As I know, with game theory we can compute the equilibrium of a game (i.e. the best strategy that each player uses according to other players best strategy) and in dynamic games, the utility function of player can be changed during game period.
I want to know is there a branch of game theory (perhaps dynamic games) where players could attack each other and make their opponents weaker and weaker (or even extirpate them) which leads to changing the payoff matrix?
Indeed, I want to analyze a game where players can attack other players, and I want to determine the final survivor (or survivors) of the game who extirpate all other players.
One set of games are the duels, modeled with payoffs of 1 if you win, 0 if you lose--in this case, zero-sum really makes sense. Wikipedia can get you started https://en.wikipedia.org/wiki/Truel
Another set of games is known as wars of attrition, where you aim to be the last one standing. Again, Wikipedia has something, but I also like Osborne and Rubinstein's example 18.4 in A Course on Game Theory. So what I think you want is something like a multi-player game of attrition. See http://web.stanford.edu/~jdlevin/Econ%20286/Wars%20of%20Attrition for a start. There are subtleties, such as is the ultimate value known or uncertain?
Perhaps not exactly the methodology you have in mind, but Schelling's The Strategy of Conflict deals with many closely related issues such as a chapter on "reciprocal fear of surprise attack."