Thanks for playing our game! In addition to what snoutfroth told you about the Oracle algorithm it there is a statistical method for rock-paper-scissor game rock-paper-scissor article, but it requires quite a lot of input data ~200,000 examples as the article states. With that much data any decent machine learning/ deep learning system could be used for a quick implementation of a predictive system. On the other hand people that played Oracles after 100 tries were able to go under 40% accuracy (algorithm accuracy). I guess it depends on the usage of it if it will work or not.
Sorry for sharing the boring stuff :D
Cheers!