Ann and Stella want to watch a movie together this weekend. There are 8 movies playing in theatres on Saturday, and 11 movies playing in theatres on Sunday. Ann first chooses a day - either Saturday or Sunday. Then, knowing the day that Ann picked, Stella chooses one movie from those playing in theatres on that day. How many pure strategies does Stella have?
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- Alice and Julia want to spend time together but have different preferences for their activities. They can choose between two options: going to a soccer game or watching a football game. Alice prefers football and Julia prefers the football game, but they both prefer being together over being apart. If they both choose to attend a soccer game then Julia receives a payoff of 2 and Alice a payoff of 1. If they both choose to attend a football game, then Julia receive a payoff of 1 and Alice a payoff of 2. If one chooses a football game and the other a soccer game then they both receive a payoff of O. The payout matrix below illustrates these payoffs. Which of the following statements are true: I. There is a dominant strategy II. A nash equilibrium exists where Alice attends a football game and Julia attends a soccer game III. A nash equilibrium exists where both friends attend a soccer game IV. Both attending a football game is a stable outcome. Alice Soccer Football Julia Soccer Football…Amir and Beatrice play the following game. Amir offers an amount of money z € [0, 1] to Beatrice. Beatrice can either accept or reject. If Beatrice accepts, then Amir receives 1 - z dollars and Beatrice receives a dollars. If Beatrice rejects, then both receive no money. As in the Ultimatum Game, Amir cares only about maximizing the amount of money he receives. Beatrice, on the other hand, detests Amir, and therefore cares about the amount of money that both receive: if Amir receives y dollars and Beatrice receives a dollars, then Beatrice's payoff is a-ay where a > 0. (a) Find all pure strategy Nash equilibria of the game in which the two players choose simultaneously (thus Beatrice accepts or rejects without seeing Amir's offer). Solution: The NE are the strategy profiles in which Beatrice rejects and Amir's offer a satisfies 2-a(1-x) ≤0, i.e. r ≤a/(1+a). (b) Find all subgame perfect equilibria of the sequential game in which Amir first makes the offer and Beatrice observes the offer…Please no written by hand solutions Consider a game between two friends, Amy and Brenda. Amy wants Brenda to give her a ride to the mall. Brenda has no interest in going to the mall unless her favorite shoes are on sale at the large department store there. Amy likes these shoes as well, but she wants to go to the mall even if the shoes are not on sale. Only Amy subscribes to the newspaper, which carries a daily advertisement of the department store. The advertisement lists all items that are on sale, so Amy learns whether or not the shoes are on sale by showing the newspaper to Brenda. But this is costly for Amy, because she will have to take the newspaper away from her sister, who will yell at her later for doing so. In this game, nature first decides whether or not the shoes are on sale, and this information is made known to Amy (Amy observes whether nature chose S or N). Nature chooses S with probability p and N with probability 1-p. Then Amy decides whether or not to take the…
- The deciding shot in a soccer game comes down to a penalty shot. If the goal-keeper jumps in one corner and the player shots the ball in the other, then it is a goal. If the goalie jumps left and the player shoots left, then it is a goal with probability 1/3. If the goalie jumps right and the player shots right, it is goal with probability 2/3. If both players play Nash strategies, what is the expected value of goals that will follow from this penalty shot. 1/9 2/9 3/9 4/9 5/9 6/9 O7/9Sean is a community college student and has been saving his tips from his job waiting tables at a restaurant for months to see Hamilton. He is willing to pay $705 for a ticket. Anca has seen Hamilton five times already, but wants to see it again before heading to Europe for a month. She is willing to pay $1,250 for a ticket. There is one ticket left, and the seller is charging $700. Does Sean or Anca buying the ticket lead to a more economically efficient outcome?UNIT 9 CHAPTER 5 In a gambling game, Player A and Player B both have a $1 and a $5 bill. Each player selects one of the bills without the other player knowing the bill selected. Simultaneously they both reveal the bills selected. If the bills do not match, Player A wins Player B's bill. If the bills match, Player B wins Player A's bill. Develop the game theory table for this game. The values should be expressed as the gains (or losses) for Player A. Is there a pure strategy? Why or why not? Determine the optimal strategies and the value of this game. Does the game favor one player over the other? Suppose Player B decides to deviate from the optimal strategy and begins playing each bill 50% of the time. What should Player A do to improve Player A’s winnings? Comment on why it is important to follow an optimal game theory strategy.
- Suppose we have a football player taking a penalty kick. The kicker can kick right or left, and the goalie can defend right or left (assume the kicker's right). Write the game using general parameters. Find all the equilibria, pure and mixed. Draw the best response curves.John and Paul are walking in the woods one day when suddenly an angry bear emerges from the underbrush. They each can do one of two things: run away or stand and fight. If one of them runs away and the other fights, then the one who ran will get away unharmed (payoff of 0) while the one who fights will be killed (payoff -100). If they both run, then the bear will chase down one of them and eat them to death but the other one will get away unharmed. Assuming they don't know which one will escape we will call this a payoff of -50 for both. If they BOTH fight, then they will successfully drive off the bear but they may be injured in the process (payoff -10). Construct a payoff matrix for this game and identify the pure strategy Nash equilibrium. (Indicate it with words not with a circle!)Two individuals are bargaining over the distribution of $100 in which payoffs must be in increments of $5. Each player must submit a one-time bid. If the sum of the bids is less than or equal to $100, each player gets the amount of the bid and the game ends. If the sum of the bids is greater than $100, the game ends and the players get nothing. Does this game have a Nash equilibrium? What is the most likely equilibrium strategy profile for this game?
- Martin has a brother and can take a selfish action, which pays him $10 and his brother $0, or an altruistic action, which pays him $6 and his brother $6. The parents love their children equally and decide how to distribute $20 between them. The parents' payment is the minimum of their two children's payments. Assume that the game is simultaneous. Find the Nash equilibrium.You have been assigned to create a new TV game show, and you have an interesting idea that you call, “I WANT TO BE A MILLIONAIRE.” The basics are: 1) two contestants; 2) the show begins with each contestant being given $1 million (!); and then 3) they begin playing a game that can increase or decrease that $1 million. You worry that the initial outlay of $2 million will stun your producers, so you decide to prepare them with a simpler version of your game that you call: “I WANT $3.” There are four steps in this simpler game: There are two contestants/opponents (who do not know each other and cannot communicate with each other during the game). Each player is given $3 at the start of the game. Independently and simultaneously, each player must choose whether they want to add $0, $1, $2 or $3 to their initial stake of $3. Doing so reduces their opponent’s award by $0, $2, $4, or $6, respectively. Each player knows that their payoff at the end of the game is based on their initial $3,…Eva and Ethan love attending sports events, especially when they go together. However, Eva prefers watching field hockey while Ethan would rather go to a cricket match. If they go together, the payoff is higher for the person who favors the sports event more (payoff = 3), and less for the other person (payoff = 2). If they attend the events alone, the payoff is smaller and the same for each (payoff = 1). Show the payoff matrix for this couple and deter- mine the Nash equilibria. How do the payoffs from the Nash equilibria compare?