You mean something League of Legends like?
I'm neither a furtune teller nor an expert politologist, so everything I can give is a bunch of thoughts, more or less sensible... and the first one is that what you mean is a very interesting idea, and I'd love to see it deeply worked out.
The first question seems to be like: how could this be enforced, and how could this develop? And I still think this system could develop, however under certain conditions.
1. How could this develop?
Quite obviously, there should be either a major power enforcing it (like a superpower state, for example the US, or maybe a more powerful version of the United Nations, or maybe even a world government?) or simply waging a war should be a suicidal decision (since everyone hates war so much that waging it means you suddenly get completely isolated with literally everyone against yourself).
The problem is that in the second scenario waging traditional wars would probably be substituted by economic wars (for example buying out other country's businnes, or laying down embargos, or what China now does to the Western world) or cyber wars (the NSA and the likes...) The tournaments option still wouldn't make sense. Granted, countries would be forced not to wage wars, but they yet wouldn't be forced to actually give out that Crimea to Putin for loosing a tennis match...
Now we go to the second option. A powerful entity enforcing it. This would of course require an explanation how could such an entity (a country, the U.N., a world government) reach such a power, but that's a whole different story. The important part is: why would such an entity enforce that system?
2. Why would a super power enforce this system?
- For moral reasons - "Stop bloodshed!"
- As a way to channel aggressive desires of ambitious individuals (or somebody's frustration, or something...) that could otherwise challenge world peace or that powerful entity's domination
- For fun, like the Roman Colloseum
Of course the first problem is that if such an entity reaches such a power that it can enforce this system (and enforce every country to pay for loosing matches), why won't it simply rule the whole world with an iron fist of tyranny? There's no need for Colosseums in the 1984 world! These days massive invigilation of everyone starts to be possible, and this addinotialy bonuses all wanna-be tyrants. Given the NSA thingy it no longer seems absurd for a soft-totalitarian regime to evolve in Europe or the USA.
So this ruling entity should either - be ruled by a man or group moral enough not to go in this direction AND strong enough to ward off any cabals, lobbys etc; or - limited by a strict constitution and strong democratic structures; or - strong enough not to be easily overthrown, but still weak enough to have to thing twice before opressing everybody on Earth.
And then. Would it really enforce peace? Yes, countries woulnd't be able to wage traditional wars... BUT what about riots? People would not necessarily be content with sudden changes of their nationality, so social unrest could develop in disputed lands.
4. Moral issues
Would it really have to be just a tennis match? Consider the following scenario: a super mercenary guild emerges. Most armies of the world belong of this guild, so it is the guild who enforces the system. Wouldn't it probably boil down to bloody battles, just controlled enough not to escalate out of the arena? Such a system would, of course, be plain immoral.
And then. Imperialism would still be Imperialism, even in the Football match scenario. Countries would be probably obligated to accept challenges. So nothing changes - the stronger takes the weaker's property.
The system is possible I think, you just have to have a super power that enforces it. I've mentioned problems, but these are only problems IMHO, not blockers.
Sorry for the messy nature of this post, but I've been in a hurry. I'll clean it up and expand it tomorrow.