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Über dieses Buch

This monograph is a treatise on adjustment processes. We consider price adjustment processes in exchange economies and strategy adjustment processes in noncooperative games. In the most simple version of an exchange economy, i.e. a pure exchange economy, there exist markets on which prices are determined by the demand and supply created by a finite number of consumers willing to exchange their initial endowments in order to maximize their utilities. An equilibrium situation is attained if, for some price vector, demand equals supply in all markets. Starting from a situation not being an equi­ librium an adjustment process reaches an equilibrium via adaptations of prices. The advantage of the adjustment processes we will present in this monograph is that they exist and converge under far weaker assumptions than existing processes. The second subject concerns the problem of finding Nash equilibria in noncooperative games. A Nash equilibrium is a situation from which no player can improve his position by unilaterally changing his strategy. We present a new algorithm for finding such equilibria. The sequence of stra­ tegy vectors generated by the algorithm can be interpreted as the path followed by a strategy adjustment process.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract
This monograph is a treatise on adjustment processes. We consider price adjustment processes in exchange economies and strategy adjustment processes in noncooperative games.
Antoon van den Elzen

Chapter 2. Preliminaries

Abstract
In this chapter we review some mathematical concepts which are needed in the sequel. The chapter is divided into four parts. In Section 2.1 we introduce some notational conventions and present the definitions of a unit simplex and a simplotope. The latter two are the sets on which almost all problems considered later on are defined. Section 2.2 deals with the stationary point problem. In fact, all the problems considered in this monograph can be viewed upon in this manner. In Section 2.3 we consider some notions being related to simplicial algorithms. Such algorithms serve to find a stationary point of an arbitrary continuous function. In later chapters we will see how these algorithms are related to the processes considered in this monograph. Finally, in Section 2.4 we discuss some concepts from differential topology. These notions are of importance when we consider the existence of the adjustment processes.
Antoon van den Elzen

Chapter 3. Existence of adjustment processes

Abstract
In this chapter we investigate under what conditions the processes to be considered in this monograph exist and converge. For that we consider one specific process. All processes treated in this monograph, except the ones in Chapter 6, have the same properties and their existence can be studied along the same lines.
Antoon van den Elzen

Chapter 4. An adjustment process for an international trade model

Abstract
In this chapter we consider price adjustments resulting from shocks in a two country model. For that purpose we present a price adjustment process that reaches an equilibrium price vector when starting from an arbitrarily chosen price vector. The latter vector can be interpreted as the equilibrium price prevailing in the economy before the shock took place.
Antoon van den Elzen

Chapter 5. An adjustment process for an exchange economy with linear production technologies

Abstract
In this chapter we propose a process that reaches an equilibrium in an exchange economy with linear production via adaptations of prices and activity levels. The process we consider here is a generalization of the sign process for a pure exchange economy (see van der Laan and Talman [1987a] and Section 3.1 of Chapter 3). It is a generalization in the sense that for the special case of an exchange economy without production both processes are the same.
Antoon van den Elzen

Chapter 6. Finding Nash equilibria in noncooperative games

Abstract
In this chapter we consider the problem of finding Nash equilibria in mixed strategies for noncooperative games, in which the payoffs are listed in matrices or more generally in tensors. The main part of this chapter deals with bi-matrix games. Such a game is a noncooperative two-person game with a finite set of actions for each player. The Nash equilibrium is the standard equilibrium concept for a noncooperative game. It states that a strategy is an equilibrium when no player can improve upon his situation by deviating from his strategy while all other players keep on playing their strategies.
Antoon van den Elzen

Backmatter

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