- 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 obviously max-dominated as well.
A strategy of player i is max-dominated if for every strategy profile of the other players there is a strategy such that . This definition means that si 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 si for player i.
It is easy to see that if a stategy is strictly dominated by strategy then it is also max-dominated, 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 si is strictly dominated by a mixed strategy it is also max-dominated.
Weakly max-dominated strateges
A strategy of player i is weakly max-dominated if for every strategy profile of the other players there is a strategy such that . This definition means that si 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 si for player i.
It is easy to see that if a stategy is weakly dominated by strategy then it is also weakly max-dominated, 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 si is weakly dominated by a mixed strategy it is also weakly max-dominated.
A game G 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 G is max-solvable if there exists a sequence of games G0,...,Gr such that:
- G0 = G
- Gk + 1 is obtained by removing a single max-dominated strategy from the strategy space of a single player in Gk.
- There is only one strategy profile left in Gr.
Obviously every max-solvable game has a unique pure Nash equilibrium which is the strategy profile left in Gr.
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.
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 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 s1,s2,s3,...,sk is an elimination sequence of the game (meaning that first s1 is eliminated from the strategy space of some player since it is max-dominated, then s2 is eliminated, and so on), then in the best-response dynamics s1 will be never played by its player after one iteration of best responses, s2 will never be played by its player after two iterations of best responses and so on. The reason for this is that s1 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 s1 in any iteration of the best responses, we can treat the game after one iteration of best responses as if s1 has been eliminated from the game, and complete the proof by induction.
A weakly max-solvable game 1, 1 0, 0 1, 0 0, 1 0, 1 1, 0
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).
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