Exam 5: Game Theory

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)

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

Best-response functions

The game of Matching Pennies

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?

Two games that differ only in the timing of moves-one simultaneous,the other sequential move-can sometimes have completely different subgame-perfect equilibria.Why?

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?

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)

A subgame-perfect equilibrium is a Nash equilibrium that

The beauty of Nash's equilibrium concept is that

In what way or ways can strategies more complicated than simple actions?

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?

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?

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?

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?

The Prisoners' Dilemma is so named because

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

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_B for fisherman A and 100 - F_A - 2F_B for fisherman B.What is the total number of fish by both in the Nash equilibrium?

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

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: