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Mathematical Appendix Appendix A Mathematical Appendix This appendix explains very briefly some of the mathematical terms used in the text. There is no claim of completeness and the presentation is sketchy. It cannot substitute for a text on mathematical methods used in game theory (e.g. Aliprantis and Border 1990). But it can serve as a reminder for the reader who does not have the relevant definitions at hand. A.1 Sets, Relations, and Functions To begin with we review some of the basic definitions of set theory, binary relations, and functions and correspondences. Most of the material in this section is elementary. A.1.1 Sets Intuitively a set is a list of objects, called the elements of the set. In fact, the elements of a set may themselves be sets. The expression x ∈ X means that x is an element of the set X,andx ∈ X means that it is not. Two sets are equal if they have the same elements. The symbol0 / denotes the empty set, the set with no elements. The expression X \ A denotes the elements of X that do not belong to the set A, X \ A = {x ∈ X | x ∈ A},thecomplement of A in (or relative to) X. The notation A ⊆ B or B ⊇ A means that the set A is a subset of the set A or that B is a superset of A,thatis,x ∈ A implies x ∈ B. In particular, this allows for equality, A = B. If this is excluded, we write A ⊂ B or B ⊃ A and refer to A as a proper subset of B or to B as a proper superset of A. © Springer-Verlag Berlin Heidelberg 2016 223 C. Alós-Ferrer, K. Ritzberger, The Theory of Extensive Form Games, Springer Series in Game Theory, DOI 10.1007/978-3-662-49944-3 224 A Mathematical Appendix The set of all elements that belong to A or B is the union of A and B, denoted A∪B. The set of elements that belong both to A and to B is the intersection of A and B, denoted A ∩ B. { } λ ∈ Λ Asetofsets Aλ λ ∈Λ may be indexed by “names” or “labels” .Inthat case λ ∈Λ Aλ denotes the union of all indexed sets, that is, the elements that belong to Aλ for some λ ∈ Λ;and λ ∈Λ Aλ denotes the intersection of all indexed sets. A familiar set of sets is the power set of a given set X which is denoted by 2X. This is the collection of all subsets of X (including the empty set). Nonempty subsets of 2X are called families of sets. For an indexed family of subsets of a given set X the following useful identities, known as de Morgan’s laws, hold: X \ Aλ = (X \ Aλ ) and X \ Aλ = (X \ Aλ ).(A.1) λ ∈Λ λ ∈Λ λ ∈Λ λ ∈Λ The Cartesian product ×λ ∈Λ Aλ of a family {Aλ }λ ∈Λ of sets is the collection of all tuples {xλ }λ ∈Λ with xλ ∈ Aλ for all λ ∈ Λ. Each set Aλ is a factor in the product. Set theory can be based on axioms. The most popular axioms are the eight known as the Zermelo-Fraenkel (ZF) axioms. They are designed so as to guarantee that operations on sets, like unions or power sets, give again sets (and to avoid Russell’s paradox). It is often useful to complement the ZF axioms with a ninth axiom, which is both consistent with and independent of ZF set theory proper: Axiom of Choice If {Aλ | λ ∈ Λ } is a nonempty set of nonempty sets, then there is a function f : Λ →∪λ ∈Λ Aλ such that f (λ) ∈ Aλ for each λ ∈ Λ. In other words, the Cartesian product of a nonempty set of nonempty sets is itself a nonempty set. Though apparently innocuous, this axiom has a lot of hidden power. The axiom system emerging from adding the Axiom of Choice to ZF is known as Zermelo-Fraenkel cum Axiom of Choice (ZFC). There are a few equivalent formulations of the Axiom of Choice, two of which are mentioned below. A.1.2 Binary Relations A binary relation ≥ on a nonempty set A assigns to each ordered pair (a1,a2) ∈ A × A exactly one of the two statements “a1 is in relation ≥ to a2”or“a1 is not in relation ≥ to a2.” Since all ordered pairs (a1,a2) ∈ A × A that satisfy the first statement constitute a subset of the Cartesian product A × A, the relation ≥ can be identified with the set of all such pairs. Hence, a binary relation may be thought of as a subset of A × A. There are many important properties that a binary relation might satisfy. A binary relation ≥ on a set A is • reflexive if a ≥ a for all a ∈ A; • irreflexive if there is no a ∈ A such that a ≥ a; A.1 Sets, Relations, and Functions 225 • symmetric if, for all a1,a2 ∈ A, a1 ≥ a2 implies a2 ≥ a1; • asymmetric if there are no a1,a2 ∈ A such that a1 ≥ a2 and a2 ≥ a1; • antisymmetric if, for all a1,a2 ∈ A, a1 ≥ a2 and a2 ≥ a1 imply a1 = a2; • transitive if, given a1,a2,a3 ∈ A, a1 ≥ a2 and a2 ≥ a3 imply a1 ≥ a3; • complete (or connected) if, for all a1,a2 ∈ A, either a1 ≥ a2 or a2 ≥ a1 (or both); • total (or weakly connected) if a1 = a2 implies either a1 ≥ a2 or a2 ≥ a1 or both, for all a1,a2 ∈ A. A complete relation is necessarily reflexive and total, but a total relation may or may not be reflexive. Note also that an antisymmetric relation may or may not be reflexive. In the following a few prominent classes of binary relations (defined by the combination or properties from the list above that they fulfill) are discussed. A.1.2.1 Partial Orders A binary relation ≥ on a set A is a partial order if it is reflexive, transitive, and antisymmetric. For a partial order ≥ the notation > is often used to refer to its asymmetric part, that is, a1 > a2 if and only if a1 ≥ a2 and a1 = a2 for a1,a2 ∈ A.The set A endowed with a partial order ≥ is denoted (A,≥) and referred to as a partially ordered set or a poset. A relation ≥ on A is a linear order if it is total, transitive, and antisymmetric. If ≥ is a linear order, then for every pair (a1,a2) ∈ A×A exactly one of the statements a1 > a2, a2 > a1,ora1 = a2 holds. A preorder (or quasiorder) is reflexive and transitive. An antisymmetric preorder is a partial order. A set endowed with a preorder ≥ is referred to as a preordered set.. A chain in a poset (A,≥) is a subset that is totally ordered. Hence, any two elements of a chain are comparable. For a subset X ⊆ A an upper bound resp. lower bound is an element a ∈ A such that a ≥ x resp. x ≥ a for all x ∈ X.Agreatest resp. least element of X is an x ∈ X such that x ≥ x resp. x ≥ x for all x ∈ X.Anelement a ∈ A is a maximal resp. minimal element of A if there is no a ∈ A such that a > a resp. a > a . A nonempty subset X of A has at most one greatest resp. least element and, if it exists, it is maximal resp. minimal. For a subset X ⊆ A of a poset (A,≥) its supremum resp. infimum is its least upper bound resp. its greatest lower bound. That is, a ∈ A is the supremum of X ⊆ A if and only if a ≥ x for all x ∈ X and a ≥ x for all x ∈ X implies a ≥ a and analogously for the infimum. Note that supremum and infimum need not exist. The following two assertions are equivalent to the Axiom of Choice. That is, both are theorems if the axiom of choice is assumed; but if one of them is taken as an axiom, then the Axiom of Choice becomes a theorem. Zorn’s Lemma. If every chain in a poset A has an upper bound, then A has a maximal element. Hausdorff Maximality Principle. If c is a chain in a poset A, then there is a maximal chain in A that contains c. 226 A Mathematical Appendix As an example, the Hausdorff Maximality Principle states that plays exist for a tree, because plays are maximal chains. A.1.2.2 Lattices and Directed Sets A poset (A,≥) is a lattice if every pair of elements a,a ∈ A has a supremum and an infimum. A sublattice of a lattice is a subset that contains the suprema and infima for all its pairs of elements. A sublattice of a lattice (A,≥) is necessarily a lattice under the restricted partial order, but the converse statement is not true. A lattice is complete if every nonempty subset has a supremum and an infimum. A direction ≥ on a set A is a reflexive and transitive binary relation such that each pair has an upper bound. That is, for each a,b ∈ A there is some c ∈ A such that c ≥ a and c ≥ b. (Beware: A direction need not be antisymmetric.) A directed set (A,≥) is any set A endowed with a direction ≥. In Chap. 3, a subset A of a poset P was called directed if every finite subset of A had an upper bound in A; this is equivalent to stating that the partial order restricted to A is a direction. If (Aλ ,≥λ ) is a family of directed sets, for some index set Λ, then their Cartesian product ×λ ∈Λ Aλ is also a directed set under the product direction,definedby(aλ )λ ∈Λ ≥ (bλ )λ ∈Λ whenever aλ ≥λ bλ for each λ ∈ Λ.
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