Notes on a stochastic game with information structure

The game described below appeared in conversations after a talk given by Lloyd Shapley at the Second International Workshop in Game Theory in Berkeley, Summer 1970. This was one of the problems circulating around the conference to which several participants contributed. The general problem is to study two-person zero-sum games with stochastic movement among subgames in which the subgame being played is not precisely known, but in which information about this aspect may be gathered or partially revealed by the strategy choices of the players, the actual game being played, and chance. Without stochastic movement among the subgames, such a game would be a repeated game of incomplete information defined and studied by the Mathematica Inc. group at Princeton 1966–1968; see the book of Aumann and Maschler, ‘‘Repeated Games with Incomplete Information’’, MIT Press, 1995. Granted that the problem is interesting, one searches for the simplest non-trivial example of the class and is led to the following game. Subgames. There are two basic subgames, G1 with matrix ð0 1Þ, and G2