## ABSTRACT

We shall now discard some, but not all, of the restrictive assumptions made in chapter 3. The generalized theory represents a major step forward. It was first formulated by John von Neumann in 1932 and first published (in German) in the late thirties (von Neumann 1937); it was reprinted in English translation after World War II (von Neumann 1945). This publication history explains why the theory took so long to acquire its amply deserved general recognition. A second reason is the style of von Neumann’s paper; it is typical of papers in mathematics, extremely condensed, and the reader needs to have considerable mathematical knowledge before he can even start to comprehend it. Mathematically simpler proofs have been found later (Gale 1956, Karlin 1959), but these also are well beyond the mathematical level of this book. We therefore confine ourselves to quoting the important results and explaining their economic meaning, and we omit all proofs. (For a “diagrammatic” explanation of the von Neumann theory, see Koopmans 1964; for a very interesting “inside” view of the history of this theory, see Morgenstern 1977).