By now the reader will have suspected that matrix games must play an important role in the theory of zero-sum two-person games and that they are much more inclusive than they may have appeared at first sight. Indeed, it may be asserted that all finite zero-sum two-person games can be reduced to matrix games. The critical reader may justly complain that this statement is meaningless since wehave never defined what constitutes a game. Furthermore, although he understands vaguely what is meant by the word “game,” his ideas on the subject are nonmathematical and give no clue as to how to verify the connection between these and the mathematical concept of a matrix game. The heart of the problem is close to this confusion; in fact, our assertion above was a non-mathematical assertion. To verify that it is true, we must provide an axiom system with its interpretation for finite games in general and show that this coincides, as an interpreted system, with the notion of a matrix game for zero-sum two-person games. We will be concerned only with the coincidence of certain aspects, indeed, exactly the strategic aspects which formed the basis of our definition of a solution. We have already solved this problem in the special instance of Simplified Poker; although the payoff matrix contained none of the verbal apparatus of the game given by its rules, we were convinced that it carried enough of the strategic possibilities to find a solution.
Therefore, our main problem in this chapter will be to give an axiom system with its interpretation which formalizes our intuitive notion of a game. The strict purist will not be satisfied with these axioms since they are framed in geometric language which could be axiomatized in turn. However, this is a mere quibble; we will make constant use of the intuitive insight given by the geometric model.
Throughout this chapter, passages that serve as motivation, interpretation, or heuristic discussion will be placed in square brackets […]. This is done to emphasize the independence of the mathematical deductions of these sections.
[In hunting for a geometrical scheme that will represent the combinatorial possibilities of an arbitrary game adequately, one might proceed by imagining the various