The Mathematical Theory of Minority Games. Statistical Mechanics of Interacting Agents
Minority games are statistical mechanical models where individuals, “agents in the market”, try to optimize their strategy in an economical game. They change their behaviour on the basis of knowledge of the previous history of (the global state of) the market. The goal of the game is to become a member of the “minority which wins”. There is a random noise (“irrationality”) in the decision of each individual acting in the game and no direct interaction between individual participants of the market is assumed. Thus the situation resembles physical models studied in (nonequilibrium) statistical mechanics, in systems of the mean field type and in spin glasses. Phenomena like phase transitions (i.e. abrupt changes of the “macroscopic” characteristics of the system) have to be expected in such models and the aim is to mathematically analyze their nature. Of course, this is usually a very difficult problem (of nonequilibrium statistical mechanics of disordered models). So, in most cases, only numerical simulations can give some insight into the behaviour of such complicated systems. The book aims to give an introduction to the relevant mathematics so that the reader need not consult specialized texts on statistical mechanics. In short, this book offers a self-contained introduction to the applications of powerful methods, originally developed by physicists, to socio-economic systems.