Question 1

A football team has the chance of scoring the game-winning touchdown on the last play of the game.It can either run or pass.The defense can play for the run or play for the pass.The following normal form lists the payoffs from the game made up by this last play.The payoffs are probabilities of winning the game.  The Nash equilibrium is (equilibria are)

Accepted Answer

D

Question 2

A teacher curves the final exam such that the top half of students get an A and the bottom half an F (so their grade depends only on relative and not absolute performance).Suppose that there are equal numbers of two groups,the Brainiacs and the Numbskulls.If they both study or they both party,the Brainiacs will get the As but if the Brainiacs party and the Numbskulls study,the Numbskulls will get the As.Suppose further that they both dislike studying and both like good grades.Suppose all students of a type choose the same action (so we can view it as a two-player game).The payoff matrix is  Characterize the Nash equilibrium or equilibria:

Accepted Answer

A

Question 3

Consider the game between the teens from the previous question.The pure-strategy Nash equilibrium is (equilibria are)

Accepted Answer

In a pure-strategy Nash equilibrium, each player is doing the best they can given the other player’s action. In this game, if one player Declares, the other player is better off by Declaring as well to avoid being seen as untrustworthy. However, if one player Rebuffs/Ignores, the other player is also better off by doing the same to avoid being taken advantage of. Thus, both players declaring and both players ignoring/rebuffing are both Nash equilibria. Therefore, the answer is C.

Question 4

The beauty of Nash's equilibrium concept is that

Accepted Answer

The beauty of Nash's equilibrium concept is that it ensures that all games have at least one equilibrium, which is a key concept in game theory. This equilibrium represents a stable outcome where no player has an incentive to change their strategy, given the strategies of the other players. So, option A is the correct choice.

Question 5

Consider the game between the teens from the previous question.Instead of being a simultaneous game,suppose it is sequential,with teen A moving first.What is the subgame-perfect equilibrium of this new game?

Accepted Answer

In a sequential game, Teen A has the advantage of moving first. If Teen A declares, it puts Teen B in a tough spot because it has to decide whether to declare or ignore/rebuff. If Teen B ignores/rebuffs, it gets nothing, but if it declares, it risks sharing the prize with Teen A. Therefore, the subgame-perfect equilibrium of this new game is for Teen A to declare and for Teen B to follow A's action. This ensures that both teens share the prize without any risk of losing it.

Question 6

A football team has the chance of scoring the game-winning touchdown on the last play of the game.It can either run or pass.The defense can play for the run or play for the pass.The following normal form lists the payoffs from the game made up by this last play.The payoffs are probabilities of winning the game.  What is the defense's dominant strategy?

Accepted Answer

The answer of A football team has the chance of...

Question 7

Consider the same football setting as in the previous question,but now suppose the payoffs (probabilities of winning)are given by the following normal form:  The Nash equilibrium is (equilibria are)

Accepted Answer

The answer of Consider the same football setting as in...

Question 8

If the Prisoners' Dilemma is repeated over and over again with the same two players having an indefinite time horizon,

Accepted Answer

If the Prisoners' Dilemma is repeated over and over again with the same two players having an indefinite time horizon, players can cooperate on Silent if they are patient enough. This is because players have more to gain from cooperating on the Silent strategy in the long run as it leads to better payoffs than the Nash equilibrium of Confess each period. However, it requires trust, patience, and a willingness to forgive. If players are not patient enough, they may end up playing the Nash equilibrium of Confess each period, resulting in lower payoffs for both players. Hence, choice C is the best choice.

Question 9

Best-response functions

Accepted Answer

Best-response functions represent the strategies that each player would choose given the other player's strategy. They intersect at Nash equilibria, which are the stable outcomes of the game.

Question 10

Consider the same setup with the curved final exam as in the previous question.Now suppose the students move sequentially.Which best describes the outcome in the subgame-perfect equilibrium?

Accepted Answer

The answer of Consider the same setup with the curved...

Question 11

Consider the game between the teens from the previous question.In addition to any pure-strategy Nash equilibrium,there is another one in mixed strategies.In it,each teen chooses to declare with probability

Accepted Answer

The answer of Consider the game between the teens from...

Question 12

Consider the same game between the fishermen in the previous question.How does the total number of fish caught in the Nash equilibrium compare to the number they would catch if they belonged to same company,sharing revenues and costs equally?

Accepted Answer

In the Nash equilibrium, each fisherman is incentivized to catch as many fish as possible, which leads to overfishing and a lower total number of fish caught compared to if they were working in the same company and sharing revenues and costs equally. When working together in the same company, they would have more of an incentive to conserve the fish population and work efficiently to maximize profits, leading to a higher total number of fish caught in the long run.

Question 13

Return to the version of the game between the fishermen in which they fish independently.If the marginal cost for just fisherman A went up,what would be the likely effect on the Nash equilibrium?

Accepted Answer

When the marginal cost for fisherman A increases, it becomes more expensive for A to catch each additional fish, leading A to reduce their fishing effort. As a result, there will be more fish available for fisherman B, who can catch more without changing their own costs. This adjustment leads to a new Nash equilibrium where A catches fewer fish and B catches more.

Question 14

The game of Matching Pennies

Accepted Answer

The game of Matching Pennies has two players, each of whom can choose to play either Heads or Tails. The payoffs are such that if both players play the same move, the first player wins and gets 1 dollar, while the second player loses and gets -1 dollar. If the players play different moves, the first player loses and gets -1 dollar, while the second player wins and gets 1 dollar.

It can be shown that there is no pure-strategy Nash equilibrium in this game, as each player has a dominant strategy of randomly choosing Heads or Tails with equal probability.

However, there is a mixed strategy Nash equilibrium in which both players randomize their moves with equal probability. In this equilibrium, both players expect to win 0 dollars on average. Any other strategy would lead to a lower expected payoff for at least one of the players.

Question 15

Consider the same football situation as in the previous question,but now suppose the probabilities of winning are less extreme than before,given by the following normal form:  What is the defense's dominant strategy?

Accepted Answer

In this normal form, there is no dominant strategy for the defense. This can be seen by comparing the payoff for each of the defense's possible strategies (defending the run, defending the pass, or defending both at random with equal probability) against the offense's strategies (running or passing). Depending on the probabilities of winning, defending one way or the other may lead to a higher payoff, so the defense will have to choose its strategy based on the situation at hand.

Question 16

A lake supports a delicious variety of fish.The land around the lake is owned by two fisherman,so by state law both are free to fish as much as they like.Let F_A and F_B be the number of fish each catches,respectively.Suppose the price of fish is 100 - F_A - F_B.Given this demand curve,it turns out that marginal revenue is 100 - 2F_A - F_Bfor fisherman A and 100 - F_A - 2F_Bfor fisherman B.What is the total number of fish by both in the Nash equilibrium?

Accepted Answer

The total number of fish caught in the Nash equilibrium is 30. This is because the marginal revenue for each fisherman is equal to the price of fish, which is 100 - F_A - F_B. Therefore, each fisherman will catch 15 fish, for a total of 30 fish.

Question 17

When a game has multiple equilibria,a useful method to sort out which one would be the "best" prediction is to

Accepted Answer

All the methods mentioned can be useful in determining the "best" prediction in games with multiple equilibria: A) looks for mutual benefit, B) looks for symmetry, and C) looks for a focal point.

Question 18

Consider the same football situation as in the previous question,but now suppose the payoffs (probabilities of winning)are as given in the following normal form:  Which team has a dominant strategy?

Accepted Answer

The answer of Consider the same football situation as in...

Question 19

The Prisoners' Dilemma is so named because

Accepted Answer

The name of the game reflects the fact that the Nash equilibrium, in which both players defect, leads to a suboptimal outcome for both players compared to if they had cooperated. This can be seen as analogous to a situation where two prisoners could collectively benefit from not confessing to a crime, but end up confessing due to mistrust and lack of cooperation.

Question 20

Teens A and B are smitten with each other but neither knows of the other's feelings.Suppose the teachers at their school organize a dance.The "payoff" is based on whether their advances are rebuffed or accepted.If they both Declare,they get positive utility but if they are Rebuffed they face humiliation (significantly negative payoff).Rebuffing an advance slightly elevates the teen's standing with peers.  What is a teen's dominant strategy?

Accepted Answer

The answer of Teens A and B are smitten with...

Quiz 5: Game Theory