Consider that the purpose for the Hunger Games in Panem is two-fold. First, it is reminder to all citizens of the power the Capitol holds over everyone. But second, it is a primary mode of entertainment for the people who live in the Capitol (and arguably Districts 1 and 2 as well).
As such, it is very much a sport. And a sport would completely lose its meaning if there was no winner. Think about the way the tributes are built up and given TV identities in the days leading up to the games. Just like our society roots for one American Idol and Survivor winner, the citizens of the Capitol are tuned in, making bets, and rooting for one Hunger Games victor.
Now, if your question is also taking in to account the moment at the end of the book when Katniss proposes a dual suicide with Peeta, the answer is that this act of rebellion would have shown that the Capitol (and the Games creators) did not have complete control. For a tribute to change the rules, so to speak, at the end of the game, would have given citizens the idea that the Capitol, in this case, was the loser. As you read on in the series you will realize exactly how dangerous this scene really was.
It is a sport to find the fittest, and the best. Also one person winning shows the power the capitol has over the people.
There must always be at least 1 victor from the Hunger Games because to the Capitol, this is a sport to find the fittest in the group of tributes. It serves as entertainment to them. Also, it shows the districts that the Capitol has the upper hand over the districts.
There can only be one victor of the Hunger Games. The whole reason for this Games is survival of the fittest and whoever achieves that title is the ultimate winner. Even after so much bloodshed, they use this as oh someone survived its worth it when in fact they view the Hunger Games as entertainment. It's like a sport and what's a sport without a winner, pointless right? This is their reasoning.