Payoff matrix

Definition

A payoff matrix is a table that represents the potential outcomes of a strategic interaction between players, showing the payoffs each player receives based on the combination of strategies they choose. This matrix is essential in analyzing competitive situations, helping to identify strategies that lead to equilibrium and informing decisions about whether to adopt pure or mixed strategies. It is also useful for determining dominant and dominated strategies among players.

5 Must Know Facts For Your Next Test

1. The payoff matrix displays the different outcomes based on each player's chosen strategies, allowing for easy comparison of payoffs.
2. It can show both cooperative and non-cooperative game scenarios, depending on how players interact within the game.
3. In a two-player game, each player's strategies are listed along the rows and columns, making it straightforward to visualize their interactions.
4. Identifying Nash equilibria often involves analyzing the payoff matrix to find stable outcomes where players do not want to change their strategies.
5. The presence of dominant and dominated strategies can be quickly assessed through the information provided in the payoff matrix.

Review Questions

• How does a payoff matrix help in identifying a Nash equilibrium in a game?
  • A payoff matrix aids in identifying a Nash equilibrium by clearly displaying the payoffs for each player based on their chosen strategies. By examining the matrix, players can determine if any strategy combination exists where neither player has an incentive to unilaterally change their strategy. This stable outcome is critical in understanding how strategic choices lead to predictable results in competitive situations.
• Discuss how dominant and dominated strategies can be identified using a payoff matrix.
  • A payoff matrix allows for the identification of dominant and dominated strategies by comparing the payoffs across different strategy combinations. A dominant strategy is one that yields a higher payoff for a player regardless of what the opponent chooses. Conversely, a dominated strategy is one that provides a lower payoff than another strategy available to the player. By analyzing the matrix, players can quickly see which strategies lead to better outcomes and eliminate those that do not.
• Evaluate the significance of using a mixed strategy in relation to the information presented in a payoff matrix.
  • Using a mixed strategy becomes significant when analyzing a payoff matrix because it allows players to randomize their choices when no pure strategy guarantees a favorable outcome. The matrix highlights potential payoffs for various strategies, helping players decide how best to allocate probabilities among them. This approach can be crucial in competitive environments where opponents might adjust their strategies based on predictable behavior, making randomness an effective tool for maintaining an advantage.

Related terms

Nash Equilibrium: A situation in a game where no player can benefit by changing their strategy while the other players keep theirs unchanged, leading to stable outcomes.

Dominant Strategy:

A strategy that results in a higher payoff for a player, regardless of what strategies other players choose.

Mixed Strategy: A strategy where a player randomizes over two or more possible moves, often used when no pure strategy is dominant.