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FIXED POINT THEOREMS and APPLICATIONS to GAME THEORY Contents 1. Introduction 1 2. Convexity and Simplices 2 3. Sperner's Lemm

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FIXED POINT THEOREMS and APPLICATIONS to GAME THEORY Contents 1. Introduction 1 2. Convexity and Simplices 2 3. Sperner's Lemm FIXED POINT THEOREMS AND APPLICATIONS TO GAME THEORY ALLEN YUAN Abstract. This paper serves as an expository introduction to fixed point m theorems on subsets of R that are applicable in game theoretic contexts. We prove Sperner's Lemma, Brouwer's Fixed Point Theorem, and Kakutani's Fixed Point Theorem, and apply these theorems to demonstrate the conditions for existence of Nash equilibria in strategic games. Contents 1. Introduction 1 2. Convexity and Simplices 2 3. Sperner's Lemma 4 4. Brouwer's Fixed Point Theorem 6 5. Kakutani's Fixed Point Theorem 11 6. Nash Equilibria of Pure Strategic Games 13 7. Nash Equilibria of Finite Mixed Strategic Games 16 Acknowledgments 19 References 19 1. Introduction Game theory is a subfield of economics that describes how decision-makers in- teract. Although it makes many of the same assumptions as traditional models of economics (for example, rationality | the assumption that decision-makers pur- sue well-defined objectives), game theory differs from traditional economic models in that decision-makers attempt to obtain information about what other decision- makers will choose and take into account this information or the expectation of this information in making their own decision. We call this assumption reasoning strategically. In other traditional economic models (a competitive equilibrium model, for example), decision-makers generally only consider a set of parameters (e.g. prices) as given when making their decision, while in game theory, decision- makers seek to optimize their strategy based on whether their decision is optimal given what other decision-makers are expected to do. Game theory is commonly described using mathematics, which offers a flexible and abstract model to describe a variety of situations. A game is a description of a strategic interaction between different parties, called players, that describes constraints on what the players can do but not what they actually do. A solution systematically describes an outcome that might occur in a game | game theory predicts reasonable solutions for games. 1 2 ALLEN YUAN In this paper, we will investigate a basic solution concept in game theory, the Nash equilibrium. A Nash equilibrium is a solution in which each player correctly predicts what the other players will do and responds optimally, so that no player can improve their position by choosing differently. We seek to understand the conditions under which a Nash equilibrium is guaranteed to exist. To do this, we must return to mathematics to describe our conditions concisely and to prove that these conditions truly will yield a Nash equilibrium. To begin with, we define a mathematical structure, the simplex, in section 2 and explore some properties of this structure. In section 3, we prove Sperner's Lemma, a key result about simplices and their subdivisions that is crucial to the proof of Brouwer's Fixed Point Theorem, in section 4. We then extend Brouwer's Theorem for point- valued functions to Kakutani's Theorem for set-valued functions in section 5. In section 6, we apply the mathematics we have covered to game theory by defining the basic components of game theory mathematically. We then establish a set of conditions for the existence of a Nash equilibrium and prove that these conditions are sufficient. Finally, in section 7, we consider an non-deterministic extension of strategic games and show that any such extension of a game with a finite number of outcomes must have a Nash equilibrium. For this paper, we only assume some familiarity with introductory real analysis and microeconomic theory. 2. Convexity and Simplices We first define the notion of convexity and use convexity to describe the set of points that make up a simplex. We then describe simplicial subdivisions, which will be important in the proof of Sperner's Lemma. Definition 2.1. A set S ⊆ Rm is convex if for all x; y 2 S and λ 2 [0; 1] we have λx + (1 − λ)y 2 S. One way to think about this definition is if S is convex, then we can take any two vectors in S and connect their tips with a straight line segment, and all the vectors with tips on that line segment will also be in S. i 1 n Notation 2.2.x will denote the ith vector in a set of vectors x ;:::; x , while xi will denote the ith component of the vector x.[n] denotes the set f1; : : : ; ng. n X i 1 n Definition 2.3. λix is a convex combination of x ;:::; x if λi ≥ 0 for all i=1 n n X X i i 2 [n] and λi = 1. We call λix strictly positive if λi > 0 for all i 2 [n]. i=1 i=1 Definition 2.4. For A ⊆ Rm, the convex hull of A, denoted co(A), is the set of all finite convex combinations of points in A: ( n n ) X i i X co(A) = λix j x 2 A; λi ≥ 0 8i 2 [n]; λi = 1 i=1 i=1 n 1 n m X i Definition 2.5.x ;:::; x 2 R are affinely independent if λix = 0 and i=1 n X λi = 0 implies that λ1 = ··· = λn = 0. In other words, any zero linear i=1 combination of these vectors with coefficients that sum to 0 must be trivial. FIXED POINT THEOREMS AND APPLICATIONS TO GAME THEORY 3 x0 x0 x1 x2 x1 x2 Figure 1. A 2-simplex on the left and a closed 2-simplex on the right. Definition 2.6. An n-simplex is the set of all strictly positive convex combina- tions of an (n + 1)-element affinely independent set. An n-simplex T with affinely- independent vertices x0;:::; xn is defined by ( n n ) 0 n X i X T = x ··· x = λix j λi > 0 8i 2 f0; : : : ; ng; λi = 1 i=0 i=0 The standard n-simplex is the n-simplex formed by the n + 1 standard basis vectors in Rn+1. Example 2.7. A 0-simplex is a single point. A 1-simplex is a line segment (minus the endpoints). A 2-simplex is a triangle (minus the boundary). A 3-simplex is a tetrahedron (minus the boundary). See Figure 1 for an example of a 2-simplex. Definition 2.8. A closed n-simplex with vertices at the affinely independent vectors x0;:::; xn is the convex hull of the set of vectors fx0;:::; xng: ( n n ) 0 n X i X T = x ··· x = λix j λi ≥ 0 8i 2 f0; : : : ; ng; λi = 1 i=0 i=0 Note that the closure of an n-simplex is a closed n-simplex. The closure of the standard n-simplex is denoted ∆n, or ∆ if the dimension is evident. Definition 2.9. If k ≤ n, then the k-simplex xi0 ··· xik is a face of x0 ··· xn, where i0; : : : ; ik 2 f0; ··· ; ng and i0 < ··· < ik. Note that a closed n-simplex is also the union of all faces of an n-simplex with the same vertices. Definition 2.10. Let P(S) denote the power set of S. For y 2 co(fx0;:::; xng) n X i 0 n such that y = λix , let the set-valued function χ : co(fx ;:::; x g) ! P(f0; : : : ; ng) i=0 be defined by χ(y) = fi j λi > 0g. Note that if χ(y) = fi0; : : : ; ikg, then y 2 xi0 ··· xik . This face is called the carrier of y. Definition 2.11. If T = x0 ··· xn is an n-simplex, a simplicial subdivision of T is a finite collection of simplices fTi j i 2 [m]g (called subsimplices) such that m [ Ti = T and for all i; j 2 [m] we have Ti \ Tj = Ø or Ti \ Tj = Tij, where Tij is i=1 a common face of Ti and Tj. The mesh of a simplicial subdivision is the diameter of the largest subsimplex in the subdivision. See Figure 2 for an example. 4 ALLEN YUAN x0 x0 x3 x3 x1 x2 x1 x2 x4 x4 Figure 2. The closed 2-simplex to the left is not simplicially sub- divided because the subsimplices x0x1x4 and x0x2x3 do not have a common face as the intersection of their closures. The closed 2-simplex to the right is simplicially subdivided. Definition 2.12. For any m 2 N, the set of vertices n ki X V = fv 2 n+1 j v = where k 2 for all i 2 f0; : : : ; ng and k = mg R i m i N0 i i=0 forms a simplicial subdivision of ∆n. (Note here that N0 is the set of natural num- bers and 0.) This subdivision is called an equilateral subdivision. By increasing m, we can make the mesh of this subdivision arbitrarily small. See Figure 3. Definition 2.13. Given a simplex T = x0 ··· xn, the barycenter of T is given n 1 X by b(T ) = xi. For simplices T ;T , we say that T > T if T is a face n + 1 1 2 1 2 2 i=0 of T1 and T1 6= T2. Given a simplex T , the family of all simplices b(T0) ··· b(Tk) such that T ≥ T0 > T1 > ··· > Tk is a simplicial subdivision of T called the first barycentric subdivision of T . Further barycentric subdivisions are defined recursively and barycentric subdivisions can also have arbitrarily small mesh. See Figure 3. 3. Sperner's Lemma Definition 3.1. Given a simplicially subdivided closed n-simplex T = x0 ··· xn, let V be the set of all vertices of all subsimplices. A labeling function f : V ! f0; : : : ; ng is called a proper labeling of this subdivision if f(v) 2 χ(v) for all v 2 V (where χ is defined as in Definition 2.10).
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