PLEASE CHECK THIS HOW TO SOLVE Three players bargain over the division of 1 dollar. There are at most three rounds of bargain-ing. In the first round, player 1 proposes a division x = (₁, 72, 73), with x₁ + x2 + x3 = 1.After observing the proposed z, first player 2 chooses whether to accept or reject it, thenplayer 3 does. If both accept the proposal, it is implemented and each player i receives payoffI. If either rejects the proposal, then we move to the second round of bargaining, in whichplayer 2 proposes a division y = (y₁, 92, 93) with y₁ + y2+y3 = 1, which player 3 first decideswhether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal isimplemented, and each player i receives payoff dyi, where 8 € (0,1) is a common discountfactor. If either rejects the proposal, we move to the third round, in which player 3 makesa proposal z = (21, 22, 23) with 2₁ +22+z3 = 1 that player 1 accepts or rejects, followed byplayer 2. If both 1 and 2 accept, each player i receives payoff 6²%, whereas if either rejects,all players receive payoffs of 0.(a) Find all subgame perfect equilibrium outcomes of this game. (Note that you do not haveto describe the full equilibrium strategies.)Solution: Proceed by backward induction. By the same logic as in the Ultimatum Game,the subgame beginning with Player 3's proposal in the third round has a unique SPE givenby z = (0,0,1) and the other players accepting all offers. Given this, in the second round,Player 1 will accept any offer and Player 3 will accept any offer of at least d. Player 2will therefore offer y = (0,1-6,8). Given this, in the first round, Player 2 will acceptany offer of at least 6(1-5) and Player 3 will accept any offer of at least 82. Thus thereis a unique SPE outcome, which involves Player 1 offering x = (1-6, 6(1-6), 6²) andboth other players accepting.(b) Now suppose that, in each round of bargaining, a proposal is adopted if at least oneof the non-proposing players agrees to it (instead of both having to agree). Find allsubgame perfect equilibrium outcomes of this game.Solution: Proceed by backward induction. By similar logic to that of the UltimatumGame, the subgame beginning with Player 3's proposal in the third round has manysubgame perfect equilibria-differing in whether Player 1 accepts proposals that Player2 will accept, and in which proposals Players 1 and 2 choose to accept when they areoffered nothing but the outcome of every equilibrium is that Player 3 offers z = (0,0,1)

Backward induction is a technique for figuring out the best course of action by going backward in…

ing. In the first round, player 1 proposes a division z = (₁, 72, 73), with 7₁ +1₂ + 13 = 1. After observing the proposed z, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff I¡. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y₁, 32, 33) with y₁ + y2+y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dy,, where 8 € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with 21 +22+ 23 = 1 that player 1 accepts or rejects, followed by 2

ing. In the first round, player 1 proposes a division z = (₁, 72, 73), with 7₁ +1₂ + 13 = 1. After observing the proposed z, first player 2 chooses whether to accept or reject it, then player 3 does. If both accept the proposal, it is implemented and each player i receives payoff I¡. If either rejects the proposal, then we move to the second round of bargaining, in which player 2 proposes a division y = (y₁, 32, 33) with y₁ + y2+y3 = 1, which player 3 first decides whether to accept or reject, followed by player 1. If both 3 and 1 accept, the proposal is implemented, and each player i receives payoff dy,, where 8 € (0,1) is a common discount factor. If either rejects the proposal, we move to the third round, in which player 3 makes a proposal z = (21, 22, 23) with 21 +22+ 23 = 1 that player 1 accepts or rejects, followed by 2

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter8: Game Theory

Section: Chapter Questions

Problem 8.8P

See similar textbooks

Similar questions

Two individuals each receive fifty dollars to play the following game. Independently of each other, they decide how much money to put in a common pot. They keep the rest for themselves. As for the money in the pot, it is increased by 80% and then distributed equally among the two individuals. For instance, suppose that the first individual puts $10 in the pot while the second individual puts $20. Increasing the total pot of $30 by 80% gives $54 to share equally between the two individuals. So the first individual’s payoff in this case is $(40 + 27) = $67, while the second individual’s payoff is $(30 + 27) = $57. (a) Compute the Nash equilibrium. (b) Is the Nash equilibrium Pareto efficient? Explain
Two players bargain over 1 unit of a divisible object. Bargaining starts with an offer of player 1, which player 2 either accepts or rejects. If player 2 rejects, then player 1 makes another offer; if player 2 rejects once more, then player 2 makes an offer. If player 1 rejects the offer of player 2, then once more it is the turn of player 1 where he makes two consecutive offers. As long as an agreement has not been reached this procedure continues. For example, suppose that agreement is reached at period 5, it follows that player 1 makes offers in period 1,2 then player 2 makes an o er in period 3, then player 1 makes offers in 4,5. Negotiations can continue indefinitely, agreement in period 't' with a division (x, 1- x) leads to payoffs ( , (1-x)).(The difference from Rubinstein's alternating offer bargaining is that player one makes two consecutive offers, whereas player 2 makes a single offer in her turn.) a. Show that there is a subgame perfect equilibrium in which player 2's…
In a classroom election, two presidential candidates, namely, Lisa and Teddy, both garnered the same number of total votes. As such, they decided to play a custom dice game to determine the winner of the election. In this game, a player needs to roll a pair of dice. Teddy will win the game if the sum is odd whereas Lisa will be declared winner if the sum results to even. However, prior to the start of the game, Teddy complained that the custom dice game is biased because according to him, the probability of an even result is 6/11, and for odd - only 5/11. Verify Teddy's claim and evaluate the fairness of the game. ANALYZE IT CLEARLY AND SOLVE
We have studied the 2-period alternating offer bargaining model with constant costs of delay. In other words, in that model, there were two agents 1 and 2 , each with a constant delay cost c1 and c2 respectively. That is the cost to each player i of each period of delay is ci . Now consider the 3 -period version of the model. In this model there are 3 periods t = 0, 1, 2 . As before the cost of each additional period of delay is ci to player i, i.e., if the period t = 2 proposal z = (z1, z2) is accepted then player i’s payoff is zi − 2ci and if the period t = 2 proposal is rejected, then it is (−2c1, −2c2). Assume c1 > c2. Find a SPE of the game?
Two players bargain over $20. Player 1 first proposes a split of(n, 20 - n), where n is an integer in {0, 1, ..., 20}. Player 2 can either accept or reject this proposal. If player accepts it, player 1 obtains $n and player 2 obtains $(20 - n). If player 2 rejects it, the money is taken away from them and both players will get $0. Question: Find two subgame perfect Nash equilibria of this game and state clearly each player's equilibrium strategies (recall that in a dynamic game, a player's strategy is a complete-contingent plan). Explain why the strategy profiles form a subgame perfect equilibrium.
Consider the game of Chicken in which each player has the option to “get out of the way” and “hang tough” with payoffs: Get out of the way Hang tough Get out of the way 2,2 1,3 Hang tough 3,1 00 a. Find all pure strategy Nash equilibria, if they exist b. Let k be the probability that player 1 chooses “hang tough” and u be the probability that player two chooses “hang tough.” Find the mixed stragety Nash equilibria, if they exist
Tom and Jerry are each given one of two cards. One card is blank, while the other has a circle on it. A player can draw a circle on the blank card or erase the circle on one that has already been drawn. Tom and Jerry make their decision separately and hand in the card at the same time.Nobody wins anything unless the two cards are handed in with one and only one circle on them. The player who hands in the card with the circle receives $20, while the one who hands in the blank card receives $10. Answer the following questions. (1) Represent the game in strategic form.(2) Find the Nash equilibria of the game (in pure strategies).
Q56 A Nash equilibrium is an outcome... a. Achieved by cooperation between players in the game. b. That is achieved by collusion where no party has an incentive to change their behaviour. c. Where each player's strategy depends on the behaviour of its opponents. d. That is achieved when players in the game have jointly maximized profits and divided those profits according to market share of each player. e. Where each player's best strategy is to maintain its present behaviour given the present behaviour of the other players.
Firm A Firm B Low Price High Price Low Price (2, 2) (10, −8) High Price (−8, 10) (15, 15) Suppose the game is infinitely repeated, and the interest rate is 10 percent. The firms are allowed to collude and make joint decisions. Both firms agree to charge a high price, provided no player has charged a low price in the past. This collusive outcome will be implemented with a trigger strategy that states that if any firm cheats (by charging a low price), then the agreement is no longer valid and each firm may make their own independent decisions. Will the trigger strategy be effective in implementing the collusive agreement? Please explain and show all necessary calculations.
Consider the following two person game. Player 1 begins the game by choosing A or B. If Player 1 chooses A the game ends and Player 1 receives $100 and Player 2 receives $100. If Player 1 chooses B then Player 2 must choose C or D. If Player 2 chooses C then Player 1 receives $150 and Player 2 receives $250. If Player 2 chooses D then Player 1 receives $0 and Player 2 receives $400. Draw the complete game tree for this situation. Be sure to accurately label the tree and include the payoffs. Using backwards induction (look forward and reason backwards) determine the rational outcome to this game. Given how experimental subjects have behaved in the Ultimatum game, provide a behavioral explanation for why an “average” Player 1 and 2 might deviate from the rational prediction.
if Y = 4 (a) If ⟨a,d⟩ is played in the first period and ⟨b,e⟩ is played in the second period, what is the resulting (repeated game) payoff for the row player? (b) What is the highest payoff any player can receive in any subgame perfect Nash equilibrium of the repeated game?
Consider the “trust game” discussed in class. The first player starts with a $100 endowment and chooses how much to give to the second player. The gift triples in value (i.e. if $20 is given, the second player receives $60). The second player then chooses how much to give back. The first player receives exactly how much is returned (i.e. if $40 is returned, the first player receives $40). The Nash equilibrium of the game is: Group of answer choices: -First player gives $100, second player returns nothing. -First player gives $50, second player returns $50. -First player gives $100, second player returns $300. -There is no Nash equilibrium of this game. -First player gives nothing, second player returns nothing.