 Maxdominated strategy

In game theory a maxdominated 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 obviously maxdominated as well.
Contents
Definition
Maxdominated strategies
A strategy of player i is maxdominated if for every strategy profile of the other players there is a strategy such that . This definition means that s_{i} is not a best response to any strategy profile s _{− i}, since for every such strategy profile there is another strategy which gives higher utility than s_{i} for player i.
It is easy to see that if a stategy is strictly dominated by strategy then it is also maxdominated, since for every strategy profile of the other players we will pick to be the strategy for which .
It is also notable that even if s_{i} is strictly dominated by a mixed strategy it is also maxdominated.
Weakly maxdominated strateges
A strategy of player i is weakly maxdominated if for every strategy profile of the other players there is a strategy such that . This definition means that s_{i} is either not a best response or not the only best response to any strategy profile s _{− i}, since for every such strategy profile there is another strategy which gives at least the same utility as s_{i} for player i.
It is easy to see that if a stategy is weakly dominated by strategy then it is also weakly maxdominated, since for every strategy profile of the other players we will pick to be the strategy for which .
It is also notable that even if s_{i} is weakly dominated by a mixed strategy it is also weakly maxdominated.
Maxsolvable games
Definition
A game G is said to be maxsolvable if by iterated elimination of maxdominated strategies only one strategy profile is left at the end.
More formally we say that G is maxsolvable if there exists a sequence of games G_{0},...,G_{r} such that:
 G_{0} = G
 G_{k + 1} is obtained by removing a single maxdominated strategy from the strategy space of a single player in G_{k}.
 There is only one strategy profile left in G_{r}.
Obviously every maxsolvable game has a unique pure Nash equilibrium which is the strategy profile left in G_{r}.
As in the previous part one can define respectively the notion of weakly maxsolvable games, which are games for which a game with a single strategy profile can be reached by eliminating weakly maxdominated strategies. The main difference would be that weakly maxdominated games may have more than one pure Nash equilibrium, and that the order of elimination might result in different Nash equilibria.
Example
Cooperate Defect Cooperate 1, 1 5, 0 Defect 0, 5 3, 3 Fig. 1: payoff matrix of the prisoner's dilemma The prisoner's dilemma is an example of a maxsolvable game (as it is also dominance solvable). The strategy cooperate is maxdominated 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.
Maxsolvable games and bestreply dynamics
In any maxsolvable game, bestreply 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 s_{1},s_{2},s_{3},...,s_{k} is an elimination sequence of the game (meaning that first s_{1} is eliminated from the strategy space of some player since it is maxdominated, then s_{2} is eliminated, and so on), then in the bestresponse dynamics s_{1} will be never played by its player after one iteration of best responses, s_{2} will never be played by its player after two iterations of best responses and so on. The reason for this is that s_{1} is not a best response to any strategy profile of the other players s _{− i} so after one iteration of best responses its player must have chosen a different strategy. Since we understand that we will never return to s_{1} in any iteration of the best responses, we can treat the game after one iteration of best responses as if s_{1} has been eliminated from the game, and complete the proof by induction.
A weakly maxsolvable game 1, 1 0, 0 1, 0 0, 1 0, 1 1, 0 It may come by surprise then that weakly maxsolvable games do not necessarily converge to a pure Nash equilibrium when using the bestreply 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
External links and references
 Nisan, Noam; Schapira, Michael; Zohar, Aviv (2009), Asynchronus best reply dynamics, Berlin: SpringerVerlag, http://www.springerlink.com. Asynchronus bestreply dynamics. [1].
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