Consider the extensive game with 3 players from before. Assume that all players have the exact same payoff. This is determined primarily by the action chosen by player 3. Players are better off the later in the alphabet that player 3's action is. For instance, they are better off if player 3's action is H than F and better off if the action is F than E. If player 3's action is F, players are better off when player 2 pick D than C. The same is true if player 3's action is E. Use backwards induction to solve the game. Which of the following strategy profiles is the solution? (B,D,J) (B,DD,FFHJ) (B,CD,EFGJ) None of the above

Consider the extensive game with 3 players from before. Assume that all players have the exact same payoff. This is determined primarily by the action chosen by player 3. Players are better off the later in the alphabet that player 3's action is. For instance, they are better off if player 3's action is H than F and better off if the action is F than E. If player 3's action is F, players are better off when player 2 pick D than C. The same is true if player 3's action is E. Use backwards induction to solve the game. Which of the following strategy profiles is the solution? (B,D,J) (B,DD,FFHJ) (B,CD,EFGJ) None of the above

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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Consider a variation of the television station broadcast game of Exercise 4 in Chapter 7. Suppose the stations interact sequentially. First, MBC chooses between 6:00 and 7:00. Then, after observing MBC’s choice, RBC decides between 6:00 and 7:00. Finally, after observing the behavior of both MBC and RBC, CBC chooses either 6:00 or 7:00. Payoffs are as given before. Draw the extensive form of this sequential game and compute the subgame perfect equilibrium.
Consider the following game. You will roll a fair, 6-sided die either once or twice. You decide whether to do the second roll after you see how the first one lands. The payoff is $n, where n is the outcome of the last roll. For example, if the first roll lands 4 and the second lands 2, you win $2. If you only do one roll and it lands 4, you win $4. Suppose you make your decision about whether to go for a second roll based on expected monetary value. Then you will go for a second roll if (and only if) the first roll lands x or lower. What is x?
a) Find the Nash equilibria in the game (in pure and mixed strategies) and the associated payoffs for the players. b) Now assume that the game is extended in the following way: in the beginning Player 1 can decide whether to opt out (this choice is denoted by O) or whether to play the simultaneous-move game in a) (this choice is denoted by G). If Player 1 opts out (plays O) then both Player 1 and Player 2 get a payoff of 4 each and the game ends. If Player 1 decides to play G, then the simultaneous-move game is played. Find the pure-strategy Nash equilibria in this extended version of the game. (Hint: note that Player 1 now has 4 strategies and write the game up in a 4x2 matrix.) c) Write the game in (b) up in extensive form (a game tree). Identify the subgames 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 Answer
Suppose Antonio and Trinity are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Antonio chooses Up and Trinity chooses Down, Antonio will receive a payoff of 7 and Trinity will receive a payoff of 5. Trinity Up Down Up 4,8 7,5 Antonio Down 3,2 5,6 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Antonio chooses, and Trinity chooses Grade It Now Save & Continue Continue without saving @ 2 F2 #3 80 Q F3 MacBook Air 44 F7 Dll F8 44 F10 74 $ 4 05 Λ & % 5 6 7 8 * 0 Q W E R T Y U 1 A N S X 9 0 -O O D F G H J K L on را H command C > B N M Λ - - P [ H Λ command opti
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the Consider a simultaneous-move integer game between two players: Marilyn and Noah. Each of player is required to announce a positive integer between 1 to 4. In other words, a player can announce 1, 2, 3, or 4. Two players announce their integers simultaneously. Notice that this game is different from the games we learned in class in that each player has four actions to take. The payoffs of the players in the game are specified as follows: (1) when the two announced integers are different, whoever reports the lower number pays $1 to the other player, so that the loser of the game has payoff -1 and the winner of the game has payoff 1; (2) when the two players announce the same integer, their payoffs are both O. What is Marilyn's maximin strategy? 01 02 03 O
“To be or not to be, that is the question.” Imagine that in answering this question Hamlet had the following data to consider. Hamlet must choose either “To Be” or “Not to Be.” If Hamlet chooses “To Be” then his enemy the King will make a decision to either “Kill” Hamlet or let him “Live.” If Hamlet is killed by the King his payoff in this game is 200 since he will have been killed by the King who is his uncle and who also killed Hamlet’s father, the King’s brother, who was the prior king. If the King decides to let Hamlet Live then Hamlet will be able to avenge his father’s death and his payoff will be +300. Hamlet also considers suicide an option but given the uncertainty of the afterlife if he chooses not to be then his payoff is +100 since his doesn’t what the afterlife holds for one who kills himself. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. Assume Hamlet will choose the course of action that offers the highest…
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…
Keith and Blake play a simultaneous one-shot game. Keith chooses between Top and Bottom, whereas Blake chooses between left and right. Payoffs are given by (Keith's payoffs are listed first and Blake's payoffs second): Top+left: 3,-6. Top+right:0,-7 Bottom+left:4,0. Bottom+right:-2,6. In a mixed strategy equilibria, Keith will play Top with what probability? 1. 6/7 2. 1/4 3. 1/2 4. 1/7 5. 0 6. 3/4
An apartment building needs a new elevator, which costs $10,000. 10 residents need to decide independently whether or not to chip in. If no resident is willing to bear the cost, the elevator will not be installed and each resident will receive zero payoff. However, if n > 0 residents are willing to chip in, the elevator will be installed. If the elevator is installed, each resident will receive a benefit equal to $2,000 regardless of whether they have chosen to chip in or not. However, a resident who has chosen to chip in will also bear their share of the For example, if 5 residents choose to chip in, they will receive a net or $0, while other residents will each receive payoff $2,000. installation cost payoff equal to $2,000 $10,000 5 (1) Is every resident chipping in a Nash equilibrium of the game? Explain. (2) Is no resident chipping in a Nash equilibrium of the game? Explain. (3) Are there any other Nash equilibria? If there are, specify such a Nash equilibrium and verify that it…
Suppose Carlos and Deborah are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 6 and Deborah will receive a payoff of 5. Deborah Left Right Carlos Left 8, 4 4, 5 Right 5, 4 6, 5 The only dominant strategy in this game is for to choose . The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses and Deborah chooses .