I'm doing some research on motivational factors in games to improve the playability of games. The first one that popped up are the need to improve and the need to better than other players. This is a nice example of linked motivational factors that are at the same time different. To be better than other players a gamer has to improve him/herself to a certain point and might even have to keep improving if there's a lot of competition. If a gamer feels the need to improve, the rate of which can depend totally on the desire to improve.
In a serious game this might result in people wanting to be the smartest or people wanting to be smarter.