They just need to have direct interaction with the opponent, preferably reward concepts like spacing, positioning, etc., and preferably have sets last around as long as a typical "Win two rounds" set from a fighting game. This would rule out most puzzle and racing games, but games like GoldenEye or the versus mode from Mario Kart would be run - if they were on the Saturn.
Also trying to do this on a budget, so cheaper is better. Quality is more or less irrelevant. Odd is preferable.
Here's what I currently have: