A person that plays game is called a player.
A game is something people do for fun, it can be anything or in any form. It is not necessarily a particular type of game.
From the prompt, it is obvious that the character is a game freak. He is a player of games. He loves playing games of any kind.
Another interesting part of the story is that you cannot play a game alone, you need one or more persons to play. This is where the fun is. A contest between equals.
So, a player of games is a person that loves and enjoys playing games with people.
You can also participate by clicking on this link