COU 5: Practice with the Matrix Form

The matrix form of a 2-by-2 game concentrates a lot of information in a small package. This is fantastic because once you know how to quickly represent ideas using the matrix form, you can use it to express some very interesting and important aspects of politics quickly. And once you know how to quickly interpret any matrix form given to you, the matrix form allows you to “read” in just a small figure important ideas that would take many long and complicated sentences to express in text.

Notice the qualifications in those previous sentences: “once you know how to quickly represent ideas using the matrix form” and “once you know how to quickly interpret a matrix form given to you”. It comes down to this: The matrix form is fantastically useful if you understand how to read and write it. Otherwise, the matrix form is completely useless!

So this COU is meant to help you build your fluency in writing and reading matrix forms. It prompts you to work a long list of problems that each entail either writing a given 2-by-2 game in matrix form or correctly interpreting a given matrix form. Expect the work to be rote and a little tedious. Expect it to get easier the more problems you do. Once you begin wrestling with the concepts coming after this COU, the payoff of the tedious and initially difficult work you do here will be substantial!

Before you get started, it’s useful to have a sense of exactly what it is you’ll be practicing in the following prompts. Recall from the lesson there is really only one aspect of the matrix form that is difficult to remember when you first start learning it: The Utility Level Ordering Convention.

Utility Level Ordering Convention in the Matrix Form

In the matrix form of a 2-by-2 game, when writing the utility levels of the two players in any cell corresponding to a given profile of actions, we always write the row-person’s utility level first and the column-person’s utility level second.

To be able to use the matrix form, this is what you need to practice. Specifically, you need to practice using the Ordering Convention to:

write a matrix form;
read a matrix form;
interpret a matrix form.

Part A: Writing Matrix Forms

Before starting the prompts in this part, watch this video about how to solve the Example Problem below.

Example Problem

Do not submit an answer to this problem. It is here only to help you learn from the video linked above.

Here is a payoff function for a 2-by-2 game:

Person A’s Action	Person B’s Action	Person A’s Utility Level	Person B’s Utility Level
a_1	b_1	1	2
a_1	b_2	3	4
a_2	b_1	5	6
a_2	b_2	7	8

write a matrix form of this game.

Prompt A1

Here is a payoff function for a 2-by-2 game:

Person A’s Action	Person B’s Action	Person A’s Utility Level	Person B’s Utility Level
a_1	b_1	0	-250
a_1	b_2	1	0
a_2	b_1	0	-300
a_2	b_2	0	-300

write a matrix form of this game.

Prompt A2

Here is a payoff function for a 2-by-2 game:

Person A’s Action	Person B’s Action	Person A’s Utility Level	Person B’s Utility Level
a_1	b_1	100	1
a_1	b_2	-200	-200
a_2	b_1	99	99
a_2	b_2	1	100

write a matrix form of this game.

Prompt A3

Here is a payoff function for a 2-by-2 game:

Person A’s Action	Person B’s Action	Person A’s Utility Level	Person B’s Utility Level
a_1	b_1	0	0
a_1	b_2	-100	1
a_2	b_1	1	-100
a_2	b_2	-101	-101

write a matrix form of this game.

Prompt A4

Here is a payoff function for a 2-by-2 game:

Person A’s Action	Person B’s Action	Person A’s Utility Level	Person B’s Utility Level
a_1	b_1	-10	-10
a_1	b_2	-10	-20
a_2	b_1	-20	-10
a_2	b_2	10	10

write a matrix form of this game.

Part B: Reading a Matrix Form

Before starting the prompts in this part, watch this video about how to solve the Example Problem below.

Example Problem

Do not submit an answer to this problem. It is here only to help you learn from the video linked above.