24. Consider the following model of a bank run. Depositors may either Hold or Withdraw their funds from a bank. Both players hold deposits of 6, which earn interest of 1 if kept in the bank. If either depositor withdraws then the bank fails. If both try to withdraw then each depositor can only recover half of their deposits. The payoffs for a two-player version of this game are given below: Player 2 Player 1 Hold Withdraw Hold 7,7 6,0 (a) Find any pure-strategy Nash equilibria of this game. (b) Find any mixed-strategy Nash equilibria of this game. Withdraw 0,6 3,3
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- 12. Consider a game where each player picks a number from 0 to 60. The guess that is closest to half ofthe average of the chosen numbers wins a prize. If several peopleare equally close, then they share theprize. The game theory implies that (A) all players have dominant strategies to choose 0 (B) all players have dominant strategies to choose 30 (C) there is a Nash equilibrium where all players pick 0 (D) there is a Nash equilibrium where all players pick positive numbers 13. Behavioral data in such games suggests that (A) most subjects choose 0; (B) most subjects choose 30; (C) common answers include 30, 15, 7.5, and 0; (D) most subjects use randomization. Can you help me answer number 13 please?Game Theory a) Consider the following game represented by the game tree below. If you observe Blue Bank playing medium and Green Bank playing bottom, would you be surprised by their choices? Carefully explain why or why not. b) Now assume that Blue and Green Banks play the game with the same actions and payoffs as in a) but they make their choices simultaneously. What is the payoff matrix for this new game? What do you predict will happen in the game? Explain. c) Assume that the game from part a) is played five times, i.e. Blue and Green Banks play a repeated game with the stage game described in a). What do you predict will happen in the game? Explain.6 Two people will select a policy that affects both of them by applying a "veto" in a sequential and alternate manner, that is: person 1 begins to veto a policy and then person 2 exercises his "veto" with the remaining policies; the process repeats until only one policy remains. Assume that there are 3 policies: X,Y,Z, and that person 1 prefers X to Y to Z and person 2 prefers Z to Y to X. a. Represents the game extensively b. Give the number of subgames C. Indicate the total strategies of the players d. find all subgame perfect nash equilibria e. Find all Nash Equilibriums.
- 5) Three legislators are set to vote on a bill to raise the salary of legislators. The majority wins, so all three will receive the raise if at least two of them vote in favor of the bill. The raise is valued at R by each legislator. Voting in favor of the bill comes with political backlash from constituents, though, even if the bill fails. Let C be the cost of backlash for anyone voting in favor of the bill. Finally, suppose that 0 < C < R. There are four possible payoffs for each legislator: 0: if they vote against the bill and at least one other legislator votes against it (so the bill fails) R: if they vote against the bill and the others vote for the bill (so the bill passes) -C: if they vote for the bill and no one else votes for the bill (so the bill fails) R-C: if they vote for the bill and at least one other legislator votes for it (so the bill passes). The three legislators are named X, Y, and Z, and voting happens sequentially and orally. So X announces their vote (to…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.Consider the following variation to the Rock (R), Paper (P), Scissors (S) game:• Suppose that the Player 1 (row player) has a single type, Normal.• Player 2 (column player) has two types Normal and Simple.• A player of Normal type plays this zero-sum game as we studied in class whereas a player of type Simple always play P.• Player 2 knows whether he is Normal or Simple, but player 1does not.a) Suppose player 2 is of type Normal with probability 1/3 and of type Simple with probability (2/3). Find all pure strategy Bayesian Nash Equilibria.b) Suppose player 2 is of type Normal with probability 2/3 and of type Simple with probability (1/3). Find all pure strategy Bayesian Nash Equilibria.
- A strategy for player 1 is a value for x1 from the set X. Similarly, a strategyfor player 2 is a value for x2 from the set X. Player 1’s payoff is V1(x1, x2) =5 + x1 - 2x2 and player 2’s payoff is V2(x1, x2) = 5 + x2 - 2x1.a. Assume that X is the interval of real numbers from 1 to 4 (including 1and 4). (Note that this is much more than integers and includes such numbers as 2.648 and 1.00037). Derive all Nash equilibria.b. Now assume that the game is played infinitely often and a player’s payoff is the present value of his stream of single-period payoffs, where dis the discount factor.(i) Assume that X is composed of only two values: 2 and 3; thus, aplayer can choose 2 or 3, but no other value. Consider the followingsymmetric strategy profile: In period 1, a player chooses the value 2. In period t(≥2), a player chooses the value 2. In period a player chooses the value 2 if both players chose 2 in all previous periods; otherwise, she chooses the value 3. Derive conditions which ensure…12.3 Armed Conflict: Consider the following strategic situation: Two rival armies plan to seize a disputed territory. Each army's general can choose either to attack (A) or to not attack (N). In addition, each army is either strong (S) or weak (W) with equal probability, and the realizations for each army are independent. Furthermore the type of each army is known only to that army's general. An army can capture the territory if either (i) it attacks and its rival does not or (ii) it and its rival attack, but it is strong and the rival is weak. If both attack and are of equal strength then neither captures the territory. As for payoffs, the territory is worth m if captured and each army has a cost of fighting equal to s if it is strong and w if it is weak, where s <w. If an army attacks but its rival does not, no costs are borne by either side. Identify all 12.7 Exercises • 267 the pure-strategy Bayesian Nash equilibria of this game for the following two cases, and briefly describe…H2. One day, Sam and Ryan play odds/evens to see who gets the last doughnut. On command, they each extend one or two fingers. If the sum is odd Sam wins the doughnut, if the sum is even Ryan wins the doughnut. Suppose the payoff from winning the doughnut is 1 and the payoff from losing is 0. a) Illustrate this interaction as a game in matrix form. b) Suppose that Sam thinks that Ryan will play one finger for sure? What will Sam play? Does Sam have reason to think that Ryan will play one finger for sure? c) Do either of them have a strictly dominated strategy? d) Find the pure strategy Nash equilibria of the game, if any. e) Suppose that Sam thinks that Ryan will play one finger or two fingers with even odds. Will Ryan play one finger for sure, play two fingers for sure or play each strategy with even odds? Does Sam have good reason to believe that Ryan will play one finger or two fingers with even odds?
- 2. Consider a game that game theory people refer to as the “ultimatum game.”We will refer to our two players as the “offerer” and the “decider”. How the gameworks is that the offerer proposes a way to split $1000 between the two players.While this could be done in a variety of ways, we will assume that the offerersonly has two possible proposals: Either a 50-50 split, or she offers the decider$50 and keeps the rest. The decider can either accept or reject the offer. If the offer is accepted, the money is split as proposed. If the offer is rejected, themoney spontaneously combusts and nobody gets anything. a) List the strategies for each player and write an extensive form version of thegame with payouts. b) List all the Nash equilibria of this game. c) Explain which, if any of the Nash equilibrium are not sub-game perfect. d) Write the game out in normal form and find the pure strategy Nashequilibrium. Explain how this matches with your answers to (b) and (c) . Alsoexplain why there…9. What is the economic mechanism assuring that the consensus mechanism works so that all miners would find it optimal to add a valid block in their own copy of the blockchain? a. The system automatically updates the ledger b. Adding a valid block candidate in their blockchain is a dominant strategy for all participants in the network c. The communication among miners and the idea that such a strategy is a Nash equilibrium assure it d. The consensus mechanism works because of cryptography and does not have any economics behindConsider 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?