Gaming machines utilize the output of random number generators (RNGs) in the presentation of games, wherein one or more of the RNGs is each assigned to a single player. Each uniquely assigned RNG is used to determine the outcome of the games played only by the assigned player, such as games presented by casino gaming machines or via remote or mobile devices. An RNG may also be stopped or paused and then restarted, whereby a sequence of random data remains continuous or contiguous relative to a number of games played, such as by the single player to which the RNG is assigned. Multiple players may also time-share a gaming machine in a manner whereby the gaming machine acts as a unique device to each player.