Proof for d 1 this is obvious in a string of bits

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Proof. For d = 1, this is obvious: In a string of bits that starts with 0 and ends with 1, the number of bit flips is odd. For d = 2, we will count in two ways the set Q of pairs consisting of a small triangle and an edge labeled 12 on that triangle. Let A 12 denote the number of 12-labeled edges of small triangles that lie on the boundary of the big triangle. Let B 12 be the number of such edges in the interior. Let N abc denote the number of small triangles where the three labels are a , b and c . Note that N 012 + 2 N 112 + 2 N 122 = | Q | = A 12 + 2 B 12 , because the left-hand side counts the contribution to Q from each small triangle and the right-hand side counts the contribution to Q from each 12-labeled edge. From the case d = 1, we know that A 12 is odd, and hence N 012 is odd too. For another proof, see Figure 5.3. Remark 5.2.5 . Sperner’s Lemma can be generalized to higher dimensions. See § 5.4. 5.2.3. Brouwer’s Fixed-Point Theorem. Definition 5.2.6 . A set S R d has the fixed-point property (abbreviated f.p.p. ) if for any continuous function T : S S , there exists x S such that T ( x ) = x . Brouwer’s Theorem asserts that every closed, bounded, convex set K R d has the f.p.p. Each of the hypotheses on K in the theorem is needed, as the following examples show: (1) K = R (closed, convex, not bounded) with T ( x ) = x + 1. (2) K = (0 , 1) (bounded, convex, not closed) with T ( x ) = x/ 2. (3) K = x R : | x | ∈ [1 , 2] (bounded, closed, not convex) with T ( x ) = - x . Remark 5.2.7 . On first reading of the following proof, the reader should take n = 2. In two dimensions, simplices are triangles. To understand the proof for n > 2, § 5.4 should be read first. Theorem 5.2.8 ( Brouwer’s Fixed-Point Theorem for the Simplex ) . The standard n -simplex Δ = { x | n i =0 x i = 1 , i x i 0 } has the fixed-point property. Proof. Let Γ be a subdivision (as in Sperner’s Lemma) of Δ where all triangles (or, in higher dimension, simplices) have diameter at most . Given a continuous
106 5. EXISTENCE OF NASH EQUILIBRIA AND FIXED POINTS * Figure 5.3. Sperner’s Lemma: The left side of the figure shows a la- beling and the three fully labeled subtriangles it induces. The right side of the figure illustrates an alternative proof of the case d = 2: Construct a graph G with a node inside each small triangle, as well as a vertex outside each 1-3 labeled edge on the outer right side of the big trian- gle. Put an edge in G between each pair of vertices separated only by a 1-3 labeled edge. In the resulting graph G (whose edges are shown in PURPLE), each vertex has degree either 0, 1 or 2, so the graph con- sists of paths and cycles. Moreover, each vertex outside the big triangle has degree 0 or 1, and an odd number of these vertices have degree 1. Therefore, at least one (in fact an odd number) of the paths starting at these degree 1 vertices must end at a vertex interior to the large triangle. Each of the latter vertices lies inside a properly labeled small triangle.

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