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Vaniver comments on Measures, Risk, Death, and War - Less Wrong
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I stopped at 45 rounds because I had hit a pretty 15-15-15. I think the most I was ahead was when it was about 10-7-6. I found that I could do well in the short run by swapping patterns- 'play what would have lost to what he played', and then when he picks up on that switching to what would beat him if he believes that about me. It then got harder / I stopped putting as much effort into it.
If you click on the "what he's thinking" thing, it looks like he just has a 3^8 lookup table based on the last four rounds. Given that game state, he throws against whatever the most likely human action was- which suggests it might be possible to infer that lookup table from his behavior then use it to find a stable loop you can mine (until you dominate that part of the lookup table). It would probably be unethical to write an AI to beat their AI, though, since that would be screwing with their data about humans.
As a way to conserve effort, you can just never throw rock, and try to pick scissors/paper at random. This is sufficiently unusual behavior that the 3^8 lookup table should fail enough to give you a small but stable edge. I went 10-7-7 doing this.