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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.)
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- 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 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 only10) 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?
- For the game in FIGURE 7.6, derive players’ best-reply functions, plot them as shown in Figure 7.5, and find the Nash equilibrium.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 gameUsing the payoff matrix, suppose this game is infinitely repeated and that the interest rate is sufficiently “low.” Identify trigger strategies that permit players 1 and 2 to earn equilibrium payoffs of 140 and 180, respectively, in each period
- Consider the following Cournot model. • The inverse demand function is given by p = 30 –Q, where Q = q1 + q2. • Firm 1’s marginal cost is $6 (c1 = 6). Firm 2 uses a new technology so that its marginal cost is $3 (c2 = 3). There is no fixed cost. • The two firms choose their quantities simultaneously and compete only once. (So it’s a one-shot simultaneous game.) d. Suppose there is a market for the technology used by Firm 2. What is the highest price that Firm 1 is willing to pay for this new technology? e. Now let’s change the setup from Cournot competition to Bertrand competition, while maintaining all other assumptions. What is the equilibrium price? f. Suppose the two firms engage in Bertrand competition. What is the highest price that Firm 1 is willing to pay for the new technology?Which of the following is FALSE for the grim trigger strategy and the infinite horizon repeated Prisoner's Dilemma game illustrated above? A. In the grim trigger strategy profile, if a player chooses D in a period, then both players chooses D forever after that period B. The threshold discount factor for sustaining cooperation under grim trigger strategy depends on the utility numbers in the stage game C. If all utility numbers remain the same but 3 is replaced by 5 in the stage game, then cooperation CANNOT be sustained in this game for all possible values of the discount factor.two players, a and b are playing an asymmetrical game. there are n points on the game board. each turn player a targets a pair of points and player b says whether those two points are connected or unconnected. a can target each pair only once and the game ends when all pairs have been targeted. player b wins if a point is connected with all other points on the very last turn, while player a wins if any point is connected with all other points on any turn but the very last one or if no point is connected to all other points after the last turn. for what values of n does either player have a winning strategy?
- Suppose 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?Player 1 and Player 2 are trying to agree on how to split a pie of size 1 in a two-stage bargaining game. If no agreement is reached after the two stages are complete, the pie is split for them according to a pre-arranged agreement that gives Player 1 and Player 2 one-quarter and three quarters of the pie, respectively. In the first stage, Player 1 makes an offer (x1, x2), where x1 + x2 = 1. Player 2 can either accept this offer (at which point the game ends and the pie is split according to Player 1’s offer), or can make a counter-offer. When Player 2 makes a counter offer, Player 1 can either accept (in which case the pie is split according to Player 2’s offer) or can reject, in which case the pie is split according to the pre-arranged agreement. Both players have a discount factor d – getting dx in the first stage (after Player 1’s proposal) is as good as getting x in the second stage (after Player 2’s proposal). a) In the last stage of the game, Player 1 will accept any offer…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.
