An approach to formalization of decision-makers awareness conditions is that a vector of uncontrolled variables should be defined for each participant of a game. In this paper, we consider random uncontrolled factors that determine the information situation in the game. Much attention is given to an introduction of subjective representation of the game situation for each player. This subjective representation affects the vector of uncontrolled variables that determines the information situation and, consequently, a choice of strategy of each participant of the game. Also, approaches to formalization of problems with two decision-makers depending on changes of the vector of information parameters are proposed. We demonstrate formalization techniques for the game with the vector of informational parameters of random nature. Such techniques are based on reduction of the game problem to one-dimensional and multi-dimensional variational calculus problems. Sets of player’s strategies in similar problems can be described by some area in functional spaces. We also present an approach to problem formalization with two decision-makers as a Bayesian game with an asymmetric awareness of players as a special case of the game information situation definition.