Here's an example I used with my students to show how altruism could be included in self-interest. Feel free to borrow:
Let's say you're watching South Park. It's funny. But you enjoy it more if you're watching it with a friend. Of course, if your friend is incredibly bored and thinks the show is stupid, then you enjoy it less: even if you don't have to deal with your friend's complaining, you still feel stupid rolling on the floor laughing all by yourself in front of someone else. But of course you don't want your friend to be faking enjoyment; you want him to actually enjoy it, as much as possible, so that you can both enjoy watching the show. So your enjoyment is maximized only when your friend's enjoyment is maximized.