Exam 15: Game Theory: the Mathematics of Competition

Use the following information to answer the Questions: In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to the column player II. $\left[ \begin{array} { c c c c } - 3 & 4 & 1 & 2 \\6 & - 1 & 2 & 1 \\3 & 2 & 3 & 6 \\1 & - 4 & 2 & 5\end{array} \right]$ -The minimax strategy of player II is:

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The maximin strategy of player I is:

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Which of the following statements is true?

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to the column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? $\left[ \begin{array} { l l l l } 7 & 3 & 5 & 1 \\3 & 4 & 6 & 4 \\2 & 8 & 2 & 5 \\9 & 7 & 7 & 7\end{array} \right]$

Describe a Nash equilibrium in a Vickrey auction.

In a truel where the choices are simultaneous, what is the optimal strategy for each player?

The minimax strategy of a column player is the strategy that corresponds to the minimum value of the maximum numbers in the columns.

Construct the game tree for a truel played sequentially, for which the outcomes have the following payoffs. What are the optimal strategies for this game? Sole survivor 1 point One of two survivors 4 point One of three survivors 3 point Nonsurvivor 2 point

Use the following information to answer the Questions: -Assume that the game is played continually, and players can change their choices at noon each day. If the game begins with both players selecting choice B, what should happen at the first opportunity to change?

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second is to player II. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Player II }$ Player I Choice A Choice B Choice A (4,4) (3,5) Choice B (5,3) (2,2) What outcomes constitute a Nash equilibrium?

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. The minimax strategy of player II is:

Describe a zero-sum game.

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix: Solve the game determining the best mix of selections for both batter and pitcher.

If you cheat on your income tax return you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Statistically, how often should the government audit you?

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second is to player II. $\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\quad$$\text { Player II }$ Player I Choice A Choice B Choice A (2,4) (3,6) Choice B (6,3) (4,2) What outcomes constitute a Nash equilibrium?

You are considering cheating on your income tax return. If you cheat you will pay $1000 in taxes; if you don't cheat you will pay $2000. If you are audited and you didn't cheat, your auditor will be kind and reduce your taxes from $2000 to $1500. If you are audited and you did cheat, you will be caught and have to pay a total of $2500, including penalties. Statistically, which is the best option for you to choose?

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. $\left[ \begin{array} { c c } 1 & - 4 \\- 2 & - 6\end{array} \right]$ Which of the following statements is true?

You can choose to buy an extended warranty for your computer for $200. During this period, you will experience either a minor ($100) or major ($500) repair bill. If you buy the warranty, you pay 50% of any repair bill: that is, you will pay either $50 or $250 for the subsequent repair. Assume the company wants to make as much money as possible, and you wish to spend as little as possible. Which of the following is a true statement?

In the game of matching pennies, player I wins a penny if the coins match and player II wins if the coins do not match. Is this a zero-sum game? Is this a fair game?

In American football the "third down and short" situation occurs often. The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. In such situations, what is the optimal solution for the defense?