Max-dominated strategies

A strategy ${\displaystyle s_{i}\in S_{i}}$ of player ${\displaystyle i}$ is max-dominated if for every strategy profile of the other players ${\displaystyle s_{-i}\in S_{-i}}$ there is a strategy ${\displaystyle s_{i}^{\prime }\in S_{i}}$ such that ${\displaystyle u_{i}(s_{i}^{\prime },s_{-i})>u_{i}(s_{i},s_{-i})}$. This definition means that ${\displaystyle s_{i}}$ is not a best response to any strategy profile ${\displaystyle s_{-i}}$, since for every such strategy profile there is another strategy ${\displaystyle s_{i}^{\prime }}$ which gives higher utility than ${\displaystyle s_{i}}$ for player ${\displaystyle i}$.

If a strategy ${\displaystyle s_{i}\in S_{i}}$ is strictly dominated by strategy ${\displaystyle s_{i}^{\prime }\in S_{i}}$ then it is also max-dominated, since for every strategy profile of the other players ${\displaystyle s_{-i}\in S_{-i}}$, ${\displaystyle s_{i}^{\prime }}$ is the strategy for which ${\displaystyle u_{i}(s_{i}^{\prime },s_{-i})>u_{i}(s_{i},s_{-i})}$.

Even if ${\displaystyle s_{i}}$ is strictly dominated by a mixed strategy it is also max-dominated.

Weakly max-dominated strategies

A strategy ${\displaystyle s_{i}\in S_{i}}$ of player ${\displaystyle i}$ is weakly max-dominated if for every strategy profile of the other players ${\displaystyle s_{-i}\in S_{-i}}$ there is a strategy ${\displaystyle s_{i}^{\prime }\in S_{i}}$ such that ${\displaystyle u_{i}(s_{i}^{\prime },s_{-i})\geq u_{i}(s_{i},s_{-i})}$. This definition means that ${\displaystyle s_{i}}$ is either not a best response or not the only best response to any strategy profile ${\displaystyle s_{-i}}$, since for every such strategy profile there is another strategy ${\displaystyle s_{i}^{\prime }}$ which gives at least the same utility as ${\displaystyle s_{i}}$ for player ${\displaystyle i}$.

If a strategy ${\displaystyle s_{i}\in S_{i}}$ is weakly dominated by strategy ${\displaystyle s_{i}^{\prime }\in S_{i}}$ then it is also weakly max-dominated, since for every strategy profile of the other players ${\displaystyle s_{-i}\in S_{-i}}$, ${\displaystyle s_{i}^{\prime }}$ is the strategy for which ${\displaystyle u_{i}(s_{i}^{\prime },s_{-i})\geq u_{i}(s_{i},s_{-i})}$.

Even if ${\displaystyle s_{i}}$ is weakly dominated by a mixed strategy it is also weakly max-dominated.

Definition

A game ${\displaystyle G}$ 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 ${\displaystyle G}$ is max-solvable if there exists a sequence of games ${\displaystyle G_{0},...,G_{r}}$ such that:

• ${\displaystyle G_{0}=G}$
• ${\displaystyle G_{k+1}}$ is obtained by removing a single max-dominated strategy from the strategy space of a single player in ${\displaystyle G_{k}}$.
• There is only one strategy profile left in ${\displaystyle G_{r}}$.

Obviously every max-solvable game has a unique pure Nash equilibrium which is the strategy profile left in ${\displaystyle G_{r}}$.

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 ${\displaystyle s_{1},s_{2},s_{3},...,s_{k}}$ is an elimination sequence of the game (meaning that first ${\displaystyle s_{1}}$ is eliminated from the strategy space of some player since it is max-dominated, then ${\displaystyle s_{2}}$ is eliminated, and so on), then in the best-response dynamics ${\displaystyle s_{1}}$ will be never played by its player after one iteration of best responses, ${\displaystyle s_{2}}$ will never be played by its player after two iterations of best responses and so on. The reason for this is that ${\displaystyle s_{1}}$ is not a best response to any strategy profile of the other players ${\displaystyle s_{-i}}$ so after one iteration of best responses its player must have chosen a different strategy. Since we understand that we will never return to ${\displaystyle s_{1}}$ in any iteration of the best responses, we can treat the game after one iteration of best responses as if ${\displaystyle s_{1}}$ 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).