3.2 Nash Equilibria in Two-Player Matrix Games

For a two-player matrix game, we can set up a matrix with each element containing a reward for each joint action pair. Then the reward function

A two-player matrix game is called a zero-sum game if the two player are fully competitive. In this way, we have general-sum matrix game refers to all types of matrix games. In a general-sum matrix game, the NE is no longer unique and the game might have multiple NEs.

For a two-player matrix game, we define

An NE for a two-player matrix game is the strategy pair

where

Given that each player has two actions in the game, we can define a two-player two-action general-sum game as

where strict NE in pure strategies if

where

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