Game Theory and Strategy Quiz 1: 7 July 2024 (May 2024 term)

An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0, 1 or 2. If player i bids more than player j then i wins the good and pays his bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to determine who the winner is, who gets the good and pays his bid while the loser pays nothing. Formulate the game in matrix form and answer the given subquestions.

Which player does not have strictly dominated strategy?

An item is up for auction. Player 1 values the item at 3 while player 2 values the item at 5. Each player can bid either 0, 1 or 2. If player i bids more than player j then i wins the good and pays his bid, while the loser does not pay. If both players bid the same amount, then a coin is tossed to determine who the winner is, who gets the good and pays his bid while the loser pays nothing. Formulate the game in matrix form and answer the given subquestions.

Which strategies survive Iterated Elimination of Strictly Dominated Strategy?

(Hawk-Dove) The following game has been widely used in evolutionary biology to understand how “fighting” and “display” strategies by animals could coexist in a population. For a typical Hawk- Dove game there are resources to be gained (i.e. food, mates, territories, etc.) denoted as v. Each of two players can chooses to be aggressive, called “Hawk” (H), or can be compromising, called “Dove” (D). If both players choose H then they split the resources, but loose some payoff from injuries, denoted as k. Assume that k > v/2. If both choose D then they split the resources, but engage in some display of power that a display cost d, with d < v/2. Finally, if player i chooses H while j chooses D, then i gets all the resources while j leaves with no benefits and no costs. Describe the game as a game matrix and answer the following question.
The outcomes cannot be supported as pure strategy Nash equilibrium, given v = 10, k = 6, and d = 4, is (are)

Determined to crack down on the drug trade, a city mayor puts more officers out on patrol to disrupt the business of drug dealers. A drug dealer in a neighborhood can work his trade either on a street corner or in the park. Each day, he decides where to set up shop, knowing that word about his location will travel among users. Because a good snitch is lacking, word does not travel to the police. The police officer on the beat then needs to decide whether she will patrol the park or the street corner, while not knowing where the drug dealer is hanging out that day. The decision of the officer and the dealer determine the extent of drug trades that day. Let suppose the total number of potential trades in the market is 100. A dealer's payoff is determined by the number of trades he consummates, while the officer’s payoff is based on the number of trades she disrupts. The table below illustrates the payoffs for both parties according to their respective strategies in the game.

Based on the above data answer the given subquestions.

Choose the correct statement

Determined to crack down on the drug trade, a city mayor puts more officers out on patrol to disrupt the business of drug dealers. A drug dealer in a neighborhood can work his trade either on a street corner or in the park. Each day, he decides where to set up shop, knowing that word about his location will travel among users. Because a good snitch is lacking, word does not travel to the police. The police officer on the beat then needs to decide whether she will patrol the park or the street corner, while not knowing where the drug dealer is hanging out that day. The decision of the officer and the dealer determine the extent of drug trades that day. Let suppose the total number of potential trades in the market is 100. A dealer's payoff is determined by the number of trades he consummates, while the officer’s payoff is based on the number of trades she disrupts. The table below illustrates the payoffs for both parties according to their respective strategies in the game.

Based on the above data answer the given subquestions.

Choose the correct statement

Determined to crack down on the drug trade, a city mayor puts more officers out on patrol to disrupt the business of drug dealers. A drug dealer in a neighborhood can work his trade either on a street corner or in the park. Each day, he decides where to set up shop, knowing that word about his location will travel among users. Because a good snitch is lacking, word does not travel to the police. The police officer on the beat then needs to decide whether she will patrol the park or the street corner, while not knowing where the drug dealer is hanging out that day. The decision of the officer and the dealer determine the extent of drug trades that day. Let suppose the total number of potential trades in the market is 100. A dealer's payoff is determined by the number of trades he consummates, while the officer’s payoff is based on the number of trades she disrupts. The table below illustrates the payoffs for both parties according to their respective strategies in the game.

Based on the above data answer the given subquestions.

What are the randomization probabilities of a police officer for patrolling the streets and patrolling the park respectively under equilibrium?

Determined to crack down on the drug trade, a city mayor puts more officers out on patrol to disrupt the business of drug dealers. A drug dealer in a neighborhood can work his trade either on a street corner or in the park. Each day, he decides where to set up shop, knowing that word about his location will travel among users. Because a good snitch is lacking, word does not travel to the police. The police officer on the beat then needs to decide whether she will patrol the park or the street corner, while not knowing where the drug dealer is hanging out that day. The decision of the officer and the dealer determine the extent of drug trades that day. Let suppose the total number of potential trades in the market is 100. A dealer's payoff is determined by the number of trades he consummates, while the officer’s payoff is based on the number of trades she disrupts. The table below illustrates the payoffs for both parties according to their respective strategies in the game.

Based on the above data answer the given subquestions.

The randomization probabilities of the drug dealer choosing between street and park respectively under equilibrium are

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

This is a game of

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

How many terminal nodes does this game have?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

How many information sets does this game have?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

How many pure strategies does player 1 have?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

How many pure strategies does player 2 have?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

How many pure strategy Nash equilibria does this game have?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

Which one of the Nash equilibria is also a subgame perfect equilibrium?

Consider a two-player game in which player 1 can choose A or B. The game ends if he chooses A while it continues to player 2 if he chooses B. Player 2 can either choose C or D, with the game ending after C and continuing again with player 1 after D. Player 1 then can choose E or F, and then the game ends after each of these choices.
Imagine that the payoffs following choice A by player 1 are (2, 0), following C by player 2 are (3, 1), following E by player 1 are (0, 0) and following F by player 1 are (1, 2)
Model this as an extensive form game and answer the given subquestions:

No. of sub games this game has

A firm (player 1) is considering entering an established industry with one incumbent firm (player 2). Player 1 must choose whether to enter or to not enter the industry. If player 1 enters the industry, then player 2 can either accommodate the entry, or fight the entry with a price war. Player 1’s most preferred outcome is entering with player 2 not fighting, and his least preferred outcome is entering with player 2 fighting. Player 2’s most preferred outcome is player 1 not entering, and his least preferred outcome is player 1 entering with player 2 fighting.
Model this as an extensive form game tree (choose ordinal payoffs that represent the preferences) and answer the following questions.
Strategies of Player 1:
E: Entering the Industry, N: Not entering the Industry
Strategies of Player 2:
A: Accommodate, F: Fight.
What is the subgame perfect equilibrium in this game?