Rock-Paper-Scissors remains a popular children's game. It's a simple 0-sum game with a mixed Nash equilibrium.
In practice, even if that is your goal, it's hard to generate a uniformly random choice from {rock,paper,scissors} which is independent from what you and your opponent have chosen before. While the unexploitable strategy is simple in theory, exploiting people is complicated, and can involve statistics and hidden-Markov models.
There is an gambling site which lets you play rock-paper-scissors against an opponent, charging a rake so that the Nash equilibrium strategy will lose on average.
Cryptographic issues arise if you want to be confident that a distant opponent's choice was not made with knowledge of yours.
