Sometimes ago, one of my friend sent me a story about the power of collaboration and the weakness of selfishness. The story is:
Suppose there are $n$ person out of a room and there are $n$ balls in the room such that on each ball there is a name of those $n$ person (we have $n$ balls with $n$ names on them, each ball has one name). Now the people goes to the room and the object is that each person obtain the ball with his name on it. We have two scenarios:
each person try to find the ball with his name separately,
each person take a ball and say aloud the name on ball and transfer it to the right case.
It is obvious that the case two is more efficient than case one. But what is the logic of these differences? How can we prove it by mathematics?
For modeling this game I used a complete graph such that it's vertices labeled randomly by the numbers $1,...,n$. Also the people have labels $1,...,n$ and randomly occupy the vertices of the labeled complete graph. Now, it seems that by this terminology, case two is a sorting with one move along each edge and case one is a random sorting along several moves on several edges. So, we can show that the case two is more better than case one.
My question is: Is there a better and more simple way to describe this result for high-school students?