Introduction: Have you ever wondered why companies choose certain pricing strategies or how politicians plan their election campaigns? The answer lies in a fascinating field of study called game theory. Game theory provides a framework for understanding how people behave and make choices by examining how individuals and organizations make strategic decisions in various scenarios. From simple games like rock-paper-scissors to complex economic and political interactions, game theory is a powerful tool for predicting outcomes and understanding human behavior. In this blog post, we will explore the fundamentals of game theory, including the famous "Prisoner's Dilemma" game and the concept of Nash Equilibrium, which has important applications in various real-world scenarios. Join us as we dive into the world of game theory and discover how it can help us better understand our choices and interactions with others.
Game theory studies mathematical models of strategic interactions between rational decision-makers. It is a field that seeks to understand how people make decisions in situations where their choices affect themselves and others. For example, in rock-paper-scissors, each player's outcome depends on their and their opponent's actions. Using game theory, we can analyze how people behave when faced with multiple options and identify the best strategies for achieving desired outcomes. Game theory has wide-ranging applications, from predicting stock market behavior to understanding political negotiations and modeling biological interactions. So, let us explore some of the key concepts of game theory and see how they can help us understand the world.
The Prisoner's Dilemma:
Suppose two criminals, Alice and Bob, have been arrested for committing a crime together. The police do not have enough evidence to convict them of the crime, but they suspect the two have something to do with it. The police separate them and offer them a deal: if they both stay silent and do not confess, they will get a minimal sentence of one year in jail for a lesser charge. Nevertheless, if one confesses and the other does not, the one who confesses will get a lighter sentence (let us say six months) or go free, while the other will get a much harsher sentence (let us say five years) for refusing to cooperate.
In this scenario, Alice and Bob face a classic prisoner's dilemma. If both cooperate and stay silent, they both get a minimal sentence. (1 year) However, if one defects and confesses, they can get a much lighter sentence or go free, while the other gets a much harsher sentence(3 years). If both defect and confess, they get a harsher sentence, but each gets a slightly lighter sentence than if they had stayed silent(2 years).
Bob - cooperate Bob - defect
Alice - Cooperate 1,1 3,0
Alice - Defect 0,3 2,2
From a purely rational perspective, each criminal's best strategy is to defect and confess, regardless of what the other criminal does. This action is because if one cooperates and stays silent, they risk getting a much harsher sentence if the other defects and confesses. However, if both defect and confess, they end up with a harsher sentence than if they had cooperated.
Thus, the dilemma arises because each criminal's rationality (i.e., the desire to minimize their sentence) conflicts with the group's rationality (i.e., the desire to minimize the total sentence of both criminals). Players' preferences, strategies, and beliefs, as well as the information they have available, determine the outcome of the dilemma.
The prisoner's dilemma can illustrate many economic situations where individual self-interest can conflict with the common good.
For example, in the context of competition between firms, each firm is incentivized to cut prices and increase market share, even if it leads to a price war that ultimately harms all firms in the industry. Similarly, in environmental regulation, each firm is incentivized to pollute more than its competitors, even if it harms the environment and ultimately affects everyone.
These situations exemplify what economists call "negative externalities," where one agent's actions negatively impact others. In the case of the prisoner's dilemma, the negative externality is the total sentence of both criminals, which increases if both criminals defect and confess.
Game theory provides a framework for analyzing these situations and identifying strategies for achieving a desirable outcome. One strategy is for the players to cooperate and agree to rules or regulations that prevent them from taking actions that harm the common good. Another strategy is for a trusted third party to intervene and enforce such rules or regulations. In this way, the prisoner's dilemma can help us understand the challenges of achieving cooperation and coordination in economic and social contexts and the potential benefits of institutions and policies that promote collective action and cooperation.
Nash Equilibrium:
Let us take an example of two gas stations competing for customers in a small town. The two gas stations are across the street, and each must choose between two strategies: setting high or low prices. Based on both stations' choices, the following table shows the payoff for each:
Station 2 - High Station 2 - Low
Station 1 - High 2,2 0,3
Station 1 - Low 3,0 1,1
In this scenario, the numbers represent the profits for each gas station. For example, if Station 1 chooses to set high prices and Station 2 also chooses to set high prices, then both stations earn a profit of 2. If Station 1 sets low prices and Station 2 sets high prices, Station 1 earns a profit of 3 while Station 2 earns a profit of 0, and so on.
To determine the Nash Equilibrium in this game, we need to find a strategy profile where neither gas station has the incentive to deviate from their chosen strategy. In this case, the Nash Equilibrium occurs when both gas stations choose to set low prices. If Station 1 sets high prices and Station 2 sets low prices, Station 1 earns a profit of 0 while Station 2 earns a profit of 3. If Station 1 sets low prices and Station 2 sets high prices, Station 1 earns a profit of 1 while Station 2 earns a profit of 0. Therefore, both gas stations are incentivized to stick to their strategy of setting low prices since any deviation would result in a lower profit for that station.
This example illustrates how the Nash Equilibrium can help us predict outcomes when multiple parties make strategic decisions. By identifying the strategy profile where no player has the incentive to deviate, we can make predictions about the behavior of each player and the likely outcome of the game.
The relationship between the Nash Equilibrium and the Prisoner's Dilemma is that the Nash Equilibrium can be used to analyze the strategies players might adopt in the Prisoner's Dilemma game. The Nash Equilibrium provides a solution concept for strategic games, such as the Prisoner's Dilemma, in which each player's strategy is optimal given the other players' strategies.
In the case of the Prisoner's Dilemma case, the Nash Equilibrium occurs when both players choose to defect, even though this results in a sub-optimal outcome for both players. This concept highlights the game's tension between individual and collective rationality and the challenge of coordinating a cooperative outcome. In the economy, the Nash Equilibrium can be used to analyze the strategies that firms might adopt in a competitive market or that countries might adopt in a trade war. The Nash Equilibrium can help identify the stable outcomes in such a situation, and the strategies firms might adopt to achieve them.
In conclusion, game theory is a fascinating subject that provides valuable insights into the strategic decision-making of individuals and groups. The Nash Equilibrium is a crucial concept in game theory that helps us understand how players in a game can make optimal decisions. The prisoner's dilemma is a classic example that illustrates the challenges of cooperation in game theory.
Furthermore, game theory has important applications in various fields, including economics, politics, and psychology. By applying game theory principles, we can better understand how individuals and organizations make decisions and use this knowledge to create better strategies and outcomes.
We hope this introduction to game theory and the Nash Equilibrium has piqued your interest in this exciting field of study. Watch for more blog posts on game theory in the future!