Hoja de repaso: Strategic Interactions in Game Theory

📋 Course Outline

1. Players and Strategies
2. Payoff Structures
3. Game Types
4. Nash Equilibrium
5. Dominated Strategies
6. Mixed Strategies
7. Extensive Form Games
8. Game Tree Analysis
9. Backward Induction
10. Applications in Economics

📖 1. Players and Strategies

🔑 Key Concepts & Definitions

• Players: The decision-makers in a game, which can be individuals, groups, or organizations, each aiming to maximize their own payoff.

• Strategies: Complete plans of action that specify a player's choices in every possible situation within the game. Strategies can be pure (a specific action) or mixed (probabilistic combination of actions).

• Payoffs: The outcomes or rewards received by players based on the combination of strategies chosen, often represented numerically in utility or monetary terms.

• Normal Form: A representation of a game using a matrix that displays players, their strategies, and corresponding payoffs, suitable for analyzing simultaneous-move games.

• Strategy Profile: A set of strategies, one for each player, representing a specific outcome in the game.

📝 Essential Points

• Players are rational and aim to optimize their payoffs based on the strategies of others.
• Strategies encompass all possible actions a player might take; in complex games, mixed strategies are used to introduce randomness.
• Payoffs depend on the combination of strategies chosen by all players, influencing their decision-making.
• The normal form provides a static snapshot of the game, facilitating analysis of strategic interactions.
• Understanding the distinction between pure and mixed strategies is crucial for analyzing equilibrium concepts like Nash Equilibrium.

💡 Key Takeaway

Players select strategies—either pure or mixed—to maximize their payoffs, and analyzing these strategies through representations like the normal form helps predict stable outcomes such as Nash Equilibria in strategic interactions.

📖 2. Payoff Structures

🔑 Key Concepts & Definitions

• Payoff: The numerical reward or outcome a player receives from a particular strategy profile, often represented in utility or monetary terms.
• Payoff Matrix: A table that displays the payoffs for each player based on the combination of strategies chosen by all players in a game.
• Pure Strategy: A strategy where a player consistently chooses a specific action or plan without randomness.
• Mixed Strategy: A strategy where a player assigns probabilities to different actions, effectively randomizing their choices.
• Payoff Function: A mathematical function that assigns a payoff value to each possible combination of strategies for all players.

📝 Essential Points

• Payoffs determine players' preferences and influence strategic choices; higher payoffs are generally more desirable.
• The structure of payoffs influences the existence and nature of equilibria, such as Nash Equilibrium.
• In a payoff matrix, the strategic interactions are visualized, allowing identification of best responses and dominated strategies.
• Payoff structures can be symmetric or asymmetric, affecting strategic considerations.
• Understanding whether players use pure or mixed strategies is crucial for analyzing complex games, especially when no pure strategy equilibrium exists.

💡 Key Takeaway

Payoff structures are fundamental to game theory as they quantify players' preferences and shape strategic interactions, ultimately guiding the analysis of equilibrium outcomes.

📖 3. Game Types

🔑 Key Concepts & Definitions

• Cooperative Games: Games where players can form binding agreements and coalitions to achieve shared objectives; outcomes depend on collective strategies.
• Non-Cooperative Games: Games where players make decisions independently without binding commitments, focusing on individual strategies.
• Symmetric Games: Games where payoffs depend solely on strategies employed, not on who is playing; identical strategies yield identical payoffs.
• Asymmetric Games: Games where players have different strategies, information, or payoff structures, leading to unequal roles or advantages.
• Zero-Sum Games: Competitive situations where one player's gain equals another's loss; total payoff remains constant.
• Non-Zero-Sum Games: Situations where players' payoffs can simultaneously increase or decrease, allowing for mutual gains or losses.
• Simultaneous Games: Players choose strategies at the same time without knowledge of others' choices; decisions are made concurrently.
• Sequential Games: Players make decisions in sequence, with later players observing earlier moves; represented through game trees.

📝 Essential Points

• The classification of game types influences strategic analysis and solution methods.
• Cooperative vs. Non-Cooperative distinction affects whether binding agreements are considered.
• Symmetry simplifies analysis; asymmetric games require more complex strategies.
• Zero-sum games focus on pure competition; non-zero-sum allows for cooperation and negotiation.
• Simultaneous games often involve mixed strategies; sequential games are analyzed via backward induction.
• Understanding the nature of the game guides the choice of solution concepts like Nash Equilibrium or Subgame Perfect Equilibrium.

💡 Key Takeaway

Different game types—cooperative, symmetric, zero-sum, simultaneous, etc.—shape the strategic landscape and determine the appropriate analytical tools for predicting outcomes. Recognizing these distinctions is essential for effective strategic decision-making.

📖 4. Nash Equilibrium

🔑 Key Concepts & Definitions

• Nash Equilibrium: A set of strategies where no player can improve their payoff by unilaterally changing their own strategy, assuming other players' strategies remain constant.
• Best Response: The strategy that yields the highest payoff for a player given the strategies chosen by other players.
• Pure Strategy: A strategy where a player chooses a specific action with certainty.
• Mixed Strategy: A strategy where a player assigns probabilities to different actions, randomizing their choices.
• Dominant Strategy: A strategy that is better for a player regardless of what others do; a special case where the strategy is always optimal.
• Equilibrium Stability: The property that, once reached, players have no incentive to deviate from their strategies.

📝 Essential Points

• A Nash Equilibrium may involve pure or mixed strategies.
• Multiple Nash Equilibria can exist within a game.
• Not all stable outcomes are Nash Equilibria; the concept specifically relates to mutual best responses.
• Finding Nash Equilibria involves analyzing players' best responses to each other's strategies.
• In symmetric games, equilibria often involve identical strategies; in asymmetric games, strategies differ.
• The Nash Equilibrium concept assumes rationality and complete information among players.

💡 Key Takeaway

A Nash Equilibrium represents a stable strategic state where no player benefits from changing their strategy unilaterally, serving as a fundamental solution concept in understanding strategic interactions.

📖 5. Dominated Strategies

🔑 Key Concepts & Definitions

• Dominated Strategy: A strategy that results in a worse payoff for a player regardless of the strategies chosen by other players. It can be eliminated from consideration because it is never optimal.
• Strictly Dominated Strategy: A strategy that yields a strictly lower payoff than another strategy for all possible strategies of opponents.
• Weakly Dominated Strategy: A strategy that yields a payoff less than or equal to another strategy in all cases and strictly less in at least one case.
• Dominant Strategy: A strategy that is the best choice for a player regardless of what others do; it may or may not be dominated.
• Iterative Elimination of Dominated Strategies: A process where dominated strategies are sequentially removed from the game to simplify analysis and identify potential equilibria.

📝 Essential Points

• Dominated strategies are suboptimal and can be eliminated to simplify game analysis.
• Removing dominated strategies does not affect the set of Nash equilibria, making it a useful step in solving games.
• The process of iterated elimination involves repeatedly removing dominated strategies until no further elimination is possible.
• Not all games have dominated strategies; their presence depends on the payoff structure.
• Recognizing dominated strategies helps in predicting rational behavior and narrowing down strategic options.

💡 Key Takeaway

Identifying and eliminating dominated strategies streamlines strategic analysis by focusing on rational choices, often leading to the discovery of Nash equilibria and simplifying complex games.

📖 6. Mixed Strategies

🔑 Key Concepts & Definitions

• Mixed Strategy: A strategy where a player assigns probabilities to each possible pure strategy, effectively randomizing their choices to keep opponents uncertain.
• Pure Strategy: A deterministic choice of a single action or plan of action in a game.
• Mixed Strategy Nash Equilibrium: A set of probability distributions over strategies such that no player can improve their expected payoff by unilaterally changing their own mixed strategy.
• Expected Payoff: The average payoff a player expects to receive when using a mixed strategy, calculated as the sum of payoffs weighted by the probabilities of opponents' strategies.
• Indifference Condition: A situation where a player is indifferent between strategies because they yield the same expected payoff, often used to solve for equilibrium probabilities.

📝 Essential Points

• Mixed strategies are essential when no pure strategy Nash equilibrium exists or when players want to keep their actions unpredictable.
• They are particularly useful in zero-sum games and games with no pure strategy equilibrium.
• To find a mixed strategy equilibrium, set the expected payoffs of the strategies that are being mixed over equal, ensuring the player is indifferent.
• Mixed strategies can stabilize outcomes in games where pure strategies lead to cycling or dominated responses.
• Many real-world strategic interactions, such as bidding in auctions or sports tactics, involve mixed strategies.

💡 Key Takeaway

Mixed strategies allow players to randomize their actions, creating equilibrium in games where pure strategies fail to produce stable outcomes, thus broadening the strategic options and ensuring stability in complex interactions.

📖 7. Extensive Form Games

🔑 Key Concepts & Definitions

• Game Tree: A graphical representation of an extensive form game, illustrating sequential moves, decision nodes, and possible outcomes. It captures the order of play and information available at each decision point.

• Backward Induction: A solution method for extensive form games where players analyze the game from the end (terminal nodes) backward to determine optimal strategies at each decision point, assuming rationality throughout.

• Subgame: A portion of a game that begins at a decision node and includes all subsequent moves. It must be a complete, independent game within the larger game, with its own starting point and outcomes.

• Subgame Perfect Equilibrium (SPE): A refinement of Nash Equilibrium applicable to extensive form games, where strategies constitute a Nash Equilibrium in every subgame, ensuring credibility of strategies at all stages.

• Information Set: A collection of decision nodes that a player cannot distinguish between when making a move, representing imperfect information. All nodes within an information set are treated as a single decision point.

📝 Essential Points

• Extensive form games model sequential decision-making, capturing the timing and information structure of moves.

• Game trees visually depict the sequence of actions, chance events, and payoffs, facilitating analysis of strategic choices.

• Backward induction is used to solve perfect information games by iteratively determining optimal strategies from the end of the game to the beginning.

• Subgame perfect equilibrium eliminates non-credible threats by requiring strategies to be optimal at every subgame, ensuring consistent and credible plans.

• Information sets account for imperfect information, where players may not observe all previous moves, affecting their strategic choices.

💡 Key Takeaway

Extensive form games provide a detailed framework for analyzing sequential and dynamic strategic interactions, with backward induction and subgame perfect equilibrium serving as fundamental tools for identifying credible and optimal strategies.

📖 8. Game Tree Analysis

🔑 Key Concepts & Definitions

• Game Tree: A graphical representation of a sequential game, illustrating the order of moves, decision points (nodes), and possible outcomes (branches). It visually maps out strategic interactions over time.

• Nodes: Points in the game tree where a player makes a decision. Each node indicates a specific point in the game where choices are made.

• Branches: The lines connecting nodes, representing possible actions or strategies a player can take at each decision point.

• Backward Induction: A method for solving sequential games by analyzing the game from the end (terminal nodes) backward to determine optimal strategies at each decision point.

• Subgame: A portion of the game tree that can be considered a game in itself, starting from a single node and including all subsequent nodes and branches.

• Subgame Perfect Equilibrium (SPE): A refinement of Nash Equilibrium applicable to extensive form games, where strategies constitute a Nash Equilibrium in every subgame, ensuring credible threats and promises.

📝 Essential Points

• Game trees are used to analyze sequential interactions where players move one after another, capturing the timing and information structure of decisions.

• Backward induction involves solving the game by starting at terminal nodes and determining optimal strategies for the players at each preceding node.

• Subgames allow for the decomposition of complex games into smaller, manageable parts, facilitating the analysis of strategic credibility.

• The concept of subgame perfect equilibrium ensures that strategies form a Nash Equilibrium in every part of the game, eliminating non-credible threats.

• Proper understanding of information sets (not explicitly listed here) is crucial when players have imperfect information, affecting how game trees are constructed and analyzed.

💡 Key Takeaway

Game tree analysis, combined with backward induction and subgame perfection, provides a systematic approach to solving sequential games, ensuring strategies are credible and optimal at every stage of decision-making.

📖 9. Backward Induction

🔑 Key Concepts & Definitions

• Backward Induction: A method used to solve sequential (extensive form) games by analyzing the game from the end (terminal nodes) backwards to determine optimal strategies at each decision point.

• Subgame: A portion of a game that constitutes a game itself, starting from a decision node and including all subsequent nodes. Backward induction applies to subgames to find subgame perfect equilibria.

• Subgame Perfect Equilibrium (SPE): A refinement of Nash Equilibrium where strategies constitute a Nash Equilibrium in every subgame, achieved through backward induction.

• Terminal Node: The end point of a game tree where payoffs are realized. Backward induction begins with analyzing these nodes.

• Backward Reasoning: The process of starting from the last move(s) of the game and reasoning backward to determine the optimal strategies at earlier decision points.

📝 Essential Points

• Backward induction is applicable only in sequential games with a clear order of moves.

• It involves solving the game from the end to the beginning, ensuring strategies are optimal at every stage.

• The method guarantees finding subgame perfect equilibrium, which eliminates non-credible threats or promises.

• In each subgame, players choose strategies that maximize their payoffs given future actions, leading to a backward reasoning process.

• It is a powerful tool for analyzing dynamic strategic interactions, such as bargaining, entry deterrence, or bargaining scenarios.

• Limitations: Backward induction assumes players are rational and have perfect information about the game structure.

💡 Key Takeaway

Backward induction systematically determines optimal strategies in sequential games by analyzing moves from the end, ensuring strategies form a subgame perfect equilibrium and credible decision-making throughout the game.

📖 10. Applications in Economics

🔑 Key Concepts & Definitions

• Oligopoly: A market structure characterized by a small number of firms whose decisions are interdependent, often analyzed using game theory to predict strategic behavior such as pricing and output decisions.

• Strategic Interaction: Situations where the outcome for each participant depends on the actions of others, modeled effectively through game theory to understand competitive and cooperative behaviors.

• Payoff Matrix: A table representing the outcomes (payoffs) for each player based on their chosen strategies, used to analyze strategic choices in economic models like duopoly or cartel formation.

• Nash Equilibrium in Economics: A set of strategies where no firm can improve its payoff by unilaterally changing its strategy, often used to predict stable market outcomes such as price setting or collusion.

• Repeated Games: Strategic interactions that occur over multiple periods, allowing for the possibility of reputation-building and cooperation, crucial in understanding long-term economic relationships like trade or cartel stability.

• Signaling and Commitment: Strategies used by firms or individuals to influence others' perceptions or to commit to future actions, often analyzed through game theory to explain credible promises or deterrence.

📝 Essential Points

• Game theory models are essential for analyzing oligopolistic markets, where firms' decisions are interdependent and strategic.
• The concept of Nash equilibrium helps predict stable market outcomes where no firm benefits from deviating unilaterally.
• Repeated interactions enable cooperation, which can lead to collusive behavior or sustained competition, depending on the incentives.
• Signaling and commitment strategies are vital for resolving issues of credibility and ensuring cooperation in economic settings.
• Strategic decision-making impacts pricing, production, entry/exit, and innovation, influencing overall market efficiency and welfare.
• Many real-world economic phenomena, such as auctions, bargaining, and regulation, are effectively modeled using game-theoretic approaches.

💡 Key Takeaway

Game theory provides a vital framework for understanding strategic decision-making in economics, enabling prediction of firm behavior, market outcomes, and the stability of cooperation or competition in various market structures.

📊 Synthesis Tables

Aspect	Players & Strategies	Payoff Structures
Core Focus	Decision-makers, strategies (pure/mixed), payoffs	Numerical outcomes, payoff matrices, functions
Representation	Normal form, strategy profiles	Payoff matrix, payoff functions
Key Concepts	Rationality, best responses, equilibrium analysis	Utility, dominance, payoff optimization
Analytical Use	Predicting stable outcomes (e.g., Nash)	Identifying dominant/dominated strategies, responses

Aspect	Game Types	Nash Equilibrium & Dominated Strategies
Core Focus	Cooperative/non-cooperative, symmetry, zero-sum, sequential	Stability, best responses, strategy dominance
Key Features	Binding agreements, symmetry, competition types	Unilateral deviations, elimination, stability
Analytical Approach	Solution concepts vary: cooperative, non-cooperative, backward induction	Identifying equilibria, removing dominated strategies

⚠️ Common Pitfalls & Confusions

1. Confusing pure and mixed strategies; forgetting that mixed strategies involve probability distributions.
2. Assuming the existence of a pure strategy Nash Equilibrium in all games; some require mixed strategies.
3. Overlooking the importance of best response analysis when identifying Nash Equilibria.
4. Misidentifying dominated strategies; confusing weakly dominated with strictly dominated.
5. Ignoring the sequential nature of extensive form games; applying normal form analysis incorrectly.
6. Misapplying backward induction in sequential games, especially when multiple subgames exist.
7. Assuming all stable outcomes are Nash Equilibria without verifying the best response condition.
8. Overlooking the difference between symmetric and asymmetric games in strategy analysis.
9. Neglecting the role of payoff structures in determining the existence and nature of equilibria.
10. Confusing cooperative and non-cooperative game assumptions; binding agreements are only in cooperative games.
11. Ignoring the possibility of multiple Nash Equilibria leading to coordination problems.

✅ Exam Checklist

• Define players, strategies (pure and mixed), and payoffs.
• Explain the purpose of the normal form and how to interpret payoff matrices.
• Differentiate between cooperative and non-cooperative, symmetric and asymmetric, zero-sum and non-zero-sum games.
• Describe the concept of Nash Equilibrium and how to identify it.
• Understand best response functions and their role in equilibrium analysis.
• Identify and eliminate dominated strategies; distinguish between weak and strict dominance.
• Explain the concept of mixed strategies and when they are necessary.
• Describe extensive form games and how to analyze them using game trees.
• Apply backward induction to solve sequential games.
• Discuss the application of game theory concepts in economics, such as market competition and bargaining.
• Recognize the limitations of Nash Equilibrium and the potential for multiple equilibria.
• Use payoff structures to analyze strategic stability and predict outcomes.