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Unformatted text preview: 14.102, Math for Economists Fall 2005 Lecture Notes, 9/15/2005 These notes are primarily based on those written by George Marios Angeletos for the Harvard Math Camp in 1999 and 2000, and updated by Stavros Panageas for the MIT Math for Economists Course in 2002. I have made only minor changes to the order of presentation, and added some material from Guido Kuersteiner’s notes on linear algebra for 14.381 in 2002. The usual disclaimer applies; questions and comments are welcome. Nathan Barczi [email protected] 2.7 Continuity and Upper/Lower Hemicontinuity Kakutani’s f xed point theorem weakens the conditions of Brouwer’s theorem so that it applies to more games  indeed, to all f nite strategicform games. ’Finite’ refers to the number of players and the actions they have to choose from; Glenn will go over this, as well as the distinction between strategicform and extensiveform games, in more detail. He will also discuss how such games are interpreted to f t the conditions of the theorem. For now, our concern is to achieve an understanding of those conditions. Kakutani’s theorem is as follows: Theorem 86 (Kakutani) Let Σ be a compact, convex, nonempty subset of a f nitedimensional Euclidean space, and r : Σ ⇒ Σ a correspondence from Σ to Σ which satis f es the following: 1. r ( σ ) is nonempty for all σ ∈ Σ . 2. r ( σ ) is convex for all σ ∈ Σ . 3. r ( · ) has a closed graph. Then r has a f xed point. Everything in this theorem is familiar from our previous discussion, with the exception of the third requirement for r , that it have a closed graph. This property is also referred to as upperhemi continuity. De f nition 87 A compactvalued correspondence g : A ⇒ B is upper hemi continuous at a if g ( a ) is nonempty and if, for every sequence a n → a and every sequence { b n } such that b n ∈ g ( a n ) for all n , there exists a convergent subsequence of { b n } whose limit point b is in g ( a ) . 16 In words, this says that for every sequence of points in the graph of the correspondence that converges to some limit, that limit is also in the graph of the correspondence. This means that we don’t ’lose points’ in our graph at the limit of a convergent sequence of points in the graph, and important property for ensuring that we have a f xed point. There is also a property called lower hemicontinuity: De f nition 88 A correspondence g : A ⇒ B is said to be lower hemicontinuous at a if g ( a ) is nonempty and if, for every b ∈ g ( a ) and every sequence a n → a , there exists N ≥ 1 and a sequence { b n } ∞ n = N such that b n → b and b n ∈ g ( a n ) for all n ≥ N . In words, this says that for every point in the graph of the correspondence, if there is a sequence in A converging to a point a for which g ( a ) is nonempty, then there is also a sequence in B converging to b ∈ g ( a ) , and that every point b n in that sequence is in the graph of a n ....
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 Vector Space, scalar multiplication

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