Max-dominated strategy
In game theory a max-dominated strategy is a strategy which is not a best response to any strategy profile of the other players. This is an extension to the notion of strictly dominated strategies, which are max-dominated as well. Definition. Max-dominated strategies. A strategy formula_1 of player formula_2 is "max-dominated" if for every strategy profile of the other players formula_3 there is a strategy formula_4 such that formula_5. This definition means that formula_6 is not a best response to any strategy profile formula_7, since for every such strategy profile there is another strategy  formula_8 which gives higher utility than formula_6 for player formula_2. If a strategy formula_1 is strictly dominated by strategy formula_12 then it is also max-dominated, since for every strategy profile of the other players formula_3, formula_8 is the strategy for which formula_5. Even if formula_6 is strictly dominated by a mixed strategy it is also max-dominated. Weakly max-dominated strategies. A strategy formula_1 of player formula_2 is weakly max-dominated if for every strategy profile of the other players formula_3 there is a strategy formula_4 such that formula_21. This definition means that formula_6 is either not a best  response or not the only best response to any strategy profile formula_7, since for every such strategy profile there is another strategy formula_8 which gives at least the same utility as formula_6 for player formula_2. If a strategy formula_1 is weakly dominated by strategy formula_12 then it is also weakly max-dominated, since for every strategy profile of the other players formula_3, formula_8 is the strategy for which formula_31. Even if formula_6 is weakly dominated by a mixed strategy it is also weakly max-dominated. Max-solvable games. Definition. A game formula_33 is said to be max-solvable if by  iterated elimination of max-dominated strategies only one strategy profile is left at the end. More formally we say that formula_33 is max-solvable if there exists a sequence of games formula_35 such that: Obviously every max-solvable game has a unique pure Nash equilibrium which is the strategy profile left in formula_39. As in the previous part one can define respectively the notion of weakly max-solvable games, which are games for which a game with a single strategy profile can be reached by eliminating weakly max-dominated strategies. The main difference would be that weakly max-dominated games  may have more than one pure Nash equilibrium, and that the order of elimination might result in different Nash equilibria. Example. The prisoner's dilemma is an example of a max-solvable game (as it is also dominance solvable). The strategy cooperate is max-dominated by the strategy defect for both players, since playing defect always gives the player a higher utility, no matter what the other player plays. To see this note that if the row player plays cooperate then the column player would prefer playing defect and go free than playing cooperate and serving one year in jail . If the row player plays defect then the column player would prefer playing defect and serve three years in jail rather than playing cooperate and serving five years in jail. Max-solvable games and best-reply dynamics. In any max-solvable game, best-reply dynamics ultimately leads to the unique pure Nash equilibrium of the game. In order to see this, all we need to do is notice that if formula_41 is an elimination sequence of the game (meaning that first formula_42 is eliminated from the strategy space of some player since it is max-dominated, then formula_43 is eliminated , and so on), then in the best-response dynamics formula_42 will be never played by its player after one iteration of best responses, formula_43 will never be played by its player after two iterations of best responses and so on. The reason for this is that formula_42 is not a best response to any strategy profile of the other players formula_7 so after one iteration of best responses its player must have chosen a different strategy. Since we understand that we will never return to formula_42 in any iteration of the best responses, we can treat the game after one  iteration of best responses as if formula_42 has been eliminated from the game, and complete the proof by induction. It may come by surprise then that weakly max-solvable games do not necessarily converge to a pure Nash equilibrium when using the best-reply dynamics, as can be seen in the game on the right. If the game starts of the bottom left cell of the matrix, then the following best replay dynamics is possible: the row player moves one row up to the center row, the column player moves to the right column, the row player moves back to  the bottom row, the column player moves back to the left column and so on. This obviously never converges to the unique pure Nash equilibrium of the game (which is the upper left cell in the payoff matrix). See also. Dominance (game theory)