Not a game per se, "Hugs and Kisses." My cat had to go on a special diet for a few months. I put down her morning food, and she balked at the change. I was leaving for work and feeling like I should stick with her for the day. So I grabbed her, gave her a kiss on the head and, in a silly voice, said "Hugs and kisses, HUGS AND KISSES!" I put her down and she ate her food.
From that point forward, she wouldn't eat her food unless I did the hugs and kisses thing. It had to be the full dance, the same voice, or she wouldn't eat.