence converging to x¯j and (yj,ν)ν∈N a sequence converging to yj such that yj,ν∈Kij(xj,ν) for every ν. Since Xj is compact, then yj∈Xj. Moreover, yj,ν∈Kij(xj,ν) implies that (36)Eyj,νχψjd=yj,νψjd⩾Pijχψjd,xj,ν∀d∈1,…,kj.Taking the limit for ν→∞, it follows from the continuity of the functions Pij that (37)Eyjχψjd=limν→∞⁡Eyj,νχψjd⩾limν→∞⁡Pijχψjd,xj,ν=Pijχψjd,x¯j∀d∈1,…,kj,which implies that yj∈Kij(x¯j) (in particular yj=x¯j). Therefore, Kij is a closed and hence continuous set-valued map in x¯j. Following the same steps of the proof of Proposition 7, we can prove that if Kij is continuous in x¯j then Bi is continuous in (xi,x¯j) for every xi∈Xi. As a final remark, the example emphasizes that it is more simple to check our Slater-like condition rather than applying the definition of upper and lower semicontinuity for correspondences, even in this particular case in which there is a unique point x¯j in which the Slater-like condition is not satisfied and the set-valued map Kij in this point is single valued. It is clear that, in more complex cases, calculations would have been even more demanding.Remark 9.
In order to keep the presentation of Proposition7 more simple, we did not consider the case in which there exist implicit linear equalities in the system of linear inequalities. However, the generalization to this case is rather straightforward. It consists in adding the assumption that the coefficient matrix of the system of the implicit equalities together with the equation in (23) has full rank. The proof is substantially identical to the one given above.



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*Source: 690295-2015-06-24.xml*