# Pay-off matrix

## Understanding Pay-off Matrix

• The pay-off matrix is a table or grid used in game theory which represents the outcomes of a two-player game.
• Each cell of the matrix corresponds to an outcome resulting from each possible combination of strategies used by the players.
• The pay-off of each player is represented by a number in the cell that determines the player’s reward or loss for each possible outcome.
• The row player generally comes first in the pair, while the column player is listed second.
• Pay-offs are usually represented as profits, where a bigger number is better, although they may also signify costs, where a lower number is better.
• A component of a pay-off matrix, also known as a cell, consists of two numbers. The first number represents the pay-off to the row player, while the second number represents the pay-off to the column player.

## Types of Games Represented by Pay-off Matrices

• Simultaneous Games: In these games, both players make their decisions at the same time without knowledge of the other’s choice. The game of Rock, Paper, Scissors is an example.
• Sequential Games: These represent scenarios in which the order of player actions matters. For example, Chess is a sequential game.

## Strategies and Optimal Outcomes

• A dominant strategy is one that leads to the best outcome for a player, irrespective of the strategy of the other player.
• If each player has a dominant strategy, the resulting outcome is referred to as a Dominant Strategy Equilibrium.
• In some games, no clear dominant strategy exists. Players instead aim to maximise their minimum guaranteed pay-off, referred to as the Maximin strategy.
• Player’s decisions are influenced by the expected pay-offs. An optimal strategy combination that brings the highest expected utility to all players is considered a Nash equilibrium.
• Pure strategies involve always choosing a single action while mixed strategies involve choosing different actions according to given probabilities.

## Examples and Applications of Pay-off Matrices

• Pay-off matrices are used to solve real-world problems such as auctions, voting systems, trade negotiations, business decisions, and military conflicts.
• They are practical tools for studying strategic interactions between two or more decision-makers in economics, political science, psychology, logic, computer science, and biology.
• Exercises often involve drawing a pay-off matrix from a given scenario, finding dominant strategies, and identifying Nash equilibriums.
• In some games, it may be strategically viable to opt for a less favourable short-term pay-off in anticipation of a more favourable long-term outcome, known as a sacrifice strategy.

Remember to practice interpreting and solving various game scenarios using pay-off matrices for better understanding and application. A strong grasp of the concept is integral to mastering Game Theory.