No, can't do this today, but I will investigate for a future enhancement. I will also put on my list a short-term workaround: add a layout token to display the Notes for a level. This would allow you to put the game type in the Notes field of a level on the Rounds tab, and use the token to display it in the layout.
Of course, if your game is the same throughout your tournament (you're playing Texas Hold 'Em all night, for example), you could use the Game Name field as a workaround. If you still wanted to display the game name, just hard-code it into the layout, since it doesn't change.
Another workaround, albeit a more painful one, is to make a property set for each level in the layout. For example, the contents of Game cell in the default layout looks like this:
<nobr><gametype> <gamename></nobr>
You could create a property set for round 1 with contents:
<nobr>No Limit <gamename></nobr>
And a property set for round 2 with contents:
<nobr>Spread Limit <gamename></nobr>
etc.