Take the points in Ryan's answer and Lauren's comments as good guides. There is a different and practical answer to your immediate question, though, and it relates to what complementary colors are and how our eyes react to one.
Two colors are optically/photographically complementary if when added together (as light) would give white. Red/Cyan, Green/Magenta, Blue/Yellow are strict complements. On a traditional color wheel the complementaries are slightly different, but not enough to make a practical difference here.
The big problem with complementary colors is that, placed side by side, they are very uncomfortable on the eyes. They "glare". Take a look:
These are extreme examples, to illustrate the point, but the principle applies anywhere you use complementary colors indiscriminately. Darker or less saturated versions of the colors will help (yellow on blue is slightly less uncomfortable because 100% blue appears darker to the eye), but the glare stems from the way that the eye reacts to color.
A strictly complementary scheme, therefore, is probably not a good idea for a game. It will be hard on the eyes, so users will feel uncomfortable quickly and won't want to play any more.
Monochromatic is possibly a bit dull for a game (not that it has to be -- it depends on the audience), so triadic, split complementary or a combination of triadic with occasional bits of complementary color as accents, would work well.
If you pop over to your nearest art supply store, you can find a copy of the Grumbacher color wheel, which comes with quite a decent little book explaining the schemes with good examples of their use in art. Remember that art (painting) is a very mature technology, refined over millenia. Proven color techniques from painting are fundamental to all of the visual arts, including theater, film, architecture and interior design, so it's good to know and understand them.
