Exam 15: Game Theory: the Mathematics of Competition

In a game, each player chooses one of three coins: penny, nickel, or dime. If both players choose the same coin, both players loose their coin. Otherwise, the player with the more valuable coin wins the less valuable coin from the other player. Represent this game as a three-by-three matrix of ordered pairs.

In the following game of batter-versus-pitcher in baseball, the batter's batting averages are shown in the game matrix. Which of the following statements is true?

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 offense? A) always choose to run B) always choose to pass C) choose to run more often than pass D) choose to pass more often than run

Suppose on a committee of three people Kim (the chair), Chris, and Terry each have one vote, but Kim breaks any tie. In attempting to elect someone to host the summer picnic, each person has a priority list: \mid Kim's Priorities Chris's Priorities Terry's Priorities 1st choice \mid Kim Chris Terry 2nd choice \mid Terry Kim Chris 3rd choice \mid Chris Terry Kim Which of the following are Nash equilibria? I: Every person votes for his/her first choice. II: Every person votes for his/her second choice.

Consider the following partial-conflict game played in a noncooperative manner. What outcomes constitute a Nash equilibrium?

As the buyer for your business, you can choose to buy an extended warranty for each new computer for $200 each. During this period, each computer will experience either a minor ($100) or major ($500) repair bill. Repairs for each computer under warranty are reduced by 50% to $50 or $250. Assume the computer company wants to make as much money as possible, and you wish to spend as little as possible. What is your optimal strategy?

In the following two-person zero-sum game, the payoffs represent gains to the row player I and losses to column player II. Does this game have a saddlepoint? What is each player's minimax or maximin strategy? Justify your response.

As the buyer for your business, you can choose to buy an extended warranty for each new computer for $200 each. During this period, each computer will experience either a minor ($100) or major ($500) repair bill. Repairs for each computer under warranty are reduced by 50% to $50 or $250. Assume the computer company wants to make as much money as possible, and you wish to spend as little as possible. What is the computer company's optimal strategy?

Create a three-by-three matrix that represents a two-person zero-sum game in which each player has three options and the game has a saddlepoint.

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 you cheat?

Use the following information to answer the Questions: 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. Defense Run Pass Offense Run 0.2 0.8 Pass 0.5 0.1 -What is the value of the game

You want to carry insurance for your small business. The annual policy costs $1000 this year. If you are sued and you have no insurance, you will pay $5000. If you have insurance and are sued you pay nothing, and your partner gives you $500 for your wise foresight, so that the policy costs you only $500. Represent this game as a two-by-two matrix.

A total-conflict game is a zero-sum game.

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?

Game theory is used by intelligence services to check for security violations.

In a pure strategy game, each player consistently selects one particular option.

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.

Use the following information to answer Questions The probabilities of obtaining a first down, shown below, are dependent on the choice of the offense and the anticipated choice of the defense. Defense Run Pass Offense Run 0.5 0.6 Pass 0.8 0.4 -What is the value of the game?

Consider the following partial-conflict game played in a noncooperative manner. The first payoff is to player I; the second to player II. Discuss the players' possible strategies when this game is played.

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: