2.1 Normal form (strategic form) game representation

Normal form games are the foundation of game theory, representing strategic interactions between players. These games capture the essence of decision-making in competitive situations, where each player's choices affect everyone's outcomes.

In this representation, players, strategies, and payoffs are laid out in a matrix. This format allows for easy analysis of pure and mixed strategies, helping us understand how players might behave in simultaneous-move scenarios with strategic interdependence.

Game Components

Players, Strategies, and Payoffs

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• Players are the decision-makers in the game who choose from a set of available actions
• Strategies represent the complete plan of action for each player, specifying what action to take in every possible situation
• Payoffs are the outcomes or utilities that each player receives based on the combination of strategies chosen by all players
• Payoff matrix is a tabular representation of the game, showing the payoffs for each player for every possible combination of strategies

Representing Games with a Payoff Matrix

• A payoff matrix summarizes the game by displaying the players, their strategies, and the corresponding payoffs in a table format
• Each cell in the matrix represents a unique combination of strategies chosen by the players
• The payoffs in each cell indicate the outcome for each player when that particular combination of strategies is played
• Payoff matrices provide a clear and concise way to represent and analyze strategic interactions between players (Prisoner's Dilemma, Battle of the Sexes)

Strategy Types

Pure Strategies

• A pure strategy is a complete plan of action that specifies a single action to be taken in every possible situation
• In a pure strategy, the player chooses one specific action with certainty
• Pure strategies do not involve randomization or probability distributions over actions
• Examples of pure strategies include always cooperating or always defecting in the Prisoner's Dilemma game

Mixed Strategies

• A mixed strategy is a probability distribution over the set of available pure strategies
• In a mixed strategy, the player randomizes their choice of action according to a specified probability distribution
• Mixed strategies allow players to introduce unpredictability into their decisions
• Examples of mixed strategies include playing "Rock" with 50% probability and "Paper" with 50% probability in the Rock-Paper-Scissors game

Game Characteristics

Simultaneous Move Games

• In simultaneous move games, players choose their strategies simultaneously without knowing the choices of other players
• Players make their decisions independently and without observing the actions of their opponents
• Simultaneous move games capture situations where players must anticipate and reason about the likely strategies of others
• Examples of simultaneous move games include the Prisoner's Dilemma, Battle of the Sexes, and the Matching Pennies game

Strategic Interaction and Interdependence

• Strategic interaction refers to the idea that a player's optimal strategy depends on their beliefs about the strategies of other players
• In games with strategic interaction, the payoffs of each player are influenced by the combined actions of all players
• Players must consider the incentives and likely behaviors of their opponents when making their own decisions
• The concept of strategic interaction highlights the interdependence of players' choices and the need for strategic reasoning in game theory