One-Time Games and the Prisoner's Dilemma: Videos & Practice Problems

In an oligopoly, firms must strategically consider competitors' actions when making pricing decisions, often analyzed through game theory. The prisoner's dilemma illustrates this, where two players face choices that lead to a Nash equilibrium, where both confess, resulting in suboptimal outcomes. Collusion, or cooperation among firms, can lead to better results, but the incentive to cheat exists, undermining collective benefits. Understanding dominant strategies and Nash equilibria is crucial for analyzing market behavior and competition, highlighting the complexities of economic interactions and the potential for market failure.

Game Theory, One-Time Games, and the Prisoner's Dilemma

In an oligopoly, firms must make strategic pricing decisions due to the limited number of competitors in the market. This interdependence among firms leads to the application of game theory, which examines how the outcomes of decisions are influenced by the actions of others. A key concept in game theory is the one-time game, exemplified by the famous Prisoner's Dilemma.

The Prisoner's Dilemma involves two individuals, Bad Boy Benny and Evil Eddie, who are arrested but lack sufficient evidence against them. They are separated and cannot communicate, leading to a critical decision-making scenario. Each prisoner faces the same offer from the police: if one confesses while the other remains silent, the confessor goes free, and the silent partner receives a 20-year sentence. If both confess, they each receive an 8-year sentence. However, if both remain silent, they each serve only 1 year for lesser charges.

This situation can be represented using a payoff matrix, which organizes the potential outcomes based on the decisions made by each prisoner. The matrix includes the following scenarios:

• If both prisoners remain silent, they each serve 1 year.
• If one confesses and the other does not, the confessor goes free (0 years), while the silent partner receives 20 years.
• If both confess, they each serve 8 years.

To analyze the best decision for each prisoner, they must consider the potential actions of the other. The dilemma illustrates the conflict between individual rationality and collective benefit, as both prisoners would be better off if they both chose to remain silent, yet the fear of betrayal often leads to a suboptimal outcome where both confess.

Understanding the dynamics of the Prisoner's Dilemma provides insight into strategic decision-making in competitive environments, highlighting the importance of trust and cooperation in achieving the best possible outcomes.

Dominant Strategy and the Nash Equilibrium

In game theory, understanding the concept of a dominant strategy is crucial for making optimal decisions in competitive situations. A dominant strategy is defined as the best course of action for a player, regardless of what the other player decides. This means that if one player has a dominant strategy, they will always choose that option because it yields the best outcome for them, irrespective of the opponent's actions.

To illustrate this, consider the scenario involving two players: Bad Boy Benny and Evil Eddie. Each player must decide whether to confess or not, and their decisions impact the length of their prison sentences. If Evil Eddie confesses, Bad Boy Benny can either confess and receive an 8-year sentence or remain silent and face 20 years. In this case, confessing is the better option for Benny. Conversely, if Evil Eddie chooses not to confess, Benny can confess and go free (0 years) or stay silent and receive 1 year. Again, confessing is the optimal choice for Benny.

Similarly, Evil Eddie faces the same dilemma. If Benny confesses, Eddie can either confess and get 8 years or stay silent and receive 20 years. If Benny does not confess, Eddie can confess and go free or stay silent and get 1 year. In both scenarios, Eddie's best choice is to confess. This leads to the conclusion that both players have a dominant strategy to confess, resulting in a situation known as the prisoner's dilemma.

The prisoner's dilemma highlights a critical aspect of game theory: while both players make the best individual decisions, the collective outcome is suboptimal. Instead of cooperating and potentially receiving only 1 year each (if both chose not to confess), they both end up with 8 years due to their inability to communicate and coordinate their strategies.

This situation is further explained by the concept of Nash equilibrium, named after economist John Nash. A Nash equilibrium occurs when all players in a game make their best possible decisions given the choices of their opponents. In this case, the Nash equilibrium is the outcome where both players confess, leading to the 8-year prison sentence for each. While this represents a stable outcome where no player can benefit from changing their strategy unilaterally, it does not necessarily reflect the best possible outcome for both players.

In summary, the interplay of dominant strategies and Nash equilibrium in the prisoner's dilemma illustrates the complexities of decision-making in competitive environments. Players may find themselves in a situation where their best individual choices lead to a collectively worse outcome, emphasizing the importance of cooperation and communication in achieving optimal results.

Collusion, Cartels, and Price Leadership

The concept of the prisoner's dilemma illustrates how two rational individuals might not cooperate, even if it appears that it is in their best interest to do so. In this scenario, if both players choose to remain silent, they would each serve only one year in prison. However, if one confesses while the other remains silent, the confessor goes free while the silent partner faces a much harsher sentence. This situation highlights the tension between individual incentives and collective benefit, a theme that extends into the realm of economics, particularly in oligopolies.

In an oligopoly, firms may engage in collusion, which is an agreement among competing firms to set prices or output levels. This practice is illegal in the United States as it undermines competition. For instance, companies like McDonald's and Burger King cannot legally coordinate their pricing strategies. When firms collude, they form what is known as a cartel. A prominent example of a cartel is OPEC, which consists of oil-producing nations that collaborate to control oil prices and production levels globally.

While collusion can lead to higher profits for the members of a cartel, it also creates an incentive for individual firms to cheat. If one firm knows that others will adhere to the agreed-upon production levels, it may choose to produce more than its quota to maximize its profits. This behavior can destabilize the cartel, as seen in the prisoner's dilemma where one player may betray the other for personal gain.

Additionally, there is a concept known as implicit collusion, which occurs when firms indirectly coordinate their pricing strategies without explicit communication. An example of this is price leadership, where one firm sets a price that others follow. For instance, if Walmart announces a price for a new video game console, other retailers may adopt the same price without any formal agreement. While this practice may raise ethical questions, it is not illegal as it does not involve direct collusion.

Understanding these dynamics is crucial for analyzing market behavior and the implications of competition and cooperation among firms. The interplay between collusion, competition, and individual incentives shapes the economic landscape, influencing pricing strategies and market outcomes.

One-Time Games:Finding Dominant Strategies and Nash Equilibrium with the "Check and X"Method

In game theory, understanding payoff matrices is crucial for identifying dominant strategies and Nash equilibria. A practical approach to analyze these matrices is the "check and X method." This method involves marking player strategies with checks and X's to simplify the decision-making process.

To begin, each player evaluates their best strategies based on the potential choices of their opponent. For Player 1, if Player 2 chooses option A, Player 1 can either select A for a payoff of $300 or B for $400. Since $400 is greater, Player 1 will mark a check next to option B. Conversely, if Player 2 opts for B, Player 1 can choose A for $100 or B for $200, again favoring B. Thus, Player 1's dominant strategy is to choose B, as indicated by the checks in the corresponding row.

Next, Player 2 assesses their options. If Player 1 chooses A, Player 2 can choose A for $500 or B for $400, leading them to select A. If Player 1 chooses B, Player 2 can choose A for $100 or B for $200, favoring B. Player 2's best strategies are marked with X's, but since these X's are not aligned in a single column, Player 2 does not have a dominant strategy.

To identify Nash equilibria, look for boxes in the matrix that contain both a check and an X. In this case, the bottom right box has both, indicating a Nash equilibrium where both players choose B. This outcome results in a payoff of $200 for each player, representing a stable state where neither player has an incentive to deviate from their strategy.

In summary, the check and X method provides a systematic way to analyze payoff matrices, revealing dominant strategies and Nash equilibria effectively. Understanding these concepts is essential for strategic decision-making in competitive environments.

Based on the information in the payoff matrix, which of the following is true?

• A. This situation is not a prisoner's dilemma
• B. If Walmart lowers its price, Target should keep its prices high
• C. If Target lowers its price and Walmart does not, Target will earn $20 million economic profit
• D. Both Walmart and Target would jointly be better off if they could each keep their prices high

Answer: (D)

The game above has:

• A. A Nash equilibrium:both Walmart and Target keep prices high
• B. A Nash equilibrium:both Walmart and Target lower prices
• C. A Nash equilibrium:Target keeps prices high and Walmart lowers prices
• D. No Nash equilibrium

Answer: (B)

In a cartel, the incentive to cheat is significant because

• A. Each firm has an incentive to decrease its own output
• B. Each firm has an incentive to raise its price
• C. Each firm has an incentive to expand its output
• D. Each firm's marginal cost exceeds the price that the cartel sets

Answer: (C)

Here’s what students ask on this topic:

What is the Prisoner's Dilemma in game theory?

The Prisoner's Dilemma is a fundamental concept in game theory that demonstrates why two rational individuals might not cooperate, even if it appears that it is in their best interest to do so. In this scenario, two prisoners are arrested and interrogated separately. They can either confess or remain silent. If both confess, they receive moderate sentences. If one confesses and the other remains silent, the confessor goes free while the silent prisoner gets a heavy sentence. If both remain silent, they receive light sentences. The dilemma shows that individual rationality leads to a worse collective outcome.

What is a dominant strategy in game theory?

A dominant strategy in game theory is a strategy that yields a better outcome for a player, regardless of what the other players do. In other words, it is the best course of action for a player no matter what the opponents' strategies are. For example, in the Prisoner's Dilemma, confessing is a dominant strategy for both prisoners because it results in a better personal outcome whether the other prisoner confesses or not. Not all games have dominant strategies, and sometimes only one player may have a dominant strategy.

What is Nash Equilibrium in the context of the Prisoner's Dilemma?

Nash Equilibrium, named after John Nash, is a concept in game theory where each player's strategy is optimal given the strategies of the other players. In the context of the Prisoner's Dilemma, the Nash Equilibrium occurs when both prisoners choose to confess. This is because confessing is the best response for each prisoner, given that the other prisoner might also confess. Although this leads to a suboptimal outcome (both getting 8 years in prison), it is the stable state where neither prisoner can improve their situation by unilaterally changing their decision.

How does the concept of collusion relate to the Prisoner's Dilemma?

Collusion in the context of the Prisoner's Dilemma refers to the idea that if the prisoners could communicate and agree to both remain silent, they would achieve a better collective outcome (each getting only 1 year in prison). This mirrors how firms in an oligopoly might collude to set prices or output levels to maximize joint profits. However, just like in the Prisoner's Dilemma, there is an incentive to cheat on the agreement to gain individual advantage, which can undermine the collusion and lead to worse outcomes for all parties involved.

What is the significance of the payoff matrix in game theory?

The payoff matrix is a crucial tool in game theory that organizes the potential outcomes of a game for each combination of strategies chosen by the players. It helps in visualizing and analyzing the payoffs for each player based on their decisions and the decisions of others. In the Prisoner's Dilemma, the payoff matrix shows the sentences each prisoner will receive based on whether they confess or remain silent. This matrix is essential for identifying dominant strategies and Nash Equilibria, aiding in the understanding of strategic interactions and decision-making processes.