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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.

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    $\begingroup$ This can certainly be modeled as a dynamic game. The challenge might be simply in describing the payoffs. What payoff do you receive if you are extirpated? If you're trying to model something like Risk, keep in mind that the informational assymetries and randomness in outcomes are critical. Could you explicitly model Risk as a dynamic game? Probably. Would you want to? No -- it would take way too long. Whatever game you're interested in, try to boil it down to the key strategic components and create a simpler game that focuses attention on those. $\endgroup$ – Shane Oct 13 '15 at 15:36
  • $\begingroup$ I don't want to model risk. What payoff do you receive if you are extirpated? Nothing. You would be extirpated if another player attacks you and you can not defend yourself. You will be out of the game and you cannot have any payoff in the next stages of the game. $\endgroup$ – Tail of Godzilla Oct 13 '15 at 16:34
  • $\begingroup$ In the other words, the goal of each player is to extirpate other players and survive herself! $\endgroup$ – Tail of Godzilla Oct 13 '15 at 16:39
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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."

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