It might work for trivial matters, but when the stakes are high and the chips are down human nature is always going to lead to force to get what is important.
"Ok so our robot beat your robot on some contrived fight. We're going to invade your country with a population of millions, take all of your money and possessions and force you all to work in the salt mines as slaves until you die an untimely and avoidable death. Please form an orderly line at the salt-death-mines tomorrow morning. This is totally fine since the robot your country sent to fight ours lost so you've just got to accept it without any resistance. Kthx Bai - signed your new overlords"
It’s more like proxy battles, and in real life we do this with expendable humans rather than robots. When a side decides to surrender it’s not necessarily when the last person is wiped out... usually done when the military is near or close to defeat.
OP is suggesting that one day robots could outclass flesh and bone humans so much that real humans will be like the women and children of past society that will not have to fight in real battles.
The kind of political dilemma that would be solved by a mecha duel would probably not be the same one that requires invasion and enslavement in death mines.
“Der Krieg ist eine bloße Fortsetzung der Politik mit anderen Mitteln.”
(“War is the mere continuation of politics with other means”)
For cyprus, the greeks and turks could just speed up the music until some team loses the beat. Spratly islands could be sorted with synchronised swimming?
Bonus clips:
>no loss of people's lives
Sure...
And even with robots, once you destroyed the enemy robots, what happens then if they don't surrender? Now your robots have to kill humans (or if you refuse you didn't achieve anything of consequence).
I think you'll find you mean 'precedent' :-)
EDIT: I didn't expect this comment to be controversial/downvoted, and it is interesting that it is evoking some level of passive aggressive sarcasm.