Winning and losing advice for a mom?

We just returned from taking all the kids to a bowling alley, an activity that seemed to cause my older son (Robbie, age 11) to melt down.

He didn't win. Not only that, he got no strikes or spares (each of the other children did), and his brother beat him by 6 points in the end. This caused Robbie to whine during the last half of the game (he came right over to me with his complaints--I couldn't use the ignoring tactic), and have an absolute melt down when it was over.

Perhaps I didn't handle it well. Maybe some of you moms or kids can give me an idea of how I might handle this kind of situation better.

My reactions were
--first, I asked him if he wanted to try a different ball. He continued to fuss and complain.
--then I said, well maybe the others are just having a good day. He stopped fussing briefly, but began fussing, more loudly, after the next frame.
--then I pointed out that I was doing worse than he was (which was true). He hit my arm and said I didn't understand.
--then I said, "Listen, when you play a game, someone wins and someone loses. If you are going to fuss if you lose, then I think you should just quit right now." He kept complaining. "But I got worse than Eli! It's so embarassing!" At this point, he was getting louder. "Buddy," I said. "I think it's more embarassing--for you and for me--for you to be making a fuss and crying about losing in front of all our relatives. I think it's better to be a good sport."

Eventually, Bruce had to take him out of the bowling alley while the rest of us finished up.

So. Do I not understand? Is there a better way of handling a very vocal bad sport who is eleven? If any moms or dads--or any kids who have siblings or cousins--have any ideas, post them to this page! Maybe you have some ideas based on your experience with children at tournaments . . . Let me know your thoughts.