1. Introduction

There are two game theories. The first is now sometimes referred to as matrix game theory and is the subject of the famous von Neumann and Morgenstern treatise [1944]. In matrix games, two players make simultaneous moves and a payment is made from one player to the other depending on the chosen moves. Optimal strategies often involve randomness and concealment of information. The other game theory is the combinatorial theory of Winning Ways [Berlekamp et al. 1982], with origins back in the work of Sprague [1936] and Grundy [1939] and largely expanded upon by Conway [1976]. In combinatorial games, two players move alternately. We may assume that each move consists of sliding a token from one vertex to another along an arc in a directed graph. A player who cannot move loses. There is no hidden information and there exist deterministic optimal strategies.

In the late 1980's, David Richman suggested a class of games that share some aspects of both sorts of game theory. Here is the set-up: The game is played by two players (Mr. Blue and Ms. Red), each of whom has some money. There is an underlying combinatorial game in which a token rests on a vertex of some finite directed graph.