Steinhaus played a significant part in building of the mathematical game theory. He was its precursor and co-originator. In 1925 he published the work Definitions for a theory of games and pursuits (in polish, „Myśl Akademicka”, pp. 13-14). The work was translated and published in Naval Research Logistics Quarterly in1960 (no 7, pp. 105-108). He defined the idea of strategy (before John von Neumann, he called it as “mode of play”) and developed the mathematical foundation of game theory and its application.

According to Steinhaus, the researches on game theory was connected with a trial of mathematization of probability theory. Until Steinhaus this theory hadn’t been treated as a proper mathematical theory because of insufficient mathematical tools. In the sixth problem David Hilbert postulated an axiomatization of it as a method of its mathematization. In 1923 H. Steinhaus and A. Łomnicki published in „Fundamenta Mathematicae” two papers in which they demonstrated how we can take the measure theory as a foundation of proability theory. In paper Les probabilités dénombrables et leur rapport à la théorie de la mesure Steinhaus mathematized the « heads and tails » game. In that paper important tools became « independent functions ». Those ideas were generalized and developed in construction of probality based on concept of measure by A. Kolmogorov.

Some years later Steinhaus (with J. Mycielski) formulated the axiom of determinateness (A mathematical axiom contradicting the axiom of choice, 1962). Besides game theory it plays a significant role in researches of set theory. It implies the negation of the axiom of choice but, on the other hand, it gives possibilities of building a weaker version of set theory (it allow to avoid problems connected with the use of the choice axiom).