Agon, by John Harper, is a "competitive" RPG set a fantastic ancient Greece. The heroes are soldiers on their way home from "the war", but their ship gets lost and finds itself casting about for a way home, and on the way finding a variety of strange islands. The people of the islands are functionally greek and speak the same language and worship the same gods as the PCs, but their cultures may differ in significant ways and the islands are subject to strange magical effects and fantasy beasts and monsters. Upon reaching an island, three greek gods approach the characters and order them to perform three quests. Hoping to make a good impression on the gods, and perhaps be sent home, the characters seek the glory of completing the quests.
That's the real goal of the game. Who can accumulate the most Glory before they run out of time and meet their eventual Fate? The game seeks to foster friendly rivalry, like the kind you might find at a weekly poker game, some ribbing, some trash-talk and seeing "who's best". The players don't compete against the person who runs the game (the Antagonist), but against each other. The Antagonist's job is just to set up suitable challenges for the players to compete over as they try to complete the quest. However, the game isn't entirely cut-throat either. The players still need to cooperate to some degree in order to complete the quest.
What keeps the Antagonist from just hosing all of the players? Strife. The Antagonist has a Strife budget. He can use Strife to build NPCs and monsters to fight, and he can use it to increase the difficulty of challenges, but eventually he's going to run out. He gains a little Strife back when the PCs rest, but not too much. Plus, the PCs can't die, unless they want to. Even if an encounter leads to a total party-wipe, the PCs will end up at the NPCs' mercy, tied up somewhere or something.
Back to the players, they earn Glory for successfully overcoming challenges, and gain more glory for overcoming harder challenges. They also get points for succeeding better than everybody else. Every 10 Glory gains them an Advance which they can use better their character. It also gives them a point of Legend, which is how the final score is tracked. Once your character meets his Fate, and is retired, how long he is remembered by the Greeks is determined by his Legend score. Speaking of Fate, every quest that you are assigned, every time you are defeated, and a few other circumstances increase your Fate. Once you've reached 16 Fate, your story is over, but the next hero that you create gets bonuses based on the number of quests that your previous character completed.
One really great aspect of the game is Oaths. These are "favors" that the other characters can owe you. You can call them in at any time to get a bonus to your roll or a couple of other effects. You can also help the other player without being asked, and gain a new Oath from him that you can call in later. He gets the Glory now, and he owes you help later.
The rules for resolving conflicts (skill rolls, fights, etc) are very simple and straight-forward with one exception, "Battles". The GM may set up an important conflict as a Battle. This means that the single roll from before ends up being several rounds of rolls, with characters trying to get the best position and to "wound" (physically or figuratively) the opposition until they give up. You can even, as a player, call for a Battle if you fail a normal challenge with another character. If you roll off against say, another hero to see who can get to the top of the hill first, and you lose, you can call for a Battle which will then give you more chances to roll. This is especially useful if you only lost because of a freakishly low roll or something. Battles can be a little complex, though, what with choosing which dice you assign to attack and defense, but that's probably the worst of it. Make sure that you have plenty of d4s, d6s, d8s, d10s, and d12s on hand, as the game uses pools of all of them.
If you're not big on the whole Greek thing, you can pretty easily port the game over to many different settings. Change the names of the gods and maybe the descriptions of the weapons and suddenly you've got Vikings instead. It's everything that "Rune" by Atlas Games should have been.
You can get the game as a PDF for $10 US or a "print-on-demand" book for $20 US. It's 120 pages, soft-cover with black and white art.
I've always liked the idea of a competitive RPG, and this one does a good job of it as it empowers the players and keeps the game moving without bogging down in complex rules or meaningless messing about by the players. I'd heard about Agon for a while without looking into it, but now I'm glad that I did.