# 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.

# Understanding the Pay-off Matrix

• 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.

# Identifying Strategies and Outcomes

• 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.

# Dominant and Dominated Strategies

• 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

• 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.

# Real World Applications

• 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.