The success of this idea might come down to the mental health of the player. Designing a game that seeks to create tensions akin to rage and then providing a focal point as to another player being the "element" which helps as well hinders you at the same time may not result in an enjoyment of the game. Of course that said, the reality is many games have a pvp content as well team based play options and are successful. COD for example as more light version or as mentioned DayZ for a far more gritty experience. DayZ would in truth be truer to your ideas as betrayal is emminently possible and the ability to go into another person's inventory and rob them without their immediate knowledge can lead to unfortunate situations when you find you have no bandages all of a sudden . Anecdotally though I seem to see more rage arising in CoD games than I do in DayZ where the opportunity of direct betrayal does not really exist.
I can perceive of designs such as where you might lock two enemies together to defeat a third enemy that could not be done individually by either and the resulting resources gained filched before the other player obtains them with a final "end game" being the fight between both players strengthened by whatever resources each had managed to accrue or interactive elements requiring two players to operate forcing a "cooperation" for game progression tradeoff. So your theme is viable.