Assume that (xi, yi) and (x³, yž) are pairs of optimal strategies in a zero-sum game. Is 2 3 1 a pair of optimal strategies? Justify your answer.
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- AsapConsider the following extensive form game between player 1 and player 2. T B (2, 2) L R R (3, 1) (0, 0) (5, 0) (0, 1) (a). Find the normal form representation of this game. (show the bimatrix) (b). Find all pure strategy NE. (c). Which of these equilibria are subgame perfect?(a) Find all allocations that are Pareto efficient in the Prisoners' dilemma game. Does moving from an allocation where both players defect to an allocation where one defects and the other cooperate feature a Pareto improvement?
- Suppose we have a football player taking a penalty kick. The kicker can kick right or left, and the goalie can defend right or left (assume the kicker's right). Write the game using general parameters. Find all the equilibria, pure and mixed. Draw the best response curves.Use the decision tree below to answer the following question Aggression Lee: -10; Cody: -10 Cody Aggression Cooperation Lee: 40; Cody: 0 Lee Соорeration Aggression Lee: 0, Cody: 4o Cody Cooperation Lee: 25; Cody: 25 What is the Subgame Perfect Equilibrium (SPE) of this game? O {Lee: Aggression; Cody: Aggression} O {Lee: Aggression; Cody: Cooperation} O {Lee: Cooperation; Cody: Aggression} O {Lee: Cooperation; Cody: Cooperation}Consider the following game tree. The players are P1 and P2. The strategies available to P1 are L, M, R. The strategies available to P2 are A and B. P1 L R M P2 (1,10) A В A B (0,0) (2,1) (0,0) (2,1)
- (a) Consider a ROCK PAPER SCISSOR game. Two players indicate either Rock, Paper or Scissor simultaneously. The winner is determined by: Rock crushes Scissors, Paper covers Rock, and Scissor cut Paper. In the case of a tie, there is no payoff. In the case of a win, the winner collects 5 dollars. Write the payoff matrix for this game. (b) Find the optimal row and column strategies and the value of the matrix game. 3 2 4 -2 1 -4 5Betty and John are opening a restaurant together. To proceed with the project two choices must be made: the type of restaurant and the location. Since Betty and John don't agree on where/what store to open, they've divided the decision into two parts. Betty will choose the type of restaurant and John the location. The payoff matrix is presented below. |Strip Mall 1,0 2, 1 |0, 4 Arts District |4, 4 5, 0 3, 5 Business District Steak house Italian Thai |7, 3 |1, 0 6, 9 What is the Nash-Equilibrium? a. (Italian, Strip Mall) b. (Tahi, Arts District) c. (Thai, Business District) d. (Steak house, Business District)6. Two players each pick a positive integer between 1 and 100. If the numbers are identical, noone wins. If the numbers differ by 1, the one with the lower number pays 1 to the opponent. If the difference is at least 2, the one with the higher number pays 2 to the opponent. What is the NE? Hint: Draw a sketch of the game matrix (while you can't consider all 100 strategies, at least consider the lowest 5-6 numbers) and think about dominated strategies.
- Consider the following simultaneous move game where player 1 has two types. Player 2 does not know if he is playing with type a player 1 or type b player 1. Player 2 C D Player 1 A 12,9 3,6 B 6,0 6,9 C D A 0,9 3,6 B 6,0 6,9 Type a Player 1 Prob = 2/3 Type b Player 1 Prob = 1/3 Find the all the possible Bayesian Nash Equilibriums (BNE) of this game.Suppose now we alter the game so that whenever Colin chooses "paper" the loser pays the winner 3 instead of 1: rock paper scissors rock 0. -3 1 1. раper scissors -1 -1 3 (a) Show that xT= (,) and yT= (5) together are not a Nash equilibrium 3'31 for this modified 3'3 game. (b) Formulate a linear program that can be used to calculate a mixed strategy x € A(R) that maximises Rosemary's security level for this modified game. (c) Solve your linear program using the 2-phase simplex algorithm. You should use the format given in lectures. Give a mixed strategy x E A(R) that has an optimal security level for Rosemary and a mixed strategy y E A(C) that has an optimal security level for Colin.Two workers are on a production line. They each have two actions: exert effort, E, or shirk, S. Effort costs a worker e > 0 and shirking costs them nothing. If two workers do action E a lot of output is produced and the workers earn £3 each. If only one worker chooses action E less output is produced and they both earn £1. The workers earn nothing if they both shirk. (i) Describe this situation as a strategic form game (assuming the workers do not observe each other's effort choice when making their own decision). (ii) For what values of e does this game have strictly dominant strategies? (iii) Describe the Nash equilibria of this game for e = 0,1, 2, 3. (iv) The workers now are re-arranged into a production line. First worker 1 moves and then worker 2 moves. Worker 2 can now see worker l's effort level before they choose their effort. Draw this extensive form game. (v) Find the subgame perfect equilibria of the production-line game for c = 1/2 and c = 3/2.