In the last fifty years, the classical Arrow-Debreu result on the existence of Walrasian equilibria has been generalized in many directions. Mas-Colell [24] first porved an exis-tence theorem of equilibria without assuming preferences to be total or transitive. Next, by using a maximal element existence theorem, Gale and Mas-Colell [10] gave a proof of the existence of a competitive equilibrium without ordered preferences. By using Kaku-tani’s fixed point theorem, Shafer and Sonnenschein [29] proved a powerful result on “the Arrow-Debreu lemma for abstract economies” for the case where preferences may not be total or transitive but have open graphs. In fact, they imposed a stronger condition on the preference correspondences and weaker condition on the feasible correspondences.
As in [10, 29, 42], in most results on the existence of equilibria for abstract economies, the underlying space (commodity spaces or choice sets) are usually compact and convex.
However, in recent papers [7, 19, 37, 39], the underlying spaces are not always compact nor paracompact. Moreover, many authors, motivated by Horvath’s paper [12], have investi-gated the same results in H-spaces (see [31, 34, 38, 39, 41]). Therefore, we will encounter many kinds of preferences in various economic situations; so that it is important for us to
consider different types of preferences and obtain more powerful existence results for such correspondences in non-compact (or non-paracompact) generalized convex spaces, which is a general topological space without linearity structure.
Following Shafer and Sonnenschein [29], an abstract economy or generalized game Σ = (Xα, Aα, Uα)α∈I is defined as follows: I is any (finite or infinite) set of agents
Xα , commodity space or choice set;
Aα : X −→ 2Xα , constraint correspondence (set-valued mapping);
Uα : X −→R , utility or pay-off function.
For arbitrarily fixed α ∈ I and any subset Yα ∈ Xα, we follows [7] and define Y
β6=α,β∈I
Xβ⊗ Yα := {x ∈ X | xα ∈ Yα, and xβ ∈ Xβ for each β ∈ I \ {α}} .
For simplicity of notation, we denote Q
β6=α,β∈IXβ ⊗ Yα by X−α ⊗ Yα, and the element of X−α by x−α. An abstract economy instead of being given by a family of order triples Ω = (Xα, Aα, Pα)α∈I such that for each α ∈ I,
Xα , commodity space or choice set;
Aα : X −→ 2Xα , constraint correspondence (set-valued mapping);
Pα : X −→ 2Xα , preference correspondence.
The relationship between the utility function Uα and the preference correspondence Pα can be exhibited by the definition:
Pα(x) := {yα ∈ Xα | Uα(x−α, yα) > Uα(x)}.
In the case of the economy being given by Σ = (Xα, Aα, Uα)α∈I, a point ˆx ∈ X is called an equilibrium point or a generalized Nash equilibrium point of Σ, if
Uα(ˆx) = Uα(ˆx−α, ˆxα) = sup
zα∈Aα(ˆx)
Uα(ˆx−α, zα),
for each α ∈ I, where ˆxα is the projection of ˆx onto Xα. It can be easily checked that a point ˆx ∈ X is an equilibrium point of Σ if and only if for each α ∈ I, Aα(ˆx) ∩ Pα(ˆx) = ∅ and ˆxα ∈ Aα(ˆx).
Thus, given an abstract economy Ω = (Xα, Aα, Pα)α∈I, an equilibrium point of Ω is a point ˆx ∈ X such that for each α ∈ I, ˆxα ∈ Aα(ˆx) and Aα(ˆx) ∩ Pα(ˆx) = ∅. For more references on this topic, we refer to [10, 29, 34].
In [43], Yuan proposed a model of abstract economy more general then that introduced by Borglin and Keing in [3] in the sense that the constraint mapping has been split into two parts A and B due to the fact the “small” constraint correspondence may have not enough fixed points but a “big” constraint correspondence B does so. Therefore, we will describe an abstract economy by a family of order quadruples Ω = (Xα, Aα, Bα, Pα)α∈I such that for each α ∈ I,
Xα , commodity space or choice set;
Aα, Bα : X −→ 2Xα , constraint correspondences (set-valued mapping);
Pα : X −→ 2Xα , preference correspondence.
An equilibrium point of Ω is a point ˆx ∈ X such that for each α ∈ I, ˆxα ∈ clBα(ˆx) and Aα(ˆx) ∩ Pα(ˆx) = ∅.
The equilibrium existence theory for various models have been extensively studied by many authors, and maximal element existence theorems are frequently used as the main tool for proving the existence of equilibria, e.g. see [19, 38] and references therein.
Let X be a topological space, together with a binary relation R on X, which is a subset of X × X. We read (y, x) ∈ R as “y is preferred to x”. Define the correspondence
P : X −→ 2X by
P (x) := {y ∈ X | (y, x) ∈ R}, and the correspondence P−1 : X −→ 2X by
P−1(y) := {x ∈ X | y ∈ P (x)}.
We call P (x) the upper section of R and P−1(y) the lower section of R. We say that R ⊂ X × X has an open graph, if the set
Gr := {(y, x) ∈ X × X | (y, x) ∈ R}
is open in X × X. If there exists ˆx ∈ X such that P (ˆx) = ∅, then ˆx is said to be a maximal element in X.
Further, let X and Y be two topological spaces. A set-valued mapping T from X to Y , written as T : X −→ 2Y, is simply a mapping which assigns each point x of X to a (possibly empty) subset T (x) of Y . The inverse and graph of T are defined respectively by
T−1(y) := {x ∈ X | y ∈ T (x)} for each y ∈ Y, and
Gr(T ) := {(x, y) ∈ X × Y | y ∈ T (x)}.
If we set the relation R0 on Y × X by
R0 := {(y, x) ∈ Y × X | (x, y) ∈ Gr(T )},
using a similar definition of maximal element as above, we say that a point ˆx ∈ X is a maximal element of T , if T (ˆx) = ∅.
It is well-known that each existence theorem of maximal elements has an equivalent version of a fixed point theorem. Yannelis and Prahbakar [42] used selection theorems and fixed point theorems for correspondences defined on topological vector spaces. Some authors developed the theory of continuous selections of set-valued mappings and gave numerous applications in game theory. Michaels selection theorem in [25] is well known and basic in many applications. Horvath [12], Tarafdar [34], Yannelis and Prabhakar [42], and many others established more general continuous selection theorems with applica-tions. In [4, 5], we also has established some generalized continuous selection theorems for almost lower semicontinuous set-valued mappings defined on Banach space or com-plete l.c.-metric space under a mild ECP condition.
In 1976, Borglin and Keiding [3] first introduced the majorized concept of set-valued mappings and recently many authors have proved a number of general results on the existence of maximal elements and equilibria for abstract economics in which the pref-erence correspondences are majorized (see [7, 17, 44]). In this thesis, we focus on the existence theory of maximal elements and equilibria for abstract economies. Indeed, we improve and generalized the recent many well-known existence theorems to l.c.-spaces for majorized preference correspondences. More precisely, we concern two types of majorized correspondences.
In chapter 2, we first generalize Tarafdar’s fixed point theorem [34, Theorem 2.3] with-out compactness conditions. This result is a useful tool for proving the existence theorems of maximal elements for Lθ-majorized set-valued mappings. In the sequel, we shall deduce a new existence theorem of equilibria for noncompact abstract economies. In addition, we obtain a existence theorem of solutions to a system of generalized quasi-variational inequalities.
In chapter 3, we further establish some existence theorems in the setting of Φθ -majorized preference correspondences. At the beginning, we attempt to employ the remarkable KKM principle in H-spaces [26] to derive a general fixed point theorem in l.c.-spaces. We shall use this fixed point theorem to derive some existence theorems of maximal elements and equilibria for abstract economies. We also study the existence theorem of equilibria for fuzzy abstract economies. In other words, there are extra fuzzy constraint correspondences in defining abstract economies.