Two players, both with zero wealth, bargain over how to divide £X > 0 between them. Failure to reach agreement means both get nothing. Both players are expected utility max- imisers. Player 1 has utility u(x) = x“, where 0 < a < 1. Player 2 has utility u(x) = where 0 < B < 1. Determine the Nash solution for this problem
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- 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 existConsider 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.Suppose two players play a two-period repeated game, where the stage game is the normal-form game shown below. Is there a subgame perfect Nash equilibrium in which the players select (A, X) in the first period? If so, fully describe such equilibrium. If not, explain why not. Player 1 has choice A, B; Player 2 has choice X, Y, Z. Payoff: (A,X)-(5,7), (A,Y)-(2,4), (A,Z)-(3,8), (B,X)-(1,4), (B,Y)-(3,5), (B,Z)-(1,4)
- The count is three balls and two strikes, and the bases are empty. The batter wants to maximize the probability of getting a hit or a walk, while the pitcher wants to minimize this probability. The pitcher has to decide whether to throw a fast ball or a curve ball, while the batter has to decide whether to prepare for a fast ball or a curve ball. The strategic form of this game is shown here. Find all Nash equilibria in mixed strategies.Consider a game with two players (Alice and Bob) and payoffffs Bob Bob s1 s2 Alice, s1 3, 3 0, 0 Alice, s2 0, 0 2, 2 In the equilibrium in the above game, Alice should (A) always choose the fifirst strategy s1; (B) choose the fifirst strategy s1 with probability 40% ; (C) choose the fifirst strategy s1 with probability 50% ; (D) choose the fifirst strategy s1 with probability 60% .Consider the game Ms. Bennet and Mr. Darcy play in ‘First Impressions’, Selected Set V. Suppose that Ms. Bennet prefers to meet Mr. Darcy (a = 0) with probability p. Further suppose that: - The ‘meeting Ms. Bennet’ plays Ball with probability q (and Dinner with probability 1 − q); - ‘avoiding Ms. Bennet’ plays Ball with probability r (and Dinner with probability 1 − r); M - r. Darcy plays Ball with probability s (and Dinner with probability 1 − s). Write down the strategic form game and find for all values of p ∈ (0, 1) the Bayesian-Nash equilibria in mixed strategies.
- Professor can give a TA scholarship for a maximum of 2 years. At the beginning of each year professor Hahn decides whether he will give a scholarship to Gong Yi or not. Gong Yi can get a scholarship in t=2, only if he gets it in t=1. Basically, the professor and TA will play the following game twice. TA can be a Hardworking type with probably 0.3 and can be a Lazy type with a probability of 0.7. Professor does not know TA's type. If TA is hard working, it will be X=5 and TA will always work if he gets a scholarship. If TA is lazy, it will be X= 1. There is no time discount for t=2. Find out a Perfect Bayesian Equilibrium of the game.Consider the following game. There are two payers, Player 1 and Player 2. Player 1 chooses a row (10, 20, or 30), and Player 2 chooses a column (10/20/30). Payoffs are in the cells of the table, with those on the left going to Player 1 and those on the right going to player 2. Suppose that Player 1 chooses his strategy (10, 20 or 30), first, and subsequently, and after observing Player 1’s choice, Player 2 chooses his own strategy (of 10, 20 or 30). Which of the following statements is true regarding this modified game? I. It is a simultaneous move game, because the timing of moves is irrelevant in classifying games.II. It is a sequential move game, because Player 2 observes Player 1’s choice before he chooses his own strategy.III. This modification gives Player 1 a ‘first mover advantage’. A) I and IIB) II and IIIC) I and IIID) I onlyE) II onlySuppose that 5 risk neutral competitors participate in a rent seeking game with a fixed prize of $100. Each player may invest as much as he wishes in the political contest, although those investments have an opportunity cost equal 1. The probability of winning is directly proportional to the candidate’s share of the total rent-seeking investment. What is the profit-maximizing investment by player 1 as a function of the investment by all the others? What is a Nash equilibrium investment by each player in a symmetric 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 attached extensive-form game tree, where player 1 moves first, then player 2. The top payoff accrues to player 1, the bottom payoff to player 2. (a) Draw the strategic form for this extensive form game.(b) Find all of the Nash equilibria, including any mixed.(c) Which of these Nash equilibria do you think would be actually played? Why?The mixed stratergy nash equalibrium consists of : the probability of firm A selecting October is 0.692 and probability of firm A selecting December is 0.309. The probability of firm B selecting October is 0.5 and probability of firm selecting December is 0.5. In the equilibrium you calculated above, what is the probability that both consoles are released in October? In December? What are the expected payoffs of firm A and of firm B in equilibrium?