Winning does not produce an equilibrium state, therefore there is selection pressure to tie eternally.

Losers lose and are eliminated from the game, obviously. But winners are also eliminated - because new contestants enter the game and fight them until they lose. The most stable configuration is to fight your opponent until you reach a stalemate and then hold that position as long as possible, effectively collaborating with your "enemy" to eliminate third party threats. Contestants that want to fight but do not want to win are selected. The extreme case of this is the arabs and the jews who have been "fighting" for ten thousand years.

I do not have a solution for this. It appears to be a property of the universe.