Mixed Dominance 1: Let o; be a mixed strategy of player i that puts positive weight on one strictly dominated pure strategy. Show that there exists a mixed strategy o that puts no weight on any dominated pure strategy and that dominates ơ;.
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- Consider the two-player game: a. Find the strategies that are consistent with both players being rationaland each player believing the other player is rational.b. In addition to that assumed in part (a), assume that player 2 knows player1 knows player 2 is rational. Find strategies consistent with these beliefs.Two bidders compete in a second price auction (i.e., the winning bidder pays the losing bidder’s bid, and the losing bidder does not pay anything). They submit sealed bids, and the one with the highest bid wins the contract and pays the other bidder’s bid. Each bidder i’s private valuation is vi and is distributed independently and uniformly between 0 and 50. 1. For any given bidder, prove that he has a dominant strategy bid and show what it is. 2. Assuming each bidder bids his dominant strategy noted above, if a bidder with vi = 40 wins, what price does he expect to pay?Consider the strategic voting game discussed at the endof this chapter, where we saw that the strategy profile (Bustamante, Schwarzenegger,Schwarzenegger) is a Nash equilibrium of the game. Show that (Bustamante, Schwarzeneg-ger, Schwarzenegger) is, in fact , the only rationalizable strategy profile. Do this by firstconsidering the dominated strategies of player L. (Basically, the question is asking youto find the outcome of the iterative elimination of strictly dominated strategies)
- Consider the bargaining problem of splitting a pie of size 1 with utility u(x1) = x1 for player 1 and v(x2) = 2x2 − x22 for player 2, where x1 and x2 denote the share of the pie for player 1 and 2 respectively. a) Consider the bargaining problem of the two players. Find the utility possibility frontier S. b) What is the Nash bargaining solution for this problem (i.e., on which division of the pie (?₁,?₂) will players agree), if the disagreement outcome (the utilities players obtain in case of disagreement) is d1 = d2 = 0? c) What is the Nash bargaining solution if the disagreement outcome is any d1 and d2 in S?Prove that in the variation on the centipede game given in figure 14.5(b) the unique sequential equilibrium described is, in fact, the unique Nash equilibrium. (Hint: Take some presumed Nash equilibrium and suppose information set 2n+ 1 [for player 2] is the first unreached information set. Derive an immediate contradiction. Then suppose that node (2n) t is the first unreached information set and derive a contradiction that is one degree removed from immediate.)(a) Find all the Nash Equilibria, if there is any. (no explanation needed for this part (b) Does player 1 (choosing rows) have any dominant action? If yes, which action(s)? Any dominated action(s)? If yes, which ones? Answer the same questions for player 2, too. (c) If player 1 moves first (and player 2 moves next), what would be the sequentially rational equilibrium (draw the game tree and use backward induction)?What if player 2 moves first (and then player 1 moves next)? (d) Looking at your findings in (c), would player 1 want to move first or second or is she indifferent (the order doesn’t matter)?
- In the attached table determine, for each player, whether any action is strictly or weakly dominated. Justify youranswer. Find the Nash equilibria of the game and determine whether any equilibrium is strict.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?(1) Write down each player’s best response functions and find all the Nash equilibria.(2) For each action of each player, determine whether it is strictly, weakly, or not domi-nated by any other actions. Justify your answer.(3) Find the set of rationalizable action profiles by iteratedly eliminate strictly dominatedactions. Clearly show the steps.
- 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.Explain all will rate what is always true for a pure nash equilibrium of a two-person non zero-sum game A.No player can improve his payoff with a unilateral change of strategy B. it is a Pareto maximum of the payoff matrix C. No player can worsen the payoff of his opponent with a unilateral change of strategy D. It gives worse payoffs to both players than any berge equilibriumSuppose that there are only two firms in a market in which demand is given by p = 64 - Q, where Q is the total production of the two firms. Each firm can choose either a low level of output, qL = 15, or a high level of output, qH = 20. The unit cost of production for both firms is $4. Write down the normal-form representation of the game in which the strategic variable for each firm is the quantity of output and the firms make their choices simultaneously. Find the pure strategy Nash equilibrium of this game (quantities produced and market price).