Learning in Two-Player Matrix Games

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 for player becomes a matrix.

A two-player matrix game is called a zero-sum game if the two player are fully competitive. In this way, we have . A zero-sum game has a unique NE in the sense of the expected reward. This means that, although each player may have multiple NE strategies in a zero-sum game, the value of the expected reward under these NE strategies will be the same. A 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 as the set of all probability distributions over player 's action set . Then becomes

(1)

An NE for a two-player matrix game is the strategy pair for two players such that, for

(2)

where denotes any other player than player , and is the set of all probability distributions over player 's action set .

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

(3)

where and denote the reward to the row player (player 1) and the reward to the column player (player 2), respectively. The row player chooses action and the column player chooses action . the pure strategies and are called a strict NE in pure strategies if

(4)

where and denote any row other than row and any column other than column ,respectively.