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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.)1. What are the advantages and disadvantages of collusion? Define Collusion 2. In a Stackelberg game, what is the best response that follower firm 2 can make to the choice y1 already made by the leader, firm 1? Defining the game and provide an example of the best response.Supersubs and Hotwich are the only two sandwich shops in town. They have differentiated their products so that Supersubs sells cold sub sandwiches and Hotwich sells hot sandwiches. At any given time, however, Supersubs could expand its selection by purchasing a toaster and Hotwich could also easily start offering cold sandwiches. Refer to the game for the expected payoffs of their possible choices. Supersubs Sell cold only Expand selection Hotwich Sell hot only A $12$12 $10$10 B $18$18 $2$2 Expand selection C $2$2 $19$19 D $5$5 $6$6 What is the unique equilibrium outcome for a one-time game?
- Suppose O2 and Vodafone are the only two telecommunicationscompanies in UK. Both companies are considering whether ornot to stop offering unlimited data plans. Each company has twostrategies: stop or don’t stop. The first entry in the brackets is the payoffsof O2 and the second entry is the payoffs of Vodafone, both in $million.What will be the dominant strategies of O2 and Vodafone and what willbe the Nash equilibrium? Explain your answers.What is the payoff for player 1 in the normal form game below: O. 3 O. 1 O. 0 O. 4Suppose 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 a modified Traveler’s Dilemma. In terms of strategy options that the players have and the dollars they earn, it is like the standard Traveler’s Dilemma, but the players do not have endless appetite for money. Up to 100 dollars, each dollar feels like a dollar. But any moneybeyond 100 is psychologically like 100 dollars. Assuming that players are maximizers of ‘psychological’ dollars instead of real dollars, describe all the Nash equilibria of this modified Traveler’s Dilemma.Consider the location game we covered in Lecture 3. Now assume there arethree players (vendors). As we assumed in the lecture, consumers in each area choosethe closest vendor and if there are multiple closest vendors then these vendors receiveequal share of consumers in the area. Notice Si = {1, 2, 3, ...., 9} for i = 1, 2, 3. Here aresome examples of payoffs: u1(1, 1, 1) = 3, u1(1, 1, 9) = u2(1, 1, 9) = 2.25, u3(1, 1, 9) =4.5, u1(1, 5, 9) = u3(1, 5, 9) = 2.5 and u2(1, 5, 9) = 4. (a) Is s′1 = 1 strictly dominated by s′′1 = 2 for player 1?(b) Is s′1 = 1 weakly dominated by s′′1 = 2 for player 1?(c) Can you find a Nash equilibrium in pure strategies?if Y = 4 (a) If ⟨a,d⟩ is played in the first period and ⟨b,e⟩ is played in the second period, what is the resulting (repeated game) payoff for the row player? (b) What is the highest payoff any player can receive in any subgame perfect Nash equilibrium of the repeated game?
- Suppose that there are two firms in a market, firm 1 and firm 2. The marketis declining in size. The game starts in period 0, and the firms can compete in periods 0, 1,2, 3, ... (i.e., indefinitely) if they so choose. Duopoly profits in period t for firm 1 are equalto 105 −10t, and they are 10.5 −t for firm 2. Monopoly profits (those if a firm is the onlyone left in the market) are 510 −25t for firm 1 and 51 −2t for firm 2. At the start of eachperiod, each firm must decide either to “stay in” or “exit” if it is still active (they do sosimultaneously if both are still active). Once a firm exits, it is out of the market forever andearns zero in each period thereafter. Firms maximize their (undiscounted) sum of profits.What is this game’s subgame perfect Nash equilibrium?Consider the two-round bargaining game. The minimum the seller will sell his home for 188,000 and the maximum the buyer is willing to pay is $200,000. Both players know these two amounts and are bargaining over the difference (M=$1200). Assume the disagreement values are zero for both players. Player 1 moves first by making a proposal and Player 2 can accept and reject. If player 2 rejects Player’s 1 proposal, then Player 2 gets to make a proposal, which Player 1 can reject and accept. The game is then over. Suppose the both players discount the future income at the rate d=0.2 per period. That is, $0.20 now is equivalent to $1,00 received next round. Find the equilibrium for this 2-round game. a) Draw the game tree b) What is the sale price of home? c)Which player gets the larger share of M?For the operating systems game, let us now assume the intrinsic superiorityof Mac is not as great and that network effects are stronger for Windows.These modifications are reflected in different payoffs. Now, the payoff fromadopting Windows is 50 X w and from adopting Mac is 15 + 5 X m;n consumers are simultaneously deciding between Windows and Mac.a. Find all Nash equilibria.b. With these new payoffs, let us now suppose that a third option exists,which is to not buy either operating system; it has a payoff of 1,000.Consumers simultaneously decide among Windows, Mac, and nooperating system. Find all Nash equilibria.