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    Game Theory

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    • Introduction to Game Theory
      • 1.1What is Game Theory?
      • 1.2History and Importance of Game Theory
      • 1.3Understanding Basic Terminology
    • Two-Person Zero-Sum Games
      • 2.1Defining Zero-Sum Games
      • 2.2Solving Simple Zero-Sum Games
      • 2.3Strategies and Dominance in Zero-Sum Games
    • Non-Zero-Sum and Cooperative Games
      • 3.1Introduction to Non-Zero-Sum Games
      • 3.2Cooperative Games and the Core
      • 3.3Bargaining & Negotiation Techniques
    • Game Theory in Business and Economics
      • 4.1Market Analysis via Game Theory
      • 4.2Strategic Moves in Business
      • 4.3Auctions and Bidding Strategies
    • Game Theory in Politics
      • 5.1Electoral Systems and Voting Strategies
      • 5.2Power and Conflict Resolution
      • 5.3Foreign Policy and International Relations
    • Psychological Game Theory
      • 6.1Perception, Belief, and Strategic Interaction
      • 6.2Emotions and Decision-Making
      • 6.3Behavioral Biases in Strategic Thinking
    • Games of Chance and Risk
      • 7.1Probability Analysis and Risk Management
      • 7.2Gambler's Fallacy
      • 7.3Risk Tolerance and Decision Making
    • Evolutionary Game Theory
      • 8.1The Origin and Motivation for Evolutionary Game Theory
      • 8.2Evolutionary Stability Strategies
      • 8.3Application of Evolutionary Game Theory
    • Games with Sequential Moves
      • 9.1Extensive Form Representation
      • 9.2Backward Induction
      • 9.3Credible Threats and Promises
    • Game Theory in Social Interactions
      • 10.1Social Rules and Norms as Games
      • 10.2Role of Reputation and Signals
      • 10.3Social Network Analysis
    • Ethics in Game Theory
      • 11.1Fairness Concepts
      • 11.2Moral Hazards and Incentives
      • 11.3Social Dilemmas and Collective Action
    • Technological Aspects of Game Theory
      • 12.1Digital Trust and Security Games
      • 12.2AI and Machine Learning in Game Theory
      • 12.3Online Marketplaces and Digital Economy
    • Applying Game Theory in Everyday Life
      • 13.1Practical Examples of Game Theory at Work
      • 13.2Thinking Strategically in Personal Decisions
      • 13.3Final Recap and Strategizing Your Life

    Two-Person Zero-Sum Games

    Solving Simple Zero-Sum Games

    stationary point that is not a local extremum

    Stationary point that is not a local extremum.

    Zero-sum games are a fundamental concept in game theory. They represent situations where the gain of one player is exactly balanced by the losses of the other player. In other words, the sum of total gains and losses in the game is zero. This article will guide you through the process of solving simple zero-sum games.

    Understanding the Matrix Form of Zero-Sum Games

    Zero-sum games can be represented in a matrix form, where each cell represents the outcome of a particular combination of strategies. The rows represent the strategies of one player, while the columns represent the strategies of the other player. The value in each cell is the payoff for the row player, which is the loss for the column player.

    The Concept of Saddle Points

    A saddle point in a zero-sum game is a position where the row player's minimum payoff is maximized, and the column player's maximum loss is minimized. In other words, it's a position where neither player would benefit from changing their strategy unilaterally. If a saddle point exists in a zero-sum game, it represents the optimal strategy for both players.

    The Minimax and Maximin Strategies

    The minimax and maximin strategies are fundamental to zero-sum games. The minimax strategy involves minimizing the maximum possible loss, while the maximin strategy involves maximizing the minimum possible gain. In a zero-sum game, these two strategies coincide at the saddle point.

    Pure and Mixed Strategies in Zero-Sum Games

    A pure strategy involves always choosing the same action, regardless of what the other player does. A mixed strategy involves choosing different actions with certain probabilities. In some zero-sum games, a mixed strategy can provide a better outcome than any pure strategy.

    Solving Simple Zero-Sum Games Using Graphical Method

    The graphical method is a simple way to solve zero-sum games with two strategies for each player. The steps are as follows:

    1. Plot the payoffs for each strategy combination on a graph.
    2. Identify the points where the row player's minimum payoff is maximized (the maximin point) and where the column player's maximum loss is minimized (the minimax point).
    3. If the maximin and minimax points coincide, that's the saddle point and the optimal strategy for both players. If they don't coincide, there's no saddle point and the game has no pure strategy solution.

    By understanding these concepts and methods, you can effectively analyze and solve simple zero-sum games. This will provide a solid foundation for understanding more complex games and strategies in game theory.

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    Next up: Strategies and Dominance in Zero-Sum Games