COU 3: Nash Equilibria in 2-by-2 Games

In this COU you’ll identify nash equilibria in 2-by-2 games.

Example

Before getting started, watch this video which demonstrates how to solve the following problem.

Prompt E1
Is the profile (\text{top},\text{left}) a Nash Equilibrium?
Prompt E2
If you answered “yes” to Prompt E1, explain how you know that to be correct by explaining how you know that “top” is a best response for the Row Person to the expectation that the Column Person will take action “left”, and how you know that “left” is a best response for the Column Person to the expectation that the Row Person will take action “top”. If you answered “no” to Prompt E1, explain how you know that to be correct by explaining how you know either that “top” is not a best response for the Row Person to the expectation that the Column Person will take action “left”, or how you know that “left” is not a best response for the Column Person to the expectation that the Row Person will take action “top”.
Prompt E3
Is the profile (\text{top},\text{right}) a Nash Equilibrium?
Prompt E4
If you answered “yes” to Prompt E3, explain how you know that to be correct by explaining how you know that “top” is a best response for the Row Person to the expectation that the Column Person will take action “right”, and how you know that “right” is a best response for the Column Person to the expectation that the Row Person will take action “top”. If you answered “no” to Prompt E3, explain how you know that to be correct by explaining how you know either that “top” is not a best response for the Row Person to the expectation that the Column Person will take action “right”, or how you know that “right” is not a best response for the Column Person to the expectation that the Row Person will take action “top”.
Prompt E5
Is the profile (\text{bottom},\text{left}) a Nash Equilibrium?
Prompt E6
If you answered “yes” to Prompt E5, explain how you know that to be correct by explaining how you know that “bottom” is a best response for the Row Person to the expectation that the Column Person will take action “left”, and how you know that “left” is a best response for the Column Person to the expectation that the Row Person will take action “bottom”. If you answered “no” to Prompt E5, explain how you know that to be correct by explaining how you know either that “bottom” is not a best response for the Row Person to the expectation that the Column Person will take action “left”, or how you know that “left” is not a best response for the Column Person to the expectation that the Row Person will take action “bottom”.
Prompt E7
Is the profile (\text{bottom},\text{right}) a Nash Equilibrium?
Prompt E8
If you answered “yes” to Prompt E7, explain how you know that to be correct by explaining how you know that “bottom” is a best response for the Row Person to the expectation that the Column Person will take action “right”, and how you know that “right” is a best response for the Column Person to the expectation that the Row Person will take action “bottom”. If you answered “no” to Prompt E7, explain how you know that to be correct by explaining how you know either that “bottom” is not a best response for the Row Person to the expectation that the Column Person will take action “right”, or how you know that “right” is not a best response for the Column Person to the expectation that the Row Person will take action “bottom”.

Protest and Repressive Violence

In Lesson 7, we modeled the interaction between a resistance organizer and the leader of a repressive regime using the following simultaneous move game:

Prompt A1
Is the profile (\text{protest},\text{limited}) a Nash Equilibrium?
Prompt A2
If you answered “yes” to Prompt A1, explain how you know that to be correct by explaining how you know that “protest” is a best response for the organizer to the expectation that the leader will take action “limited”, and how you know that “limited” is a best response for the leader to the expectation that the organizer will take action “protest”. If you answered “no” to Prompt A1, explain how you know that to be correct by explaining how you know either that “protest” is not a best response for the organizer to the expectation that the leader will take action “limited”, or how you know that “limited” is not a best response for the leader to the expectation that the organizer will take action “protest”.
Prompt A3
Is the profile (\text{protest},\text{overwhelming}) a Nash Equilibrium?
Prompt A4
If you answered “yes” to Prompt A3, explain how you know that to be correct by explaining how you know that “protest” is a best response for the organizer to the expectation that the leader will take action “overwhelming”, and how you know that “overwhelming” is a best response for the leader to the expectation that the organizer will take action “protest”. If you answered “no” to Prompt A3, explain how you know that to be correct by explaining how you know either that “protest” is not a best response for the organizer to the expectation that the leader will take action “overwhelming”, or how you know that “overwhelming” is not a best response for the leader to the expectation that the organizer will take action “protest”.
Prompt A5
Is the profile (\text{no protest},\text{limited}) a Nash Equilibrium?
Prompt A6
If you answered “yes” to Prompt A5, explain how you know that to be correct by explaining how you know that “no protest” is a best response for the organizer to the expectation that the leader will take action “limited”, and how you know that “limited” is a best response for the leader to the expectation that the organizer will take action “no protest”. If you answered “no” to Prompt A5, explain how you know that to be correct by explaining how you know either that “no protest” is not a best response for the organizer to the expectation that the leader will take action “limited”, or how you know that “limited” is not a best response for the leader to the expectation that the organizer will take action “no protest”.
Prompt A7
Is the profile (\text{no protest},\text{overwhelming}) a Nash Equilibrium?
Prompt A8
If you answered “yes” to Prompt A7, explain how you know that to be correct by explaining how you know that “no protest” is a best response for the organizer to the expectation that the leader will take action “overwhelming”, and how you know that “overwhelming” is a best response for the leader to the expectation that the organizer will take action “no protest”. If you answered “no” to Prompt A7, explain how you know that to be correct by explaining how you know either that “no protest” is not a best response for the organizer to the expectation that the leader will take action “overwhelming”, or how you know that “overwhelming” is not a best response for the leader to the expectation that the organizer will take action “no protest”.

Power in Numbers

Here is a model based on the one you developed an analyzed in the COUs for Lesson 7 that depicts the idea that persons considering participating in a protest know that there is “power in numbers”. Specifically, each person prefers to participate if she expects many others to also participate and prefers to not participate if she expects few others to participate.

Prompt B1
Is the profile in which both persons choose to not participate a Nash Equilibrium?
Prompt B2
If you answered “yes” to Prompt B1, explain how you know that to be correct by explaining how you know that “not participate” is a best response for Person B to the expectation that Person A will not participate, and how you know that “not participate” is a best response for Person A to the expectation that Person B will not participate. If you answered “no” to Prompt B1, explain how you know that to be correct by explaining how you know either that “not participate” is not a best response for Person B to the expectation that Person A will not participate, or how you know that “not participate” is not a best response for Person A to the expectation that Person B will not participate.
Prompt B3
Is the profile in which Person B does not participate and Person A participates a Nash Equilibrium?
Prompt B4
If you answered “yes” to Prompt B3, explain how you know that to be correct by explaining how you know that “not participate” is a best response for Person B to the expectation that Person A will participate, and how you know that “participate” is a best response for Person A to the expectation that Person B will not participate. If you answered “no” to Prompt B3, explain how you know that to be correct by explaining how you know either that “not participate” is not a best response for Person B to the expectation that Person A will participate, or how you know that “participate” is not a best response for Person A to the expectation that Person B will not participate.
Prompt B5
Is the profile in which Person B participates and Person A does not participate a Nash Equilibrium?
Prompt B6
If you answered “yes” to Prompt B5, explain how you know that to be correct by explaining how you know that “participate” is a best response for the Person B to the expectation that Person A will not participate, and how you know that “not participate” is a best response for Person A to the expectation that Person B will participate. If you answered “no” to Prompt B5, explain how you know that to be correct by explaining how you know either that “participate” is not a best response for Person B to the expectation that Person A will not participate, or how you know that “not participate” is not a best response for Person A to the expectation that Person B will participate.
Prompt B7
Is the profile in which both Person B and Person A participate a Nash Equilibrium?
Prompt B8
If you answered “yes” to Prompt B7, explain how you know that to be correct by explaining how you know that “participate” is a best response for Person B to the expectation that Person A will participate, and how you know that “participate” is a best response for Person A to the expectation that Person B will participate. If you answered “no” to Prompt B7, explain how you know that to be correct by explaining how you know either that “participate” is not a best response for Person B to the expectation that Person A will participate, or how you know that “participate” is not a best response for Person A to the expectation that Person B will participate.

Rubric

Each of the two sets of prompts in this COU (“Protest and Repressive Violence” and “Power in Numbers”) consist of four pairs of prompts. The first prompt in each pair asks you to identify whether a given profile is a Nash Equilibrium, and the second asks you to explain why that profile either is or is not a Nash Equilibrium.

Each “identify whether” prompt is worth 1 point. You get that one point if you give the correct answer and not otherwise.

Each “explain why” prompt is worth 3 points. If your answer to the corresponding “identify whether” question is incorrect, you get 0 of the 3 points. If your answer to the corresponding “identify whether” question is correct, your score on the “explain why” question is determined as follows:

  • If the correct answer to the corresponding “explain why” question is yes, you get 3 points if you do both of the following:

    1. Show that each action in the relevant profile is a best response for the person taking that action to the expectation that the other person takes the other action in the profile.

    2. For each action, you show that that action is a best response by correctly stating the relevant highest utility level and stating the utility level that the action gets for the relevant person, given the expectation that the other person will take the relevant action.

  • If the correct answer to the corresponding “explain why” question is yes, you get 0 points if you fail to do either (i) or (ii) above.

  • If the correct answer to the corresponding “explain why” question is no, you get 3 points if you do both of the following:

    1. Show that at least one action in the relevant profile is not a best response for the person taking that action to the expectation that the other person takes the other action in the profile.

    2. For each action you name as not a best response as in (i), you show that that action is not a best response by correctly stating the relevant highest utility level and stating the utility level that the action gets for the relevant person, given the expectation that the other person will take the relevant action.

  • If the correct answer to the corresponding “explain why” question is no, you get 0 points if you fail to do either (i) or (ii) above.