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The theory of pursuit games is obviously fragmentary at present. We know that general determinability of such games is incompatible with analysis, based on the principle of time continuity; but we also witness some reasonably successful probing on a smaller scale. The problem is one of existence of winning strategies for quite general sets and spaces. It will be shown here that in one case, where multivalued strategies are used, such strategies must necessarily be subclasses of Polish spaces and in the other, the monovalued case, the loser's set either has to be a first category set in the sense of Baire or an ideal, but in any case a kind of small set. This paper is meant to provide a common topological basis for the appreciation of more recent results.

Game‐theoretic studies1,2 have been extended and complemented by a coincidence theorem based on and closely related to the Lefschetz fixed‐point theory. It is shown (Theorem 4) that admissible Hausdorff‐spaces (e.g. Banach spaces) exhibit coincidences of pairs of maps associated with multi‐valued maps which have been introduced and discussed in previous papers in this Journal.1,2

Some of the basic ideas of pursuit‐evasion games have been presented in an earlier paper. This paper is an extension and an elaboration of these ideas with emphasis on the determinability of pursuit‐evasion games.

The result of a true strategy applied by a (perfectly informed) player in positional games represents a sequence of consecutive choices from a given set. It is, therefore, subject to the axiom of choice or some equivalent selection principle. Our attention in this study is focused on the existence of finite sequences associated with winning strategies. It will be shown in the sequel that the use of the axiom of choice may lead to sets devoid of winning strategies, while the negation of this axiom produces winning strategies for both players. A modified axiom, the axiom of determination, is discussed which no longer admits such “paradoxes” in virtue of certain inherent restrictions.