Question 1

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 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

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

Accepted Answer

In a repeated version of the Prisoners' Dilemma, strategies can evolve over time. If the players are patient enough and know that they will continue to play against each other, they can learn to cooperate on the Silent strategy. This is because the players know that if they both choose Silent, they will both receive a higher payoff than they would if they both choose Rat. In the long run, it can be beneficial for both players to cooperate on Silent, rather than continually choosing Rat and receiving lower payoffs. Therefore, option C is correct. Options A, B, and D are incorrect based on this reasoning.

Question 4

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, the subgame-perfect equilibrium is found by working backward from the final node. Teen B knows that if they ignore/rebuff A's declaration, A will retaliate by revealing embarrassing information about B. Therefore, B's best response is to follow A's action, i.e., to declare as well. Knowing this, A's best response is to declare in the first place, rather than risk retaliatory action from B. Therefore, the subgame-perfect equilibrium is for Teen A to Declare and for Teen B to follow A's action.

Question 5

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

Accepted Answer

All of the options provided are methods to determine the "best" or most likely equilibrium in a game with multiple equilibria. Finding an equilibrium where both players are better off, identifying a symmetric equilibrium, or selecting a "focal" equilibrium (one that stands out as most reasonable or salient to the players) are all valid approaches to predict the outcome of a game.

Question 6

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 scenario, the probabilities of winning are not extreme, and the optimal strategy for the defense depends on the opponent's strategy. Therefore, there is no single dominant strategy for the defense. The best choice for the defense is to adapt their strategy based on the opponent's plays and adjust their defense accordingly. Defending both plays at random (option C) with equal probability may not be optimal for all situations, and in some cases, it may lead to costly mistakes. Consequently, the defense must be flexible and adjust their strategy depending on the game situation.

Question 7

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

Accepted Answer

In this scenario, if one teen Declares, the other would benefit from Ignoring/Rebuffing. If one teen Ignores/Rebuffs, the other would benefit from Declaring. Therefore, the only stable outcomes are both teens Declaring or both Ignoring/Rebuffing, which are two different Nash equilibria.

Question 8

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 9

The game of Matching Pennies

Accepted Answer

Matching Pennies has a mixed strategy Nash equilibrium. In this game, both players have two options - to place their penny heads-up or tails-up. The payoffs for each player depend on whether or not their pennies match. If the pennies match, Player 1 wins and Player 2 loses. If the pennies do not match, Player 2 wins and Player 1 loses. Since both players have two options, there are four possible outcomes. If both players play strictly one strategy, they have a 50-50 chance of winning. Thus, neither player has an incentive to stick to one strategy. Instead, the best strategy for each player is to randomly choose between heads and tails with equal probability. Therefore, the mixed strategy Nash equilibrium for Matching Pennies is for both players to play each strategy with a probability of 0.5.

Question 10

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 11

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

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

Question 12

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

There is no dominant strategy because the outcome of the payoffs depends on the actions of both teens. The best choice for each teen depends on their beliefs about the other teen's preferences and actions. They must weigh the potential positive utility of declaring their feelings against the potential negative consequences of being rebuffed. Ultimately, it is up to each teen to make their own decision based on their individual circumstances and preferences.

Question 13

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 party moving first studies because they know that if they party, the other party will study and they will get a lower grade.

Question 14

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 15

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 16

The Prisoners' Dilemma is so named because

Accepted Answer

The Prisoners' Dilemma is named so because the Nash equilibrium, where both players confess, leads to a worse outcome for both players compared to if they both had chosen not to confess. It highlights the idea that rational individuals may make choices that lead to suboptimal outcomes for everyone involved.

Question 17

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 18

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,so shared revenues and costs equally?

Accepted Answer

In the Nash equilibrium, each fisherman chooses to catch fewer fish in order to maximize their own profit. If they belonged to the same company and shared revenues and costs equally, they would have less of an incentive to hold back on their catches and would likely catch more fish overall. Therefore, the total number of fish caught would be less in the Nash equilibrium compared to if they worked in the same company.

Question 19

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 FA and FB be the number of fish each catches,respectively. Suppose the price of fish is 100 - FA - FB. Given this demand curve,it turns out that marginal revenue is 100 - 2FA - FB for fisherman A and 100 - FA - 2FB for fisherman B. What is the total number of fish by both in the Nash equilibrium?

Accepted Answer

The Nash equilibrium is found by solving the system of equations:```MR_A = 100 - 2FA - FB = 0MR_B = 100 - FA - 2FB = 0```Solving this system of equations gives:```FA = 20FB = 10```Therefore, the total number of fish caught by both fishermen in the Nash equilibrium is 20 + 10 = 30.

Question 20

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 correct answer is A because 0.52 is the probability that each teen chooses to declare in the mixed-strategy Nash equilibrium.

Quiz 5: Game Theory