Glossary

3.1 Normal form games and payoff matrices

4 min read•july 30, 2024

Normal form games and payoff matrices are essential tools for analyzing strategic interactions. They provide a clear way to represent players, strategies, and payoffs in simultaneous move games, allowing us to visualize all possible outcomes.

Understanding these concepts is crucial for identifying Nash equilibria and dominant strategies. By examining payoff matrices, we can analyze strategic interdependence and determine best responses, key elements in predicting game outcomes and player behavior.

Normal Form Representation of Games

Representing Simultaneous Move Games

Top images from around the web for Representing Simultaneous Move Games

1 of 1

Top images from around the web for Representing Simultaneous Move Games

1 of 1
  • Normal form games represent strategic situations where players make decisions simultaneously without knowing the decisions of other players
  • A specifies:
    • The players in the game
    • The strategies available to each player
    • The payoff received by each player for each combination of strategies that could be chosen by the players
  • Normal form representation captures the key elements of a game: players, strategies, and payoffs

Using Payoff Matrices

  • Payoff matrices are a convenient way to represent normal form games, listing the players, player strategies, and payoffs in a grid format
    • Player strategies are listed on the dimensions of the matrix
      • The row player's strategies are listed on the left
      • The column player's strategies are listed on the top
    • Each cell in the matrix shows the payoffs to the row and column player for the corresponding row and column strategies
    • By convention, payoffs are listed with the row player's payoff first, followed by the column player's payoff
  • Payoff matrices provide a clear visual representation of the game, making it easier to analyze player incentives and outcomes (, )

Interpreting Payoff Matrices

Understanding Outcomes and Payoffs

  • Each cell in a represents an outcome, a combination of strategies chosen by the players and the resulting payoffs
  • To determine the payoff for a player from a particular outcome:
    • Identify the row corresponding to the row player's strategy
    • Identify the column corresponding to the column player's strategy
    • The cell at the intersection shows the payoffs to each player
  • Payoffs are the values that each player receives at the outcome of the game based on the combination of strategies chosen
    • Payoffs are often represented as numerical values but can take other forms as long as they show the rank-ordering of player preferences (utility values, win/lose, years in prison)

Analyzing All Possible Outcomes

  • The payoff matrix allows you to see all possible combinations of strategies and the resulting payoffs
  • By examining the entire matrix, you can:
    • Identify the payoffs each player receives for each combination of strategies
    • Determine which outcomes are more or less desirable for each player
    • Analyze player incentives and likely strategies based on the payoff structure
  • Seeing all potential outcomes helps determine the likely results of the game (, dominant strategies)

Strategic Interdependence in Games

Optimal Strategies and Opponent Actions

  • In simultaneous move games, the optimal strategy for a player depends on the strategy chosen by the other player(s)
    • This relationship between the best strategies of players is known as strategic interdependence
  • Each player must anticipate the actions of their opponent(s) and choose their strategy based on how they expect their opponent(s) to play the game
    • Players consider questions like: "What is my opponent likely to do?" and "What is my best strategy given what I think my opponent will do?"

Analyzing Strategic Interdependence

  • To analyze the strategic interdependence, examine how each player's payoffs change in response to changes in the other player's strategy choice
    • Look at a player's payoffs in each row (for the row player) or column (for the column player) to see how their payoffs change as the other player's strategy changes
    • If a player's best strategy depends on the other player's choice, then there is strategic interdependence
    • If a player has a single best strategy no matter what the other player does, then their optimal choice is independent of their opponent
  • The presence or absence of strategic interdependence shapes the incentives and outcomes in the game (Matching Pennies vs. Prisoner's Dilemma)

Best Responses in Simultaneous Games

Defining Best Responses

  • A player's is the strategy that gives them the highest payoff given the strategy chosen by the other player
  • The best response is the optimal strategy choice for a player assuming their opponent's action is fixed
  • Best responses indicate what each player should do to maximize their own payoff in light of their opponent's strategy

Finding Best Responses

  • To find the best response, look at the payoffs in the matrix while holding the other player's strategy constant
    • For the row player, look across each row (holding the column player's strategy constant) to find the highest payoff
      • The row player's strategy corresponding to the highest payoff is their best response to that column strategy
    • For the column player, look down each column (holding the row player's strategy constant) to find the highest payoff
      • The column player's strategy corresponding to the highest payoff is their best response to that row strategy
  • A player may have a single best response or multiple best responses to a particular strategy by their opponent

Best Responses and Nash Equilibrium

  • If both players are simultaneously playing a best response to the other player's strategy, then the combination of strategies is a Nash equilibrium
  • At a Nash equilibrium, no player can unilaterally improve their payoff by changing strategies
    • Each player is doing the best they can given the strategy of their opponent
  • Finding best responses is a key step in identifying Nash equilibria in simultaneous move games (Prisoner's Dilemma, Battle of the Sexes)

Key Terms to Review (15)

Battle of the Sexes: The Battle of the Sexes is a classic game theory scenario that highlights the coordination problem between two players who have different preferences but wish to reach a mutually beneficial outcome. It illustrates how individuals can benefit from cooperation even when their interests are not perfectly aligned, showcasing the importance of strategic decision-making and rational choice in situations where players must choose between competing options.
Best Response: In game theory, a best response is the strategy that provides the highest payoff for a player, given the strategies chosen by other players. This concept is crucial because it helps players make rational choices based on their expectations of others' behavior, connecting to the broader themes of strategic decision-making and rational choice. Understanding best responses is essential for analyzing both pure and mixed strategies, determining optimal actions in normal form games, and finding Nash equilibria within those frameworks.
Cooperative Game: A cooperative game is a type of game in game theory where players can form binding commitments and work together to achieve better outcomes than they could individually. These games focus on how players can benefit from cooperating, and they often lead to negotiations on resource allocation, profit sharing, or strategy development, emphasizing the importance of collaboration over competition.
Dominant strategy: A dominant strategy is a course of action that yields a better outcome for a player, regardless of the actions chosen by other players. This concept highlights how players can make decisions based on their own best interests, which often leads to predictable behavior in strategic settings.
Nash equilibrium: Nash equilibrium is a concept in game theory where no player can benefit from changing their strategy while the other players keep theirs unchanged. This situation arises when each player's strategy is optimal given the strategies of all other players, leading to a stable state in strategic interactions.
Normal Form Game: A normal form game is a representation of a strategic interaction among players, where each player's strategy choices and their corresponding payoffs are displayed in a matrix format. This format allows for the analysis of how players make decisions based on the strategies available to them and the potential outcomes. It highlights the strategic dependencies between players, providing insight into the best responses each player can take given the choices made by others.
Pareto efficiency: Pareto efficiency refers to a situation in which resources are allocated in such a way that no individual can be made better off without making someone else worse off. It is a key concept in understanding optimal resource allocation and plays a significant role in various strategic interactions, showing how individuals or groups can reach outcomes where any change would harm at least one party involved.
Payoff Matrix: A payoff matrix is a table that shows the possible outcomes of a strategic interaction between players in a game, detailing their payoffs based on different strategies they may choose. This matrix is fundamental in understanding how players make decisions, as it encapsulates all possible strategies and their respective outcomes, leading to insights on rational choice and strategic behavior.
Payoff Vector: A payoff vector is a mathematical representation of the outcomes achieved by each player in a strategic game, typically displayed as an array or list that corresponds to the strategies chosen by the players. In normal form games, the payoff vector captures the numerical rewards that players receive based on their strategy choices and those of their opponents. Understanding payoff vectors is crucial for analyzing the incentives and potential decisions made by players in these types of games.
Prisoner's Dilemma: The prisoner's dilemma is a standard example in game theory that illustrates a situation where two individuals must choose between cooperation and betrayal, leading to outcomes that are suboptimal for both. It showcases how rational decision-making can lead to a worse collective outcome when individuals act in their self-interest rather than cooperating.
Rational Choice: Rational choice refers to the economic principle that individuals make decisions by considering the available options and choosing the one that maximizes their utility or benefit. This concept assumes that players in a game are rational agents, meaning they act in their own self-interest based on preferences, beliefs, and available information. In the context of decision-making and strategy, rational choice helps in predicting how individuals or groups will behave when faced with different scenarios, particularly within normal form games represented by payoff matrices.
Strategy Profile: A strategy profile is a combination of strategies chosen by all players in a game, representing their actions simultaneously. It provides a complete description of how each player will act, allowing the analysis of outcomes based on different strategic choices. Understanding strategy profiles is crucial for identifying equilibria and assessing how players' decisions interact within various frameworks, including payoff matrices and equilibria concepts.
Strictly Dominant Strategy: A strictly dominant strategy is a strategy in a game that yields a higher payoff for a player, regardless of what the other players choose to do. This means that no matter how opponents act, the strictly dominant strategy will always produce better outcomes for the player compared to any other strategies they might consider. Understanding this concept is crucial in analyzing normal form games and payoff matrices, as it helps identify optimal strategies that players can adopt to maximize their benefits.
Utility Function: A utility function is a mathematical representation that assigns a real number to each possible outcome or choice, reflecting the level of satisfaction or preference that an individual derives from that outcome. This concept is crucial in understanding decision-making processes and is linked to how players evaluate their options in strategic interactions, providing insights into their preferences, strategies, and behaviors.
Zero-sum game: A zero-sum game is a situation in game theory where one player's gain is exactly balanced by the losses of another player. In such games, the total utility remains constant, meaning that any advantage gained by one participant results in an equivalent disadvantage to another. This concept is crucial for understanding competitive strategies and interactions among rational players.
© 2024 Fiveable Inc. All rights reserved.
AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.
Back
Practice QuizGlossary
Practice QuizGlossary
Next guide