Matrix Two-Person Games

1.1   The Basics

Problems

1.1 There are 100 bankers lined up in each of 100 rows. Pick the richest banker in each row. Javier is the poorest of those. Pick the poorest banker in each column. Raoul is the richest of those. Who is richer: Javier or Raoul?

1.1 Answer: Think of this as a 100 × 100 game matrix and we are looking for the upper and lower values except that we are really doing it for the transpose of the matrix.

If we take the maximum in each row and then Javier is the minimum maximum, Javier is v+. If we take the minimum in each column, and Raoul is the maximum of those, then Raoul is the maximum minimum, or v. Thus, Javier is richer.

Another way to think of this is that the poorest rich guy is wealthier than the richest poor guy. Common sense.

1.2 In a Nim game start with 4 pennies. Each player may take 1 or 2 pennies from the pile. Suppose player I moves first. The game ends when there are no pennies left and the player who took the last penny pays 1 to the other player.

(a) Draw the game as we did in 2 × 2 Nim.

1.2.a Answer: The game tree is

(b) Write down all the strategies for each player and then the game matrix.
The text passage contains a series of problems and their corresponding answers, specifically focusing on matrix two-person games and a Nim game.
multiple_questions_with_answers
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
Matrix Two-Person Games

1.1   The Basics

Problems

1.1 There are 100 bankers lined up in each of 100 rows. Pick the richest banker in each row. Javier is the poorest of those. Pick the poorest banker in each column. Raoul is the richest of those. Who is richer: Javier or Raoul?

1.1 Answer: Think of this as a 100 × 100 game matrix and we are looking for the upper and lower values except that we are really doing it for the transpose of the matrix.

If we take the maximum in each row and then Javier is the minimum maximum, Javier is v+. If we take the minimum in each column, and Raoul is the maximum of those, then Raoul is the maximum minimum, or v. Thus, Javier is richer.

Another way to think of this is that the poorest rich guy is wealthier than the richest poor guy. Common sense.

1.2 In a Nim game start with 4 pennies. Each player may take 1 or 2 pennies from the pile. Suppose player I moves first. The game ends when there are no pennies left and the player who took the last penny pays 1 to the other player.

(a) Draw the game as we did in 2 × 2 Nim.

1.2.a Answer: The game tree is

(b) Write down all the strategies for each player and then the game matrix.
user
The text passage contains a series of problems and their corresponding answers, specifically focusing on matrix two-person games and a Nim game.

TYPE: MULTIPLE_QUESTIONS_WITH_ANSWERS