Consider the two-player Rock-Paper-Scissors game. Each player can choose between three pure strategies: Rock, Paper, and Scissors. Strategy Rock beats Scissors, but loses to Paper. Strategy Paper beats Rock but loses to Scissors. Strategy Scissors beats Paper, but loses to Rock. When the game is a draw, the payoff for the players is '0'. The payoff of winning is ‘+1' and the payoff of losing is '—1'. Formulate this problem as a two-person, zero-sum game by constructing a payoff table.
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- 1. Each of two players simultaneously announces either Rock, Paper, or Scissors. Paper beats Rock, Rock beats Scissors, and Scissors beats Paper. The player who names the winning object receives $1 from her opponent. If both players make the same choice, no payment is made. Each players's preferences are represented by by the expected amount of money she receives. a) Formulate the situation as a strategic game and find all its mixed strategy equilibria. Make sure that you consider all different supports. b) Find all the mixed strategy equilibria of the modified game in which player 2 is prohibited from announcing Rock.1. A row of minus signs is written on a whiteboard (can be any number). Two players take turns in replacing either a single minus sign by a plus sign or two adjacent minus signs by two plus signs. The player who makes the last move wins. Discuss the possible strategy for the first mover to force a win.4. Find the Nash equilibrium of the following modified Rock-Paper-Scissors game: • When rock (R) beats scissors (S), the winner's payoff is 10 and the loser's payoff is –10. • When paper (P) beats rock, the winner's payoff is 5 and the loser's payoff is -5. • When scissors beats paper, the winner's payoff is 2 and the loser's payoff is -2. • In case of ties, both players receive 0 payoff.
- 5. Aaron and Betty play the following one-shot game. Aaron Up Middle Down Left 10,0 5,10 1,1 Betty Right 0,5 1,0 2,2 a. Does the game above have any (pure strategy) Nash equilibria? Find any equilibria or explain why an equilibrium doesn't exist. b. Suppose instead that Aaron moves first and Betty moves second. Represent this situation with a game tree and find the subgame perfect Nash equilibrium. Are Aaron and Betty better off playing a simultaneous game or a sequential game in which Aaron moves first? Explain. c.5. Consider a game in which two players, Fred and Barney, take turns removing matchsticks from a pile. They start with 21 matchsticks, and Fred goes first. On each turn, each player may remove either 1, 2, 3, or 4 matchsticks. The player to remove the last matchstick wins the game. 1. Suppose there are only 6 matchsticks left, and it is Barney' s turn. What move should Barney make to guarantee himself victory? Explain your reasoning. 2. Suppose there are 12 matchsticks left, and it is Barney's turn. What move should Barney make to guarantee himself victory? (Hint: Use your answer to part (a) and roll back.) 3. Now start from the beginning of the game. If both players play optimally, who will win? 4. What are the optimal (complete) strategies for each player?II. Determine which of the following two-person zero-sum games are strictly determinable and fair. Give the optimum strategies for each player in the case of it being strictly determinable. a- Player A Player B B1 B2 B3 A1 8 2 4 A2 5 -1 3 b. II II V V I -4 -2 -2 1 A II 1 -1 III -6 -5 -2 -4 4 IV 3 1 -6 0 -8 3.
- 4. Consider the following game. There are four distinct glasses of wine on a table: three are red wine, and one is white. Two players must pick a glass of wine at the same time. If they choose the exact same glass of wine, then they each earn $1,000. Otherwise, they earn nothing. Draw the payoff matrix. What is this game's Nash equilibrium (or equilibria)? What is this game's focal point? Explain. (20%) 5. Rational ignorance poses a serious problem for democracy, but can be partially resolved by implementing Skin in the Game policies. For example, consider the following law: politicians who vote for war must enlist in the military, or have a family member enlist. How does this law (a) fall under Skin in the Game; and (b) Resolve the problem of Rational Ignorance when it comes to foreign policy? (20%)The following payoff matrix represents a simultaneous-move game between two players: Kay and Jack. Each player has two choices: Black or White. The first number in each cell is the payoff to Kay, and the second number is the payoff to Jack. Jay Black White 50, 30, Black 50 30 Кay 45, 40, White 30 50 Refer to the scenario above. Which is true? a. This game has a dominant strategy equilibrium. b. This game has two dominant strategy equilibria. c. This game has two Nash equilibria. d. This game has one Nash equilibrium.Game theory Consider a simultaneous move game with two players. Player 1 has three possible actions (A, B, or C) and Player 2 has two possible actions (D or E.) In the payoff matrix below, each cell contains the payoff for Player 1 followed by the payoff for Player 2. Player 2 7. Player 1 A B C D -3, -3 0, -11 -4, 3 -11, E 0 -7, -7 -12, 0 (a) Identify any dominated strategies in this game. If there are none, state this clearly. (b) Identify any pure strategy Nash Equilibria in this game. If there are none, state this clearly
- 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.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…Table 8. This table shows a game played between two players, A and B. The payoffs are given in the table as (Payoff to A, Payoff to B). A Select one: Up Middle Down Left (4,2) a. Middle-Center O b. Down-Center (3, 1) (1,3) B Center (2,5) (5, 3) (4,4) Refer to Table 8. This table shows a game played between two players, A and B. The payoffs in the table are shown as (Payoff to A, Payoff to B). Which of the following outcomes represents a Nash equilibrium in the game? Right (3, 3) (5,2) (6, 1) O c. Up-Left O d. More than one of the above is a Nash equilibrium in this game.