Yeah that doesn't seem like it would scale up very well if you have a huge number of games.
First thing you should do is build an index on (name,id) so that the selects will be quicker.
Also if your current game starts with 'C' for example, you could
select * from games where name between 'b' and 'd';
That way you limit how many games you're pulling out.
That works for postgres anyway - not sure about mysql.