Topics in Game Theory

The first part of this book is about Decision Form Games. In the preliminaries we explain the Banach fixed point theorem. In the first chapter we introduce the Decision-form Games, the Strategy spaces and strategy base of a decision form game, the decision rules, the concept of possible reaction, some classic examples, disarming strategies, subgames, rules induced by utility functions, rules induced by preorders, the Cournot decision game, and a first price auction decision form game. Chapter 2 is about Non-Reactivity, in particular: Reactive strategies; Reduced games by elimination of non-reactive strategies; elimination of non-reactive strategies; iterated elimination of non-reactive strategies; solvability by iterated elimination; example of resolutions; iterated elimination in Cournot game; iterated elimination survival; survival of equilibria; Nested compacts lemma; Non-reactive strategy and equilibria; Reactivity. Chapter 3 is about: super-reactive strategies; comparison of reactivity; the reactivity preorder; the reactivity of a strategy; super-reactive strategies and maxima; maximally reactive strategies; sub-reactive strategies; elimination of sub-reactive strategies; iterated elimination of sub-reactivity; relative super-reactivity; dominant strategies; dominant and super-reactive strategies; the preorder of dominance; dominance and reactivity; non-reactivity and strong dominance. Chapter 4 deals with Mixed Extensions; Mixed extension of vector correspondences; Mixed extension of finite decision form games; extension of finite univocal games; extension of the finite multivocal games. In Part II the book deals with normal form games and applications; payoff space for C1 games; the critical space of the game; cooperative phase: selection of Pareto bistrategies; example in dimension 2 and 3; non-cooperative friendly phase; properly non-cooperative (egoistic phase); defensive phase; offensive phase; Selection of Pareto bistrategies. Part III is devoted to Algorithms for Normal Form Games; Algorithm for Payoff space in C1-Games; payoff trajectories in C1 parametric games.