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People have a lot of attitudes they can apply to make Paper Rock Scissors interesting. Some people think they have the gift of crafty evasion, strong enough to defeat your superior intelligence. Other people think they are generally smarter than most people they meet, and relish the chance to show it. Other people think the accidental outcomes get funnier and funnier. These are just a start.
As a computer programmer working in survey research, I was challenged by the idea that I could not choose a random sample by pulling numbers out of my head. If I did, they said I would favor some number over others. Experiments were done, and participants failed miserably (citation needed). Pointing randomly to digits in a phone book would fail likewise, as some people wanted and got a number like 123-4000.
I looked up Nash Equilibrium in Wikipedia, and it looks like there is no Nash Equilibrium point for Rock-Paper-Scissors playing against a random strategy. You cannot improve your chances by playing a strategy. As a game-writer, I figured people can't play a random strategy, so they can be beaten. The simplest test would be that they can't help favoring one of their three options, so the program should play against the option they have favored. So if you do the laziest thing, playing the same option every time, playing against my algorithm, you will soon start losing all the time.
I think it is still true that there are predictable sequences of digits in the standard builtin random number functions. Leaving that aside, once the computer abandons attempts to be random, any strategy can be beaten. If you knew my algorithm and did the calculation yourself, you could choose the option that beats it, and accumulate a lopsided winning score.