Yes. :) 

That's actually the main point of them; Capital was revolted against, so these games are to remind them what kind of cost they pay for revolt.  

From the first book:
>The Treaty of Treason gave us the new laws to guarantee peace and, as our yearly reminder that the Dark Days must never be repeated, it gave us the Hunger Games. The rules of the Hunger Games are simple. In punishment for the uprising, each of the twelve districts must provide one girl and one boy, called tributes, to participate. The twentyfour tributes will be imprisoned in a vast outdoor arena that could hold anything from a burning desert to a frozen wasteland. Over a period of several weeks, the competitors must fight to the death. The last tribute standing wins. Taking the kids from our districts, forcing them to kill one another while we watch—this is the Capitol’s way of reminding us how totally we are at their mercy. How little chance we would stand of surviving another rebellion.

It's kind of meant as a constant reminder that "Capital rules; the last revolt, 75 years ago, is STILL being paid for.. Do you really want to try to dispute our rule again?"