ZeroSum Matrix Games And their uses in the

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Zero-Sum Matrix Games And their uses in the classroom

Zero-Sum Matrix Games And their uses in the classroom

What is a Zero-Sum Game • Two player Zero-sum game is a game where

What is a Zero-Sum Game • Two player Zero-sum game is a game where whatever one player wins the other loses. • A good example of this is Rock-Paper-Scissors, where there is no way for both players to win or both players to lose. • It is possible for players to draw, but we can consider that one player winning nothing and the other losing nothing.

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice B Choice A (1, -1) (-2, 2) Choice B (2, -2) (-1, 1) • This is an example of a matrix game, composed of players, choices and results.

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice B Choice A (1, -1) (-2, 2) Choice B (2, -2) (-1, 1) The first lesson of this lesson plan is to simply teach the game to the students so that they understand the gameplay mechanics and can start to develop a little intuition in playing the game. The Game begins simply enough. Both players begin by choosing one of the two choices in this scenario.

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice

Lesson 1: How to Play the Game Player 1 Player 2 Choice A Choice B Choice A (1, -1) (-2, 2) Choice B (-1, 1) (2, -2) After both players have selected one of their choices, they reveal their choices and from here find the corresponding cell on the matrix. For example if both players choose A, then we go to the first cell on the matrix which gives us the result of (1, -1). The first number corresponds to player 1 and the second to player two, therefore player 1 wins one game piece and player 2 loses that one game piece.

Lesson 2: Step 1 Player 2 Choice A Choice B Choice A (1, -1)

Lesson 2: Step 1 Player 2 Choice A Choice B Choice A (1, -1) (-2, 2) Choice B (-1, 1) (2, -2) Now that our students understand our game we can move forward into a little math. Our first step is to transform our matrix into something usable, by changing the ordered pairs into single numbers with positive numbers representing wins for player 1 and negative numbers representing wins for player 2

Lesson 2: Step 1 Player 2 Choice A Choice B Choice A 1 -2

Lesson 2: Step 1 Player 2 Choice A Choice B Choice A 1 -2 Choice B -1 2 Now that we have our altered matrix we can now write the equations to find the average winnings and the optimal strategy

Lesson 2: equations Player 2 • Choice B Choice A 1 -2 Choice B

Lesson 2: equations Player 2 • Choice B Choice A 1 -2 Choice B -1 2 Player 1 Choice A

Lesson 2: Optimal Strategy To find what is called the optimal strategy we need

Lesson 2: Optimal Strategy To find what is called the optimal strategy we need to eliminate our opponents importance. To do this we need to equalize our two equations; then solve for x. Player 1’s optimal strategy • Player 2’s optimal strategy •

Lesson 3: Application to Real Problems Newcomb’s Problem of Freewill An application of matrix

Lesson 3: Application to Real Problems Newcomb’s Problem of Freewill An application of matrix games to a philosophical problem.

Lesson 3: The Story • There is a near perfect being who you think

Lesson 3: The Story • There is a near perfect being who you think is able to predict the outcome of the future about 90% of the time. • We have a choice that involves two boxes, Box 1 has 1000$ and box 2 has either 1, 000$ or nothing. • We have to choose to either take only the second box or take both boxes. • If the being believes you will take both boxes he will put nothing in box 2, if he believes you will only take box 2 then he will put 1, 000$ in box 2. He makes his prediction the day before you choose • The question becomes what is the best option for you to make.

Lesson 3: Game Form You Being Predicts you take both boxes Predicts you take

Lesson 3: Game Form You Being Predicts you take both boxes Predicts you take only box #2 You take both boxes 1, 000$ You take only box #2 0$ 1, 000$

Lesson 4: Moving On • From here, we can move on to matrix games

Lesson 4: Moving On • From here, we can move on to matrix games with more than two choices Player 2 Player 1 Choice A Choice B Choice C Choice A 2 -1 1 Choice B -3 0 3 Choice C -2 5 -1

Lesson 4: Moving On Player 2 Player 1 Choice A Choice B Choice C

Lesson 4: Moving On Player 2 Player 1 Choice A Choice B Choice C Choice A 2 -1 1 Choice B -3 0 3 Choice C -2 5 -1 • This lesson is to take this subject and have it advance up to higher grade level. • For this game type we have to use multiple variables pushing this lesson up to the Algebra II level.

Lesson 4: Moving On Player 2 Player 1 Choice A Choice B Choice C

Lesson 4: Moving On Player 2 Player 1 Choice A Choice B Choice C Choice A 2 -1 1 Choice B -3 0 3 Choice C -2 5 -1 • This lesson is to take this subject and have it advance up to higher grade level. • For this game type we have to use multiple variables pushing this lesson up to the Algebra II level.

Lesson 4: Moving On Player 2 Player 1 Choice A • Choice B Choice

Lesson 4: Moving On Player 2 Player 1 Choice A • Choice B Choice C Choice A 2 -1 1 Choice B -3 0 3 Choice C -2 5 -1

Sources • Game Theory, Thomas Ferguson • Game Theory and Strategy, Phillip D. Straffin

Sources • Game Theory, Thomas Ferguson • Game Theory and Strategy, Phillip D. Straffin • Thank you to my advisor Professor Ciancetta.