This is a Game Theory / Strategy question. It does not have a graph that accompanies it or any other material. To solve the problem, you have to create a type strategic form game with the information presented.

Say that Carrie and Domingo play the following word game. First, Carrie chooses either the letter A or the letter I. Then Domingo chooses either the letter P or the letter S. If the resulting two letters form a word, then both players are happy, but Carrie wants the word to be early in alphabetical order, while Domingo wants the word to be late in alphabetical order. For example, Carrie prefers the word AS over the world IS, while Domingo prefers IS over AS. If the resulting two letters do not form a word, then both get payoff 0. Note that AS and IS are words while AP and IP are not words.

a. Model this as an extensive form game. For players' payoff numbers, please use either 0, 1, or 2. Find all (pure strategy) Nash equilibria of this game. Which of these Nash equilibria are subgame perfect?

b. Now say that Carrie can choose either the letter A, the letter I, or the letter U. Domingo again can choose either the letter P or the letter S. Again, Carrie prefers the word to be early in alphabetical order, while Domingo prefers the word to be late. For example, Carrie prefers UP over US, while Domingo prefers US over UP. Again, if the resulting two letters do not form a word, then both get payoff 0. Note that AS, IS, UP, and US are words, while AP and IP are not words. Model this as an extensive form game. For players' payoff numbers, plesase use either 0, 1, 2, 3, or 4. Find all (pure strategy) Nash equilibria of this game. Which of these Nash equilibria are subgame perfect?

c. Now say that Carrie goes back to choosing either the letter A or the letter I. Now Domingo can choose the letter P, the letter S, or the level T. Again, Carrie prefers the word to be early in alphabetical order, while Domingo prefers the word to be late. For example, Carrie prefers AT over IS, while Domingo prefers IS over AT. Again, if the resulting two letters do not form a word, then both get payoff 0. Note that AS, AT, IS, and IT are words, while AP and IP are not words. Model this as an extensive form game. For players' payoff numbers, please use either 0, 1, 2, 3, or 4. Find all (pure strategy) Nash equilibria of this game. Which of these Nash equilibria are subgame perfect?