Index and Stability in Bimatrix Games: A Geometric-Combinatorial Approach (Lecture Notes in Economics and Mathematical Systems Book 560)

The index of an equilibrium in a game gives information about the stability of the equilibrium, for example with respect to game dynamics. Unfortunately, index theory is often very technical. This book presents a new geometric construction that visualises the index in an intuitive way. For example, a 3×n game, for any n, can be represented by a figure in the plane, from which one can read off any equilibrium, and its index as a geometric orientation. With this insight, the index can be characterised in strategic terms alone. Moreover, certain hyperstable equilibrium components are seen to have nonzero index. The construction gives an elementary proof that two-player games have a Nash equilibrium, and, in an unusual direction, the powerful fixed point theorem of Brouwer. Contents......Page 7 Introduction......Page 9 1 Equilibrium Components with Arbitrary Index......Page 14 1.1 Preliminaries......Page 15 1.2 The Lemke-Howson Algorithm......Page 22 1.3 Index Theory......Page 25 1.4 Construction of Equilibrium Components with Arbitrary Index......Page 31 2 A Reformulation of the Index for Equilibria in Bimatrix Games......Page 38 2.1 The Dual Construction......Page 39 2.2 Labelling and Characterisation of Nash Equilibria......Page 47 2.3 The Lemke-Howson Algorithm in the Labelled Dual Construction......Page 52 2.4 An Orientation for Nash Equilibria......Page 57 3 Sperner’s Lemma and Labelling Theorems......Page 66 3.1 Sperner’s Lemma......Page 67 3.2 The Application to Bimatrix Games......Page 79 3.3 A Topological Interpretation of the Dual Construction......Page 86 4 A Strategic Characterisation of the Index......Page 92 4.1 A Geometric Interpretation......Page 93 4.2 Some Technical Requisites......Page 98 4.3 A Game Theoretic Characterisation of the Index......Page 102 5 Outside Option Equilibrium Components......Page 107 5.1 A Generalised Version of Sperner’s Lemma......Page 108 5.2 The Index for Outside Option Equilibrium Components......Page 111 5.3 Degenerate Games and General Equilibrium Components......Page 119 6 Index Zero and Hyperstability......Page 122 6.1 Index Zero Labellings......Page 123 6.2 Index Zero Outside Option Equilibrium Components......Page 129 6.3 Restricted Duplication of Strategies and Index Zero: An Example......Page 143 References......Page 148 Symbolindex......Page 151 F......Page 153 P......Page 154 V......Page 155 The Index Of An Equilibrium In A Game Gives Information About The Stability Of The Equilibrium, For Example With Respect To Game Dynamics. Unfortunately, Index Theory Is Often Very Technical. This Book Presents A New Geometric Construction That Visualises The Index In An Intuitive Way. For Example, A 3a-n Game, For Any N, Can Be Represented By A Figure In The Plane, From Which One Can Read Off Any Equilibrium, And Its Index As A Geometric Orientation. With This Insight, The Index Can Be Characterised In Strategic Terms Alone. Moreover, Certain Hyperstable Equilibrium Components Are Seen To Have Nonzero Index. The Construction Gives An Elementary Proof That Two-player Games Have A Nash Equilibrium, And, In An Unusual Direction, The Powerful Fixed Point Theorem Of Brouwer. The index of an equilibrium in a game gives information about the "stability" of the equilibrium, for example with respect to game dynamics. Unfortunately, index theory is often very technical. This book presents a new geometric construction that visualises the index in an intuitive way. For example, a 3C B C B C B C B C B C B C B C B n game, for any n, can be represented by a figure in the plane, from which one can read off any equilibrium, and its index as a geometric orientation. With this insight, the index can be characterised in strategic terms alone. Moreover, certain "hy Projektmanagement ist in Zeiten von enger werdenden Projektbudgets und Projektzeitplanen eine zunehmende Herausforderung geworden. Ein weiterer Schwerpunkt des Buches ist der konkrete Einsatz eines Werkzeuges, das den gesamten Prozess aus Sicht des Projektleiters steuert.