2.1 Normal Form Games and Payoff Matrices
2 min read•july 22, 2024
Normal form games are a fundamental concept in game theory, representing simultaneous-move scenarios where make decisions without knowing others' choices. These games consist of players, , and , which are organized into for analysis.
is a crucial aspect of game theory, where outcomes depend on all players' decisions. This concept distinguishes game theory from decision theory, as players must anticipate and respond to others' actions, making it essential in various real-world scenarios.
Normal Form Games
Components of normal form games
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- Normal form games represent simultaneous-move games where players make decisions without knowing the other players' choices
- Players are the decision-makers in the game (firms, individuals, countries)
- Strategies are the actions available to each player (pricing, quantity, investment levels)
- Payoffs are the outcomes for each player based on the combination of strategies chosen (profits, market share, )
Construction of payoff matrices
- Payoff matrices represent the outcomes of a in a tabular format
- Rows represent the strategies of one player (Firm A's pricing options)
- Columns represent the strategies of the other player (Firm B's pricing options)
- Each cell contains the payoffs for both players given the corresponding strategy combination (Firm A's profit, Firm B's profit)
- To construct a payoff matrix:
- Identify the players and their available strategies
- List the strategies of one player along the rows and the other player along the columns
- Fill in the payoffs for each player in each cell based on the corresponding strategy combination
Elements of game structure
- To analyze a normal form game, identify its key components
- Players are the decision-makers in the game (consumers, producers, governments)
- Strategies can be pure (choosing a single action) or mixed (a probability distribution over actions)
- Payoffs are the outcomes for each player based on the combination of strategies chosen
- Payoffs can be represented as numerical values (profits in dollars), rankings (1st, 2nd, 3rd), or verbal descriptions (high, medium, low)
Strategic Interdependence
Strategic interdependence in games
- Strategic interdependence is a key feature of game theory where the outcome depends on the decisions of all players involved
- Each player's optimal strategy depends on their beliefs about the other players' strategies ()
- Players must anticipate and respond to the actions of others (strategic thinking)
- The presence of strategic interdependence distinguishes game theory from decision theory, which focuses on individual decision-making without strategic interactions
- Examples of strategic interdependence include pricing decisions by competing firms (Coke vs Pepsi), arms races between nations (US vs USSR), and negotiations and bargaining situations (labor unions vs management)
Key Terms to Review (16)
Battle of the Sexes: The Battle of the Sexes is a classic game theory scenario where two players (often represented as a man and a woman) must choose between two options, each preferring a different option while still wanting to coordinate with each other. This situation illustrates the challenges of achieving mutual agreement when preferences conflict, showcasing the dynamics of cooperation and negotiation in decision-making.
Best Response: A best response is the strategy that yields the highest payoff for a player, given the strategies chosen by other players in a game. Understanding best responses is crucial because it helps players determine their optimal strategies based on the actions of others, highlighting the interdependence of decisions in strategic interactions.
Dominant strategy: A dominant strategy is a strategy that yields a higher payoff for a player, regardless of what the other players choose. This concept is central to understanding decision-making in strategic interactions, where players assess their options based on the potential responses of others, leading to predictable outcomes in competitive environments.
Mixed strategy: A mixed strategy is a decision-making approach where a player chooses between different strategies randomly, assigning a probability to each possible action rather than sticking to a single strategy. This adds an element of unpredictability to the player's choices, which can be crucial in competitive situations. It connects closely to game elements such as players, their available strategies, and the associated payoffs, allowing for a more complex analysis in various game forms.
Nash Equilibrium: Nash Equilibrium is a concept in game theory where players, knowing the strategies of their opponents, choose their optimal strategies resulting in a situation where no player has anything to gain by changing their own strategy unilaterally. This balance occurs when each player's strategy is the best response to the strategies chosen by others, highlighting the interdependence of player decisions and strategic decision-making.
Normal Form Game: A normal form game is a representation of a strategic interaction between players, where each player's strategies and corresponding payoffs are organized in a payoff matrix. This format allows for a clear visualization of the players' choices and outcomes, enabling the analysis of their best responses to each other's strategies. Normal form games are essential for understanding strategic decision-making and can illustrate concepts like Nash equilibrium and dominant strategies.
Pareto Efficiency: Pareto efficiency refers to a state where resources are allocated in a way that no individual's situation can be improved without worsening someone else's situation. This concept highlights the importance of mutual benefit in various strategic interactions and economic environments, emphasizing that an optimal allocation exists when it is impossible to make any participant better off without making at least one other participant worse off.
Payoff matrices: Payoff matrices are tools used in game theory to illustrate the outcomes of strategic interactions between players. They represent the payoffs for each player based on the combination of strategies chosen, allowing for a clear visual representation of how different choices lead to different results. By mapping out these outcomes, payoff matrices help analyze decisions and predict behaviors in competitive situations.
Payoffs: Payoffs refer to the outcomes or rewards that players receive in a game based on the strategies they choose. They represent the utility or benefit associated with each possible combination of strategies, allowing players to assess their choices and make informed decisions. Understanding payoffs is crucial for analyzing how players interact, predict behavior, and determine the best strategies to maximize their outcomes.
Players: In game theory, players are the decision-makers involved in a game, who can be individuals, groups, or organizations that make strategic choices to maximize their outcomes. Players interact with one another, often competing or cooperating, and their decisions directly influence the game's structure, strategies, and payoffs. Understanding the role of players is crucial for analyzing how different strategies can lead to various outcomes based on the interactions among those involved.
Prisoner's dilemma: The prisoner's dilemma is a fundamental concept in game theory that illustrates how two rational individuals may not cooperate, even if it appears that it is in their best interest to do so. This situation arises when both players have a choice to either cooperate or betray each other, leading to outcomes where mutual betrayal results in a worse payoff than if both had chosen to cooperate. Understanding this concept is crucial in various strategic decision-making scenarios, as it highlights the tension between individual rationality and collective benefit.
Pure Strategy: A pure strategy is a specific plan of action that a player in a game consistently follows, making a definitive choice in every situation they encounter. This concept is essential as it relates to decision-making where a player adopts one course of action without randomness or variation, allowing for predictable outcomes based on the player's choices. The idea of pure strategy is crucial for understanding normal form games and payoff matrices, differentiating between strategies, and calculating equilibria in competitive scenarios.
Strategic Interdependence: Strategic interdependence refers to a situation in which the decisions and outcomes of one player in a game directly influence the strategies and payoffs of other players. In the context of normal form games, this concept highlights how players must consider the potential actions of others when formulating their own strategies, making the outcomes dependent on the interplay between their choices. This interconnectedness is crucial for understanding equilibrium concepts and payoff matrices, as it shapes how players anticipate and react to each other's moves.
Strategies: In game theory, strategies refer to the complete plans of action that a player can take in a game, outlining how they will respond to different scenarios and the actions of other players. These strategies can be either pure, where a player consistently chooses one action, or mixed, where they randomize over possible actions. Understanding strategies is crucial as they shape the decisions made by players in both normal and extensive form games, influencing the outcomes based on the interactions among participants.
Subgame Perfection: Subgame perfection is a refinement of Nash equilibrium used in dynamic games, where players' strategies must be optimal not only for the game as a whole but also for every subgame that may arise. This concept ensures that players' strategies are credible and can be sustained at every possible decision point in the game. The idea is crucial for understanding how players make decisions over time and how their past actions affect future choices, especially in scenarios involving reputation effects and sequential interactions.
Utility: Utility refers to a measure of satisfaction or value that a player derives from the outcomes of a game. It is a key concept in understanding how players make decisions based on their preferences, as they aim to maximize their own utility when selecting strategies. In the context of games, utility connects to the payoff matrices, where the different outcomes are represented numerically, reflecting how much each player values those outcomes in their decision-making process.