Results and problems in games of timing

Abstract

This paper presents history of investigations concerning a subclass of zero-sum two-person games called duels, which were initiated by David Blackwell and other mathematicians in reports of RAND Corporation in 1948-52. The second part of paper discusses mutual rela- tionships between discrete and non-discrete duels, and gives a review of recent more general results. The paper also discusses some open problems in general theory and makes hypothesis on them (strongly suggested by previous results). In 1948 RAND Corporation collected a team of mathematicians, statisticians, economists, and social scientists to analyze the uncertainties in global world situation and to construct a blue-print for optimal operation against that. One of results of this study, achieved within a component of program, was solution of many problems formulated in form of some zero-sum two-person games, called duels, or games of timing when considered in a more general sense. Particularly, David Black- well, one of members of that team, was very instrumental in formulating and solving several versions of such games. He, together with M. Shiffman, M. A. Girshick, L. S. Shapley, R. Bellman, I. Glicksberg and others, initi- ated a new topic within zero-sum games at that time, and recognized wide scope of possible applications of games of timing, particularly in description and explaining of some conflict situations in economics. Since then many new general problems in games of timing have been formulated and many important and interesting results have been achieved. However, to say more in detail about it, at first, we must give a definition of games of timing in a sufficiently general form to include whole rich collection of all different duel-models studied in literature. We will do it in a slightly different (but equivalent) convention in comparison to that adopted by pioneers of this topic mentioned above. Consider following model of a zero-sum game: There are two Players 1 and 2 with initial amounts ?? and Mi of some homogeneous resources, respectively. It is assumed that they should distribute some or all of their