Game theory: Pay-off matrix

Introduction to Game Theory and Pay-off Matrix

Game Theory is a mathematical model of conflicts and cooperation between intelligent rational decision-makers.
A key tool in game theory is the pay-off matrix, representing the outcomes of decisions made by players in a game.
Games in this context are situations in which the outcome for a player depends on the choices of all players, not just their own.

A pay-off matrix is a table that contains information about gains or losses of players in a strategic game.
A strategic game, also known as a normal form game, is where players make their decisions simultaneously and independently.
The matrix has rows for each of the strategies of the first player and columns for each of the strategies of the second player.
At the intersection of each row and column, there are two numbers. The first represents the pay-off to the row player (Player 1) and the second represents the pay-off to the column player (Player 2).
The pay-off is the reward or penalty received by a player at the end of the game, often represented by a numerical value.

A player’s strategy in a game is a complete plan of actions the player will take in every possible circumstance.
The combination of strategies chosen by all players in the game is called a strategy profile.
The outcome of any given strategy profile is represented in the pay-off matrix.

If one strategy results in a higher pay-off than another, regardless of what the opponent does, that strategy is dominant.
The strategy that is outperformed by another strategy is considered dominated.
Recognising dominant and dominated strategies can help simplify pay-off matrices and decision-making in games.

Nash Equilibrium represents the set of strategies where no player can improve their pay-off by unilaterally deviating from their current strategy, assuming the other players keep their strategies unchanged.
In a pay-off matrix, a Nash Equilibrium is found where the strategies intersect at the most optimal point for both players.

Game theory and pay-off matrices have extensive applications, including fields such as economics, politics, military strategy, social sciences, and AI design.
In these domains, game theory is used to analyse competition, cooperation and conflict.