Random walks are powerful tools to analyze spatial-temporal patterns produced by living organisms ranging from cells to humans. At the same time, it is evident that these patterns are not completely random but are results of a convolution of organisms' sensor-based information processing and motility. The complexity of the first component is reflected in the statistical characteristics of trajectories produced by an organism -- when it is, e.g., foraging or searching for a mate (or a pathogen) -- and therefore some knowledge about the component can be obtained by analyzing the trajectories with the standard toolbox of methods used for random walks. Here we consider trajectories which appear as the results of a game played by two players on a finite square lattice. One player wants to survive, i. e., to stay within the interior of the square, as long as possible while another one wants to reach the adsorbing boundary. A game starts from the center of the square and every next movement of the point is determined by independent strategy choices made by the players. The value of the game is the survival time that is the number of steps before the adsorption happens. We present the results of a series of experiments involving both human players and an autonomous agent (bot) and concentrate on the probability distribution of the survival time. This distribution indicate that the process we are dealing with is more complex than the standard random walks.