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1 . Consider the following two-players extensive form game: [21.0) (-110) {11.1) {4,3} {it Find the pure-strategy Nash equilibria of such a game. {ii} Find the Subgame Perfect equilibria of such a game. {iii} Discuss why some of the Nash equilibria of the game are not Subgame Perfect. 2. Consider the following Stackelberg duopoly: firm 1 chooses output first. then firm 2 observes the output chosen by firm 1 and decides its own output. The market inverse demand is p = 1 ql so. where q; is firm i's output. Each firm incurs a fixed cost of 1,18 if it produces positive output tothenvise. its costs are zero]. Once the fixed cost is paid it costs each firm zero to produce each additional unit: variable costs are zero. Find the unique Subgame Perfect equilibrium pure strategy Nash equilibria of this game 3. Consider two players, a seller and a buyer. and two dates. At date 1. the seller chooses his investment level I :3 c at cost I. At date 2, the seller may 1 sell one unit of a good and the seller has cost out of supplying it. where glib} = :x::-, c' r. o, and cm] is less than the buyer's valuation for the good. Assume no discounting, so the socially optimal level or investment 1" is given py1+dtlfl = 0. {it Suppose that at date 2 the buyer observes the investment 1 and makes a take-it-or-leave-it offer to the seller. What is this offer? What is the subgame perfect equilibrium of the game? {ii} Can you think of a contractual way of avoiding the inefficient outcome of [i]? {Assume that contracts cannot be written on the level of investment L} 4. Consider the following two players game. First player 1 can choose either Stop or continue. If she chooses Stab then the game ends with the pair of payoffs [1. 1). If she chooses Continue then the payers simultaneously and indepen- dently announce an integer number between it and 100. and each player's payoff is the product of the numbers. Formulate this situation as an extensive form game and find its subgame perfect equilibria. 5. Consider the following three-players extensive form game: R (1, 2, 0) A. B (2, 1, 0) (3, 2, 3) (1, 4, 2) (i) Construct the normal form associated with this extensive form game. (ii) Find the pure-strategy Nash equilibria of such a game. 2 (iii) Find the Subgame Perfect equilibria of such a game. 6. Two firms 1 and 2 Bertrand compete to employ a worker. The worker's expected output in firm 1 is m, while in firm 2 is my, where mi > my > 0. The two firms simultaneously and independently submit wage offers to the worker. The worker observes the wage offers and decides which one to accept. (i) Formulate this situation as a game in extensive form and find the set of Subgame-perfect equilibria of this game. Assume now that when the worker is employed by firm i (i = 1,2) he gen- erates a positive externality to firm - which increases firm -i's payoff of an amount of, where 0 by > 0. (ii) Find the set of Subgame-perfect equilibria of this new Bertrand game. (iii) Does the equilibrium choice of employer by the worker coincides with the allocation of the worker that a central planner faced with the same payoffs will choose? Discuss.1 . Consider two players A and B that bargain for no more than two periods on the division of a pie of unit size. At the beginning of each period the identity of the player who makes the offer is determined by a public randomizing device. This randomizing device has two possble realizations: e that occur with probability g and ,8 that occurs with probability 5. At the beginning of period 1 ifthe realization of the randomizing device is o it is player A's turn in make the offer {with prcbabillty :1 then player B observes player as offer and decides whether to accept it or reject it. If instead the realization of the randomizing device is if then it is player B's turn to make the offer [with probability 3} then player A observes player It's offer and decides whether to accept it or reject it. if an offer is accepted then the game ends and each player's payoff is the share of the pie specied in the accepted offer. If an offer is rejected then the game moves to period 2. At the beginning of period 2 if the realization of the randomizing device is a then it is player B's turn to make the offer {notice the difference pith pencd t; [with probability ] then player A observes player B's offer and decides whether to accept it or reject it. If instead the realization of the randomizing device is i3 then it is player A's turn to make the offer {with probability El then player 3 observes player A's offer and decides whether to accept it or reject it. If an offer is accepted then the game ends and each player's payoff is the discounted share of the pie specified in the accepted offer. If an offer is rejected then the game ends and both players receive a zero payoff. Assume that both players discount the future at the same rate ii. {i} Identify the Subgame-Perfect-eguilbrium strategies of such a game. 1 {ii} Compute the equilibrium expected payoff to each player in the subgame Perfect equilibrium identified in {i} above when the common discount factor is 5 = g. 2. Consider two players A and B that bargain for no more than three periods on the division of a pie of unit size. In period 1 player A makes an offer to player B of a possible sharing of the pie. Player B sees the offer and then decides whether to accept it or reject it. If the offer is accepted the game ends and the payoff to the two players is represented by the share of the pie that according to the accepted offer accrues to each player. If the offer is rejected instead the game moves to the following period. In period 2 player B makes an offer to player A of a possible sharing of the pie. Player A sees the offer and then decides whether to accept it or reject it. If the offer is accepted the game ends and the payoff to the two players is represented by the share of the pie that according to the accepted offer accrues to each player properly discounted. If the offer is rejected instead the game moves to the final period. In period 3 it is once again player A's turn to make an offer to player B of a possible sharing of the pie. Player B sees the offer and then decides whether to accept it or reject it. If the offer is accepted the game ends and the payoff to the two players is represented by the share of the pie that according to the accepted offer accrues to each player properly discounted. If the offer is rejected instead the game ends and each player receives a zero payoff. Assume that both players discount the future at the same rate o = 1/3. (i) Identify the Subgame-Perfect-equilibrium strategies of such a game. (ii) Compute the equilibrium expected payoff to each player in the Subgame Perfect equilibrium identified in (i) above. 2 3. In the same bargaining game above (question 2) assume now that the legal system requires the acceptance of an offer to be formalized. This process is costly for the individual that is planning to accept the offer and this cost is c = 1/7. Notice however that such a cost is only paid after and if the offer is accepted. In this new situation (ii) identify the Subgame-Perfect-equilibrium strategies of such a game; (iv) compute the equilibrium expected payoff to each player in the Subgame Perfect equilibrium identified in (iii) above.4. In the same bargaining game above [question 2} assume now that. instead of the cost r: of formalizing the offer. player B has to pay the opportunityr cost it = 1;? of showing no at the negotiation table. Notice that this cost or. incurred only by player It. is a cost of participating in the bargaining game and as such is paid before entering the negotiation. Once the negotiation starts player It does net incur any cost for accepting an offer or for making an offer. Of course. 3 may decide not to pay is and hence not to participate to the negotiation. In this case the negotiation does not take place and both A and .8 receive no share of the pie. In this third and final situation {y} identityr the Subgame-F'edect-eguilibrium strategies of such a game: [vii compute the eguilbrium expected payoff to each player in the Subgame Perfect eguilibrium identied in {or} above. 5. Consider the Ftthinstein's alternating-offer bargaining model and assume that. when a player makes an offer he is restricted to the pairs {1,0}. [3} .[%,% . [iii . [i111]. Characterize the set of sub-game perfect equilibrium payoffs when the discount factor .5 is near 1. {it Formulate this bargaining model as an extensive form game. {it} Find the Subgame-perlect equilibria of such a game