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 - 2 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.
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- Lee and Cody are playing a game in which Lee has the first move at A in the accompanying decision tree: Once Lee has chosen either aggression or cooperation, Cody, who can see what Lee has chosen, must choose either aggression or cooperation at B or C. Both players know the payoffs at the end of each branch. Lee chooses A Aggression Cooperation Cody chooses 8 Multiple Choice C Cody chooses Aggression aggression: aggression Cooperation Aggression In the equilibrium of this game, Lee chooses Cooperation aggression; cooperation -10 for Lee -10 for Cody 40 for Lee O for Cody O for Leo 40 for Cody 25 for Lee 25 for Cody and then Cody chooses.The chicken game has often been used to model crises. Recall that in this game, the two players drive straight at each other. They can choose to swerve or keep going straight. If one swerves, and the other goes straight, assume that the one that swerves gets -10 utility and the one that goes straight gets 10 utility, since the one that swerves is deemed the loser. If both swerve, both get 0 utility. If both go straight, they crash and get -50 utility. Assume both players have a discount rate of 0.9 Draw the stage game of date night List all pure strategy Nash equilibria of the single stage game Consider an infinite horizon version of Chicken. Can you get an SPNE in which the both players swerve using a grim trigger type strategy? Consider the following strategies: both players swerve, as long as neither ever went straight. If one player ever plays straight, in all subsequent rounds the player that swerved goes straight and the player that went straight swerves. Can you think…Suppose that, in an ultimatum game, the proposer may not propose less than $1 nor fractional amounts, and therefore must propose $1, $2, …, or $10 (see image attached below). The responder must Accept (A) or Reject (R). Suppose, first, that this game is played by two egoists, for whom u(x,y)=x. Find all subgame-perfect equilibria in this game. Suppose, second, that this game is played by two altruists, for whom u(x,y) = ⅔(x)1/2 + ⅓(y)1/2. Find all subgame-perfect equilibria in this game.
- Consider the following two-player game.First, player 1 selects a number x≥0. Player 2 observes x. Then, simultaneously andindependently, player 1 selects a number y1 and player 2 selects a number y2, at which pointthe game ends.Player 1’s payoff is: u1(x; y1) = −3y21 + 6y1y2 −13x2 + 8xPlayer 2’s payoff is: u2(y2) = 6y1y2 −6y22 + 12xy2Draw the game tree of this game and identify its Subgame Perfect Nash Equilibrium.Suppose players A and B play a discrete ultimatum game where A proposes to split a $5 surplus and B responds by either accepting the offer or rejecting it. The offer can only be made in $1 increments. If the offer is accepted, the players' payoffs resemble the terms of the offer while if the offer is rejected, both players get zero. Also assume that players always use the strategy that all strictly positive offers are accepted, but an offer of $0 is rejected. A. What is the solution to the game in terms of player strategies and payoffs? Explain or demonstrate your answer. B. Suppose the ultimatum game is played twice if player B rejects A's initial offer. If so, then B is allowed to make a counter offer to split the $5, and if A rejects, both players get zero dollars at the end of the second round. What is the solution to this bargaining game in terms of player strategies and payoffs? Explain/demonstrate your answer. C. Suppose the ultimatum game is played twice as in (B) but now there…$11. Anne and Bruce would like to rent a movie, but they can't decide what kind of movie to get: Anne wants to rent a comedy, and Bruce wants to watch a drama. They decide to choose randomly by playing "Evens or Odds." On the count of three, each of them shows one or two fingers. If the sum is even, Anne wins and they rent the comedy; if the sum is odd, Bruce wins and they rent the drama. Each of them earns a payoff of 1 for winning and 0 for losing "Evens or Odds." (a) Draw the game table for "Evens or Odds." (b) Demonstrate that this game has no Nash equilibrium in pure strategies.
- Sophia is a contestant on a game show and has selected the prize that lies behind door number 3.The show’s host tells her that there is a 50% chance that there is a $15,000 diamond ring behindthe door and a 50% chance that there is a goat behind the door (which is worth nothing to Sophia,who is allergic to goats). Before the door is opened, someone in the audience shouts, “I will giveyou the option of selling me what is behind the door for $8,000 if you will pay me $4,500 for thisoption.” [Assume that the game show allows this offer.]a. If Sophia cares only about the expected dollar values of various outcomes, will she buythis option?b. Explain why Sophia’s degree of risk aversion might affect her willingness to buy thisoptionUNIT 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.10. Player A and Player B are playing a game . First , Player A chooses to either " Keep " or " Pass " . Second , Player B observes A's choice and Player B then chooses to either Keep or Pass . This process continues which creates the sequential game below . Please mark decisions that rational and selfish players will choose at every decision node ( 3 decisions by player A and 3 decisions by player B ) - mark them on the Figure . What is the equilibrium of this game ?
- Two friends are deciding where to go for dinner. There are three choices, which we label A, B, and C. Max prefers A to B to C. Sally prefers B to A to C. To decide which restaurant to go to, the friends adopt the following procedure: First, Max eliminates one of three choices. Then, Sally decides among the two remaining choices. Thus, Max has three strategies (eliminate A, eliminate B, and eliminate C). For each of those strategies, Sally has two choices (choose among the two remaining). a.Write down the extensive form (game tree) to represent this game. b.If Max acts non-strategically, and makes a decision in the first period to eliminate his least desirable choice, what will the final decision be? c.What is the subgame-perfect equilibrium of the above game? d. Does your answer in b. differ from your answer in c.? Explain why or why not. Only typed AnswerIn 'the dictator' game, one player (the dictator) chooses how to divide a pot of $10 between herself and another player (the recipient). The recipient does not have an opportunity to reject the proposed distribution. As such, if the dictator only cares about how much money she makes, she should keep all $10 for herself and give the recipient nothing. However, when economists conduct experiments with the dictator game, they find that dictators often offer strictly positive amounts to the recipients. Are dictators behaving irrationally in these experiments? Whether you think they are or not, your response should try to provide an explanation for the behavior.Consider the following game - one card is dealt to player 1 ( the sender) from a standard deck of playing cards. The card may either be red (heart or diamond) or black (spades or clubs). Player 1 observes her card, but player 2 (the receiver) does not - Player 1 decides to Play (P) or Not Play (N). If player 1 chooses not to play, then the game ends and the player receives -1 and player 2 receives 1. - If player 1 chooses to play, then player 2 observes this decision (but not the card) and chooses to Continue (C) or Quit (Q). If player 2 chooses Q, player 1 earns a payoff of 1 and player 2 a payoff of -1 regardless of player 1's card - If player 2 chooses continue, player 1 reveals her card. If the card is red, player 1 receives a payoff of 3 and player 2 a payoff of -3. If the card is black, player 1 receives a payoff of 2 and player 2 a payoff of -1 a. Draw the extensive form game b. Draw the Bayesian form game