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- Which one of the following descriptions is WRONG, according to this Extensive Form shown below? 1.) There are 8 strategy profiles in this game. 2.) If Player 1 chooses strategy IA and Player 2 chooses strategy O, then player 1's payoff will be 4. 3.) If Player 1 chooses strategy OB and Player 2 chooses strategy I, then Player 2's payoff will be 2. 4.) Game over if Player 1 chooses O initially.Suppose 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).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.
- 4 Consider an extensive game where player 1 starts with choosing of two actions, A or B. Player 2 observes player 1’s move and makes her move; if the move by player 1 is A, then player 2 can take three actions, X, Y or Z, if the move by player 1 is B, then player 2 can take of of two actions, U or V. Write down all teminal histories, proper subhistories, the player function and strategies of players in this game.Q14. Do players have perfect information in the above game? Yes, all of them have perfect information No, player 2 has imperfect information No, player 3 has imperfect information No, no player has perfect information Q15. If we want to describe the above game with a strategic form representation, what would the strategy sets for the three players be? Player 1={a, b, c} ; Player 2={x, y}; Player 3={r, s} Player 1={a, b, c} ; Player 2={xx, xy, yx, yy}; Player 3={r, s}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 equilibrium
- 10) Consider a repeated game in which the following one-shot game between two players is repeated "infinitely." Denote by "d" the discount factor. (Assume both players have the same discount factor.) Player 2 Cooperate Defect Player 1 Cooperate 5, 4 0, 5 Defect 6, 0 1, 1 The two players try to support cooperation by using "grim trigger strategy" as discussed in class. Answer YES or NO to each of the following three questions. (a) Can they support cooperation if the common discount factor, d, is equal to 0.23? (b) Can they support cooperation if the common discount factor, d, is equal to 0.4? (c) Can they support cooperation if the common discount factor, d, is equal to 0.15?Consider the game tree below (attached). In this game, each player can mover either Up (U) or Down (D). Player 1 moves first, then Player 2, and Player 3 moves last. Using backwards induction, clearly identify the choice that each player will make at each stage in the tree circle the ultimate outcome (payoff) to the gameTwo 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.
- 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)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 onlyPlayers 1, 2, and 3 are playing a game in which the strategy of player i isdenoted yi and can be any nonnegative real number. The payoff function for player 1 is V1(y1,y2,y3) = y1 + y1y2 - (y1)2,for player 2 is V2(y1,y2,y3) = y2 + y1y2 - (y2)2,and for player 3 is V3(y1,y2,y3) = (10 - y1 - y2 - y3)y3.These payoff functions are hill shaped. Find a Nash equilibrium. (Hint: Thepayoff functions are symmetric for players 1 and 2.)