Well, kind of a game, anyway. You remember Six Degrees of Kevin Bacon, right? If you dig into IMDb’s advanced search functions you can enjoy a sort of solitaire version.
On the homepage of imdb.com, select the pulldown menu next to the search box and choose Advanced Search. From there you can search by Collaborations and Overlaps by entering the names of two actors. How else would I have known that Rob Lowe and Tracy Morgan both appeared on the 2004 ESPYs? (Works for crew as well as cast.)
For more fun — or research, or time killing — the Advanced Name Search lets you search actors by place of birth (or death), height and star sign. Jennifer Grey and I are both 5-foot-3 Aries from New York. (I think we went to the same high school, too, actually. Bonus: when I sort by ascending height, the title of the page becomes “Shortest Aries Females….”) With the Name Text Search you “select a section to search within (mini biography, trivia, quotes) and enter a word to search for (e.g. “arrested”).”
That one, by the by, turns up 483 names if you search within bios and 526 if you pick trivia. Have at it.
No Happy Medium 
It appears that I am the Asian Alfonso Ribeiro. That sounds about right.