Make sure you clearly write your name andsubmission:.1. Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player choosesone of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices.The outcome is determined as follows. If both players choose the same strategy,neither player wins or loses anything. Otherwise:• "paper covers rock": if one player chooses paper and the other chooses rock,the player who chose paper wins and is paid 1 by the other player.• "scissors cut paper": if one player chooses scissors and the other chooses paper,the player who chose scissors wins and is paid 1 by the other player.• "rock breaks scissors": if one player chooses rock and the other player choosesscissors, the player who chose rock wins and is paid 1 by the other player.We can write the payoff matrix for this game as follows:rock paper-101rock0paper1scissors -1(a) Show that this game does not have a pure Nash equilibrium.1X3(b) Show that the pair of mixed strategies x¹ = (,,) and y¹ = (3) togetherare a Nash equilibrium.10 Coursework Questions.2. Suppose now we alter the game so that whenever Colin chooses "paper" the loserpays the winner 3 instead of 1:scissors1-10rock01rockpaperscissors -1paper scissors1-30-130X(a) Show that x¹ = (3) and y¹ = (3) together are not a Nash equilibriumfor this modified game.MTH5114 (Spring 2023)

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1. Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player chooses one of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices. The outcome is determined as follows. If both players choose the same strategy, neither player wins or loses anything. Otherwise: • "paper covers rock": if one player chooses paper and the other chooses rock, the player who chose paper wins and is paid 1 by the other player. • "scissors cut paper": if one player chooses scissors and the other chooses paper, the player who chose scissors wins and is paid 1 by the other player. plr brooks coiccors" if one player chooses rock and the other player chooses

1. Consider the 2-player, zero-sum game "Rock, Paper, Scissors". Each player chooses one of 3 strategies: rock, paper, or scissors. Then, both players reveal their choices. The outcome is determined as follows. If both players choose the same strategy, neither player wins or loses anything. Otherwise: • "paper covers rock": if one player chooses paper and the other chooses rock, the player who chose paper wins and is paid 1 by the other player. • "scissors cut paper": if one player chooses scissors and the other chooses paper, the player who chose scissors wins and is paid 1 by the other player. plr brooks coiccors" if one player chooses rock and the other player chooses

ENGR.ECONOMIC ANALYSIS

14th Edition

ISBN:9780190931919

Author:NEWNAN

Publisher:NEWNAN

Chapter1: Making Economics Decisions

Section: Chapter Questions

Problem 1QTC

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7. Solving for dominant strategies and the Nash equilibrium Suppose Carlos and Deborah are playing a game in which both must simultaneously choose the action Left or Right. The payoff matrix that follows shows the payoff each person will earn as a function of both of their choices. For example, the lower-right cell shows that if Carlos chooses Right and Deborah chooses Right, Carlos will receive a payoff of 5 and Deborah will receive a payoff of 1. Carlos Deborah Left Left 4,4 Right 2,4 Right 6,7 5,1 The only dominant strategy in this game is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Carlos chooses and Deborah chooses
1. Consider the following normal-form game: Player 2 Left Right 30,20 10,10 10,30 20,20 Player 1 How many pure-strategy Nash Equilibria are there in this game? (a) 0 (b) 1 Up Down (c) 2 (d) 3 (e) 4 (f) None of the above Answer: 1b. The only Nash Equilirbium is (Up,Left).
4. Assume a Hawk -Dove game with the following payoff matrix, where the first entry is Animal A's payoff and the second entry is Animal B's payoff: Animal A Hawk (rows)/Animal B (columns) Hawk Dove (-10,-10) (0,20) Dove (20,0) (8,8) An animal that plays Hawk will always fight until it wins or is badly hurt. An animal that plays Dove makes a bold display but retreats if his opponent starts to fight. If two Dove animals meet they share. (a) Explain why there cannot be an equilibrium where all animals act as Doves. (b) Explore whether there are any Nash equilibria in pure strategies and explain which these are and why they are equilibria.
2. Consider the two-player strategic form game represented in the following payoff table: t 1 m b 7,2 1,2 3,7 2 M 2,4 3,4 1,6 R 1,3 7,1 4,2 Player 1 chooses between rows, S₁ = {t, m, b), and receives the first listed payoff in each cell. Player 2 chooses between columns, S₂ = {L, M, R), and receives the second listed payoff in each cell. a. Does either player have a strictly dominant strategy? Explain. b. Explaining your steps carefully, identify the set of rationalizable strategies in this game. c. Making sure to explain your solution method, solve for all pure strategy Nash equilibria in this game.
3.Students A and B decide whether to clean the dormitory. Regardless of whether the other party participates, the cleaning cost for each student to clean the dormitory is $ 8; and the benefit for each student from cleaning is $5 times the number of participants. (hint: if both of them clean the dormitory together, A and B benefit. If one cleans while the other does not, A and B benefit too) (a) Please use a game theory to briefly describe the above scenario, (that is to calculate the revenue for different strategies) (b) Draw the payoff matrix, find the Nash equilibrium.
2. Consider the following "centipede game." The game starts with player 1 choosing be- tween terminate (T) and continue (C). If player 1 chooses C, the game proceeds with player 2 choosing between terminate (t) and continue (c). The two players choose be- tween terminate and continue in turn if the other player chooses continue until the terminal nodes with (player l's payoff, player 2's payoff) are reached as shown below. TTTT Player 1 Player 2 Player 1 Player 2 (3, 3) t (1, 1) (0, 3) (2, 2) (1, 4) (a) List all possible strategies of each player. (b) Transform the game tree into a normal-form matrix representation. (c) Find all pure-strategy Nash equilibria. (d) Find the unique pure-strategy subgame-perfect equilibrium.
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7. Solving for dominant strategies and the Nash equilibrium Suppose Gabriel and Nia are playing a game that requires both to simultaneously choose an action: Up or Down. The payoff matrix that follows shows the earnings of each person as a function of both of their choices. For example, the upper-right cell shows that if Gabriel chooses Up and Nia chooses Down, Gabriel will receive a payoff of 6 and Nia will receive a payoff of 4. Gabriel Up Down Up 6,3 3,3 Nia Down 6, 4 7,4 In this game, the only dominant strategy is for to choose The outcome reflecting the unique Nash equilibrium in this game is as follows: Gabriel chooses and Nia chooses
11. The following payoff table involves two players each of whom has three possible moves. L M N L 30,15 45,35 55,25 M 25,35 20,20 30,50 N 15,35 35,40 15,15 a. Are there any dominant strategies in this game?
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1. Consider a game of chicken: two players approach a narrow bridge with room only for one, and each chooses either to go through and hope the other guy backs off (call this action H, for "hawk"), or backs off to let the other go through (call this action D, for "dove"). If both players are normal, then each gets a payoff of -1 from a collision (i.e. if both choose H), 1 if they choose H against D, 0 if they choose D (regardless of what the other player chooses). Now perturb the game as follows: there is a chance p > 0 that P2 is a “violent type", who gets 1 from playing H regardless of Pl's action, and -1 from playing D, again regardless of P1's action. (a) Illustrate the game tree corresponding to the Harsanyi transformation of this game (i.e. the dynamic game of imperfect information in which nature moves first). (b) Illustrate the extended game (i.e. the normal form game corresponding to your game tree in part (a)). (c) Find all values of p for which there is a BNE in which P1…
Consider the following game: