A topological characterization of the non-emptiness of the Banks set and the uncovered set

Journal Pre-proof A topological characterization of the non-emptiness of the Banks Set and the Uncovered Set

Athanasios Andrikopoulos, Katerina Zontou

PII:

S0166-8641(19)30432-8

DOI:

https://doi.org/10.1016/j.topol.2019.107026

Reference:

TOPOL 107026

To appear in:

Topology and its Applications

Received date:

31 January 2019

Revised date:

29 March 2019

Accepted date:

22 April 2019

Please cite this article as: A. Andrikopoulos, K. Zontou, A topological characterization of the non-emptiness of the Banks Set and the Uncovered Set, Topol. Appl. (2019), 107026, doi: https://doi.org/10.1016/j.topol.2019.107026.

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A topological characterization of the non-emptiness of the Banks Set and the Uncovered Set Athanasios Andrikopoulos, Katerina Zontou Abstract Banks set and the uncovered set are two of the most important concepts that underlie in the intersection of game theory and social choice theory, since these sets provide a generalization of the core when core does not exist. These sets, named general solution concepts, have been extensively studied in case there is a finite set of alternatives. However, the process which leads to the elements of these sets has not been studied in more detail, when we treat an infinite set of alternatives. In this paper, we give a topological characterization of the non-emptyness of Banks set and the uncovered set on an infinite space of alternatives. Keywords: Banks set, Uncovered set, lower (upper) semicontinuous, compactness. MSC2010: 54D30, 91A10, 91A12, 91A13, 91B06, 91B12, 91B15. JEL classification: C6, C7, D6, D7

1

Introduction

In neoclassical economic theory a choice process is presented by a choice function that associates with each feasible set of alternatives a certain subset of it. This subset consists the maximal elements accordingly to the viewpoint of a binary relation. Due to this foundational axiom of economic and game theories, the existence of maximal elements is a useful and important tool of proving the existence of non-empty choice sets or equilibrium existence theorems of mathematical economies and generalized games in different underline spaces. However, the set of maximal elements is often empty.1 In this case, the crucial question arises, is what to count as a choice. Thus, we are wondering which sets of alternatives may be considered as reasonable solutions. To answer this question, several methods, named General Solution Concepts have been proposed for the construction of non-empty choice sets. The most important of these solutions include (i) The Uncovered Set; (ii) The Banks Set. Given an arbitrary binary relation, a covering relation is a subrelation defined in terms of nested upper sections. The set of undominated elements of this relation is the uncovered set. The notion of uncovered set is fundamental in a number of analyses in both noncooperative and cooperative game theory. Following Gillies [13] analysis of transferable utility cooperative games, an alternative x covers y if x is strictly preferred to y and every alternative strictly preferred to x is also strictly preferred to y. In contrast to Gillies, Bordes [4] define covering relation as follows: An alternative x covers y if it is strictly preferred to y and every alternative weakly preferred to x is also weakly preferred to y. The McKelvey [15] covering relation is the conjunction of Gillies and Bordes. The notion of covering relation, introduced by Bordes, has often been attributed to Miller [19], but the formal definition used by Miller is slightly different.2 In fact, according to Miller, an alternative x covers y if it is weakly preferred to y and every alternative weakly preferred to x is also weakly preferred to y. Fishburn weakens Gillies covering as follows: x covers y if x is weakly preferred to y and every alternative strictly preferred to x is also strictly preferred to y. The Richelson uncovered set is essentially the weak dominance relation defined by McKelvey [15]. The undominated elements of Richelson covering comprise the weakly undominated set, as defined by McKelvey. In more recent work, Duggan [6] introduced two concepts that bound existing uncovered sets from above and below: the deep uncovered set and the shallow uncovered set. The deep uncovered set is the largest choice set of those defined, in 1

This problem occurs when the social preference yielded by an aggregation process is cyclic. For example, this is common in the analysis of pairwise majority voting, in the aggregation of multiple choice criteria, in committee selection, in the choice under uncertainty, etc. 2 Miller [19] originally examined the uncovered set in majority preference tournaments that result when an odd number n of voters have strong preferences over discrete alternatives.

contrast to the shallow uncovered set which is the smallest. The set of undominated elements of all covering relations, discussed above, is the corresponding uncovered set. Generally, the uncovered set is a general solution concept which is useful, essentialy in the absence of a core. In many cases, the uncovered set coincides with the core when it exists (for example, in tournaments or in case preferences are discriminating), and it is a subset of the Pareto set when the core is empty. Miller [19] observed that an electoral strategy is undominated if and only if it is an element of the uncovered set. Furthermore, he demonstrates that in case of tournaments, the uncovered set coincides with the core. McKelvey [15] underlined that the uncovered set contains the support set of any mixed strategy equilibria. He also extended some of Miller’s results to a more general setting of alternative spaces where voters have weak preferences over alternatives. Duggan [6], [7] and Duggan and Jackson [9] searched some properties of a weaker version of the uncovered set with nice continuity properties. Dutta, Jackson, and Le Breton [11] provided some characterizations of the uncovered set within the broad class of budget allocation problems. Banks, Duggan, and Le Breton [3] showed that for a two candidate symmetric zero sum election, the support of mixed strategy equilibria of the game lies in the uncovered set. Laslier and Picard [14] proved that the uncovered set is equal to the Pareto set in Pareto constant games, where the covering relation is defined in terms of pluralities instead of majorities. Penn [17] showed that within three-player distributive games the uncovered set is equal to the set of sophisticated voting outcomes. Banks [18] provided a thorough characterization of the set of alternatives that a committee may adopt operating under the standard amendment procedure, when its members vote in a sophisticated fashion. He demonstrated that this set of alternatives is a subset, sometimes proper, of the uncovered set defined by Miller [19]. That is, even if all voting outcomes belong to the uncovered set, this does not imply that all the elements of the uncovered set are voting outcomes. According to this characterization, an alternative can be a sophisticated voting outcome under the standard amendment procedure (also known as voting by successive elimination), if and only if it is the maximal element of some Banks chain.3 The set of all maximal elements of Banks’ chains is the Banks set. In fact, Banks’ set create bounds on the monopoly power of an agenda setter in the context of the amendment agendas. Banks’ set has been investigated extensively in the case of a finite set of alternatives. However, the explicit computation of these sets is not that simple, particularly when the feasible set of alternatives is a subset of a multi-dimensional Euclidean space. For example, Feld, Godfrey and Grofman [12] considered to 2-dimensional Lebesgue measurable Euclidean spaces X, equivalent topologically to the real plane and endowed with an inner product (i.e., the dot or scalar product of vectors), and defined infinite agendas A ⊆ X referred to as continuous agendas. In this case, note that the agenda is assumed to include both endpoints, i.e., the agenda is compact. The authors defined the Banks set for the infinite case as the set of all policies that are the maximal element of a Banks chain in a spatial voting game. Penn [16] extended the definition of Banks set on infinite space of alternatives with voters’ preferences being quasi-concave, continuous utility functions. According to Penn, “defining the Banks set over an infinite space requires us to think more deeply about the process of agenda formation, with agendas being not simply orderings of alternatives, but subsets of the alternative space”. Dutta, Jackson, and LeBreton [11] defined a voting procedure over a countably infinite set of alternatives. An amendment procedure over an uncountably infinite agenda would require voting in continuous time. Penn’s definition of Banks’ set in infinite settings is directly in terms of maximal chains rather than in terms of an agenda, and her tie-breaking rule is different from Dutta, Jackson and Le Breton [11]. Duggan [8] proposed a procedure in which one deletes particular instances of preferences until the resulting subrelation is acyclic (alternatively transitive or negatively transitive), and collects the maximal elements of all such maximal subrelations. A related procedure is to collect the maximal elements of all maximal chains, i.e., subsets of alternatives on which the majority preference is a linear order. If the chains consist of the sophisticated voting outcomes of some binary agenda, then, this procedure yields the Banks set. Based on the procedure previously mentioned, Andrikopoulos and Zacharias [1] and Andrikopoulos [2] presented a topological and a binary characterization of Smith set, Duggan set, Schwartz set and the generalized stable sets solution of an arbitrary binary relation over non-finite sets. Generally, the concept of compactness is an extension of the benefits of finiteness to infinite sets.4 In economic and game theories, the Banks set and the uncovered set are always non-empty when the set of alternatives is finite. In this paper, we show that the same statement remains true if finiteness is replaced by compactness and the binary relation is lower (upper) semicontinuous. We also give a characterization of non-emptiness of the Banks set and the uncovered set for an infinite space of alternatives. 3

We call a Banks chain the set of possible sophisticated outcomes under the amendment procedure. Any topology on a finite set is compact, but a finite set is usually considered to be a discrete topological space. Here is why: (i) if a set D has a discrete topology, then D is compact if and only if D is finite; (ii) a topology on a finite set D is Hausdorff if and only if it is the discrete topology; (iii) any function from a space with the discrete topology is continuous. 4

2

2

Notations and definitions

Neoclassical economic theory assumes that each decision-maker is able to compare two alternatives x and y in the choice set X using a preference-indifference-incomparability operator (binary relation) R ⊆ X × X.5 According to this fact, three binary relations can be defined on X: the strict preference relation P (R) = {(x, y)|(x, y) ∈ R and (y, x) ∈ / R}, the indifference relation I(R) = {(x, y)|(x, y) ∈ R and (y, x) ∈ R} and the incomparability relation N (R) = {(x, y)|(x, y) ∈ / R and (y, x) ∈ / R}. Let also Δ = {(x, x)|x ∈ X} denotes the diagonal of X. We sometimes abbreviate (x, y) ∈ R as xRy. We say that R is (i) asymmetric if (x, y) ∈ R implies that (y, x) ∈ / R; (ii) complete if for each x, y ∈ X, x = y we have xRy or yRx. A tournament T = (X, R) consists of a non-empty finite set X and a complete asymmetric binary relation R defined on X (xRy means x beats y). Such a tournament can arise, for example, as the majority preference relation of an odd number of voters with linear preferences. The transitive closure of a relation R is denoted by R, that is for all x, y ∈ X, (x, y) ∈ R if there exists k ∈ N and x0 , ..., xK ∈ X such that x = x0 , (xk−1 , xk ) ∈ R for all k ∈ {1, ..., K} and xK = y. Clearly, R is transitive, also the case where K = 1 is included, therefore R ⊆ R. A subset Y ⊆ X is an R-cycle if, for all x, y ∈ Y , we have (x, y) ∈ R and (y, x) ∈ R. We say that R is acyclic (P -acyclic) if there does not exist an R-cycle (resp. a P (R)-cycle). Suzumura [23] provides the following definition, which generalizes the notions of transitivity and acyclicity: The binary relation R is consistent, if for all x, y ∈ X, (x, y) ∈ R implies (y, x) ∈ / P (R). If in the definition of the consistent binary relation above we replace R with Rc , then we get the notion of a negative consistent binary relation. Negative consistency is defined by Duggan [10]. Let M(R) denote the elements of X that are R-maximal in X, i.e., M(R) = {x ∈ X| for all y ∈ X, yRx implies xRy}. For each asymmetric binary relation P , Schwartz [20, Page 143] gives the definition of the weak binary relation S corresponded to R as follows: for all x, y ∈ X, xSy if and only if (y, x) ∈ / P . The generalized weak binary relation S corresponded to an arbitrary binary relation R is defined as follows: for all x, y ∈ X, xSy if and only if (y, x) ∈ / P (R)[2, Definition 4]. If R is asymmetric, then the notions of generalized weak binary relation and weak binary relation coincide. One way of completing a binary relation is to indifference between the alternatives which are not comparable. That is, we have: For all x, y ∈ X, xSy ⇔ (y, x) ∈ / P (R), xP (S)y ⇔ xP R)y and xI(S)y = xN (R)y.

(1)

In cases where no difference is made between indifference and incomparability, asymmetry is a sufficient condition.6 A subset Y ∈ X is (i) R-undominated if and only if for no x ∈ Y there is a y ∈ X \ X such that yRx; (ii) R-dominant if and only if xRy for each x ∈ Y and each y ∈ X \ Y . An R-undominated (R-dominated) set is minimal if none of its proper subsets has this property. The Smith set SM(R) of a binary relation R is equivalent to the union of all minimal R-dominant subsets of X. A subset Y ⊂ X is defined to be externally stable if for all x ∈ X \ Y , there exists a y ∈ Y such that yP (R)x. A chain, denoted C, is a class of relations such that B, B  ∈ C implies B ⊆ B  or B  ⊆ B. Zorn’s Lemma: If every chain of a partially ordered set has an upper bound, then E has a maximal element. We now give eight primary definitions of covering in the literature (these definitions have been given.....asymmetric...). Gillies covering x C(G)y if and only if xP (R)y and P (R)(x) ⊆ P (R)(y) Bordes covering x C(B)y if and only if xP (R)y and S(x) ⊆ S(y)

(2)

McKelvey covering x C(K)y if and only if xC(G)y and xC(B)y. There’s also three other notions of covering appear in the literature:7 Fishburn covering x C(F )y if and only if xSy and P (R)(x) ⊆ P (R)(y) Miller covering x C(M )y if and only if xSy and S(x) ⊆ S(y)

(3)

Richelson covering x C(R)y if and only if xC(F )y and xC(M )y. 5 That is, given two alternatives, the decision maker can act in one of the following three ways: i) he/she clearly prefers one to the other; ii) the two alternatives are indifferent to him/her; and iii) he/she is unable to compare the two alternatives. 6 This is a direct consequence of the real numbers’ definition as a complete ordered field. This definition states that for any two real numbers x and y, exactly one of the following must be true: Either x < y, x = y, or x > y (this axiom is also known as trichotomy law). Hence, x ≮ y and y ≮ x is equivalent to x = y. 7 Duggan uses the term “shading”relation, rather than “covering”relation, to distinguish relations of this type with the other covering relations.

3

Duggan in [6] also introduce two notions of covering, which have crucial meaning for the other notions. Duggan deep covering x Dy if and only if xP (R)y and S(x) ⊆ P (R)(y) Duggan shallow covering x Sy if and only if xSy andP (R)(x) ⊆ S(y).

(4)

The sets of undominated elements of C(G), C(B), C(K), C(F ), C(M ), C(R), D, S

(5)

U(G), U(B), U(K), U(F ), U(M ), U(R), U(D), U(S),

(6)

denoted by are the Gillies, Bordes, McKelvey, Fishburn, Miller, Richelson, Duggan deep and Duggan shallow uncovered set, respectively. An alternative x is uncovered in the sense of the above definitions, if for each y ∈ X the following “two-step principle”holds: x U(G)y ⇔ xSy or ∃z ∈ X : xSzP (R)y x U(B)y ⇔ xSy or ∃z ∈ X : xP (R)Sy x U(K)y ⇔ xSy or ∃z ∈ X|xP (R)zSy or ∃z ∈ X|xSzP (R)y x U(F )y ⇔ xP (R)y or ∃z ∈ X|xSzP (R)y x U(M )y ⇔ xP (R)y or ∃z ∈ X|xP (R)zSy

(7)

x U(R)y ⇔ xP (R)y or ∃z ∈ X|xP (R)zSy or ∃z ∈ X|xSzP (R)y x U(D)y ⇔ xSy or ∃z ∈ X|xSzSy x U(S)y ⇔ xP (R)y or ∃z ∈ X|xP (R)zP (R)y Given an alternative x ∈ X and a binary relation R ∈ X, an x-chain of R is a R-chain C with x ∈ C such that, xP (R)y for all y ∈ C. The set for all x-chains is denoted C(x, R). Thus, an x-chain is a chain where x wins all the other alternatives in the chain according to R. Banks set [18] constitutes a solution for sophisticated voting under the amendment procedure. The intuition behind the Banks Solution concept is that the alternatives in the chain are those who temporarily win at some stage in the voting, the remaining of the alternatives are those who are eliminated at the stages. Formally: Given an agenda A = {x1 , ..., xt }, the sophisticated equivalent agenda A∗ = {x∗1 , ..., x∗t } is defined as, (i) x∗1 = x1 (ii) for 1 < i ≤ t, x∗i =



xi x∗i−1

if xi ∈ R(x∗j ), ∀j < i otherwise

and the sophisticated outcome of agenda A∗ is defined to be the last element in A∗ , or x∗t . An agenda A is defined to be stable if for all x ∈ X \ A, there is a sophisticated equivalent x∗i ∈ A such that x∗i Sx. A binary relation R defined on a topological space (X, τ ) is upper (resp. lower) semicontinuous (see [5]), if for all x ∈ X, the set {y ∈ X|(x, y) ∈ P (R)} (resp. {y ∈ X|(y, x) ∈ P (R)}) is open in X; and it is continuous if it is both upper and lower semicontinuous. The topological space (X, τ ) is compact if for each collection of open sets which cover X, there exists a finite subcollection that also covers X.

3

The main results

While the properties of Banks set for finite set of alternatives have been extensively studied, little is known about how to find members of this set for spaces involving possibly infinite agendas, i.e. in spatial voting games with a continuum of policies [16]. Penn and Feld, Godfrey and Grofman extended the definition of Banks set, over infinite space of alternatives. This extension is important since defining Banks set over infinite space requires us to think more deeply the process of agenda formation, with agendas being not just orderings of alternatives, but also subsets of the alternative space. Penn provides a definition of Banks set, or set of sophisticated voting outcomes, over an infinite policy space and when individual preferences are weak. More preciselly, in the finite case, the Banks set is equivalent to the set of maximal elements of maximal chains, and this set is always a subset of the uncovered set (see [18]). However, this definition does not apply to the case of an uncountably infinite policy space because it is

4

unclear how an amendment procedure would progress over the entire space. To deal with this problem Penn [16] have extended the definition of Banks set over an infinite policy space as follows (see [16, Page 534]): Let R be a binary relation on a set of alternatives X and let C be the set of all finite chains with respect to generalized weak binary relation S corresponded to R. Let also C∗ ⊆ C be the set of finite externally stable chains with respect to S in C, and for all C ∗ ∈ C, define M(X, C ∗ ) to be the set of maximal elements of C ∗ , so that for all x ∈ M(X, C ∗ ) and all y ∈ C ∗ we have xSy. Every finite chain with respect to S has at least one maximal element. Then, Pen’s definition of Banks set in (X, R) is BP (R) = {x ∈ X|∃C ∗ ∈ C∗ with x ∈ M(X, C ∗ )}.

(8)

In words, Pen’s definition of Banks set over an infinite policy space X, equals the set of maximal elements of all finite, externally stable chains with respect to S in X. If the set of alternatives is finite, then this definition coincides with Banks’ original definition. A second generalization of Banks set over infinite set of alternatives is given by Feld, Godfrey and Grofman ([12, Page 50]) as follows: Definition 3.1. An agenda (A, t) is a subset A ⊂ X with order parameter λ : A → [0, 1]. Individual elements of A on the agenda are represented as x(λ), with a strict order  defined by λ > λ implies x(λ)  x(λ ). Note that the agenda is assumed to include both endpoints, i.e., the agenda is compact. Definition 3.2. A sophisticated agenda (A,t) is an agenda such that x(λ)  x(λ ) iff λ > λ . A sophisticated agenda is considered stable if there is no alternative off the agenda that dominates all alternatives on the agenda. The element λ(1) of a sophisticated agenda is called the maximal element and the the element λ(0) is called the minimal element. Definition 3.3. A Banks chain is a stable sophisticated agenda. The F GG-Banks set BF GG (R) is the set of all alternatives that are the maximal element of a Banks chain. Comparing the two generalizations for the Banks set, we observe two differences: 1. Penn’s definition restricts consideration to finite agendas, where Feld, Godfrey and Grofman’s definition do not. 2. Penn’s definition uses in agenda a weak binary relation, i.e., a tie favor the challenger, while Feld, Godfrey and Grofman’s definition requires strict dominance relation, i.e., the status quo cannot be defeated by a tie. 3. According to Penn’s definition, a sophisticated agenda is considered externally stable if there is no alternative off the agenda that dominates all alternatives on the agenda while in Feld, Godfrey and Grofman’s definition there is no alternative off the agenda that weak dominates all alternatives on the agenda. Obviously, by definition, F GG-Banks set is a subset of Bordes, McKelvey and Duggan Deep uncovered set respectivelly, and similarly P -Banks set is a subset of Fishburn, Gillies, McKelvey, Richelson and Duggan Deep uncovered set respectivelly. The next proposition establishes this fact. Proposition 3.1. Let R be a binary relation on a set of alternatives X. Then, (i) BF GG (R) ⊆ U(B) ⊆ U(K) ⊆ U(D), (ii) BP (R) ⊆ U(F ) ⊆ U(G) ⊆ U(K) ⊆ U(D), (iii) BP (R) ⊆ U(F ) ⊆ U(R) ⊆ U(K) ⊆ U(D). Theorem 3.2. Let (X, τ ) be a compact topological space and let R be a lower semicontinuous relation on X. Then, the F GG-Banks set is non-empty. Proof. To prove that BF GG (R) = ∅, we have two cases to consider: (A) There exists x∗ ∈ X such that for each y ∈ X we have (y, x∗ ) ∈ / P (R); (B) For each x ∈ X there exists y ∈ X such that (y, x) ∈ P (R). In case (A), we have that: for each y ∈ X \ x∗ we have (x∗ , y) ∈ S. (9) Therefore, x∗ ∈ BF GG (R). In case (B), fix an x0 ∈ X. Then, there exists x1 ∈ X such that (x1 , x0 ) ∈ P (R). Let Ξ0 = {z ∈ X|(x0 , z) ∈ S}.

(10)

x1 P (R)x0 SΞ0

(11)

Then, we have

5

by which notation we mean that x1 P (R)x0 Sy for each y ∈ Ξ0 .

(12)

Ξ1 = {z ∈ X|(x1 , z) ∈ S}.

(13)

We call equation (12), Γ(1)-Property. Let

If Ξ0 ∪ Ξ1 = X 8 (x0 ∈ S(x0 ), x1 ∈ S(x1 )), then x1 ∈ BF GG (R). Otherwise, there exists x2 ∈ X such that (x2 , x1 ) ∈ P (R) and (x2 , x0 ) ∈ P (R). It follows that x2 P (R)x1 SΞ1 and x2 P (R)x0 SΞ0 .

(14)

We call equation (14), Γ(2)-Property. The Well-Ordering Principle asserts that every set X can be well-ordered; that is, if X is any set, then there exists a well-ordered set Λ which serves as an index set for the elements of X, so we may write X = {xλ |λ ∈ Λ}.

(15)

This principle is logically equivalent to the Axiom of Choice, or Zorn’s Lemma. We now proceed by transfinite induction. We may assume that a well ordered set Λ, cofinal to Λ, enumerates the steps of the procedure described in (10)-(14). Let λ ∈ Λ be an ordinal which has the Γ(λ)-Property, that is: xλ P (R)xν SΞν for each ν < λ where Ξν = {z ∈ X|(xν , z) ∈ S}.

(16)

In this case, we refer to the construction in the step λ ∈ Λ. According to transfinite induction, if T (λ) is true whenever T (λ ) is true for all λ < λ, then T (λ) is true for all λ. We follow two steps: (i) Successor case: Prove that for any successor ordinal λ + 1, T (λ + 1) follows from T (λ). (ii) Limit case: Prove that for any limit ordinal λ, T (λ) follows from [T (λ ) for all λ < λ].  Step (i). Let λ be a successor ordinal. If X = Ξν , then xλ ∈ BF GG (R) ({xν |ν ∈ {0, 1, ..., λ}} is a chain with ν≤λ

gratest element xλ and for each y ∈ X \ {xν |ν ∈ {0, 1, ..., λ}} we have xν P (R)Sy for some ν ∈ {0, 1, ..., λ}). Otherwise, there exists μ ∈ Λ, λ < μ such that xμ ∈ X \ Ξν . It follows that ν≤λ

xμ P (R)xλ SΞλ for each λ < μ where Ξλ = {z ∈ X|(xλ , z) ∈ S}.

(17)

Therefore, T (λ + 1) = T (μ) holds. Step (ii). Let λ be a limit ordinal. As it is well-known, an ordinal λ is a limit ordinal if and only if there is an ordinal     less than λ, and whenever λ is an ordinal less than λ, then there exists an ordinal λ such that λ < λ < λ. If  X= Ξν for some λ , then the procedure will eventually stop since xλ ∈ BF GG (R). Otherwise, ν<λ

xλ P (R)xν SΞν for each ν < λ where Ξν = {z ∈ X|(xν , z) ∈ S}.

(18)

Then, we put xλ = xλ and therefore T (λ ) holds. Since T (λ ) is true for all λ < λ] we conclude that T (λ) holds. It follows that  X= Ξλ and for each λ ∈ Λ, T (λ) holds. (19) λ∈Λ

There are two cases to consider: (ω1 ) There exists a x∗ ∈ X such that (x∗ , xλ ) ∈ P (R) for all λ ∈ Λ;

(20)

(ω2 ) For each x ∈ X there exists λ ∈ Λ such that (x, xλ ) ∈ / P (R) or equivalently (xλ , x) ∈ S. 8

For example, Ξ0 and Ξ1 are P (R)-cycles with x1 ∈ Ξ1 .

6

(21)

In case (ω1 ), by (19) and (20) we have that for each y ∈ X \ {xλ |λ ∈ Λ} there holds x∗ P (R)xμ Sy for some μ ∈ Λ,

(22)

which jointly to (x∗ , xλ ) ∈ P (R) for all λ ∈ Λ implies that x∗ ∈ BF GG (R). In case (ω2 ) we first show that  (X \ {x ∈ X|xλ Sx}). X = 

Suppose to the contrary that

(23)

λ∈Λ

(X \ {x ∈ X|xλ Sx}) = X. Since

λ∈Λ

X \ {x ∈ X|xλ Sx} = {x ∈ X|xP (R)xλ }

(24)

and R is lower semicontinuous we conclude that the family {X \ {x ∈ X|xλ Sx} | λ ∈ Λ}

(25)

is an open cover of (X, τ ). By compactness, there exist xλ , ..., xλn ∈ X such that 1

X=



({X \ {x ∈ X|xμ Sx}).

(26)

μ∈{λ1 ,...,λn }

Consider the finite set {xλ , ..., xλn }. Since xλ ∈ X, then there exist i ∈ {1, ..., n} such that xλ P (R)xλ . 1 1 1 i If i = 1, then we have a contradiction. Otherwise, call this element xλ . We have xλ P (R)xλ . Similarly, 2 2 1 xλ P (R)xλ P (R)xλ . As {xλ , ..., xλn } is finite, by an induction argument based on this logic, we obtain the 3 2 1 1 existence of a P (R)-cycle. Therefore, there exists at least one pair (xλ , xλ ), i ∈ {1, ..., n − 1} such that i+1

i

, a contradiction (by the construction of (xλ )λ∈Λ , λ > λ implies xλ P (R)xλ ).   The last contradiction shows that (X \ {x ∈ X|xλ Sx} = X. Let x∗ ∈ X \ (X \ {x ∈ X|xλ Sx}. Then, λi+1 > λi and xλ P (R)xλ i

i+1

λ∈Λ

λ∈Λ

/ X \ {x ∈ X|xλ Sx}. It follows that for each λ ∈ Λ there holds x∗ P (R)xλ . for each λ ∈ Λ, we have that x∗ ∈ We now prove that the set A = {xλ |λ ∈ Λ} is an agenda of X, that is, A is a linear ordered compact subset of X with first and last element. The last conclusion is evident. To prove that A is compact, let O be a covering of A and let x0 ∈ O0 , x∗ ∈ O∗ where O0 , O∗ ∈ O. Let A∗ = A \ (O0 ∪ O∗ ). Since A∗ is a closed subset of X we conclude that it is compact. Therefore, there exists O1 , ..., On ∈ O such that A∗ = Om . It follows that A =



m∈{1,...,n}

Om ∪ O∗ which implies that A is a compact subset of X. Finally, for each z ∈ A, x∗ P (R)z

m∈{0,1,...,n}

holds and for each y ∈ X \ A we have zSy for some z ∈ A. The last conclusion shows that x∗ ∈ BF GG (R) and thus the FGG-Banks set is non-empty. The existence theorems of maximal elements are useful and important tools to prove the existence of nonempty choice sets or equilibrium existence theorems of mathematical economies and generalized games in different underline spaces. In this direction, the notion of quasi-transitivity is an important notion in social science and game theory. According to Sen [21], a quasi-transitive social preference relation under a social decision process ensures the existence of at least one socially best alternative for every nonempty finite subset of the given set of alternatives under consideration. In the following, we generalize the well known notion of quasi-transitivity in order to give a topological characterization of the non-emptyness of the F GG-Banks set. Definition 3.4. Let R be a binary relation on X. We say that R is generalized quasi-transitive if for all x, y, z ∈ X, (x, z) ∈ P (R) and (z, y) ∈ P (R) implies (x, y) ∈ / [N (R) ∪ I(R)]. The previous definition generalizes in a natural way the usual definition of quasi-transitivity since P (R) ⊂ [N (R) ∪ I(R)]c = P (R) ∪ (P (R))−1 . Notice that this definition does not preclude the possibility of R being non-quasi-transitive since it allows P (R)-cycles. In case where R is P -acyclic, then the two definitions coincide. Theorem 3.3. Let R be a generalized quasi-transitive binary relation on X. The following conditions are equivalent: (i) The F GG-Banks set is non-empty, (ii) there exists a compact topology τ on X such that R is lower semicontinuous.

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Proof. To prove the implication ((i) ⇒ (ii)), let us suppose that the F GG-Banks set is non-empty. Then, there exists a Banks chain C which has a minimal element, let m. Let τ be the excluded point topology generated by m [22, Page 47] (it has as open sets all those subsets of X which are disjoint from {m}, together with X itself). Then, X is compact under τ since every open cover of X includes X itself. Hence, {X} is always a finite subcover. It remains to show that R is lower semicontinuous. To prove it, we must show that for each x ∈ X, the set {y ∈ X|yP (R)x} is open in τ or equivalently {y ∈ X|yP (R)x} ∩ {m} = ∅. Indeed, suppose to the contrary that it is not the case. Then, there exists x∗ ∈ X such that {y ∈ X|yP (R)x∗ } ∩ {m} = ∅. Therefore, (m, x∗ ) ∈ P (R). By definition, for each a ∈ A we have that (a, m) ∈ P (R). Since R is a generalized quasi-ordering, we have that (a, x∗ ) ∈ P (R) or (x∗ , a) ∈ P (R). It follows that x∗ ∈ A, a contradiction to the minimal character of m. The last contradiction shows that for each x ∈ X, we have {y ∈ X|yP (R)x} ∩ {m} = ∅ (m ∈ / {y ∈ X|yP (R)m}). Therefore, R is lower semicontinuous. The converse is an immediate consequence of Theorem 3.2. The following corollary is an immediate consequence of Proposition 3.1 and Theorem 3.2. Corollary 3.4. Let (X, τ ) be a compact topological space and let R be a lower semicontinuous relation on X. Then, the Bordes, McKelvey and Duggan deep uncovered set are non-empty. Although Penn’s definition of Banks set restricts on finite agendas, we proceed by giving a topological characterization of the P -Banks set in a general form, where agendas are allowed to be infinite. Theorem 3.5. Let (X, R) be a compact topological space and let R be an upper semicontinuous relation on X. Then, the P -Banks set is non-empty. Proof. To prove that BP (X) = ∅, we have two cases to consider: (A) There exists x∗ ∈ X such that for each y ∈ X we have x∗ P (R)y; (B) For each x ∈ X there exists y ∈ X such that (x, y) ∈ / P (R) or equivalently ySx. In case (A), we have that: for each y ∈ X \ x∗ we have (x∗ , y) ∈ P (R). (27) Therefore, x∗ ∈ BP (R). In case (B), fix an x0 ∈ X. Then, there exists x1 ∈ X such that (x1 , x0 ) ∈ S. Let  0 = {z ∈ X|(x0 , z) ∈ P (R)}. Ξ

(28)

0 x1 Sx0 P (R)Ξ

(29)

x1 Sx0 P (R)y for each y ∈ Ξ0 .

(30)

Then, we have

by which notation we mean that We now repeat the procedure described by equation (11)-(19) of Theorem 3.2, and we construct a net ( xλ )λ∈Λ and  the sets Ξ = {z ∈ X|(x , z) ∈ P (R)}, λ ∈ Λ, such that λ

λ

X=



 μ when μ < λ.  and for each λ ∈ Λ, x λ S xμ P (R)Ξ Ξ λ

(31)

λ∈Λ

There are two cases to consider: (ω1∗ ) There exists a x ∗ ∈ X such that

( x∗ , x λ ) ∈ S for all λ ∈ Λ;

(32)

∗

(ω2 ) For each x ∈ X there exists λ ∈ Λ such that (x, x λ ) ∈ / S or equivalently ( xλ , x) ∈ P (R).

(33)

In case (ω1∗ ), for each y ∈ X we have x ∗ S xλ P (R)y for some λ ∈ Λ.

(34)

∗

In case (ω2 ) we arrived at a contradiction. Indeed, suppose that for each x ∈ X there exists λ ∈ Λ such that ( xλ , x) ∈ P (R). Then, we have that X=



({x ∈ X| xλ P (R)x}.

x λ ∈X

8

(35) (36)

By compactness, there exist x λ , ..., x λn ∈ Y such that 1  {x ∈ X| xλ P (R)x}. X= x λ ∈{ xλ ,..., xλ 1

n

(37)

}

λn }. Then, as in the proof of Theorem 3.2, we obtain the existence of a P (R)-cycle. Consider the finite set { xλ , ..., x 1 Thus, there exists at least one pair (xλ , xλ ), i ∈ {1, ..., n − 1} such that λi+1 > λi and xλ P (R)xλ , a i

i+1

i

i+1

contradiction (by the construction of ( xλ )λ∈Λ , λ > λ implies x λ S xλ or equivalently ( xλ , x λ ) ∈ / P (R)). The last contradiction shows that only the case (ω1∗ ) needs consideration. In this case, the set C = { x∗ } ∪ ∗ ∗ { xλ |λ ∈ Λ} is an external stable chain with respect to S and x ∈ M(X, C). It follows that x  ∈ BP (R) and thus BP (R) = ∅. The following corollary is an immediate consequence of Proposition 3.1 and Theorem 3.5. Corollary 3.6. Let (X, τ ) be a compact topological space and let R be a upper semicontinuous relation on X. Then, the Fishburn, Gillies, McKelvey, Richelson and Duggan deep uncovered set are non-empty. By [2, Theorem 5(c)], the Smith set SM(R) is equivalent to the set of maximal elements of S, i.,e., SM(R) = {x ∈ X|for all y ∈ X, ySx implies xSy}. Proposition 3.7. Let R be a binary relation on a set of alternatives X. Then, all versions of the Banks set and all versions of the uncovered set are subsetes of the Smith set. Proof. Let x belongs in one of the two versions of Banks set or in one of the eight versions of uncovered set. Suppose that ySx for some y ∈ X. Then, we have four cases to consider: (i) xP (R)y, (ii) xSy; (iii) xP (R)z1 Sy for some z1 ∈ X and (iv) xSz2 P (R)y for some z2 ∈ X. In any case, we have (x, y) ∈ S. Therefore, x belongs to the Smith set. The utility and choice theories are supported by the results on the functions that represent the decision-makers’ preferences. The notion of cosistency for a preference relation (binary relation) R on a set of alternatives X is connected with the existence of a representation of R, that is, a continuous function u such that xRy if and only if u(x) ≥ u(y), x, y ∈ X (utility function). In fact, if we have xRx1 R...Rxn y for some x1 , ..., xn ∈ X, then u(x) ≥ u(y) holds. Therefore, u(y) ≯ u(x) which is equivalent to (y, x) ∈ / P (R). Since xI(R)y if and only u(x) = u(y) and (x, y) ∈ I(R) implies that (y, x) ∈ / P (R) we conclude that consistency allows indifference (ties). Clearly, by definition, negative consistency allows indecisiveness ((x, y) ∈ N (R)). We give a general definition of negative consistency which allow indifference and indecisiveness. c Definition 3.5. A binary relation R is a generalized  negative P -consistent, if for all x, y ∈ X, (x, y) ∈ P (R) c implies (y, x) ∈ / P (P (R) ) ((y, x) ∈ P (R) I(R) N (R)).

If in the definition of the negative consistent binary relation defined by Duggan [6], we replace R with P (R), then we get the notion of a generalized negative P -consistent binary relation. Clearly, generalized negative P -consistency of R is equivalent to the consistency of the weak relation S of R. Theorem 3.8. Let R be a generalized negative P -consistent binary relation on X. The following conditions are equivalent: (i) The P -Banks set is non-empty, (ii) there exists a compact topology τ on X such that R is upper semicontinuous. Proof. To prove the implication ((i) ⇒ (ii)), let us suppose that the P -Banks set is non-empty. By Proposition 3.7 we have that the Smith set SM(R) is non-empty. Let τ be the excluded set topology generated by the Smith set SM(R) [22, Page 48] (it has as open sets all those subsets of X which are disjoint from SM(R), together with X itself). Then, X is compact under τ since every open cover of X includes X itself. Hence, {X} is always a finite subcover. It remains to show that R is upper semicontinuous.To prove this, we must show that for each x ∈ X, the set {y ∈ X|xP (R)y} is open in τ or equivalently {y ∈ X|xP (R)y} ∩ SM(R) = ∅. Indeed, suppose to the contrary that it is not the case. Then, there exists x∗ ∈ X such that {y ∈ X|x∗ P (R)y} ∩ SM(R) = ∅. Therefore, (x∗ , y ∗ ) ∈ P (R) for some y ∗ ∈ SM(R). Since P (R) ⊆ S ⊆ S, by [2, Theorem 5(c)] we have that (x∗ , y ∗ ) ∈ I(S). It follows that x∗ ∈ SM(R). Since R is negative consistent, (y ∗ , x∗ ) ∈ S implies that (x∗ , y ∗ ) ∈ / P (S). But then, (x∗ , y ∗ ) ∈ P (R) ⊆ S implies that (y ∗ , x∗ ) ∈ S, a contradiction to (x∗ , y ∗ ) ∈ P (R). The last contradiction shows that for each x ∈ X, we have {y ∈ X|xP (R)y} ∩ SM(R) = ∅. Therefore, R is upper semicontinuous. The converse is an immediate consequence of Theorem 3.5.

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The following corollary is an immediate consequence of Proposition 3.1 and Theorem 3.8. Corollary 3.9. Let R be a generalized negative P -consistent binary relation on X. The following conditions are equivalent: (i) One of versions of the Fishburn, Gillies, McKelvey, Richelson and Deep uncovered set is non-empty, (ii) there exists a compact topology τ on X such that R is upper semicontinuous.

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