What ML algorithm can I use for building a "recommended" list for players?
Before jumping into machine learning solutions, it would be good to think more about the problem you're solving. If there are only 20 games and some are unavailable at any given time, then a well laid-out menu with good navigation is superior to a recommender system. Recommender systems are only appropriate when people cannot adequately parse all of the available options.

If you do want personalized recommendations, you don't even have to start with machine learning models. You can simply recommend that players keep playing the same games or the most popular games.

And if it turns out that machine learned models are best, I suggest looking at association rule mining based on unary data (which gives you shopping-basket recommendations: people who played games A, B, and C also played games D and E) or some variety of collaborative filtering based on ratings data (which gives you a user-item preference space). That totally depends on what sort of feedback you get from users about their individual game experiences.