Fixed Point Theorems with Applications to Economics and Game TheoryCambridge University Press, 1985 - 129 páginas One of the problems in economics that economists have devoted a considerable amount of attention in prevalent years has been to ensure consistency in the models they employ. Assuming markets to be generally in some state of equilibrium, it is asked under what circumstances such equilibrium is possible. The fundamental mathematical tools used to address this concern are fixed point theorems: the conditions under which sets of assumptions have a solution. This book gives the reader access to the mathematical techniques involved and goes on to apply fixed point theorems to proving the existence of equilibria for economics and for co-operative and noncooperative games. Special emphasis is given to economics and games in cases where the preferences of agents may not be transitive. The author presents topical proofs of old results in order to further clarify the results. He also proposes fresh results, notably in the last chapter, that refer to the core of a game without transitivity. This book will be useful as a text or reference work for mathematical economists and graduate and advanced undergraduate students. |
Contenido
Introduction models and mathematics | 1 |
Convexity | 9 |
Simplexes | 19 |
Sperners lemma | 23 |
The KnasterKuratowskiMazurkiewicz lemma | 26 |
Brouwers fixed point theorem | 28 |
Maximization of binary relations | 31 |
Variational inequalities price equilibrium and complementarity | 38 |
Selection theorems for correspondences | 69 |
Fixed point theorems for correspondences | 71 |
Sets with convex sections and a minimax theorem | 74 |
The FanBrowder theorem | 78 |
Equilibrium of excess demand correspondences | 81 |
Nash equilibrium of games and abstract economies | 88 |
Walrasian equilibrium of an economy | 95 |
More interconnections | 104 |
Some interconnections | 44 |
What good is a completely labeled subsimplex | 50 |
Continuity of correspondences | 53 |
The maximum theorem | 63 |
Approximation of correspondences | 67 |
The KnasterKuratowskiMazurkiewiczShapley lemma | 109 |
Cooperative equilibria of games | 112 |
| 122 | |
| 127 | |
Términos y frases comunes
abstract economy assume assumptions balanced family binary relation Brouwer Theorem budget set Chapter characteristic function coalition commodity compact and convex compact and nonempty compact convex set compact convex values compact-valued completely labeled subsimplex consumer consumer's consumption vectors continuous function contradiction convex cone convex set Corollary correspondence with nonempty Debreu Define Definition Let denote dist equilibrium price Exercise family ẞ finite fixed point theorem follows free disposal equilibrium Ichiishi implies intersection K-K-M lemma Kakutani Let EC Rm Let f Let KC lower hemi-continuous maximal elements minimax theorem n-simplex Nash equilibrium neighborhood nonempty compact convex nonnegative open graph outcome partition of unity player preferences price vector problem Proposition quasi-concave satisfy the hypotheses selection theorem sequence simplexes Sperner's lemma strategic form strategy sets strategy vector subdivision subset Suppose Theorem 7.2 Theorem Let upper hemi-continuous utility Walras Walrasian equilibrium