Strictly Competitive Games

In short words, strictly competitive games are a special type of strategic games whose preferences are completely opposed. The formal definition is shown as following.

• A strategic game $\langle {1,2}, (A_{i}), (\succsim_{i}) \rangle$ is strictly competitive if for any $a \in A$ and $b \in A$ we have $a \succsim_{1} b$ if and only if $b \succsim_{2} a$.

A strictly competitive game is sometimes called zero-sum because if player 1’s preference relation $\succsim_{1}$ is represented by the payoff function $u_{1}$ then player 2’s preference relation is represented by $u_{2}$ with $u_{1} + u_{2} = 0$.

Literally, the procedure of the strictly competitive games can be explained that player i maxminimizes if he chooses an action that is best for him on the assumption that whatever he does, player j will choose her action to hurt him as much as possible. The formal definition of a strictly competitive game is shown as following.

• Let $\langle {1,2}, (A_{i}), (\succsim_{i}) \rangle$ be a strictly competitive strategic game. The action $x^{*} \in A_{1}$ is a maxminimizer for player 1 if

Similarly, the action $y^{*} \in A_{2}$ is a maxminimizer for player 2 if

In words, a maxminimizer for player i is an action that maximizes the payoff that player i can guarantee. There are some important properties of strictly competitive games shown as following.

• Let $G = \langle {1,2}, (A_{i}), (u_{i}) \rangle$ be a strictly competitive strategic game.
• If $(x^{*}, y^{*})$ is a Nash equilibrium of G then $x^{*}$ is a maxminimizer for player 1 and $y^{*}$ is a maxminimizer for player 2.
• If $(x^{*}, y^{*})$ is a Nash equilibrium of G then $\max_{x} \min_{y} u_{1}(x, y) = \min_{y} \max_{x} u_{1}(x, y) = u_{1}(x^{*}, y^{*})$, and thus all Nash equilibria of G yield the same payoffs.
• If $\max_{x} \min_{y} u_{1}(x, y) = \min_{y} \max_{x} u_{1}(x, y)$, $x^{*}$ is a maxminimizer for player 1, and $y^{*}$ is a maxminimizer for player 2, then $(x^{*}, y^{*})$ is a Nash equilibrium of G.

By the third statement above, we can find the players’ Nash equilibrium strategies by solving the problems $\max_{x} \min_{y} u_{1}(x, y)$ and $\max_{y} \min_{x} u_{2}(x, y)$. Because of the first and the third statements above, we can conclude that the Nash equilibria of a strictly competitive game are interchangeable: if $(x, y)$ and $(x’, y’)$ are equilibria then so are $(x, y’)$ and $(x’, y)$. If $\max_{x} \min_{y} u_{1}(x, y) = \min_{y} \max_{x} u_{1}(x, y)$ then we say that this payoff, the equilibrium payoff of player 1, is the value of the game.

Written on December 18, 2018