18.9: Prisoner&#39s Dilemma II

The prisoner's dilemma is a classic game theory model where two crime suspects must decide whether to betray each other or cooperatively remain silent. The choices they make determine their respective sentences. The Nash equilibrium occurs when each suspect chooses the best option based on the other's likely decision.

For Suspect A:

1. If Suspect B stays silent, Suspect A benefits most by betraying. This is because betrayal results in no prison time versus one year if A were to also remain silent.
2. If Suspect B betrays, Suspect A's best option is still to betray, as betrayal leads to two years in prison instead of three if A were to remain silent.

For Suspect B:

1. If Suspect A stays silent, B should choose to betray. This is because betrayal allows B to walk free rather than serving a one-year sentence if B were to also remain silent.
2. If Suspect A betrays, B's best move is also to betray, as betrayal results in two years in prison instead of three if B were to remain silent.

The Nash equilibrium in this situation is when both suspects choose to betray each other. This outcome is stable because neither suspect can improve their situation by changing their decision unilaterally, as doing so would lead to a worse outcome if the other chooses betrayal. The fear of receiving a harsher sentence encourages both to betray rather than cooperate.

This equilibrium is also a dominant strategy equilibrium because, for both suspects, betrayal is the best choice regardless of what the other chooses. The dilemma highlights how rational decision-making based on self-interest can lead to a worse collective outcome, as both suspects would have received shorter sentences if they had trusted each other and remained silent.

The matrix provided represents the payoffs of a Prisoner's Dilemma.

To find the Nash Equilibrium, first consider Suspect A's options.

If B remains silent, A benefits more by betraying, as it results in no prison time compared to one year.

If B betrays, A again finds betraying better, serving two years instead of three.

Next, examine Suspect B's options.

If A remains silent, B will choose to betray since this results in zero years instead of one. Similarly, if A betrays, B will betray as well, preferring two years in prison over three.

The Nash Equilibrium is reached when both suspects betray each other. They choose betrayal to avoid the worst outcome of trusting and being betrayed. This risk aversion makes the Nash Equilibrium stable.

Neither suspect gains by changing their strategy alone. If one remains silent and the other betrays, the silent one faces a harsher penalty. This ensures that both are compelled into betraying.

This equilibrium is also a dominant strategy equilibrium. It's important to note that while all dominant strategy equilibria are Nash Equilibria, not all Nash Equilibria involve dominant strategies.

• Prisoner's Dilemma - A game theory model used to understand decision making between two entities.
• Nash Equilibrium - A state of equilibrium where each player's decision is optimal, considering others' choices.
• Betray - In prisoner's dilemma, a choice to not cooperate leading to specific outcomes.
• Cooperation - Contrary to betray, it's a choice to work together towards mutual benefit.
• Dominant Strategy Equilibrium - A state where betrayal is the best choice regardless of the other's action.

• Define Prisoner's Dilemma - Explain what it is (e.g., Prisoner's Dilemma).
• Contrast Betray vs Cooperation - Explain key differences (e.g., penalties and rewards).
• Explore Nash Equilibrium - Explain the scenario where the Nash Equilibrium occurs in the Prisoners Dilemma.
• Explain Decision making process - Briefly explain suspects' choices and their outcomes.
• Apply in Economics Context - Understand when and how the prisoner's dilemma is used in microeconomics.

• What is the Prisoner's Dilemma and how does it apply to decision making?
• What are the potential outcomes of the prisoner's dilemma?
• What is Nash Equilibrium in the context of Prisoner's Dilemma?

• Economics Students - Understand how the prisoner's dilemma concept supports understanding of strategic interactions.
• Educators - Provides a clear model for teaching game theory and strategic decision making.
• Researchers - Relevance for research in game theory, economics, psychology, and political science.
• Game theoretic enthusiasts - Offers insights and sparks deeper interest in understanding strategic interactions.