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I played multiple times and I found two strategies, and I think it depends on what you want to get out of this which is better to use. My first strategy, which is great if you don't care that you tie with Serendip or if it's better to tie with as much a possible. In this case, just cooperate in all 10 rounds, what I found is that the computer usually did whatever your previous move was, starting with cooperate in the first round every time, so if you cooperate every time you tie with 30 coins each.
My other option if you don't want to tie, but you want as many coins as possible I would cooperate for the first 9 rounds and compete in the last, in that case you would win with 32 coins and Serendip would lose with 27.