In the paper several infinite two-person games are studied, all having the following common structure: Player 1 (Emitter) produces a binary periodic sequence; Player 2 (Predictor) observes some initial segment of this sequence and then tries to predict the next digit. The payoff to Emitter is zero if the prediction is a correct one. The games differ in additional assumptions--those are in particular: Predictor required to make his prediction after observing a prescribed number of digits of the sequence; Predictor allowed to observe any number of digits but earning a decreasing amount for each correct prediction as the number increases; The period of the emitted sequence being chosen by random from some fixed distribution; Emitter allowed to choose the period but being paid a decreasing amount for incorrect prediction as the period increases. Combining these assumptions two zero-sum and two nonzero-sum games are obtained. It is shown that all these games possess a solution, some are at least partially solved and their further properties investigated. (Author)