How does a payoff matrix organize the possible outcomes of a two-player game?
The normal-form table illustrates the concept of a saddlepoint, or entry, in a payoff matrix at which the expected gain of each participant (row or column) has the highest guaranteed payoff.
In game theory, a payoff matrix is a mathematical tool used to organize and analyze the interdependent decisions of players in a game. It provides a structured way to visualize all possible outcomes based on the strategies chosen by each participant.1
The matrix is organized as follows:
* Rows: Each row represents one possible strategy available to the first player.1
* Columns: Each column represents one possible strategy available to the second player.1
* Cells: The intersection of a specific row and column contains the payoff or outcome resulting from that particular combination of strategies.12
By mapping these combinations, the matrix allows players to evaluate their options systematically. For instance, in two-person constant-sum games, the matrix helps identify a saddlepointāan outcome where rational players can determine their optimal strategy by calculating maximin and minimax values. Ultimately, the payoff matrix serves as a foundational framework for determining the Nash equilibrium, the state in which no player can improve their outcome by unilaterally changing their strategy.12